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arXiv:math/0211159v1 [math.DG] 11 Nov 2002 The entropy formula for the Ricci flow and its geometric applications Grisha...

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arXiv:math/0211159v1 [math.DG] 11 Nov 2002

The entropy formula for the Ricci flow and its geometric applications Grisha Perelman∗ February 1, 2008

Introduction 1. The Ricci flow equation, introduced by Richard Hamilton [H 1], is the evolution equation dtd gij (t) = −2Rij for a riemannian metric gij (t). In his seminal paper, Hamilton proved that this equation has a unique solution for a short time for an arbitrary (smooth) metric on a closed manifold. The evolution equation for the metric tensor implies the evolution equation for the curvature tensor of the form Rmt = △Rm + Q, where Q is a certain quadratic expression of the curvatures. In particular, the scalar curvature R satisfies Rt = △R + 2|Ric|2 , so by the maximum principle its minimum is non-decreasing along the flow. By developing a maximum principle for tensors, Hamilton [H 1,H 2] proved that Ricci flow preserves the positivity of the Ricci tensor in dimension three and of the curvature operator in all dimensions; moreover, the eigenvalues of the Ricci tensor in dimension three and of the curvature operator in dimension four are getting pinched pointwisely as the curvature is getting large. This observation allowed him to prove the convergence results: the evolving metrics (on a closed manifold) of positive Ricci curvature in dimension three, or positive curvature operator St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia. Email: [email protected] or [email protected] ; I was partially supported by personal savings accumulated during my visits to the Courant Institute in the Fall of 1992, to the SUNY at Stony Brook in the Spring of 1993, and to the UC at Berkeley as a Miller Fellow in 1993-95. I’d like to thank everyone who worked to make those opportunities available to me. ∗

1

in dimension four converge, modulo scaling, to metrics of constant positive curvature. Without assumptions on curvature the long time behavior of the metric evolving by Ricci flow may be more complicated. In particular, as t approaches some finite time T, the curvatures may become arbitrarily large in some region while staying bounded in its complement. In such a case, it is useful to look at the blow up of the solution for t close to T at a point where curvature is large (the time is scaled with the same factor as the metric tensor). Hamilton [H 9] proved a convergence theorem , which implies that a subsequence of such scalings smoothly converges (modulo diffeomorphisms) to a complete solution to the Ricci flow whenever the curvatures of the scaled metrics are uniformly bounded (on some time interval), and their injectivity radii at the origin are bounded away from zero; moreover, if the size of the scaled time interval goes to infinity, then the limit solution is ancient, that is defined on a time interval of the form (−∞, T ). In general it may be hard to analyze an arbitrary ancient solution. However, Ivey [I] and Hamilton [H 4] proved that in dimension three, at the points where scalar curvature is large, the negative part of the curvature tensor is small compared to the scalar curvature, and therefore the blow-up limits have necessarily nonnegative sectional curvature. On the other hand, Hamilton [H 3] discovered a remarkable property of solutions with nonnegative curvature operator in arbitrary dimension, called a differential Harnack inequality, which allows, in particular, to compare the curvatures of the solution at different points and different times. These results lead Hamilton to certain conjectures on the structure of the blow-up limits in dimension three, see [H 4,§26]; the present work confirms them. The most natural way of forming a singularity in finite time is by pinching an (almost) round cylindrical neck. In this case it is natural to make a surgery by cutting open the neck and gluing small caps to each of the boundaries, and then to continue running the Ricci flow. The exact procedure was described by Hamilton [H 5] in the case of four-manifolds, satisfying certain curvature assumptions. He also expressed the hope that a similar procedure would work in the three dimensional case, without any a priory assumptions, and that after finite number of surgeries, the Ricci flow would exist for all time t → ∞, and be nonsingular, in the sense that the normalized curvatures ˜ Rm(x, t) = tRm(x, t) would stay bounded. The topology of such nonsingular solutions was described by Hamilton [H 6] to the extent sufficient to make sure that no counterexample to the Thurston geometrization conjecture can 2

occur among them. Thus, the implementation of Hamilton program would imply the geometrization conjecture for closed three-manifolds. In this paper we carry out some details of Hamilton program. The more technically complicated arguments, related to the surgery, will be discussed elsewhere. We have not been able to confirm Hamilton’s hope that the solution that exists for all time t → ∞ necessarily has bounded normalized curvature; still we are able to show that the region where this does not hold is locally collapsed with curvature bounded below; by our earlier (partly unpublished) work this is enough for topological conclusions. Our present work has also some applications to the Hamilton-Tian conjecture concerning Kähler-Ricci flow on Kähler manifolds with positive first Chern class; these will be discussed in a separate paper. 2. The Ricci flow has also been discussed in quantum field theory, as an approximation to the renormalization group (RG) flow for the two-dimensional nonlinear σ-model, see [Gaw,§3] and references therein. While my background in quantum physics is insufficient to discuss this on a technical level, I would like to speculate on the Wilsonian picture of the RG flow. In this picture, t corresponds to the scale parameter; the larger is t, the larger is the distance scale and the smaller is the energy scale; to compute something on a lower energy scale one has to average the contributions of the degrees of freedom, corresponding to the higher energy scale. In other words, decreasing of t should correspond to looking at our Space through a microscope with higher resolution, where Space is now described not by some (riemannian or any other) metric, but by an hierarchy of riemannian metrics, connected by the Ricci flow equation. Note that we have a paradox here: the regions that appear to be far from each other at larger distance scale may become close at smaller distance scale; moreover, if we allow Ricci flow through singularities, the regions that are in different connected components at larger distance scale may become neighboring when viewed through microscope. Anyway, this connection between the Ricci flow and the RG flow suggests that Ricci flow must be gradient-like; the present work confirms this expectation. 3. The paper is organized as follows. In §1 we explain why Ricci flow can be regarded as a gradient flow. In §2, 3 we prove that Ricci flow, considered as a dynamical system on the space of riemannian metrics modulo diffeomorphisms and scaling, has no nontrivial periodic orbits. The easy (and known) 3

case of metrics with negative minimum of scalar curvature is treated in §2; the other case is dealt with in §3, using our main monotonicity formula (3.4) and the Gaussian logarithmic Sobolev inequality, due to L.Gross. In §4 we apply our monotonicity formula to prove that for a smooth solution on a finite time interval, the injectivity radius at each point is controlled by the curvatures at nearby points. This result removes the major stumbling block in Hamilton’s approach to geometrization. In §5 we give an interpretation of our monotonicity formula in terms of the entropy for certain canonical ensemble. In §6 we try to interpret the formal expressions , arising in the study of the Ricci flow, as the natural geometric quantities for a certain Riemannian manifold of potentially infinite dimension. The Bishop-Gromov relative volume comparison theorem for this particular manifold can in turn be interpreted as another monotonicity formula for the Ricci flow. This formula is rigorously proved in §7; it may be more useful than the first one in local considerations. In §8 it is applied to obtain the injectivity radius control under somewhat different assumptions than in §4. In §9 we consider one more way to localize the original monotonicity formula, this time using the differential Harnack inequality for the solutions of the conjugate heat equation, in the spirit of Li-Yau and Hamilton. The technique of §9 and the logarithmic Sobolev inequality are then used in §10 to show that Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. The results of sections 1 through 10 require no dimensional or curvature restrictions, and are not immediately related to Hamilton program for geometrization of three manifolds. The work on details of this program starts in §11, where we describe the ancient solutions with nonnegative curvature that may occur as blow-up limits of finite time singularities ( they must satisfy a certain noncollapsing assumption, which, in the interpretation of §5, corresponds to having bounded entropy). Then in §12 we describe the regions of high curvature under the assumption of almost nonnegative curvature, which is guaranteed to hold by the Hamilton and Ivey result, mentioned above. We also prove, under the same assumption, some results on the control of the curvatures forward and backward in time in terms of the curvature and volume at a given time in a given ball. Finally, in §13 we give a brief sketch of the proof of geometrization conjecture. The subsections marked by * contain historical remarks and references. See also [Cao-C] for a relatively recent survey on the Ricci flow.

4

1

Ricci flow as a gradient flow

R 1.1. Consider the functional F = M (R + |∇f |2)e−f dV for a riemannian metric gij and a function f on a closed manifold M. Its first variation can be expressed as follows: Z δF (vij , h) = e−f [−△v + ∇i ∇j vij − Rij vij M

=

Z

−vij ∇i f ∇j f + 2 < ∇f, ∇h > +(R + |∇f |2 )(v/2 − h)]

M

e−f [−vij (Rij + ∇i ∇j f ) + (v/2 − h)(2△f − |∇f |2 + R)],

where δgij = vij , δf = h, v = g ij vij . Notice that v/2 − h vanishes identically iff the measure dm = e−f dV is kept fixed. Therefore, theR symmetric tensor −(Rij +∇i ∇j f ) is the L2 gradient of the functional F m = M (R + |∇f |2 )dm, where now f denotes log(dV /dm). Thus given a measure m , we may consider the gradient flow (gij )t = −2(Rij + ∇i ∇j f ) for F m . For general m this flow may not exist even for short time; however, when it exists, it is just the Ricci flow, modified by a diffeomorphism. The remarkable fact here is that different choices of m lead to the same flow, up to a diffeomorphism; that is, the choice of m is analogous to the choice of gauge. 1.2 Proposition. Suppose that the gradient flow for F m exists for t ∈ [0, T ]. R n Then at t = 0 we have F m ≤ 2T M dm. R Proof. We may assume M dm = 1. The evolution equations for the gradient flow of F m are (gij )t = −2(Rij + ∇i ∇j f ), ft = −R − △f, and F m satisfies

Ftm

=2

Z

|Rij + ∇i ∇j f |2 dm

(1.1) (1.2)

Modifying by an appropriate diffeomorphism, we get evolution equations (gij )t = −2Rij , ft = −△f + |∇f |2 − R,

(1.3)

and retain (1.2) in the form Ft = 2

Z

|Rij + ∇i ∇j f |2 e−f dV 5

(1.4)

Now we compute Z Z 2 2 2 2 −f (R + △f ) e dV ≥ ( (R + △f )e−f dV )2 = F 2 , Ft ≥ n n n and the proposition follows. 1.3 Remark. The functional F m has a natural interpretation in terms of Bochner-Lichnerovicz formulas. The classical formulas of Bochner (for one-forms) and Lichnerovicz (for spinors) are ∇∗ ∇ui = (d∗ d + dd∗ )ui − Rij uj and ∇∗ ∇ψ = δ 2 ψ − 1/4Rψ. Here the operators ∇∗ , d∗ are defined using the riemannian volume form; this volume form is also implicitly used in the definition of the Dirac operator δ via the requirement δ ∗ = δ. A routine computation shows that if we substitute dm = e−f dV for dV , we get m modified Bochner-Lichnerovicz formulas ∇∗m ∇ui = (d∗m d + dd∗m )ui − Rij uj ∗m m 2 m m m and ∇ ∇ψ = (δ ) ψ − 1/4R ψ, where δ ψ = δψ − 1/2(∇f ) · ψ , Rij = m Rij +∇i ∇j f , Rm = 2△f −|∇f |2 +R. Note that g ij Rij = R+△f 6= Rm . How∗m m m ever, we do have the Bianchi identity ∇i Rij = ∇i Rij −Rij ∇i f = 1/2∇j Rm . R R m Now F m = M Rm dm = M g ij Rij dm. 1.4* The Ricci flow modified by a diffeomorphism was considered by DeTurck, who observed that by an appropriate choice of diffeomorphism one can turn the equation from weakly parabolic into strongly parabolic, thus considerably simplifying the proof of short time existence and uniqueness; a nice version of DeTurck trick can be found in [H 4,§6]. The functional F and its first variation formula can be found in the literature on the string theory, where it describes the low energy effective action; the function f is called dilaton field; see [D,§6] for instance. m The Ricci tensor Rij for a riemannian manifold with a smooth measure has been used by Bakry and Emery [B-Em]. See also a very recent paper [Lott].

2

No breathers theorem I

2.1. A metric gij (t) evolving by the Ricci flow is called a breather, if for some t1 < t2 and α > 0 the metrics αgij (t1 ) and gij (t2 ) differ only by a diffeomorphism; the cases α = 1, α < 1, α > 1 correspond to steady, shrinking and expanding breathers, respectively. Trivial breathers, for which the metrics gij (t1 ) and gij (t2 ) differ only by diffeomorphism and scaling for each pair of

6

t1 and t2 , are called Ricci solitons. (Thus, if one considers Ricci flow as a dynamical system on the space of riemannian metrics modulo diffeomorphism and scaling, then breathers and solitons correspond to periodic orbits and fixed points respectively). At each time the Ricci soliton metric satisfies an equation of the form Rij + cgij + ∇i bj + ∇j bi = 0, where c is a number and bi is a one-form; in particular, when bi = 12 ∇i a for some function a on M, we get a gradient Ricci soliton. An important example of a gradient shrinking soliton is the Gaussian soliton, for which the metric gij is just the euclidean metric on Rn , c = 1 and a = −|x|2 /2. In this and the next section we use the gradient interpretation of the Ricci flow to rule out nontrivial breathers (on closed M). The argument in the steady case is pretty straightforward; the expanding case is a little bit more subtle, because our functional F is not scale invariant. The more difficult shrinking case is discussed in section 3. 2.2. DefineR λ(gij ) = inf F (gij , f ), where infimum is taken over all smooth f, satisfying M e−f dV = 1. Clearly, λ(gij ) is just the lowest eigenvalue of the operator −4△+R. Then formula (1.4) implies that λ(gij (t)) is nondecreasing in t, and moreover, if λ(t1 ) = λ(t2 ), then for t ∈ [t1 , t2 ] we have Rij +∇i ∇j f = 0 for f which minimizes F . Thus a steady breather is necessarily a steady soliton. 2.3. To deal with the expanding case consider a scale invariant version ¯ ij ) = λ(gij )V 2/n (gij ). The nontrivial expanding breathers will be ruled λ(g out once we prove the following ¯ is nondecreasing along the Ricci flow whenever it is nonpositive; Claim λ moreover, the monotonicity is strict unless we are on a gradient soliton. (Indeed, on an expanding breather we would necessarily have dV R /dt > 0 1 d for some t∈[t1 , t2 ]. On the other hand, for every t, − dt logV = V RdV ≥ ¯ can not be nonnegative everywhere on [t1 , t2 ], and the claim apλ(t), so λ plies.) Proof of the claim. ¯ dλ(t)/dt ≥ 2V 2/n

R

|Rij + ∇i ∇j f |2 e−f dV + n2 V (2−n)/n λ

R

R 2V 2/n [ |Rij + ∇i ∇j f − n1 (R + △f )gij |2 e−f dV + R

1 n(

−RdV ≥

R (R + △f )2 e−f dV − ( (R + △f )e−f dV )2 )] ≥ 0,

where f is the minimizer for F .

7

2.4. The arguments above also show that there are no nontrivial (that is with non-constant Ricci curvature) steady or expanding Ricci solitons (on closed M). Indeed, the equality case in the chain of inequalities above requires that R+△f be constant on M; on the other hand, the Euler-Lagrange equation for the minimizer f is 2△f − |∇f |2 + R = const. Thus, △f − |∇f |2 = const = 0, R because (△f − |∇f |2)e−f dV = 0. Therefore, f is constant by the maximum principle. 2.5*. A similar, but simpler proof of the results in this section, follows im2 mediately from [H 6,§2], where Hamilton checks that the minimum of RV n is nondecreasing whenever it is nonpositive, and monotonicity is strict unless the metric has constant Ricci curvature.

3

No breathers theorem II

3.1. In order to handle the shrinking case when λ > 0, we need to replace our functional F by its generalization, which contains explicit insertions of the scale parameter, to be denoted by τ. Thus consider the functional Z n W(gij , f, τ ) = [τ (|∇f |2 + R) + f − n](4πτ )− 2 e−f dV , (3.1) M

restricted to f satisfying Z

n

(4πτ )− 2 e−f dV = 1,

(3.2)

M

τ > 0. Clearly W is invariant under simultaneous scaling of τ and gij . The evolution equations, generalizing (1.3) are (gij )t = −2Rij , ft = −△f + |∇f |2 − R +

n , τt = −1 2τ

(3.3)

The evolution equation for f can also be written as follows: 2∗ u = 0, where n u = (4πτ )− 2 e−f , and 2∗ = −∂/∂t − △ + R is the conjugate heat operator. Now a routine computation gives Z n 1 dW/dt = 2τ |Rij + ∇i ∇j f − gij |2 (4πτ )− 2 e−f dV . (3.4) 2τ M Therefore, if we let µ(gij , τ ) = inf W(gij , f, τ ) over smooth f satisfying (3.2), and ν(gij ) = inf µ(gij , τ ) over all positive τ, then ν(gij (t)) is nondecreasing 8

along the Ricci flow. It is not hard to show that in the definition of µ there always exists a smooth minimizer f (on a closed M). It is also clear that limτ →∞ µ(gij , τ ) = +∞ whenever the first eigenvalue of −4△ + R is positive. Thus, our statement that there is no shrinking breathers other than gradient solitons, is implied by the following Claim For an arbitrary metric gij on a closed manifold M, the function µ(gij , τ ) is negative for small τ > 0 and tends to zero as τ tends to zero. Proof of the Claim. (sketch) Assume that τ¯ > 0 is so small that Ricci n flow starting from gij exists on [0, τ¯]. Let u = (4πτ )− 2 e−f be the solution of the conjugate heat equation, starting from a δ-function at t = τ¯, τ (t) = τ¯ − t. Then W(gij (t), f (t), τ (t)) tends to zero as t tends to τ¯, and therefore µ(gij , τ¯) ≤ W(gij (0), f (0), τ (0)) < 0 by (3.4). Now let τ → 0 and assume that f τ are the minimizers, such that 1 1 W( τ −1 gij , f τ , ) = W(gij , f τ , τ ) = µ(gij , τ ) ≤ c < 0. 2 2 The metrics 21 τ −1 gij ”converge” to the euclidean metric, and if we could extract a converging subsequence from f τ , we would get a function f on Rn , R n such that Rn (2π)− 2 e−f dx = 1 and Z n 1 [ |∇f |2 + f − n](2π)− 2 e−f dx < 0 Rn 2 The latter inequality contradicts the Gaussian logarithmic Sobolev inequality, due to L.Gross. (To pass to its standard form, take f = |x|2 /2 − 2 log φ and integrate by parts) This argument is not hard to make rigorous; the details are left to the reader. 3.2 Remark. Our monotonicity formula (3.4) can in fact be used to prove a version of the logarithmic Sobolev inequality (with description of the equality cases) on shrinking Ricci solitons. Indeed, assume that a metric gij satisfies Rij − gij − ∇i bj − ∇j bi = 0. Then under Ricci flow, gij (t) is isometric to (1 − 2t)gij (0), µ(gij (t), 12 − t) = µ(gij (0), 21 ), and therefore the monotonicity formula (3.4) implies that the minimizer f for µ(gij , 12 ) satisfies Rij + ∇i ∇j f − gij = 0. Of course, this argument requires the existence of minimizer, and justification of the integration by parts; this is easy if M is closed, but can also be done with more efforts on some complete M, for instance when M is the Gaussian soliton. 9

3.3* The no breathers theorem in dimension three was proved by Ivey [I]; in fact, he also ruled out nontrivial Ricci solitons; his proof uses the almost nonnegative curvature estimate, mentioned in the introduction. Logarithmic Sobolev inequalities is a vast area of research; see [G] for a survey and bibliography up to the year 1992; the influence of the curvature was discussed by Bakry-Emery [B-Em]. In the context of geometric evolution equations, the logarithmic Sobolev inequality occurs in Ecker [E 1].

4

No local collapsing theorem I

In this section we present an application of the monotonicity formula (3.4) to the analysis of singularities of the Ricci flow. 4.1. Let gij (t) be a smooth solution to the Ricci flow (gij )t = −2Rij on [0, T ). We say that gij (t) is locally collapsing at T, if there is a sequence of times tk → T and a sequence of metric balls Bk = B(pk , rk ) at times tk , such that rk2 /tk is bounded, |Rm|(gij (tk )) ≤ rk−2 in Bk and rk−n V ol(Bk ) → 0. Theorem. If M is closed and T < ∞, then gij (t) is not locally collapsing at T. Proof. Assume that there is a sequence of collapsing balls Bk = B(pk , rk ) at times tk → T. Then we claim that µ(gij (tk ), rk2 ) → −∞. Indeed one can take fk (x) = − log φ(disttk (x, pk )rk−1 ) + ck , where φ is a function of one variable, equal 1 on [0, 1/2], decreasing on [1/2, 1], and very close to 0 on [1, ∞), and ck is a constant; clearly ck → −∞ as rk−n V ol(Bk ) → 0. Therefore, applying the monotonicity formula (3.4), we get µ(gij (0), tk + rk2 ) → −∞. However this is impossible, since tk + rk2 is bounded. 4.2. Definition We say that a metric gij is κ-noncollapsed on the scale ρ, if every metric ball B of radius r < ρ, which satisfies |Rm|(x) ≤ r −2 for every x ∈ B, has volume at least κrn . It is clear that a limit of κ-noncollapsed metrics on the scale ρ is also κ-noncollapsed on the scale ρ; it is also clear that α2 gij is κ-noncollapsed on the scale αρ whenever gij is κ-noncollapsed on the scale ρ. The theorem above essentially says that given a metric gij on a closed manifold M and T < ∞, one can find κ = κ(gij , T ) > 0, such that the solution gij (t) to the Ricci flow starting at gij is κ-noncollapsed on the scale T 1/2 for all t ∈ [0, T ), provided it exists on this interval. Therefore, using the convergence theorem of Hamilton, we obtain the following 10

Corollary. Let gij (t), t ∈ [0, T ) be a solution to the Ricci flow on a closed manifold M, T < ∞. Assume that for some sequences tk → T, pk ∈ M and some constant C we have Qk = |Rm|(pk , tk ) → ∞ and |Rm|(x, t) ≤ CQk , whenever t < tk . Then (a subsequence of ) the scalings of gij (tk ) at pk with factors Qk converges to a complete ancient solution to the Ricci flow, which is κ-noncollapsed on all scales for some κ > 0.

5

A statistical analogy

In this section we show that the functional W, introduced in section 3, is in a sense analogous to minus entropy. 5.1 Recall that the partition R function for the canonical ensemble at temperature β −1 is given by Z = exp(−βE)dω(E), where ω(E) is a ”density of states” measure, which does not depend on β. Then one computes the ∂ average energy < E >= − ∂β log Z, the entropy S = β < E > + log Z, and 2

∂ the fluctuation σ =< (E− < E >)2 >= (∂β) 2 log Z. Now fix a closed manifold M with a probability measure m, and suppose that our system is described by a metric gij (τ ), which depends on the temperature τ according to equation (gij )τ = 2(Rij + ∇i ∇j f ), where Rdm = udV, u = n (4πτ )− 2 e−f , and the partition function is given by log Z = (−f + n2 )dm. (We do not discuss here what assumptions on gij guarantee that the corresponding ”density of states” measure can be found) Then we compute Z n 2 < E >= −τ (R + |∇f |2 − )dm, 2τ M Z S=− (τ (R + |∇f |2) + f − n)dm, M

σ = 2τ 4

Z

M

|Rij + ∇i ∇j f −

1 gij |2 dm 2τ

Alternatively, we could prescribe the evolution equations by replacing the t-derivatives by minus τ -derivatives in (3.3 ), and get the same formulas for Z, < E >, S, σ, with dm replaced by udV. Clearly, σ is nonnegative; it vanishes only on a gradient shrinking soliton. < E > is nonnegative as well, whenever the flow exists for all sufficiently small τ > 0 (by proposition 1.2). Furthermore, if (a) u tends to a δ-function as τ → 0, or (b) u is a limit of a sequence of functions ui , such that each ui 11

tends to a δ-function as τ → τi > 0, and τi → 0, then S is also nonnegative. In case (a) all the quantities < E >, S, σ tend to zero as τ → 0, while in case (b), which may be interesting if gij (τ ) goes singular at τ = 0, the entropy S may tend to a positive limit. If the flow is defined for all sufficiently large τ (that is, we have an ancient solution to the Ricci flow, in Hamilton’s terminology), we may be interested in the behavior of the entropy S as τ → ∞. A natural question is whether we have a gradient shrinking soliton whenever S stays bounded. 5.2 Remark. Heuristically, this statistical analogy is related to the description of the renormalization group flow, mentioned in the introduction: in the latter one obtains various quantities by averaging over higher energy states, whereas in the former those states are suppressed by the exponential factor. 5.3* An entropy formula for the Ricci flow in dimension two was found by Chow [C]; there seems to be no relation between his formula and ours. The interplay of statistical physics and (pseudo)-riemannian geometry occurs in the subject of Black Hole Thermodynamics, developed by Hawking et al. Unfortunately, this subject is beyond my understanding at the moment.

6

Riemannian formalism in potentially infinite dimensions

When one is talking of the canonical ensemble, one is usually considering an embedding of the system of interest into a much larger standard system of fixed temperature (thermostat). In this section we attempt to describe such an embedding using the formalism of Rimannian geometry. ˜ = M × SN × R+ with the following metric: 6.1 Consider the manifold M g˜ij = gij , g˜αβ = τ gαβ , g˜00 =

N + R, g˜iα = g˜i0 = g˜α0 = 0, 2τ

where i, j denote coordinate indices on the M factor, α, β denote those on the SN factor, and the coordinate τ on R+ has index 0; gij evolves with τ by the backward Ricci flow (gij )τ = 2Rij , gαβ is the metric on SN of 1 constant curvature 2N . It turns out that the components of the curvature tensor of this metric coincide (modulo N −1 ) with the components of the matrix Harnack expression (and its traces), discovered by Hamilton [H 3]. One can also compute that all the components of the Ricci tensor are equal 12

to zero (mod N −1 ). The heat equation and the conjugate heat equation on ˜ for functions and volume M can be interpreted via Laplace equation on M forms respectively: u satisfies the heat equation on M iff u˜ (the extension of ˜ u = 0 mod N −1 ; similarly, u ˜ constant along the SN fibres) satisfies △˜ u to M N−1 ˜ u∗ = satisfies the conjugate heat equation on M iff u˜∗ = τ − 2 u˜ satisfies △˜ −1 ˜. 0 mod N on M ˜ isometric 6.2 Starting from g˜, we can also construct a metric g m on M, −1 to g˜ (mod N ), which corresponds to the backward m-preserving Ricci flow ( given by equations (1.1) with t-derivatives replaced by minus τ -derivatives, n dm = (4πτ )− 2 e−f dV ). To achieve this, first apply to g˜ a (small) diffeomorphism, mapping each point (xi , y α , τ ) into (xi , y α, τ (1 − 2f )); we would get a N m −1 metric g˜ , with components (mod N ) m = (1 − g˜ijm = g˜ij , g˜αβ

2f f m m m m )˜ gαβ , g˜00 = g˜00 − 2fτ − , g˜i0 = −∇i f, g˜iα = g˜α0 = 0; N τ

then apply a horizontal (that is, along the M factor) diffeomorphism to get g m satisfying (gijm )τ = 2(Rij + ∇i ∇j f ); the other components of g m become (mod N −1 ) m gαβ = (1 −

1 N 2f m m )˜ gαβ , g00 = g˜00 − |∇f |2 = ( − [τ (2△f − |∇f |2 + R) + f − n]), N τ 2 m m m gi0 = gα0 = giα =0

Note that the hypersurface τ =const in the metric g m has the volume form τ N/2 e−f times the canonical form on M and SN , and the scalar curvature of this hypersurface is τ1 ( N2 + τ (2△f − |∇f |2 + R) + f ) mod N −1 . Thus the entropy S multiplied by the inverse temperature β is essentially minus the total scalar curvature of this hypersurface. 6.3 Now we return to the metric g˜ and try to use its Ricci-flatness by interpreting the Bishop-Gromov relative volume comparison theorem. Con˜ g˜) centered at some point p where τ = 0. Then sider a metric ball in (M, clearly the shortest geodesic between p and an arbitrary point q is always orthogonal to the SN fibre. The length of such curve γ(τ ) can be computed as Z τ (q) r N + R + |γ˙ M (τ )|2 dτ 2τ 0 Z τ (q) p √ 3 1 = 2Nτ (q) + √ τ (R + |γ˙ M (τ )|2 )dτ + O(N − 2 ) 2N 0 13

R τ (q) √ Thus a shortest geodesic should minimize L(γ) = 0 τ (R + |γ˙ M (τ )|2 )dτ , an expression defined entirely in terms of M. Let L(qM ) denote the p corre˜ of radius 2Nτ (q) sponding infimum. It follows that a metric sphere in M centered at p is O(N −1 )-close τ = τ (q), and its volume pthe hypersurface R to 1 N can be computed as V (S ) M ( τ (q) − 2N L(x) + O(N −2 ))N dx, so the ratio p N +n n is just constant times N − 2 times of this volume to 2Nτ (q) Z n 1 L(x))dx + O(N −1 ) τ (q)− 2 exp(− p 2τ (q) M

The computation suggests that this integral, which we will call the reduced volume and denote by V˜ (τ (q)), should be increasing as τ decreases. A rigorous proof of this monotonicity is given in the next section. 6.4* The first geometric interpretation of Hamilton’s Harnack expressions was found by Chow and Chu [C-Chu 1,2]; they construct a potentially degenerate riemannian metric on M × R, which potentially satisfies the Ricci soliton equation; our construction is, in a certain sense, dual to theirs. Our formula for the reduced volume resembles the expression in Huisken monotonicity formula for the mean curvature flow [Hu]; however, in our case the monotonicity is in the opposite direction.

7

A comparison geometry approach to the Ricci flow

7.1 In this section we consider an evolving metric (gij )τ = 2Rij on a manifold M; we assume that either M is closed, or gij (τ ) are complete and have uniformly bounded curvatures. To each curve γ(τ ), 0 < τ1 ≤ τ ≤ τ2 , we associate its L-length Z τ2 √ τ (R(γ(τ )) + |γ(τ ˙ )|2 )dτ L(γ) = τ1

(of course, R(γ(τ )) and |γ(τ ˙ )|2 are computed using gij (τ )) Let X(τ ) = γ(τ ˙ ), and let Y (τ ) be any vector field along γ(τ ). Then the first variation formula can be derived as follows: δY (L) = 14

Z

τ2

τ1

√ Z

τ (< Y, ∇R > +2 < ∇Y X, X >)dτ = τ2

Z

τ2 τ1

√ τ (< Y, ∇R > +2 < ∇X Y, X >)dτ

√ d τ (< Y, ∇R > +2 < Y, X > −2 < Y, ∇X X > −4Ric(Y, X))dτ dτ τ1 Z τ2 τ2 √ √ 1 τ < Y, ∇R − 2∇X X − 4Ric(X, ·) − X > dτ = 2 τ < X, Y > τ1 + τ τ1 (7.1) Thus L-geodesics must satisfy =

1 1 ∇X X − ∇R + X + 2Ric(X, ·) = 0 2 2τ

(7.2)

Given two points p, q and τ2 > τ1 > 0, we can always find an L-shortest curve γ(τ ), τ ∈ [τ1 , τ2 ] between them, and every such L-shortest curve √ is Lgeodesic. It is easy to extend this to the case τ1 = 0; in this case τ X(τ ) has a limit as τ → 0. From now on we fix p and τ1 = 0 and denote by L(q, τ¯) the L-length of the L-shortest curve γ(τ ), 0 ≤ τ ≤ τ¯, connecting p and q. In the computations below we pretend that shortest L-geodesics between p and q are unique for all pairs (q, τ¯); if this is not the case, the inequalities that we obtain are still valid when understood in the barrier sense, or in the sense of distributions. √ The first variation formula (7.1) implies that ∇L(q, τ¯) = 2 τ¯X(¯ τ ), so 2 2 2 that |∇L| = 4¯ τ |X| = −4¯ τ R + 4¯ τ (R + |X| ). We can also compute √ √ √ Lτ¯ (q, τ¯) = τ¯(R + |X|2 )− < X, ∇L >= 2 τ¯R − τ¯(R + |X|2) To evaluate R + |X|2 we compute (using (7.2)) d (R(γ(τ )) + |X(τ )|2 ) = Rτ + < ∇R, X > +2 < ∇X X, X > +2Ric(X, X) dτ 1 1 = Rτ + R + 2 < ∇R, X > −2Ric(X, X) − (R + |X|2 ) τ τ 1 = −H(X) − (R + |X|2), (7.3) τ where H(X) is the Hamilton’s expression for the trace Harnack inequality (with t = −τ ). Hence, 3 1 τ¯ 2 (R + |X|2 )(¯ τ ) = −K + L(q, τ¯), 2

15

(7.4)

where K = K(γ, τ¯) denotes the integral a few times below. Thus we get

R τ¯ 0

3

τ 2 H(X)dτ, which we’ll encounter

√ 1 1 Lτ¯ = 2 τ¯R − L + K 2¯ τ τ¯

(7.5)

2 4 |∇L|2 = −4¯ τR + √ L − √ K (7.6) τ¯ τ¯ Finally we need to estimate the second variation of L. We compute Z τ¯ √ 2 δY (L) = τ (Y · Y · R + 2 < ∇Y ∇Y X, X > +2|∇Y X|2 )dτ 0

=

Z

τ¯ 0

√

τ (Y · Y · R + 2 < ∇X ∇Y Y, X > +2 < R(Y, X), Y, X > +2|∇X Y |2 )dτ

Now d < ∇Y Y, X >=< ∇X ∇Y Y, X > + < ∇Y Y, ∇X X > +2Y ·Ric(Y, X)−X·Ric(Y, Y ), dτ so, if Y (0) = 0 then δY2 (L) = 2 < ∇Y Y, X > Z

τ¯ 0

√

τ¯+

√ τ (∇Y ∇Y R + 2 < R(Y, X), Y, X > +2|∇X Y |2 + 2∇X Ric(Y, Y ) − 4∇Y Ric(Y, X))dτ, (7.7)

where we discarded the scalar product of −2∇Y Y with the left hand side of (7.2). Now fix the value of Y at τ = τ¯, assuming |Y (¯ τ )| = 1, and construct Y on [0, τ¯] by solving the ODE ∇X Y = −Ric(Y, ·) +

1 Y 2τ

We compute d 1 < Y, Y >= 2Ric(Y, Y ) + 2 < ∇X Y, Y >= < Y, Y >, dτ τ 16

(7.8)

so |Y (τ )|2 = ττ¯ , and in particular, Y (0) = 0. Making a substitution into (7.7), we get HessL (Y, Y ) ≤ Z τ¯ √ τ (∇Y ∇Y R + 2 < R(Y, X), Y, X > +2∇X Ric(Y, Y ) − 4∇Y Ric(Y, X) 0

1 2 )dτ +2|Ric(Y, ·)|2 − Ric(Y, Y ) + τ 2τ τ¯ To put this in a more convenient form, observe that

d Ric(Y (τ ), Y (τ )) = Ricτ (Y, Y ) + ∇X Ric(Y, Y ) + 2Ric(∇X Y, Y ) dτ

so

where

1 = Ricτ (Y, Y ) + ∇X Ric(Y, Y ) + Ric(Y, Y ) − 2|Ric(Y, ·)|2, τ Z τ¯ √ √ 1 τ H(X, Y )dτ , HessL (Y, Y ) ≤ √ − 2 τ¯Ric(Y, Y ) − τ¯ 0

(7.9)

H(X, Y ) = −∇Y ∇Y R−2 < R(Y, X)Y, X > −4(∇X Ric(Y, Y )−∇Y Ric(Y, X)) 1 −2Ricτ (Y, Y ) + 2|Ric(Y, ·)|2 − Ric(Y, Y ) τ is the Hamilton’s expression for the matrix Harnack inequality (with t = −τ ). Thus √ 1 n (7.10) △L ≤ −2 τ R + √ − K τ τ A field Y (τ ) along L-geodesic γ(τ ) is called L-Jacobi, if it is the derivative of a variation of γ among L-geodesics. For an L-Jacobi field Y with |Y (¯ τ )| = 1 we have d |Y |2 = 2Ric(Y, Y ) + 2 < ∇X Y, Y >= 2Ric(Y, Y ) + 2 < ∇Y X, Y > dτ Z τ¯ 1 1 1 1 = 2Ric(Y, Y ) + √ HessL (Y, Y ) ≤ − √ τ 2 H(X, Y˜ )dτ , (7.11) τ¯ τ¯ τ¯ 0 where Y˜ is obtained by solving ODE (7.8) with initial data Y˜ (¯ τ ) = Y (¯ τ ). ˜ Moreover, the equality in (7.11) holds only if Y is L-Jacobi and hence d |Y |2 = 2Ric(Y, Y ) + √1τ¯ HessL (Y, Y ) = τ1¯ . dτ 17

Now we can deduce an estimate for the jacobian J of the L-exponential map, given by LexpX (¯ τ ) = γ(¯ τ ), where γ(τ ) is the L-geodesic, starting at p √ ˙ ) as τ → 0. We obtain and having X as the limit of τ γ(τ n 1 3 d logJ(τ ) ≤ − τ¯− 2 K, dτ 2¯ τ 2

(7.12)

with equality only if 2Ric + √1τ¯ HessL = τ1¯ g. Let l(q, τ ) = 2√1 τ L(q, τ ) be the reduced distance. Then along an L-geodesic γ(τ ) we have (by (7.4)) d 1 1 1 3 l(τ ) = − l + (R + |X|2 ) = − τ¯− 2 K, dτ 2¯ τ 2 2 n

so (7.12) implies that τ − 2 exp(−l(τ ))J(τ ) is nonincreasing in τ along γ, and monotonicity is strict unless we are on a gradient shrinking soliton. Integrating M, we get monotonicity of the reduced volume function R over −n ˜ 2 V (τ ) = M τ exp(−l(q, τ ))dq. ( Alternatively, one could obtain the same monotonicity by integrating the differential inequality lτ¯ − △l + |∇l|2 − R +

n ≥ 0, 2¯ τ

(7.13)

which follows immediately from (7.5), (7.6) and (7.10). Note also a useful inequality l−n ≤ 0, (7.14) 2△l − |∇l|2 + R + τ¯ which follows from (7.6), (7.10).) ¯ τ ) = 2√τ L(q, τ ), then from (7.5), On the other hand, if we denote L(q, (7.10) we obtain ¯ τ¯ + △L ¯ ≤ 2n L (7.15) ¯ τ¯) − 2n¯ Therefore, the minimum of L(·, τ is nonincreasing, so in particular, the minimum of l(·, τ¯) does not exceed n2 for each τ¯ > 0. (The lower bound for l is much easier to obtain since the evolution equation Rτ = −△R − 2|Ric|2 implies R(·, τ ) ≥ − 2(τ0n−τ ) , whenever the flow exists for τ ∈ [0, τ0 ].) 7.2 If the metrics gij (τ ) have nonnegative curvature operator, then Hamilton’s differential Harnack inequalities hold, and one can say more about the behavior of l. Indeed, in this case, if the solution is defined for τ ∈ [0, τ0 ], then H(X, Y ) ≥ −Ric(Y, Y )( τ1 + τ01−τ ) ≥ −R( τ1 + τ01−τ )|Y |2 and 18

H(X) ≥ −R( τ1 + τ01−τ ). Therefore, whenever τ is bounded away from τ0 (say, τ ≤ (1 − c)τ0 , c > 0), we get (using (7.6), (7.11)) |∇l|2 + R ≤

Cl , τ

(7.16)

and for L-Jacobi fields Y 1 d log|Y |2 ≤ (Cl + 1) dτ τ

(7.17)

7.3 As the first application of the comparison inequalities above, let us give an alternative proof of a weakened version of the no local collapsing theorem 4.1. Namely, rather than assuming |Rm|(x, tk ) ≤ rk−2 for x ∈ Bk , we require |Rm|(x, t) ≤ rk−2 whenever x ∈ Bk , tk − rk2 ≤ t ≤ tk . Then the 1 proof can go as follows: let τk (t) = tk −t, p = pk , ǫk = rk−1 V ol(Bk ) n . We claim n that V˜k (ǫk rk2 ) < 3ǫk2 when k is large. Indeed, using the L-exponential map we can integrate over Tp M rather than M; the vectors in Tp M of length at most 1 1 −2 ǫ 2 k

give rise to L-geodesics, which can not escape from Bk in time ǫk rk2 , so n

their contribution to the reduced volume does not exceed 2ǫk2 ; on the other −1

hand, the contribution of the longer vectors does not exceed exp(− 12 ǫk 2 ) by the jacobian comparison theorem. However, V˜k (tk ) (that is, at t = 0) stays bounded away from zero. Indeed, since min lk (·, tk − 12 T ) ≤ n2 , we can pick a point qk , where it is attained, and obtain a universal upper bound on lk (·, tk ) by considering only curves γ with γ(tk − 12 T ) = qk , and using the fact that all geometric quantities in gij (t) are uniformly bounded when t ∈ [0, 21 T ]. Since the monotonicity of the reduced volume requires V˜k (tk ) ≤ V˜k (ǫk rk2 ), this is a contradiction. A similar argument shows that the statement of the corollary in 4.2 can be strengthened by adding another property of the ancient solution, obtained as a blow-up limit. Namely, we may claim that if, say, this solution is defined for t ∈ (−∞, 0), then for any point p and any t0 > 0, the reduced volume function V˜ (τ ), constructed using p and τ (t) = t0 − t, is bounded below by κ. 7.4* The computations in this section are just natural modifications of those in the classical variational theory of geodesics that can be found in any textbook on Riemannian geometry; an even closer reference is [L-Y], where they use ”length”, associated to a linear parabolic equation, which is pretty much the same as in our case. 19

8

No local collapsing theorem II

8.1 Let us first formalize the notion of local collapsing, that was used in 7.3. Definition. A solution to the Ricci flow (gij )t = −2Rij is said to be κ-collapsed at (x0 , t0 ) on the scale r > 0 if |Rm|(x, t) ≤ r −2 for all (x, t) satisfying distt0 (x, x0 ) < r and t0 − r 2 ≤ t ≤ t0 , and the volume of the metric ball B(x0 , r 2 ) at time t0 is less than κrn . 8.2 Theorem. For any A > 0 there exists κ = κ(A) > 0 with the following property. If gij (t) is a smooth solution to the Ricci flow (gij )t = −2Rij , 0 ≤ t ≤ r02 , which has |Rm|(x, t) ≤ r0−2 for all (x, t), satisfying dist0 (x, x0 ) < r0 , and the volume of the metric ball B(x0 , r0 ) at time zero is at least A−1 r0n , then gij (t) can not be κ-collapsed on the scales less than r0 at a point (x, r02 ) with distr02 (x, x0 ) ≤ Ar0 . Proof. By scaling we may assume r0 = 1; we may also assume dist1 (x, x0 ) = A. Let us apply the constructions of 7.1 choosing p = x, τ (t) = 1 − t. Arguing as in 7.3, we see that if our solution is collapsed at x on the scale r ≤ 1, then the reduced volume V˜ (r 2 ) must be very small; on the other hand, V˜ (1) can 1 is large. not be small unless min l(x, 12 ) over x satisfying dist 1 (x, x0 ) ≤ 10 2 ¯ Thus all we need is to estimate l, or equivalently L, in that ball. Recall that ¯ satisfies the differential inequality (7.15). In order to use it efficiently in L a maximum principle argument, we need first to check the following simple assertion. 8.3 Lemma. Suppose we have a solution to the Ricci flow (gij )t = −2Rij . (a) Suppose Ric(x, t0 ) ≤ (n − 1)K when distt0 (x, x0 ) < r0 . Then the distance function d(x, t) = distt (x, x0 ) satisfies at t = t0 outside B(x0 , r0 ) the differential inequality 2 dt − △d ≥ −(n − 1)( Kr0 + r0−1 ) 3 (the inequality must be understood in the barrier sense, when necessary) (b) (cf. [H 4,§17]) Suppose Ric(x, t0 ) ≤ (n − 1)K when distt0 (x, x0 ) < r0 , or distt0 (x, x1 ) < r0 . Then d 2 distt (x0 , x1 ) ≥ −2(n − 1)( Kr0 + r0−1 ) at t = t0 dt 3 R Proof of Lemma. (a) Clearly, dt (x) = γ −Ric(X, X), where γ is the shortest geodesic between x0 and X is its unit tangent vector, On the other Pn−1x and ′′ hand, △d ≤ k=1 sYk (γ), where Yk are vector fields along γ, vanishing at 20

x0 and forming an orthonormal basis at x when complemented by X, and s′′Yk (γ) denotes the second variation along Yk of the length of γ. Take Yk to be parallel between x and x1 , and linear between x1 and x0 , where d(x1 , t0 ) = r0 . Then △d ≤ =

Z

γ

n−1 X

s′′Yk (γ)

=

Z

d(x,t0 )

r0

k=1

−Ric(X, X)+

Z

0

−Ric(X, X)ds+

r0

(Ric(X, X)(1 −

Z

0

r0

(

s2 n−1 (−Ric(X, X)) + )ds 2 r0 r02

n−1 2 s2 )+ )ds ≤ dt +(n−1)( Kr0 +r0−1 ) 2 2 r0 r0 3

The proof of (b) is similar. Continuing the proof of theorem, apply the maximum principle to the ¯ 1 − t) + 2n + 1), where d(y, t) = function h(y, t) = φ(d(y, t) − A(2t − 1))(L(y, 1 ), and distt (x, x0 ), and φ is a function of one variable, equal 1 on (−∞, 20 1 1 rapidly increasing to infinity on ( 20 , 10 ), in such a way that 2(φ′ )2 /φ − φ′′ ≥ (2A + 100n)φ′ − C(A)φ,

(8.1)

¯ + 2n + 1 ≥ 1 for t ≥ 1 by the for some constant C(A) < ∞. Note that L 2 remark in the very end of 7.1. Clearly, min h(y, 1) ≤ h(x, 1) = 2n + 1. On 1 the other hand, min h(y, 12 ) is achieved for some y satisfying d(y, 12 ) ≤ 10 . Now we compute ′′ ¯ ¯ > +(L ¯ t −△L)φ ¯ 2h = (L+2n+1)(−φ +(dt −△d−2A)φ′ )−2 < ∇φ∇L (8.2)

¯ + 2n + 1)∇φ + φ∇L ¯ ∇h = (L

(8.3)

At a minimum point of h we have ∇h = 0, so (8.2) becomes ¯ + 2n + 1)(−φ′′ + (dt − △d − 2A)φ′ + 2(φ′ )2 /φ) + (L ¯ t − △L)φ ¯ 2h = (L (8.4) 1 1 whenever φ′ 6= 0, and since Ric ≤ n − 1 in B(x0 , 20 ), Now since d(y, t) ≥ 20 we can apply our lemma (a) to get dt − △d ≥ −100(n − 1) on the set where φ′ 6= 0. Thus, using (8.1) and (7.15), we get

¯ + 2n + 1)C(A)φ − 2nφ ≥ −(2n + C(A))h 2h ≥ −(L This implies that min h can not decrease too fast, and we get the required estimate. 21

9

Differential Harnack inequality for solutions of the conjugate heat equation

9.1 Proposition. Let gij (t) be a solution to the Ricci flow (gij )t = −2Rij , 0 ≤ n t ≤ T, and let u = (4π(T − t))− 2 e−f satisfy the conjugate heat equation 2∗ u = −ut − △u + Ru = 0. Then v = [(T − t)(2△f − |∇f |2 + R) + f − n]u satisfies 1 2∗ v = −2(T − t)|Rij + ∇i ∇j f − gij |2 (9.1) 2(T − t) Proof. Routine computation. Clearly, this proposition immediately implies the monotonicity formula (3.4); its advantage over (3.4) shows up when one has to work locally. 9.2 Corollary. Under the same assumptions, on a closed manifold M,or whenever the application of the maximum principle can be justified, min v/u is nondecreasing in t. 9.3 Corollary. Under the same assumptions, if u tends to a δ-function as t → T, then v ≤ 0 for all t < T. Proof. If h satisfies the ordinary heat Requation ht = △h R with respect to d d the evolving metric gij (t), then we have dt hu = 0 and dt hvR≥ 0. Thus we only need to check that for everywhere positive h the limit of hv as t → T is nonpositive. But it is easy to see, that this limit is in fact zero. 9.4 Corollary. Under assumptions of the previous corollary, for any smooth curve γ(t) in M holds −

1 1 d 2 f (γ(t), t) ≤ (R(γ(t), t) + |γ(t)| ˙ )− f (γ(t), t) dt 2 2(T − t)

(9.2)

Proof. From the evolution equation ft = −△f + |∇f |2 − R + 2(Tn−t) and v ≤ 0 we get ft + 12 R− 12 |∇f |2 − 2(Tf−t) ≥ 0. On the other hand,− dtd f (γ(t), t) = ˙ 2 . Summing these two inequalities, −ft − < ∇f, γ(t) ˙ >≤ −ft + 21 |∇f |2 + 12 |γ| we get (9.2). 9.5 Corollary. If under assumptions of the previous corollary, p is the point where the limit δ-function is concentrated, then f (q, t) ≤ l(q, T − t), where l is the reduced distance, defined in 7.1, using p and τ (t) = T − t. 22

Proof. Use (7.13) in the form 2∗ exp(−l) ≤ 0. 9.6 Remark. Ricci flow can be characterized among all other evolution equations by the infinitesimal behavior of the fundamental solutions of the conjugate heat equation. Namely, suppose we have a riemannian metric gij (t) evolving with time according to an equation (gij )t = Aij (t). Then we have ∂ ∂ − △ and its conjugate 2∗ = − ∂t − △ − 21 A, so that theR heat operator 2 = ∂t R d uv = ((2u)v − u(2∗ v)). (Here A = g ij Aij ) Consider the fundamental dt n solution u = (−4πt)− 2 e−f for 2∗ , starting as δ-function at some point (p, 0). R ¯ Then for general Aij the function (2f¯+ ft )(q, t), where f¯ = f − f u, is of the order O(1) for (q, t) near (p, 0). The Ricci flow Aij = −2Rij is characterized ¯ by the condition (2f¯ + ft )(q, t) = o(1); in fact, it is O(|pq|2 + |t|) in this case. 9.7* Inequalities of the type of (9.2) are known as differential Harnack inequalities; such inequality was proved by Li and Yau [L-Y] for the solutions of linear parabolic equations on riemannian manifolds. Hamilton [H 7,8] used differential Harnack inequalities for the solutions of backward heat equation on a manifold to prove monotonicity formulas for certain parabolic flows. A local monotonicity formula for mean curvature flow making use of solutions of backward heat equation was obtained by Ecker [E 2].

10

Pseudolocality theorem

10.1 Theorem. For every α > 0 there exist δ > 0, ǫ > 0 with the following property. Suppose we have a smooth solution to the Ricci flow (gij )t = −2Rij , 0 ≤ t ≤ (ǫr0 )2 , and assume that at t = 0 we have R(x) ≥ −r0−2 and V ol(∂Ω)n ≥ (1 − δ)cn V ol(Ω)n−1 for any x, Ω ⊂ B(x0 , r0 ), where cn is the euclidean isoperimetric constant. Then we have an estimate |Rm|(x, t) ≤ αt−1 + (ǫr0 )−2 whenever 0 < t ≤ (ǫr0 )2 , d(x, t) = distt (x, x0 ) < ǫr0 . Thus, under the Ricci flow, the almost singular regions (where curvature is large) can not instantly significantly influence the almost euclidean regions. Or , using the interpretation via renormalization group flow, if a region looks trivial (almost euclidean) on higher energy scale, then it can not suddenly become highly nontrivial on a slightly lower energy scale. Proof. It is an argument by contradiction. The idea is to pick a point ¯ (¯ x, t) not far from (x0 , 0) and consider the solution u to the conjugate heat equation, starting as δ-function at (¯ x, t¯), and the corresponding nonpositive function v as in 9.3. If the curvatures at (¯ x, t¯) are not small compared to 23

R t¯−1 and are larger than at nearby points, then one can show that v at time t is bounded away from zero for (small) time intervals t¯ − t of the order of R −1 |Rm| (¯ x, t¯). By monotonicity we conclude that v is bounded away from zero at t = 0. In fact, using (9.1) and an appropriate cut-off function, we can show that at t = 0 already the integral of v over B(x0 , r) is bounded away from zero, whereas the integral of u over this ball is close to 1, where r can be made as small as we like compared to r0 . Now using the control over the scalar curvature and isoperimetric constant in B(x0 r0 ), we can obtain a contradiction to the logarithmic Sobolev inequality. Now let us go into details. By scaling assume that r0 = 1. We may also 1 . From now on we fix α and denote by assume that α is small, say α < 100n Mα the set of pairs (x, t), such that |Rm|(x, t) ≥ αt−1 . Claim 1.For any A > 0, if gij (t) solves the Ricci flow equation on 0 ≤ 1 , and |Rm|(x, t) > αt−1 + ǫ−2 for some (x, t), satisfying 0 ≤ t ≤ ǫ2 , Aǫ < 100n 2 t ≤ ǫ , d(x, t) < ǫ, then one can find (¯ x, t¯) ∈ Mα , with 0 < t¯ ≤ ǫ2 , d(¯ x, t¯) < (2A + 1)ǫ, such that |Rm|(x, t) ≤ 4|Rm|(¯ x, t¯), (10.1) whenever

1

(x, t) ∈ Mα , 0 < t ≤ t¯, d(x, t) ≤ d(¯ x, t¯) + A|Rm|− 2 (¯ x, t¯)

(10.2)

Proof of Claim 1. We construct (¯ x, t¯) as a limit of a (finite) sequence (xk , tk ), defined in the following way. Let (x1 , t1 ) be an arbitrary point, satisfying 0 < t1 ≤ ǫ2 , d(x1 , t1 ) < ǫ, |Rm|(x1 , t1 ) ≥ αt−1 + ǫ−2 . Now if (xk , tk ) is already constructed, and if it can not be taken for (¯ x, t¯), because there is some (x, t) satisfying (10.2), but not (10.1), then take any such (x, t) for (xk+1 , tk+1 ). Clearly, the sequence, constructed in such a way, satisfies |Rm|(xk , tk ) ≥ 4k−1|Rm|(x1 , t1 ) ≥ 4k−1ǫ−2 , and therefore, d(xk , tk ) ≤ (2A + 1)ǫ. Since the solution is smooth, the sequence is finite, and its last element fits. Claim 2. For (¯ x, t¯), constructed above, (10.1) holds whenever 1 1 1 t¯ − αQ−1 ≤ t ≤ t¯, distt¯(x, x¯) ≤ AQ− 2 , 2 10

(10.3)

where Q = |Rm|(¯ x, t¯). Proof of Claim 2. We only need to show that if (x, t) satisfies (10.3), then it must satisfy (10.1) or (10.2). Since (¯ x, t¯) ∈ Mα , we have Q ≥ αt¯−1 , so 1 1 −1 t¯− 2 αQ ≥ 2 t¯. Hence, if (x, t) does not satisfy (10.1), it definitely belongs to 24

1 1 AQ− 2 . On the other Mα . Now by the triangle inequality, d(x, t¯) ≤ d(¯ x, t¯) + 10 hand, using lemma 8.3(b) we see that, as t decreases from t¯ to t¯ − 12 αQ−1 , 1 the point x can not escape from the ball of radius d(¯ x, t¯) + AQ− 2 centered at x0 . Continuing the proof of the theorem, and arguing by contradiction, take sequences ǫ → 0, δ → 0 and solutions gij (t), violating the statement; by reducing ǫ, we’ll assume that

|Rm|(x, t) ≤ αt−1 + 2ǫ−2 whenever 0 ≤ t ≤ ǫ2 and d(x, t) ≤ ǫ

(10.4)

1 → ∞, construct (¯ x, t¯), and consider solutions u = (4π(t¯ − Take A = 100nǫ n − 2 −f x, t¯), t)) e of the conjugate heat equation, starting from δ-functions at (¯ and corresponding nonpositive functions v. −1 ¯ ˜ ¯ 1 Claim R 3.As ǫ, δ → 0, one can find times t ∈ [t − 2 αQ , t], such that the integral B v stays bounded away from zero, where B is the ball at time t˜ of p radius t¯ − t˜ centered at x¯. Proof of Claim 3(sketch). The statement is invariant under scaling, so we can try to take a limit of scalings of gij (t) at points (¯ x, t¯) with factors ¯ Q. If the injectivity radii of the scaled metrics at (¯ x, t) are bounded away from zero, then a smooth limit exists, it is complete and has |Rm|(¯ x, t¯) = 1 1 and |Rm|(x, t) ≤ 4 when t¯ − 2 α ≤ t ≤ t¯. It is not hard to show that the fundamental solutions u of the conjugate heat equation converge to such a R solution on the limit manifold. But on the limit manifold, B v can not be zero for t˜ = t¯ − 21 α, since the evolution equation (9.1) would imply in this case that the limit is a gradient shrinking soliton, and this is incompatible with |Rm|(¯ x, t¯) = 1. If the injectivity radii of the scaled metrics tend to zero, then we can change the scaling factor, to make the scaled metrics converge to a flat manifold with finite injectivity radius; in this case it is not hard to choose t˜ in R such a way that B v → −∞. R The positive lower bound for − B v will be denoted by β. Our next goal is to construct an appropriate cut-off function. We choose √ ˜ ˜ t) = d(y, t) + 200n t, and φ is ), where d(y, it in the form h(y, t) = φ( d(y,t) 10Aǫ a smooth function of one variable, equal one on (−∞, 1] and decreasing to zero on [1, 2]. Clearly, h vanishes at t = 0 outside B(x0 , 20Aǫ); on the other hand, it is equal to one near (¯ x, t¯). 1 100n 100n 1 ′′ √ ≥ 0 Now 2h = 10Aǫ (dt − △d + √t )φ′ − (10Aǫ) 2 φ . Note that dt − △t + t ′ on the set where φ 6= 0 − this follows from the lemma 8.3(a) and our

25

assumption (10.4). WeR may alsoRchoose φ so that φ′′R≥ −10φ, (φ′R)2 ≤ 10φ. 1 Now we can compute ( M hu)t = M (2h)u ≤ (Aǫ) 2 , so M hu |t=0 ≥ M hu |t=t¯ R R R 1 t¯ −2 −hv, − (Aǫ)2 ≥ 1 − A . Also, by (9.1), ( M −hv)t ≤ M −(2h)v ≤ (Aǫ) 2 M R t¯ −2 so by Claim 3, − M hv |t=0 ≥ βexp(− (Aǫ)2 ) ≥ β(1 − A ). From now on we”ll work at t = 0 only. Let u˜ = hu and correspondingly ˜ f = f − logh. Then Z Z −2 β(1 − A ) ≤ − hv = [(−2△f + |∇f |2 − R)t¯ − f + n]hu M

=

Z

M

R

˜2

M

[−t¯|∇f| − f˜ + n]˜ u+ ≤

Z

M

Z

M

[t¯(|∇h|2 /h − Rh) − hlogh]u

[−t¯|∇f˜|2 − f˜ − n]˜ u + A−2 + 100ǫ2

( Note that M −uh log h does Rnot exceed theR integral of u over ¯ ≥ 1 − A−2 , B(x0 , 20Aǫ)\B(x0 , 10Aǫ), and B(x0 ,10Aǫ) u ≥ M hu ¯ = φ( d˜ )) where h 5Aǫ Now scaling the metric by the factor 21 t¯−1 and sending ǫ, δ to zero, we get a sequence of metric balls with radii going to infinity, and a sequence R n of compactly supported nonnegative functions u = (2π)− 2 e−f with u → 1 R and [− 21 |∇f |2 − f + n]u bounded away from zero by a positive constant. We also have isoperimetric inequalities with the constants tending to the euclidean one. This set up is in conflict with the Gaussian logarithmic Sobolev inequality, as can be seen by using spherical symmetrization. 10.2 Corollary(from the proof) Under the same √ assumptions, √ n we also 2 have at time t, 0 < t ≤ (ǫr0 ) , an estimate V olB(x, t) ≥ c t for x ∈ B(x0 , ǫr0 ), where c = c(n) is a universal constant. 10.3 Theorem. There exist ǫ, δ > 0 with the following property. Suppose gij (t) is a smooth solution to the Ricci flow on [0, (ǫr0 )2 ], and assume that at t = 0 we have |Rm|(x) ≤ r0−2 in B(x0 , r0 ), and V olB(x0 , r0 ) ≥ (1 − δ)ωn r0n , where ωn is the volume of the unit ball in Rn . Then the estimate |Rm|(x, t) ≤ (ǫr0 )−2 holds whenever 0 ≤ t ≤ (ǫr0 )2 , distt (x, x0 ) < ǫr0 . The proof is a slight modification of the proof of theorem 10.1, and is left to the reader. A natural question is whether the assumption on the volume of the ball is superfluous. 10.4 Corollary(from 8.2, 10.1, 10.2) There exist ǫ, δ > 0 and for any A > 0 there exists κ(A) > 0 with the following property. If gij (t) is a 26

smooth solution to the Ricci flow on [0, (ǫr0 )2 ], such that at t = 0 we have R(x) ≥ −r0−2 , V ol(∂Ω)n ≥ (1 − δ)cn V ol(Ω)n−1 for any x, Ω ⊂ B(x0 , r0 ), and (x, t) satisfies A−1 (ǫr0 )2 ≤ t ≤ (ǫr0 )2 , distt (x, x√0 ) ≤ Ar0 , then gij (t) can not be κ-collapsed at (x, t) on the scales less than t. 10.5 Remark. It is straightforward to get from 10.1 a version of the Cheeger diffeo finiteness theorem for manifolds, satisfying our assumptions on scalar curvature and isoperimetric constant on each ball of some fixed radius r0 > 0. In particular, these assumptions are satisfied (for some controllably smaller r0 ), if we assume a lower bound for Ric and an almost euclidean lower bound for the volume of the balls of radius r0 . (this follows from the LevyGromov isoperimetric inequality); thus we get one of the results of Cheeger and Colding [Ch-Co] under somewhat weaker assumptions. 10.6* Our pseudolocality theorem is similar in some respect to the results of Ecker-Huisken [E-Hu] on the mean curvature flow.

11

Ancient solutions with nonnegative curvature operator and bounded entropy

11.1. In this section we consider smooth solutions to the Ricci flow (gij )t = −2Rij , −∞ < t ≤ 0, such that for each t the metric gij (t) is a complete non-flat metric of bounded curvature and nonnegative curvature operator. Hamilton discovered a remarkable differential Harnack inequality for such solutions; we need only its trace version Rt + 2 < X, ∇R > +2Ric(X, X) ≥ 0

(11.1)

and its corollary, Rt ≥ 0. In particular, the scalar curvature at some time t0 ≤ 0 controls the curvatures for all t ≤ t0 . We impose one more requirement on the solutions; namely, we fix some κ > 0 and require that gij (t) be κ-noncollapsed on all scales (the definitions 4.2 and 8.1 are essentially equivalent in this case). It is not hard to show that this requirement is equivalent to a uniform bound on the entropy S, defined as in 5.1 using an arbitrary fundamental solution to the conjugate heat equation. 11.2. Pick an arbitrary point (p, t0 ) and define V˜ (τ ), l(q, τ ) as in 7.1, for τ (t) = t0 − t. Recall that for each τ > 0 we can find q = q(τ ), such that l(q, τ ) ≤ n2 . 27

Proposition.The scalings of gij (t0 − τ ) at q(τ ) with factors τ −1 converge along a subsequence of τ → ∞ to a non-flat gradient shrinking soliton. Proof (sketch). It is not hard to deduce from (7.16) that for any ǫ > 0 one can find δ > 0 such that both l(q, τ ) and τ R(q, t0 − τ ) do not exceed τ )) ≤ ǫ−1 τ¯ for some τ¯ > 0. δ −1 whenever 12 τ¯ ≤ τ ≤ τ¯ and dist2t0 −¯τ (q, q(¯ Therefore, taking into account the κ-noncollapsing assumption, we can take ¯ ij . We may gij )τ = 2R a blow-down limit, say g¯ij (τ ), defined for τ ∈ ( 21 , 1), (¯ assume also that functions l tend to a locally Lipschitz function ¯l, satisfying (7.13),(7.14) in the sense of distributions. Now, since V˜ (τ ) is nonincreasing and bounded away from zero (because the scaled metrics are not collapsed near q(τ )) the limit function V¯ (τ ) must be a positive constant; this constant n is strictly less than limτ →0 V˜ (τ ) = (4π) 2 , since gij (t) is not flat. Therefore, on the one hand, (7.14) must become an equality, hence ¯l is smooth, and on the other hand, by the description of the equality case in (7.12), g¯ij (τ ) must ¯ ij + ∇ ¯ i∇ ¯ j ¯l − 1 g¯ij = 0. If this soliton be a gradient shrinking soliton with R 2τ is flat, then ¯l is uniquely determined by the equality in (7.14), and it turns n out that the value of V¯ is exactly (4π) 2 , which was ruled out. 11.3 Corollary. There is only one oriented two-dimensional solution, satisfying the assumptions stated in 11.1, - the round sphere. Proof. Hamilton [H 10] proved that round sphere is the only non-flat oriented nonnegatively curved gradient shrinking soliton in dimension two. Thus, the scalings of our ancient solution must converge to a round sphere. However, Hamilton [H 10] has also shown that an almost round sphere is getting more round under Ricci flow, therefore our ancient solution must be round. 11.4. Recall that for any non-compact complete riemannian manifold M of nonnegative Ricci curvature and a point p ∈ M, the function V olB(p, r)r −n is nonincreasing in r > 0; therefore, one can define an asymptotic volume ratio V as the limit of this function as r → ∞. Proposition.Under assumptions of 11.1, V = 0 for each t. Proof. Induction on dimension. In dimension two the statement is vacuous, as we have just shown. Now let n ≥ 3, suppose that V > 0 for some t = t0 , and consider the asymptotic scalar curvature ratio R = lim supR(x, t0 )d2 (x) as d(x) → ∞. (d(x) denotes the distance, at time t0 , from x to some fixed point x0 ) If R = ∞, then we can find a sequence of points xk and radii rk > 0, such that rk /d(xk ) → 0, R(xk )rk2 → ∞, and 28

R(x) ≤ 2R(xk ) whenever x ∈ B(xk , rk ). Taking blow-up limit of gij (t) at (xk , t0 ) with factors R(xk ), we get a smooth non-flat ancient solution, satisfying the assumptions of 11.1, which splits off a line (this follows from a standard argument based on the Aleksandrov-Toponogov concavity). Thus, we can do dimension reduction in this case (cf. [H 4,§22]). If 0 < R < ∞, then a similar argument gives a blow-up limit in a ball of finite radius; this limit has the structure of a non-flat metric cone. This is ruled out by Hamilton’s strong maximum principle for nonnegative curvature operator. Finally, if R = 0, then (in dimensions three and up) it is easy to see that the metric is flat. 11.5 Corollary. For every ǫ > 0 there exists A < ∞ with the following property. Suppose we have a sequence of ( not necessarily complete) solutions (gk )ij (t) with nonnegative curvature operator, defined on Mk × [tk , 0], such that for each k the ball B(xk , rk ) at time t = 0 is compactly contained in Mk , 1 R(x, t) ≤ R(xk , 0)√= Qk for all√(x, t), tk Qk → −∞, rk2 Qk → ∞ as k → ∞. 2 Then V olB(xk , A/ Qk ) ≤ ǫ(A/ Qk )n at t = 0 if k is large enough. Proof. Assuming the contrary, we may take a blow-up limit (at (xk , 0) with factors Qk ) and get a non-flat ancient solution with positive asymptotic volume ratio at t = 0, satisfying the assumptions in 11.1, except, may be, the κ-noncollapsing assumption. But if that assumption is violated for each κ > 0, then V(t) is not bounded away from zero as t → −∞. However, this is impossible, because it is easy to see that V(t) is nonincreasing in t. (Indeed, Ricci √ flow decreases the volume and does not decrease the distances faster than C R per time unit, by lemma 8.3(b)) Thus, κ-noncollapsing holds for some κ > 0, and we can apply the previous proposition to obtain a contradiction. 11.6 Corollary. For every w > 0 there exist B = B(w) < ∞, C = C(w) < ∞, τ0 = τ0 (w) > 0, with the following properties. (a) Suppose we have a (not necessarily complete) solution gij (t) to the Ricci flow, defined on M × [t0 , 0], so that at time t = 0 the metric ball B(x0 , r0 ) is compactly contained in M. Suppose that at each time t, t0 ≤ t ≤ 0, the metric gij (t) has nonnegative curvature operator, and V olB(x0 , r0 ) ≥ wr0n . Then we have an estimate R(x, t) ≤ Cr0−2 + B(t − t0 )−1 whenever distt (x, x0 ) ≤ 14 r0 . 29

(b) If, rather than assuming a lower bound on volume for all t, we assume it only for t = 0, then the same conclusion holds with −τ0 r02 in place of t0 , provided that −t0 ≥ τ0 r02 .

Proof. By scaling assume r0 = 1. (a) Arguing by contradiction, consider a sequence of B, C → ∞, of solutions gij (t) and points (x, t), such that distt (x, x0 ) ≤ 41 and R(x, t) > C + B(t − t0 )−1 . Then, arguing as in the proof of claims 1,2 in 10.1, we can find a point (¯ x, t¯), satisfying distt¯(¯ x, x0 ) < 1 −1 ′ ′ ¯ ¯ , Q = R(¯ x, t) > C + B(t − t0 ) , and such that R(x , t ) ≤ 2Q whenever 3 1 −1 ¯ t − AQ ≤ t′ ≤ t¯, distt¯(x′ , x¯) < AQ− 2 , where A tends to infinity with B, C. Applying the previous corollary at (¯ x, t¯) and using the relative volume comparison, we get a contradiction with the assumption involving w. (b) Let B(w), C(w) be good for (a). We claim that B = B(5−n w), C = C(5−n w) are good for (b) , for an appropriate τ0 (w) > 0. Indeed, let gij (t) be a solution with nonnegative curvature operator, such that V olB(x0 , 1) ≥ w at t = 0, and let [−τ, 0] be the maximal time interval, where the assumption of (a) still holds, with 5−n w in place of w and with −τ in place of t0 . Then at time t = −τ we must have V olB(x0 , 1) ≤ 5−n w. On the other hand, from lemma 8.3 (b) we see that the ball B(x0 , 41 ) at time t = −τ contains the ball √ √ B(x0 , 14 − 10(n − 1)(τ C + 2 Bτ )) at time t = 0, and the volume of the former is at least as large as the volume of the latter. Thus, it is enough to choose τ0 = τ0 (w) in such a way that the radius of the latter ball is > 51 . Clearly, the proof also works if instead of assuming that curvature operator is nonnegative, we assumed that it is bounded below by −r0−2 in the (time-dependent) metric ball of radius r0 , centered at x0 . 11.7. From now on we restrict our attention to oriented manifolds of dimension three. Under the assumptions in 11.1, the solutions on closed manifolds must be quotients of the round S3 or S2 × R - this is proved in the same way as in two dimensions, since the gradient shrinking solitons are known from the work of Hamilton [H 1,10]. The noncompact solutions are described below. Theorem.The set of non-compact ancient solutions , satisfying the assumptions of 11.1, is compact modulo scaling. That is , from any sequence of such solutions and points (xk , 0) with R(xk , 0) = 1, we can extract a smoothly converging subsequence, and the limit satisfies the same conditions. Proof. To ensure a converging subsequence it is enough to show that whenever R(yk , 0) → ∞, the distances at t = 0 between xk and yk go to infinity as well. Assume the contrary. Define a sequence zk by the requirement 30

that zk be the closest point to xk (at t = 0), satisfying R(zk , 0)dist20 (xk , zk ) = 1. 1 We claim that R(z, 0)/R(zk , 0) is uniformly bounded for z ∈ B(zk , 2R(zk , 0)− 2 ). Indeed, otherwise we could show, using 11.5 and relative volume comparison 1 in nonnegative curvature, that the balls B(zk , R(zk , 0)− 2 ) are collapsing on the scale of their radii. Therefore, using the local derivative estimate, due to W.-X.Shi (see [H 4,§13]), we get a bound on Rt (zk , t) of the order of R2 (zk , 0). Then we can compare 1 = R(xk , 0) ≥ cR(zk , −cR−1 (zk , 0)) ≥ cR(zk , 0) for some small c > 0, where the first inequality comes from the Harnack inequality, obtained by integrating (11.1). Thus, R(zk , 0) are bounded. But now the existence of the sequence yk at bounded distance from xk implies, via 11.5 and relative volume comparison, that balls B(xk , c) are collapsing - a contradiction. It remains to show that the limit has bounded curvature at t = 0. If this was not the case, then we could find a sequence yi going to infinity, such 1 that R(yi , 0) → ∞ and R(y, 0) ≤ 2R(yi, 0) for y ∈ B(yi, Ai R(yi , 0)− 2 ), Ai → ∞. Then the limit of scalings at (yi, 0) with factors R(yi, 0) satisfies the assumptions in 11.1 and splits off a line. Thus by 11.3 it must be a round infinite cylinder. It follows that for large i each yi is contained in a round 1 cylindrical ”neck” of radius ( 12 R(yi , 0))− 2 → 0, - something that can not happen in an open manifold of nonnegative curvature. 11.8. Fix ǫ > 0. Let gij (t) be an ancient solution on a noncompact oriented three-manifold M, satisfying the assumptions in 11.1. We say that a point x0 ∈ M is the center of an ǫ-neck, if the solution gij (t) in the set {(x, t) : −(ǫQ)−1 < t ≤ 0, dist20 (x, x0 ) < (ǫQ)−1 }, where Q = R(x0 , 0), is, after scaling with factor Q, ǫ-close (in some fixed smooth topology) to the corresponding subset of the evolving round cylinder, having scalar curvature one at t = 0. Corollary (from theorem 11.7 and its proof) For any ǫ > 0 there exists C = C(ǫ, κ) > 0, such that if gij (t) satisfies the assumptions in 11.1, and Mǫ denotes the set of points in M, which are not centers of ǫ-necks, then 1 Mǫ is compact and moreover, diamMǫ ≤ CQ− 2 , and C −1 Q ≤ R(x, 0) ≤ CQ whenever x ∈ Mǫ , where Q = R(x0 , 0) for some x0 ∈ ∂Mǫ . 11.9 Remark. It can be shown that there exists κ0 > 0, such that if an ancient solution on a noncompact three-manifold satisfies the assumptions in 11.1 with some κ > 0, then it would satisfy these assumptions with κ = κ0 . This follows from the arguments in 7.3, 11.2, and the statement (which is not hard to prove) that there are no noncompact three-dimensional gradient 31

shrinking solitons, satisfying 11.1, other than the round cylinder and its Z2 quotients. Furthermore, I believe that there is only one (up to scaling) noncompact three-dimensional κ-noncollapsed ancient solution with bounded positive curvature - the rotationally symmetric gradient steady soliton, studied by R.Bryant. In this direction, I have a plausible, but not quite rigorous argument, showing that any such ancient solution can be made eternal, that is, can be extended for t ∈ (−∞, +∞); also I can prove uniqueness in the class of gradient steady solitons. 11.10* The earlier work on ancient solutions and all that can be found in [H 4, §16 − 22, 25, 26].

12

Almost nonnegative curvature in dimension three

12.1 Let φ be a decreasing function of one variable, tending to zero at infinity. A solution to the Ricci flow is said to have φ-almost nonnegative curvature if it satisfies Rm(x, t) ≥ −φ(R(x, t))R(x, t) for each (x, t). Theorem. Given ǫ > 0, κ > 0 and a function φ as above, one can find r0 > 0 with the following property. If gij (t), 0 ≤ t ≤ T is a solution to the Ricci flow on a closed three-manifold M, which has φ-almost nonnegative curvature and is κ-noncollapsed on scales < r0 , then for any point (x0 , t0 ) with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , the solution in {(x, t) : dist2t0 (x, x0 ) < (ǫQ)−1 , t0 − (ǫQ)−1 ≤ t ≤ t0 } is , after scaling by the factor Q, ǫ-close to the corresponding subset of some ancient solution, satisfying the assumptions in 11.1. Proof. An argument by contradiction. Take a sequence of r0 converging to zero, and consider the solutions gij (t), such that the conclusion does not hold for some (x0 , t0 ); moreover, by tampering with the condition t0 ≥ 1 a little bit, choose among all such (x0 , t0 ), in the solution under consideration, the one with nearly the smallest curvature Q. (More precisely, we can choose (x0 , t0 ) in such a way that the conclusion of the theorem holds for all (x, t), satisfying R(x, t) > 2Q, t0 − HQ−1 ≤ t ≤ t0 , where H → ∞ as r0 → 0) Our goal is to show that the sequence of blow-ups of such solutions at such points with factors Q would converge, along some subsequence of r0 → 0, to an ancient solution, satisfying 11.1.

32

¯ Claim 1. For each (¯ x, t¯) with t0 − HQ−1 ≤ t¯ ≤ t0 we have R(x, t) ≤ 4Q 1 ¯ = Q + R(¯ ¯ −1 ≤ t ≤ t¯ and distt¯(x, x¯) ≤ cQ ¯ − 2 , where Q x, t¯) whenever t¯ − cQ and c = c(κ) > 0 is a small constant. Proof of Claim 1. Use the fact ( following from the choice of (x0 , t0 ) and the description of the ancient solutions) that for each (x, t) with R(x, t) > 2Q and t0 − HQ−1 ≤ t ≤ t0 we have the estimates |Rt (x, t)| ≤ CR2 (x, t), 3 |∇R|(x, t) ≤ CR 2 (x, t). Claim 2. There exists c = c(κ) > 0 and for any A > 0 there exist D = D(A) < ∞, ρ0 = ρ0 (A) > 0, with the following property. Suppose that r0 < ρ0 , and let γ be a shortest geodesic with endpoints x¯, x in gij (t¯), for some t¯ ∈ [t0 − HQ−1 , t0 ], such that R(y, ¯t) > 2Q for each y ∈ γ. Let z ∈ γ satisfy ¯ Then distt¯(¯ ¯ − 12 whenever R(x, t¯) ≥ DQ. ¯ cR(z, t¯) > R(¯ x, t¯) = Q. x, z) ≥ AQ Proof of Claim 2. Note that from the choice of (x0 , t0 ) and the description of the ancient solutions it follows that an appropriate parabolic (backward in time) neighborhood of a point y ∈ γ at t = t¯ is ǫ-close to the evolving round ¯ ≤ R(y, ¯t) ≤ cR(x, t¯) for an appropriate c = c(κ). cylinder, provided c−1 Q Now assume that the conclusion of the claim does not hold, take r0 to zero, ¯ R(x, t¯) - to infinity, and consider the scalings around (¯ x, t¯) with factors Q. We can imagine two possibilities for the behavior of the curvature along γ in the scaled metric: either it stays bounded at bounded distances from x¯, or not. In the first case we can take a limit (for a subsequence) of the scaled metrics along γ and get a nonnegatively curved almost cylindrical metric, with γ going to infinity. Clearly, in this case the curvature at any point of the limit does not exceed c−1 ; therefore, the point z must have escaped to infinity, and the conclusion of the claim stands. In the second case, we can also take a limit along γ; it is a smooth nonnegatively curved manifold near x¯ and has cylindrical shape where curvature is large; the radius of the cylinder goes to zero as we approach the (first) singular point, which is located at finite distance from x¯; the region beyond the first singular point will be ignored. Thus, at t = t¯ we have a metric, which is a smooth metric of nonnegative curvature away from a single singular point o. Since the metric is cylindrical at points close to o, and the radius of the cylinder is at most ǫ times the distance from o, the curvature at o is nonnegative in Aleksandrov sense. Thus, the metric near o must be cone-like. In other words, the scalings of our metric at points xi → o with factors R(xi , t¯) converge to a piece of nonnegatively curved non-flat metric cone. Moreover, using claim 1, we see that we actually have the convergence 33

of the solutions to the Ricci flow on some time interval, and not just metrics at t = t¯. Therefore, we get a contradiction with the strong maximum principle of Hamilton [H 2]. Now continue the proof of theorem, and recall that we are considering scalings at (x0 , t0 ) with factor Q. It follows from claim 2 that at t = t0 the curvature of the scaled metric is bounded at bounded distances from x0 . This allows us to extract a smooth limit at t = t0 (of course, we use the κ-noncollapsing assumption here). The limit has bounded nonnegative curvature (if the curvatures were unbounded, we would have a sequence of cylindrical necks with radii going to zero in a complete manifold of nonnegative curvature). Therefore, by claim 1, we have a limit not only at t = t0 , but also in some interval of times smaller than t0 . We want to show that the limit actually exists for all t < t0 . Assume that this is not the case, and let t′ be the smallest value of time, such that the blowup limit can be taken on (t′ , t0 ]. From the differential Harnack inequality of Hamilton [H 3] we have an estimate Rt (x, t) ≥ −R(x, t)(t−t′ )−1 , therefore, if ˜ denotes the maximum of scalar curvature at t = t0 , then R(x, t) ≤ Q ˜ t0 −t′′ . Q t−t Hence by lemma 8.3(b) dist (x, y) ≤ dist (x, y) + C for all t, where C = t t 0 q ˜ 10n(t0 − t′ ) Q. The next step is needed only if our limit is noncompact. In this case there exists D > 0, such that for any y satisfying d = distt0 (x0 , y) > D, one can find x satisfying distt0 (x, y) = d, distt0 (x, x0 ) > 32 d. We claim that the scalar curvature R(y, t) is uniformly bounded for all such y and all t ∈ (t′ , t0 ]. Indeed, if R(y, t) is large, then the neighborhood of (y, t) is like in an ancient solution; therefore, (long) shortest geodesics γ and γ0 , connecting at time t the point y to x and x0 respectively, make the angle close to 0 or π at y; the former case is ruled out by the assumptions on distances, if D > 10C; in the latter case, x and x0 are separated at time t by a small neighborhood of y, 1 with diameter of order R(y, t)− 2 , hence the same must be true at time t0 , which is impossible if R(y, t) is too large. Thus we have a uniform bound on curvature outside a certain compact set, which has uniformly bounded diameter for all t ∈ (t′ , t0 ]. Then claim 2 gives a uniform bound on curvature everywhere. Hence, by claim 1, we can extend our blow-up limit past t′ - a contradiction. 12.2 Theorem. Given a function φ as above, for any A > 0 there exists K = K(A) < ∞ with the following property. Suppose in dimension three we have a solution to the Ricci flow with φ-almost nonnegative curvature, which 34

satisfies the assumptions of theorem 8.2 with r0 = 1. Then R(x, 1) ≤ K whenever dist1 (x, x0 ) < A. Proof. In the first step of the proof we check the following Claim. There exists K = K(A) < ∞, such that a point (x, 1) satisfies the conclusion of the previous theorem 12.1 (for some fixed small ǫ > 0), whenever R(x, 1) > K and dist1 (x, x0 ) < A. The proof of this statement essentially repeats the proof of the previous theorem (the κ-noncollapsing assumption is ensured by theorem 8.2). The only difference is in the beginning. So let us argue by contradiction, and suppose we have a sequence of solutions and points x with dist1 (x, x0 ) < A and R(x, 1) → ∞, which do not satisfy the conclusion of 12.1. Then an argument, similar to the one proving claims 1,2 in 10.1, delivers points (¯ x, t¯) 1 x, x0 ) < 2A, with Q = R(¯ x, t¯) → ∞, and such that with 2 ≤ t¯ ≤ 1, distt¯(¯ (x, t) satisfies the conclusion of 12.1 whenever R(x, t) > 2Q, t¯ − DQ−1 ≤ t ≤ 1 t¯, distt¯(¯ x, x) < DQ− 2 , where D → ∞. (There is a little subtlety here in the application of lemma 8.3(b); nevertheless, it works, since we need to apply it only when the endpoint other than x0 either satisfies the conclusion of 12.1, or has scalar curvature at most 2Q) After such (¯ x, t¯) are found, the proof of 12.1 applies. Now, having checked the claim, we can prove the theorem by applying the claim 2 of the previous theorem to the appropriate segment of the shortest geodesic, connecting x and x0 . 12.3 Theorem. For any w > 0 there exist τ = τ (w) > 0, K = K(w) < ∞, ρ = ρ(w) > 0 with the following property. Suppose we have a solution gij (t) to the Ricci flow, defined on M × [0, T ), where M is a closed threemanifold, and a point (x0 , t0 ), such that the ball B(x0 , r0 ) at t = t0 has volume ≥ wr0n , and sectional curvatures ≥ −r0−2 at each point. Suppose that gij (t) is φ-almost nonnegatively curved for some function φ as above. Then we have an estimate R(x, t) < Kr0−2 whenever t0 ≥ 4τ r02 , t ∈ [t0 − τ r02 , t0 ], distt (x, x0 ) ≤ 14 r0 , provided that φ(r0−2 ) < ρ. Proof. If we knew that sectional curvatures are ≥ −r0−2 for all t, then we could just apply corollary 11.6(b) (with the remark after its proof) and take τ (w) = τ0 (w)/2, K(w) = C(w) + 2B(w)/τ0 (w). Now fix these values of τ, K, consider a φ-almost nonnegatively curved solution gij (t), a point (x0 , t0 ) and a radius r0 > 0, such that the assumptions of the theorem do hold whereas the conclusion does not. We may assume that any other point (x′ , t′ ) and radius r ′ > 0 with that property has either t′ > t0 or t′ < t0 − 2τ r02 , or 35

2r ′ > r0 . Our goal is to show that φ(r0−2) is bounded away from zero. Let τ ′ > 0 be the largest time interval such that Rm(x, t) ≥ −r0−2 whenever t ∈ [t0 − τ ′ r02 , t0 ], distt (x, x0 ) ≤ r0 . If τ ′ ≥ 2τ, we are done by corollary 11.6(b). Otherwise, by elementary Aleksandrov space theory, we can find at time t′ = t0 − τ ′ r02 a ball B(x′ , r ′ ) ⊂ B(x0 , r0 ) with V olB(x′ , r ′ ) ≥ 12 ωn (r ′ )n , and with radius r ′ ≥ cr0 for some small constant c = c(w) > 0. By the choice of (x0 , t0 ) and r0 , the conclusion of our theorem holds for (x′ , t′ ), r ′ . Thus we have an estimate R(x, t) ≤ K(r ′ )−2 whenever t ∈ [t′ − τ (r ′ )2 , t′ ], distt (x, x′ ) ≤ 1 ′ r . Now we can apply the previous theorem (or rather its scaled version) and 4 get an estimate on R(x, t) whenever t ∈ [t′ − 21 τ (r ′ )2 , t′ ], distt (x′ , x) ≤ 10r0 . Therefore, if r0 > 0 is small enough, we have Rm(x, t) ≥ −r0−2 for those (x, t), which is a contradiction to the choice of τ ′ . 12.4 Corollary (from 12.2 and 12.3) Given a function φ as above, for any w > 0 one can find ρ > 0 such that if gij (t) is a φ-almost nonnegatively curved solution to the Ricci flow, defined on M × [0, T ), where M is a closed three-manifold, and if B(x0 , r0 ) is a metric ball at time t0 ≥ 1, with r0 < ρ, and such that min Rm(x, t0 ) over x ∈ B(x0 , r0 ) is equal to −r0−2 , then V olB(x0 , r0 ) ≤ wr0n .

13

The global picture of the Ricci flow in dimension three

13.1 Let gij (t) be a smooth solution to the Ricci flow on M × [1, ∞), where M is a closed oriented three-manifold. Then, according to [H 6, theorem 4.1], ˜ the normalized curvatures Rm(x, t) = tRm(x, t) satisfy an estimate of the ˜ ˜ ˜ t), where φ behaves at infinity as 1 . This form Rm(x, t) ≥ −φ(R(x, t))R(x, log estimate allows us to apply the results 12.3,12.4, and obtain the following Theorem. For any w > 0 there exist K = K(w) < ∞, ρ = ρ(w) > 0, such that for sufficiently large S times t the manifold M admits a thick-thin decomposition M = Mthick Mthin with the following properties. (a) √For ˜ ≤ K in the ball B(x, ρ(w) t). every x ∈ Mthick we have an estimate |Rm| √ 1 and the volume of this ball is at least 10 w(ρ(w) t)n . (b) For every y ∈ Mthin √ there exists r = r(y), 0 < r < ρ(w) t, such that for all points in the ball B(y, r) we have Rm ≥ −r −2 , and the volume of this ball is < wrn . Now the arguments in [H 6] show that either Mthick is empty for large t, or , for an appropriate sequence of t → 0 and w → 0, it converges to 36

a (possibly, disconnected) complete hyperbolic manifold of finite volume, whose cusps (if there are any) are incompressible in M. On the other hand, collapsing with lower curvature bound in dimension three is understood well enough to claim that, for sufficiently small w > 0, Mthin is homeomorphic to a graph manifold. The natural questions that remain open are whether the normalized curvatures must stay bounded as t → ∞, and whether reducible manifolds and manifolds with finite fundamental group can have metrics which evolve smoothly by the Ricci flow on the infinite time interval. 13.2 Now suppose that gij (t) is defined on M × [1, T ), T < ∞, and goes singular as t → T. Then using 12.1 we see that, as t → T, either the curvature goes to infinity everywhere, and then M is a quotient of either S3 or S2 × R, or the region of high curvature in gij (t) is the union of several necks and capped necks, which in the limit turn into horns (the horns most likely have finite diameter, but at the moment I don’t have a proof of that). Then at the time T we can replace the tips of the horns by smooth caps and continue running the Ricci flow until the solution goes singular for the next time, e.t.c. It turns out that those tips can be chosen in such a way that the need for the surgery will arise only finite number of times on every finite time interval. The proof of this is in the same spirit, as our proof of 12.1; it is technically quite complicated, but requires no essentially new ideas. It is likely that by passing to the limit in this construction one would get a canonically defined Ricci flow through singularities, but at the moment I don’t have a proof of that. (The positive answer to the conjecture in 11.9 on the uniqueness of ancient solutions would help here) Moreover, it can be shown, using an argument based on 12.2, that every maximal horn at any time T, when the solution goes singular, has volume at least cT n ; this easily implies that the solution is smooth (if nonempty) from some finite time on. Thus the topology of the original manifold can be reconstructed as a connected sum of manifolds, admitting a thick-thin decomposition as in 13.1, and quotients of S3 and S2 × R. 13.3* Another differential-geometric approach to the geometrization conjecture is being developed by Anderson [A]; he studies the elliptic equations, arising as Euler-Lagrange equations for certain functionals of the riemannian metric, perturbing the total scalar curvature functional, and one can observe certain parallelism between his work and that of Hamilton, especially taking into account that, as we have shown in 1.1, Ricci flow is the gradient flow for a functional, that closely resembles the total scalar curvature. 37

References [A] M.T.Anderson Scalar curvature and geometrization conjecture for three-manifolds. Comparison Geometry (Berkeley, 1993-94), MSRI Publ. 30 (1997), 49-82. [B-Em] D.Bakry, M.Emery Diffusions hypercontractives. Seminaire de Probabilites XIX, 1983-84, Lecture Notes in Math. 1123 (1985), 177-206. [Cao-C] H.-D. Cao, B.Chow Recent developments on the Ricci flow. Bull. AMS 36 (1999), 59-74. [Ch-Co] J.Cheeger, T.H.Colding On the structure of spaces with Ricci curvature bounded below I. Jour. Diff. Geom. 46 (1997), 406-480. [C] B.Chow Entropy estimate for Ricci flow on compact two-orbifolds. Jour. Diff. Geom. 33 (1991), 597-600. [C-Chu 1] B.Chow, S.-C. Chu A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow. Math. Res. Let. 2 (1995), 701-718. [C-Chu 2] B.Chow, S.-C. Chu A geometric approach to the linear trace Harnack inequality for the Ricci flow. Math. Res. Let. 3 (1996), 549-568. [D] E.D’Hoker String theory. Quantum fields and strings: a course for mathematicians (Princeton, 1996-97), 807-1011. [E 1] K.Ecker Logarithmic Sobolev inequalities on submanifolds of euclidean space. Jour. Reine Angew. Mat. 522 (2000), 105-118. [E 2] K.Ecker A local monotonicity formula for mean curvature flow. Ann. Math. 154 (2001), 503-525. [E-Hu] K.Ecker, G.Huisken In terior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105 (1991), 547-569. [Gaw] K.Gawedzki Lectures on conformal field theory. Quantum fields and strings: a course for mathematicians (Princeton, 1996-97), 727-805. [G] L.Gross Logarithmic Sobolev inequalities and contractivity properties of semigroups. Dirichlet forms (Varenna, 1992) Lecture Notes in Math. 1563 (1993), 54-88. [H 1] R.S.Hamilton Three manifolds with positive Ricci curvature. Jour. Diff. Geom. 17 (1982), 255-306. [H 2] R.S.Hamilton Four manifolds with positive curvature operator. Jour. Diff. Geom. 24 (1986), 153-179. [H 3] R.S.Hamilton The Harnack estimate for the Ricci flow. Jour. Diff. Geom. 37 (1993), 225-243. [H 4] R.S.Hamilton Formation of singularities in the Ricci flow. Surveys in Diff. Geom. 2 (1995), 7-136. 38

[H 5] R.S.Hamilton Four-manifolds with positive isotropic curvature. Commun. Anal. Geom. 5 (1997), 1-92. [H 6] R.S.Hamilton Non-singular solutions of the Ricci flow on threemanifolds. Commun. Anal. Geom. 7 (1999), 695-729. [H 7] R.S.Hamilton A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1 (1993), 113-126. [H 8] R.S.Hamilton Monotonicity formulas for parabolic flows on manifolds. Commun. Anal. Geom. 1 (1993), 127-137. [H 9] R.S.Hamilton A compactness property for solutions of the Ricci flow. Amer. Jour. Math. 117 (1995), 545-572. [H 10] R.S.Hamilton The Ricci flow on surfaces. Contemp. Math. 71 (1988), 237-261. [Hu] G.Huisken Asymptotic behavior for singularities of the mean curvature flow. Jour. Diff. Geom. 31 (1990), 285-299. [I] T.Ivey Ricci solitons on compact three-manifolds. Diff. Geo. Appl. 3 (1993), 301-307. [L-Y] P.Li, S.-T. Yau On the parabolic kernel of the Schrodinger operator. Acta Math. 156 (1986), 153-201. [Lott] J.Lott Some geometric properties of the Bakry-Emery-Ricci tensor. arXiv:math.DG/0211065.

39

arXiv:math/0303109v1 [math.DG] 10 Mar 2003

Ricci flow with surgery on three-manifolds Grisha Perelman∗ February 1, 2008 This is a technical paper, which is a continuation of [I]. Here we verify most of the assertions, made in [I, §13]; the exceptions are (1) the statement that a 3-manifold which collapses with local lower bound for sectional curvature is a graph manifold - this is deferred to a separate paper, as the proof has nothing to do with the Ricci flow, and (2) the claim about the lower bound for the volumes of the maximal horns and the smoothness of the solution from some time on, which turned out to be unjustified, and, on the other hand, irrelevant for the other conclusions. The Ricci flow with surgery was considered by Hamilton [H 5,§4,5]; unfortunately, his argument, as written, contains an unjustified statement (RMAX = Γ, on page 62, lines 7-10 from the bottom), which I was unable to fix. Our approach is somewhat different, and is aimed at eventually constructing a canonical Ricci flow, defined on a largest possible subset of space-time, - a goal, that has not been achieved yet in the present work. For this reason, we consider two scale bounds: the cutoff radius h, which is the radius of the necks, where the surgeries are performed, and the much larger radius r, such that the solution on the scales less than r has standard geometry. The point is to make h arbitrarily small while keeping r bounded away from zero.

Notation and terminology B(x, t, r) denotes the open metric ball of radius r, with respect to the metric at time t, centered at x. P (x, t, r, △t) denotes a parabolic neighborhood, that is the set of all points (x′ , t′ ) with x′ ∈ B(x, t, r) and t′ ∈ [t, t + △t] or t′ ∈ [t + △t, t], depending on the sign of △t. A ball B(x, t, ǫ−1 r) is called an ǫ-neck, if, after scaling the metric with factor −2 r , it is ǫ-close to the standard neck S2 × I, with the product metric, where S2 has constant scalar curvature one, and I has length 2ǫ−1 ; here ǫ-close refers to C N topology, with N > ǫ−1 . A parabolic neighborhood P (x, t, ǫ−1 r, r2 ) is called a strong ǫ-neck, if, after scaling with factor r−2 , it is ǫ-close to the evolving standard neck, which at each ∗ St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia. Email: [email protected] or [email protected]

1

time t′ ∈ [−1, 0] has length 2ǫ−1 and scalar curvature (1 − t′ )−1 . A metric on S2 × I, such that each point is contained in some ǫ-neck, is called an ǫ-tube, or an ǫ-horn, or a double ǫ-horn, if the scalar curvature stays bounded on both ends, stays bounded on one end and tends to infinity on the other, and tends to infinity on both ends, respectively. ¯ 3 , such that each point outside some compact A metric on B3 or RP3 \ B subset is contained in an ǫ-neck, is called an ǫ-cap or a capped ǫ-horn, if the scalar curvature stays bounded or tends to infinity on the end, respectively. We denote by ǫ a fixed small positive constant. In contrast, δ denotes a positive quantity, which is supposed to be as small as needed in each particular argument.

1

Ancient solutions with bounded entropy

1.1 In this section we review some of the results, proved or quoted in [I,§11], correcting a few inaccuracies. We consider smooth solutions gij (t) to the Ricci flow on oriented 3-manifold M , defined for −∞ < t ≤ 0, such that for each t the metric gij (t) is a complete non-flat metric of bounded nonnegative sectional curvature, κ-noncollapsed on all scales for some fixed κ > 0; such solutions will be called ancient κ-solutions for short. By Theorem I.11.7, the set of all such solutions with fixed κ is compact modulo scaling, that is from any sequence α of such solutions (M α , gij (t)) and points (xα , 0) with R(xα , 0) = 1, we can extract a smoothly (pointed) convergent subsequence, and the limit (M, gij (t)) belongs to the same class of solutions. (The assumption in I.11.7. that M α be noncompact was clearly redundant, as it was not used in the proof. Note also that M need not have the same topology as M α .) Moreover, according to Proposition I.11.2, the scalings of any ancient κ-solution gij (t) with factors (−t)−1 about appropriate points converge along a subsequence of t → −∞ to a non-flat gradient shrinking soliton, which will be called an asymptotic soliton of the ancient solution. If the sectional curvature of this asymptotic soliton is not strictly positive, then by Hamilton’s strong maximum principle it admits local metric splitting, and it is easy to see that in this case the soliton is either the round infinite cylinder, or its Z2 quotient, containing one-sided projective plane. If the curvature is strictly positive and the soliton is compact, then it has to be a metric quotient of the round 3-sphere, by [H 1]. The noncompact case is ruled out below. 1.2 Lemma. There is no (complete oriented 3-dimensional) noncompact κ-noncollapsed gradient shrinking soliton with bounded positive sectional curvature. Proof. A gradient shrinking soliton gij (t), −∞ < t < 0, satisfies the equation 1 gij = 0 2t Differentiating and switching the order of differentiation, we get ∇i ∇j f + Rij +

∇i R = 2Rij ∇j f 2

(1.1)

(1.2)

Fix some t < 0, say t = −1, and consider a long shortest geodesic γ(s), 0 ≤ s ≤ s¯; let x = γ(0), x ¯ = γ(¯ s), X(s) = γ(s). ˙ Since the curvature is bounded and R s¯ positive, it is clear from the second variation formula that 0 Ric(X, X)ds ≤ R s¯ R s¯ √ const. Therefore, 0 |Ric(X, ·)|2 ds ≤ const, and 0 |Ric(X, Y )|ds ≤ const( s¯ + 1) for any unit vector field Y along γ, orthogonal to X.√Thus by integrating s))| ≤ const( s¯ + 1). We conclude (1.1) we get X · f (γ(¯ s)) ≥ 2s¯ + const, |Y · f (γ(¯ that at large distances from x0 the function f has no critical points, and its gradient makes small angle with the gradient of the distance function from x0 . Now from (1.2) we see that R is increasing along the gradient curves of f, ¯ = lim sup R > 0. If we take a limit of our soliton about points in particular, R α ¯ then we get an ancient κ-solution, which splits (x , −1) where R(xα ) → R, off a line, and it follows from I.11.3, that this solution is the shrinking round ¯ at time t = −1. Now comparing the infinite cylinder with scalar curvature R evolution equations for the scalar curvature on a round cylinder and for the ¯ = 1. asymptotic scalar curvature on a shrinking soliton we conclude that R Hence, R(x) < 1 when the distance from x to x0 is large enough, and R(x) → 1 when this distance tends to infinity. Now let us check that the level surfaces of f, sufficiently distant from x0 , are convex. Indeed, if Y is a unit tangent vector to such a surface, then ∇Y ∇Y f = 1 R 1 2 − Ric(Y, Y ) ≥ 2 − 2 > 0. Therefore, the area of the level surfaces grows as f increases, and is converging to the area of the round sphere of scalar curvature one. On the other hand, the intrinsic scalar curvature of a level surface turns out to be less than one. Indeed, denoting by X the unit normal vector, this intrinsic curvature can be computed as R − 2Ric(X, X) + 2

det(Hessf ) (1 − R + Ric(X, X))2 ≤ R − 2Ric(X, X) + > 1), with two spherical caps, smoothly attached to its boundary components. By [H 1] we know that the flow shrinks such a metric to a point in time, comparable to one (because both the lower bound for scalar curvature and the upper bound for sectional curvature are comparable to one) , and after 3

normalization, the flow converges to the round 3-sphere. Scale the initial metric and choose the time parameter in such a way that the flow starts at time t0 = t0 (L) < 0, goes singular at t = 0, and at t = −1 has the ratio of the maximal sectional curvature to the minimal one equal to 1 + ǫ. The argument in [I.7.3] shows that our solutions are κ-noncollapsed for some κ > 0 independent of L. We also claim that t0 (L) → −∞ as L → ∞. Indeed, the Harnack inequality of 0) for t ≤ −1, Hamilton [H 3] implies that Rt ≥ t0R−t , hence R ≤ 2(−1−t t−t0 p d Rmax (t) dist (x, y) ≥ −const and then the distance change estimate t dt from [H 2,§17] implies that the diameter of g (t ) does not exceed −const · t0 , ij 0 √ which is less than L −t0 unless t0 is large enough. Thus, a subsequence of our solutions with L → ∞ converges to an ancient κ-solution on S3 , whose asymptotic soliton can not be anything but the cylinder. 1.5 The important conclusion from the classification above and the proof of Proposition I.11.2 is that there exists κ0 > 0, such that every ancient κ-solution is either κ0 -solution, or a metric quotient of the round sphere. Therefore, the compactness theorem I.11.7 implies the existence of a universal constant η, such that at each point of every ancient κ-solution we have estimates 3

|∇R| < ηR 2 , |Rt | < ηR2

(1.3)

Moreover, for every sufficiently small ǫ > 0 one can find C1,2 = C1,2 (ǫ), such that for each point (x, t) in every ancient κ-solution there is a radius r, 0 < r < 1 C1 R(x, t)− 2 , and a neighborhood B, B(x, t, r) ⊂ B ⊂ B(x, t, 2r), which falls into one of the four categories: (a) B is a strong ǫ-neck (more precisely, the slice of a strong ǫ-neck at its maximal time), or (b) B is an ǫ-cap, or (c) B is a closed manifold, diffeomorphic to S3 or RP3 , or (d) B is a closed manifold of constant positive sectional curvature; furthermore, the scalar curvature in B at time t is between C2−1 R(x, t) and 3 C2 R(x, t), its volume in cases (a),(b),(c) is greater than C2−1 R(x, t)− 2 , and in −1 case (c) the sectional curvature in B at time t is greater than C2 R(x, t).

2

The standard solution

Consider a rotationally symmetric metric on R3 with nonnegative sectional curvature, which splits at infinity as the metric product of a ray and the round 2-sphere of scalar curvature one. At this point we make some choice for the metric on the cap, and will refer to it as the standard cap; unfortunately, the most obvious choice, the round hemisphere, does not fit, because the metric on R3 would not be smooth enough, however we can make our choice as close to it as we like. Take such a metric on R3 as the initial data for a solution gij (t) to the Ricci flow on some time interval [0, T ), which has bounded curvature for each t ∈ [0, T ). Claim 1. The solution is rotationally symmetric for all t. 4

Indeed, if ui is a vector field evolving by uit = △ui + Rji uj , then vij = ∇i uj evolves by (vij )t = △vij + 2Rikjl vkl − Rik vkj − Rkj vik . Therefore, if ui was a Killing field at time zero, it would stay Killing by the maximum principle. It is also clear that the center of the cap, that is the unique maximum point for the Busemann function, and the unique point, where all the Killing fields vanish, retains these properties, and the gradient of the distance function from this point stays orthogonal to all the Killing fields. Thus, the rotational symmetry is preserved. Claim 2. The solution converges at infinity to the standard solution on the round infinite cylinder of scalar curvature one. In particular, T ≤ 1. Claim 3. The solution is unique. Indeed, using Claim 1, we can reduce the linearized Ricci flow equation to the system of two equations on (−∞, +∞) of the following type ft = f ′′ + a1 f ′ + b1 g ′ + c1 f + d1 g, gt = a2 f ′ + b2 g ′ + c2 f + d2 g, where the coefficients and their derivatives are bounded, and the unknowns f, g and their derivatives tend to zero at infinity by Claim 2. So we get uniqueness RA by looking at the integrals −A (f 2 + g 2 ) as A → ∞. Claim 4. The solution can be extended to the time interval [0, 1). Indeed, we can obtain our solution as a limit of the solutions on S3 , starting from the round cylinder S2 × I of length L and scalar curvature one, with two caps attached; the limit is taken about the center p of one of the caps, L → ∞. Assume that our solution goes singular at some time T < 1. Take T1 < T very close to T, T − T1 0, we can find ¯ D ¯ < ∞, depending on δ and T1 , such that for any point x at distance D ¯ L, ¯ from p at time zero, in the solution with L ≥ L, the ball B(x, T1 , 1) is δ-close to the corresponding ball in the round cylinder of scalar curvature (1 − T1 )−1 . We can also find r = r(δ, T ), independent of T1 , such that the ball B(x, T1 , r) is δ-close to the corresponding euclidean ball. Now we can apply Theorem I.10.1 and get a uniform estimate on the curvature at x as t → T , provided that T − T1 < ǫ2 r(δ, T )2 . Therefore, the t → T limit of our limit solution on the capped infinite cylinder will be smooth near x. Thus, this limit will be a positively curved space with a conical point. However, this leads to a contradiction via a blow-up argument; see the end of the proof of the Claim 2 in I.12.1. The solution constructed above will be called the standard solution. Claim 5. The standard solution satisfies the conclusions of 1.5 , for an appropriate choice of ǫ, η, C1 (ǫ), C2 (ǫ), except that the ǫ-neck neighborhood need not be strong; more precisely, we claim that if (x, t) has neither an ǫ-cap neighborhood as in 1.5(b), nor a strong ǫ-neck neighborhood as in 1.5(a), then x is not in B(p, 0, ǫ−1 ), t < 3/4, and there is an ǫ-neck B(x, t, ǫ−1 r), such that the solution in P (x, t, ǫ−1 r, −t) is, after scaling with factor r−2 , ǫ-close to the appropriate piece of the evolving round infinite cylinder. Moreover, we have an estimate Rmin (t) ≥ const · (1 − t)−1 .

5

Indeed, the statements follow from compactness and Claim 2 on compact subintervals of [0, 1), and from the same arguments as for ancient solutions, when t is close to one.

3

The structure of solutions at the first singular time

Consider a smooth solution gij (t) to the Ricci flow on M × [0, T ), where M is a closed oriented 3-manifold, T < ∞. Assume that curvature of gij (t) does not stay bounded as t → T. Recall that we have a pinching estimate Rm ≥ −φ(R)R for some function φ decreasing to zero at infinity [H 4,§4], and that the solution is κ-noncollapsed on the scales ≤ r for some κ > 0, r > 0 [I, §4].Then by Theorem I.12.1 and the conclusions of 1.5 we can find r = r(ǫ) > 0, such that each point (x, t) with R(x, t) ≥ r−2 satisfies the estimates (1.3) and has a neighborhood, which is either an ǫ-neck, or an ǫ-cap, or a closed positively curved manifold. In the latter case the solution becomes extinct at time T, so we don’t need to consider it any more. If this case does not occur, then let Ω denote the set of all points in M, where curvature stays bounded as t → T. The estimates (1.3) imply that Ω is open and that R(x, t) → ∞ as t → T for each x ∈ M \Ω. If Ω is empty, then the solution becomes extinct at time T and it is entirely covered by ǫ-necks and caps shortly before that time, so it is easy to see that M is diffeomorphic to either S3 , or RP3 , or S2 × S1 , or RP3 ♯ RP3 . Otherwise, if Ω is not empty, we may (using the local derivative estimates due to W.-X.Shi, see [H 2,§13]) consider a smooth metric g¯ij on Ω, which is the limit of gij (t) as t → T. Let Ωρ for some ρ < r denotes the set of points x ∈ Ω, ¯ where the scalar curvature R(x) ≤ ρ−2 . We claim that Ωρ is compact. Indeed, if −2 ¯ R(x) ≤ ρ , then we can estimate the scalar curvature R(x, t) on [T − η −1 ρ2 , T ) using (1.3), and for earlier times by compactness, so x is contained in Ω with a ball of definite size, depending on ρ. Now take any ǫ-neck in (Ω, g¯ij ) and consider a point x on one of its boundary components. If x ∈ Ω\Ωρ , then there is either an ǫ-cap or an ǫ-neck, adjacent to the initial ǫ-neck. In the latter case we can take a point on the boundary of the second ǫ-neck and continue. This procedure can either terminate when we reach a point in Ωρ or an ǫ-cap, or go on indefinitely, producing an ǫ-horn. The same procedure can be repeated for the other boundary component of the initial ǫ-neck. Therefore, taking into account that Ω has no compact components, we conclude that each ǫ-neck of (Ω, g¯ij ) is contained in a subset of Ω of one of the following types: (a) An ǫ-tube with boundary components in Ωρ , or (b) An ǫ-cap with boundary in Ωρ , or (c) An ǫ-horn with boundary in Ωρ , or (d) A capped ǫ-horn, or (e) A double ǫ-horn.

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Clearly, each ǫ-cap, disjoint from Ωρ , is also contained in one of the subsets above. It is also clear that there is a definite lower bound (depending on ρ) for the volume of subsets of types (a),(b),(c), so there can be only finite number of them. Thus we can conclude that there is only a finite number of components of Ω, containing points of Ωρ , and every such component has a finite number of ends, each being an ǫ-horn. On the other hand, every component of Ω, containing no points of Ωρ , is either a capped ǫ-horn, or a double ǫ-horn. Now, by looking at our solution for times t just before T, it is easy to see that the topology of M can be reconstructed as follows: take the components Ωj , 1 ≤ j ≤ i of Ω which contain points of Ωρ , truncate their ǫ-horns, and glue to the boundary components of truncated Ωj a collection of tubes S2 × I and caps ¯ j , 1 ≤ j ≤ i, B3 or RP3 \B3 . Thus, M is diffeomorphic to a connected sum of Ω 2 1 with a finite number of S × S (which correspond to gluing a tube to two ¯j boundary components of the same Ωj ), and a finite number of RP3 ; here Ω denotes Ωj with each ǫ-horn one point compactified.

4

Ricci flow with cutoff

4.1 Suppose we are given a collection of smooth solutions gij (t) to the Ricci flow, + + k defined on Mk × [t− ¯ij ) be the limits k , tk ), which go singular as t → tk . Let (Ωk , g + of the corresponding solutions as t → tk , as in the previous section. Suppose + k−1 k − also that for each k we have t− ¯ij ) and (Mk , gij (tk )) k = tk−1 , and (Ωk−1 , g contain compact (possibly disconnected) three-dimensional submanifolds with smooth boundary, which are isometric. Then we can identify these isometric submanifolds and talk about the solution to the Ricci flow with surgery on the + union of all [t− k , tk ). Fix a small number ǫ > 0 which is admissible in sections 1,2. In this section we consider only solutions to the Ricci flow with surgery, which satisfy the following a priori assumptions: (pinching) There exists a function φ, decreasing to zero at infinity, such that Rm ≥ −φ(R)R, (canonical neighborhood) There exists r > 0, such that every point where scalar curvature is at least r−2 has a neighborhood, satisfying the conclusions of 1.5. (In particular, this means that if in case (a) the neighborhood in question is B(x0 , t0 , ǫ−1 r0 ), then the solution is required to be defined in the whole P (x0 , t0 , ǫ−1 r0 , −r02 ); however, this does not rule out a surgery in the time interval (t0 − r02 , t0 ), that occurs sufficiently far from x0 .) Recall that from the pinching estimate of Ivey and Hamilton, and Theorem I.12.1, we know that the a priori assumptions above hold for a smooth solution on any finite time interval. For Ricci flow with surgery they will be justified in the next section. 4.2 Claim 1. Suppose we have a solution to the Ricci flow with surgery, satisfying the canonical neighborhood assumption, and let Q = R(x0 , t0 )+r−2 . Then 1 we have estimate R(x, t) ≤ 8Q for those (x, t) ∈ P (x0 , t0 , 21 η −1 Q− 2 , − 81 η −1 Q−1 ), for which the solution is defined. 7

Indeed, this follows from estimates (1.3). Claim 2. For any A < ∞ one can find Q = Q(A) < ∞ and ξ = ξ(A) > 0 with the following property. Suppose we have a solution to the Ricci flow with surgery, satisfying the pinching and the canonical neighborhood assumptions. Let γ be a shortest geodesic in gij (t0 ) with endpoints x0 and x, such that R(y, t0 ) > r−2 for each y ∈ γ, and Q0 = R(x0 , t0 ) is so large that φ(Q0 ) < ξ. Finally, let z ∈ γ be any point satisfying R(z, t0 ) > 10C2 R(x0 , t0 ). −1

Then distt0 (x0 , z) ≥ AQ0 2 whenever R(x, t0 ) > QQ0 . The proof is exactly the same as for Claim 2 in Theorem I.12.1; in the very end of it, when we get a piece of a non-flat metric cone as a blow-up limit, we get a contradiction to the canonical neighborhood assumption, because the canonical neighborhoods of types other than (a) are not close to a piece of metric cone, and type (a) is ruled out by the strong maximum principle, since the ǫ-neck in question is strong. 4.3 Suppose we have a solution to the Ricci flow with surgery, satisfying our a priori assumptions, defined on [0, T ), and going singular at time T. Choose a small δ > 0 and let ρ = δr. As in the previous section, consider the limit (Ω, g¯ij ) of our solution as t → T, and the corresponding compact set Ωρ . Lemma. There exists a radius h, 0 < h < δρ, depending only on δ, ρ and the ¯ − 12 (x) ≤ h in an ǫpinching function φ, such that for each point x with h(x) = R horn of (Ω, g¯ij ) with boundary in Ωρ , the neighborhood P (x, T, δ −1 h(x), −h2 (x)) is a strong δ-neck. Proof. An argument by contradiction. Assuming the contrary, take a seα quence of solutions with limit metrics (Ωα , g¯ij ) and points xα with h(xα ) → 0. α Since x lies deeply inside an ǫ-horn, its canonical neighborhood is a strong ǫ-neck. Now Claim 2 gives the curvature estimate that allows us to take a limit α of appropriate scalings of the metrics gij on [T − h2 (xα ), T ] about xα , for a subsequence of α → ∞. By shifting the time parameter we may assume that the limit is defined on [−1, 0]. Clearly, for each time in this interval, the limit is a complete manifold with nonnegative sectional curvature; moreover, since xα was α contained in an ǫ-horn with boundary in Ωα ρ , and h(x )/ρ → 0, this manifold has two ends. Thus, by Toponogov, it admits a metric splitting S2 × R. This implies that the canonical neighborhood of the point (xα , T − h2 (xα )) is also of type (a), that is a strong ǫ-neck, and we can repeat the procedure to get the limit, defined on [−2, 0], and so on. This argument works for the limit in any finite time interval [−A, 0], because h(xα )/ρ → 0. Therefore, we can construct a limit on [−∞, 0]; hence it is the round cylinder, and we get a contradiction. 4.4 Now we can specialize our surgery and define the Ricci flow with δ-cutoff. Fix δ > 0, compute ρ = δr and determine h from the lemma above. Given a smooth metric gij on a closed manifold, run the Ricci flow until it goes singular at some time t+ ; form the limit (Ω, g¯ij ). If Ωρ is empty, the procedure stops here, and we say that the solution became extinct. Otherwise we remove the components of Ω which contain no points of Ωρ , and in every ǫ-horn of each of the remaining components we find a δ-neck of radius h, cut it along the middle two-sphere, remove the horn-shaped end, and glue in an almost standard cap

8

in such a way that the curvature pinching is preserved and a metric ball of radius (δ ′ )−1 h centered near the center of the cap is, after scaling with factor h−2 , δ ′ -close to the corresponding ball in the standard capped infinite cylinder, considered in section 2. (Here δ ′ is a function of δ alone, which tends to zero with δ.) The possibility of capping a δ-neck preserving a certain pinching condition in dimension four was proved by Hamilton [H 5,§4]; his argument works in our case too (and the estimates are much easier to verify). The point is that we can change our δ-neck metric near the middle of the neck by a conformal factor e−f , where f = f (z) is positive on the part of the neck we want to remove, and zero on the part we want to preserve, and z is the coordinate along I in our parametrization S2 × I of the neck. Then, in the region near the middle of the neck, where f is small, the dominating terms in the formulas for the change of curvature are just positive constant multiples of f ′′ , so the pinching improves, and all the curvatures become positive on the set where f > δ ′ . Now we can continue our solution until it becomes singular for the next time. Note that after the surgery the manifold may become disconnected; in this case, each component should be dealt with separately. Furthermore, let us agree to declare extinct every component which is ǫ-close to a metric quotient of the round sphere; that allows to exclude such components from the list of canonical neighborhoods. Now since every surgery reduces the volume by at least h3 , the sequence of surgery times is discrete, and, taking for granted the a priori assumptions, we can continue our solution indefinitely, not ruling out the possibility that it may become extinct at some finite time. 4.5 In order to justify the canonical neighborhood assumption in the next section, we need to check several assertions. ¯ Lemma. For any A < ∞, 0 < θ < 1, one can find δ¯ = δ(A, θ) with the following property. Suppose we have a solution to the Ricci flow with δ-cutoff, ¯ Suppose we have a satisfying the a priori assumptions on [0, T ], with δ < δ. surgery at time T0 ∈ (0, T ), let p correspond to the center of the standard cap, and let T1 = min(T, T0 + θh2 ). Then either (a) The solution is defined on P (p, T0 , Ah, T1 − T0 ), and is, after scaling with factor h−2 and shifting time T0 to zero, A−1 -close to the corresponding subset on the standard solution from section 2, or (b) The assertion (a) holds with T1 replaced by some time t+ ∈ [T0 , T1 ), where + t is a surgery time; moreover, for each point in B(p, T0 , Ah), the solution is defined for t ∈ [T0 , t+ ) and is not defined past t+ . Proof. Let Q be the maximum of the scalar curvature on the standard solution in the time interval [0, θ], let △t = N −1 (T1 − T0 ) < ǫη −1 Q−1 h2 , and let tk = T0 + k△t, k = 0, ..., N. Assume first that for each point in B(p, T0 , A0 h), where A0 = ǫ(δ ′ )−1 , the solution is defined on [t0 , t1 ]. Then by (1.3) and the choice of △t we have a uniform curvature bound on this set for h−2 -scaled metric. Therefore we can define A1 , depending only on A0 and tending to infinity with A0 , such that the solution in P (p, T0 , A1 h, t1 − t0 ) is, after scaling and time shifting, A−1 1 close to the corresponding subset in the standard solution. In particular, the 9

scalar curvature on this subset does not exceed 2Qh−2 . Now if for each point in B(p, T0 , A1 h) the solution is defined on [t1 , t2 ], then we can repeat the procedure, defining A2 etc. Continuing this way, we eventually define AN , and it would remain to choose δ so small, and correspondingly A0 so large, that AN > A. Now assume that for some k, 0 ≤ k < N, and for some x ∈ B(p, T0 , Ak h) the solution is defined on [t0 , tk ] but not on [tk , tk+1 ]. Then we can find a surgery time t+ ∈ [tk , tk+1 ], such that the solution on B(p, T0 , Ak h) is defined on [t0 , t+ ), but for some points of this ball it is not defined past t+ . Clearly, the A−1 k+1 closeness assertion holds on P (p, T0 , Ak+1 h, t+ − T0 ). On the other hand, the solution on B(p, T0 , Ak h) is at least ǫ-close to the standard one for all t ∈ [tk , t+ ), hence no point of this set can be the center of a δ-neck neighborhood at time t+ . However, the surgery is always done along the middle two-sphere of such a neck. It follows that for each point of B(p, T0 , Ak h) the solution terminates at t+ . 4.6 Corollary. For any l < ∞ one can find A = A(l) < ∞ and θ = θ(l), 0 < θ < 1, with the following property. Suppose we are in the situation ¯ of the lemma above, with δ < δ(A, θ). Consider smooth curves γ in the set B(p, T0 , Ah), parametrized by t ∈ [T0 , Tγ ], such that γ(T0 ) ∈ B(p, T0 , Ah/2) and either Tγ = T1 < T , or Tγ < T1 and γ(Tγ ) ∈ ∂B(p, T0 , Ah). Then R Tγ 2 (R(γ(t), t) + |γ(t)| ˙ )dt > l. T0 Proof. Indeed, if Tγ = T1 , then on the standard solution we would have R Tγ Rθ −1 dt = −const · (log(1 − θ))−1 , so by choosing T0 R(γ(t), t)dt ≥ const 0 (1 − t) θ sufficiently close to one we can handle this case. Then we can choose A so large that on the standard solution distt (p, ∂B(p, 0, A)) ≥ 3A/4 for each t ∈ [0, θ]. RT 2 Now if γ(Tγ ) ∈ ∂B(p, T0 , Ah) then T0γ |γ(t)| ˙ dt ≥ A2 /100, so by taking A large enough, we can handle this case as well. 4.7 Corollary. For any Q < ∞ there exists θ = θ(Q), 0 < θ < 1 with the following property. Suppose we are in the situation of the lemma above, ¯ with δ < δ(A, θ), A > ǫ−1 . Suppose that for some point x ∈ B(p, T0 , Ah) the solution is defined at x (at least) on [T0 , Tx ], Tx ≤ T, and satisfies Q−1 R(x, t) ≤ R(x, Tx ) ≤ Q(Tx − T0 )−1 for all t ∈ [T0 , Tx ]. Then Tx ≤ T0 + θh2 . Proof. Indeed, if Tx > T0 + θh2 , then by lemma R(x, T0 + θh2 ) ≥ const · (1 − θ)−1 h−2 , whence R(x, Tx ) ≥ const · Q−1 (1 − θ)−1 h−2 , and Tx − T0 ≤ const · Q2 (1 − θ)h2 < θh2 if θ is close enough to one.

5

Justification of the a priori assumption

5.1 Let us call a riemannian manifold (M, gij ) normalized if M is a closed oriented 3-manifold, the sectional curvatures of gij do not exceed one in absolute value, and the volume of every metric ball of radius one is at least half the volume of the euclidean unit ball. For smooth Ricci flow with normalized initial data we have, by [H 4, 4.1], at any time t > 0 the pinching estimate Rm ≥ −φ(R(t + 1))R,

10

(5.1)

1 . As explained where φ is a decreasing function, which behaves at infinity like log in 4.4, this pinching estimate can be preserved for Ricci flow with δ-cutoff. Justification of the canonical neighborhood assumption requires additional arguments. In fact, we are able to construct solutions satisfying this assumption only allowing r and δ be functions of time rather than constants; clearly, the arguments of the previous section are valid in this case, if we assume that r(t), δ(t) are non-increasing, and bounded away from zero on every finite time interval. Proposition. There exist decreasing sequences 0 < rj < ǫ2 , κj > 0, 0 < ¯ δj < ǫ2 , j = 1, 2, ..., such that for any normalized initial data and any function δ(t), satisfying 0 < δ(t) < δ¯j for t ∈ [2j−1 ǫ, 2j ǫ], the Ricci flow with δ(t)-cutoff is defined for t ∈ [0, +∞] and satisfies the κj -noncollapsing assumption and the canonical neighborhood assumption with parameter rj on the time interval [2j−1 ǫ, 2j ǫ].( Recall that we have excluded from the list of canonical neighborhoods the closed manifolds, ǫ-close to metric quotients of the round sphere. Complete extinction of the solution in finite time is not ruled out.) The proof of the proposition is by induction: having constructed our sequences for 1 ≤ j ≤ i, we make one more step, defining ri+1 , κi+1 , δ¯i+1 , and redefining δ¯i = δ¯i+1 ; each step is analogous to the proof of Theorem I.12.1. First we need to check a κ-noncollapsing condition. 5.2 Lemma. Suppose we have constructed the sequences, satisfying the proposition for 1 ≤ j ≤ i. Then there exists κ > 0, such that for any r, 0 < r < ¯ > 0, which may also depend on the already constructed ǫ2 , one can find δ¯ = δ(r) sequences, with the following property. Suppose we have a solution to the Ricci flow with δ(t)-cutoff on a time interval [0, T ], with normalized initial data, satisfying the proposition on [0, 2i ǫ], and the canonical neighborhood assumption with parameter r on [2i ǫ, T ], where 2i ǫ ≤ T ≤ 2i+1 ǫ, 0 < δ(t) < δ¯ f or t ∈ [2i−1 ǫ, T ]. Then it is κ-noncollapsed on all scales less than ǫ. Proof. Consider a neighborhood P (x0 , t0 , r0 , −r02 ), 2i ǫ < t0 ≤ T, 0 < r0 < ǫ, where the solution is defined and satisfies |Rm| ≤ r0−2 . We may assume r0 ≥ r, since otherwise the lower bound for the volume of the ball B(x0 , t0 , r0 ) follows from the canonical neighborhood assumption. If the solution was smooth everywhere, we could estimate from below the volume of the ball B(x0 , t0 , r0 ) using the argument from [I.7.3]: define τ (t) = t0 − t and consider the reduced volume function using the L-exponential map from x0 ; take a point (x, ǫ) where the reduced distance l attains its minimum for τ = t0 − ǫ, l(x, τ ) ≤ 3/2; use it to obtain an upper bound for the reduced distance to the points of B(x, 0, 1), thus getting a lower bound for the reduced volume at τ = t0 , and apply the monotonicity formula. Now if the solution undergoes surgeries, then we still can measure the L-length, but only for admissible curves, which stay in the region, unaffected by surgery. An inspection of the constructions in [I,§7] shows that the argument would go through if we knew that every barely admissible curve, that is a curve on the boundary of the set of admissible curves, has reduced length at least 3/2 + κ′ for some fixed κ′ > 0. Unfortunately, at the moment I don’t see how to ensure that without imposing new restrictions on δ(t) for all t ∈ [0, T ], so we need some additional arguments. Recall that for a curve γ, parametrized by t, with γ(t0 ) = x0 , we have

11

Rt √ 2 L(γ, τ ) = t00−τ t0 − t(R(γ(t), t) + |γ(t)| ˙ )dt. We can also define L+ (γ, τ ) by √ replacing in the previous formula R with R+ = max(R, 0). Then L+ ≤ L+4T T because R ≥ −6 by the maximum principle and normalization. Now suppose we could show that every barely admissible √ curve with endpoints (x0 , t0 ) and (x, t), where t ∈ [2i−1 ǫ, T ), has L+ > 2ǫ−2 T T ; then we could argue that either there √ exists a point (x, t), t ∈ [2i−1 ǫ, 2i ǫ], such that R(x, t) ≤ ri−2 and L+ ≤ ǫ−2 T T , in which case we can take this point in place of (x, ǫ) in the argument of the previous paragraph, and obtain (using Claim 1 in 4.2) an estimate for κ in i−1 terms of r√ ǫ, t0 ], γ(t0 ) = x0 , we have L+ ≥ i , κi , T, or for any γ, defined√on [2 2 i−1 23 −2 −2 −2 min(ǫ T T , 3 (2 ǫ) ri ) > ǫ T T , which is in contradiction with the assumed bound for barely admissible curves and the bound min l(x, t0 −2i−1 ǫ) ≤ 3/2, valid in the smooth case. Thus, to conclude the proof it is sufficient to check the following assertion. ¯ r0 ) > 0 with the 5.3 Lemma. For any L < ∞ one can find δ¯ = δ(L, following property. Suppose that in the situation of the previous lemma we have a curve γ, parametrized by t ∈ [T0 , t0 ], 2i−1 ǫ ≤ T0 < t0 , such that γ(t0 ) = x0 , T0 is a surgery time, and γ(T0 ) ∈ B(p, T0 , ǫ−1 h), where p corresponds to the of the cap, and h is the radius of the δ-neck. Then we have an estimate Rcenter t0 √ 2 t − t(R+ (γ(t), t) + |γ(t)| ˙ )dt ≥ L. 0 T0 Proof. It is clear that if we take △t = ǫr04 L−2 , then either γ satisfies our estimate, or γ stays in P (x0 , t0 , r0 , −△t) for t ∈ [t0 − △t, t0 ]. In the latter 1 case our estimate follows from Corollary 4.6, for l = L(△t)− 2 , since clearly Tγ < t0 − △t when δ is small enough. 5.4 Proof of proposition. Assume the contrary, and let the sequences rα , δ¯αβ be such that rα → 0 as α → ∞, δ¯αβ → 0 as β → ∞ with fixed α, and let αβ (M αβ , gij ) be normalized initial data for solutions to the Ricci flow with δ(t)cutoff, δ(t) < δ¯αβ on [2i−1 ǫ, 2i+1 ǫ], which satisfy the statement on [0, 2i ǫ], but violate the canonical neighborhood assumption with parameter rα on [2i ǫ, 2i+1 ǫ]. Slightly abusing notation, we’ll drop the indices α, β when we consider an individual solution. Let t¯ be the first time when the assumption is violated at some point x¯; clearly such time exists, because it is an open condition. Then by lemma 5.2 we have uniform κ-noncollapsing on [0, t¯]. Claims 1,2 in 4.2 are also valid on [0, t¯]; moreover, since h 0, 3 , and it follows that M in then the solution becomes extinct in time at most 2a this case is diffeomorphic to a connected sum of several copies of S2 × S1 and metric quotients of round S3 . ( The topological description of 3-manifolds with positive scalar curvature modulo quotients of homotopy spheres was obtained by Schoen-Yau and Gromov-Lawson more than 20 years ago, see [G-L] for instance; in particular, it is well known and easy to check that every manifold that can be decomposed in a connected sum above admits a metric of positive scalar curvature.) Moreover, if the scalar curvature is only nonnegative, then by the strong maximum principle it instantly becomes positive unless the metric is (Ricci-)flat; thus in this case, we need to add to our list the flat manifolds. However, if the scalar curvature is negative somewhere, then we need to work more in order to understand the long tome behavior of the solution. To achieve this we need first to prove versions of Theorems I.12.2 and I.12.3 for solutions with cutoff. 6.2 Correction to Theorem I.12.2. Unfortunately, the statement of Theorem I.12.2 was incorrect. The assertion I had in mind is as follows: Given a function φ as above, for any A < ∞ there exist K = K(A) < ∞ and ρ = ρ(A) > 0 with the following property. Suppose in dimension three we have a solution to the Ricci flow with φ-almost nonnegative curvature, which satisfies the assumptions of theorem 8.2 for some x0 , r0 with φ(r0−2 ) < ρ. Then 13

R(x, r02 ) ≤ Kr0−2 whenever distr02 (x, x0 ) < Ar0 . It is this assertion that was used in the proof of Theorem I.12.3 and Corollary I.12.4. 6.3 Proposition. For any A < ∞ one can find κ = κ(A) > 0, K1 = K1 (A) < ∞, K2 = K2 (A) < ∞, r¯ = r¯(A) > 0, such that for any t0 < ∞ there exists δ¯ = δ¯A (t0 ) > 0, decreasing in t0 , with the following property. Suppose we have a solution to the Ricci flow with δ(t)-cutoff on time interval [0, T ], δ(t) < ¯ on [0, T ], δ(t) < δ¯ on [t0 /2, t0 ], with normalized initial data; assume that the δ(t) solution is defined in the whole parabolic neighborhood P (x0 , t0 , r0 , −r02 ), 2r02 < t0 , and satisfies |Rm| ≤ r0−2 there, and that the volume of the ball B(x0 , t0 , r0 ) is at least A−1 r03 . Then (a) The solution is κ-noncollapsed on the scales less than r0 in the ball B(x0 , t0 , Ar0 ). (b) Every point x ∈ B(x0 , t0 , Ar0 ) with R(x, t0 ) ≥ K1 r0−2 has a canonical neighborhood as √ in 4.1. (c) If r0 ≤ r¯ t0 then R ≤ K2 r0−2 in B(x0 , t0 , Ar0 ). Proof. (a) This is an analog of Theorem I.8.2. Clearly we have p κ-noncollapsing on the scales less than r(t0 ), so we may assume r(t0 ) ≤ r0 ≤ t0 /2 , and study the scales ρ, r(t0 ) ≤ ρ ≤ r0 . In particular, for fixed t0 we are interested in the scales, uniformly equivalent to one. So assume that x ∈ B(x0 , t0 , Ar0 ) and the solution is defined in the whole P (x, t0 , ρ, −ρ2 ) and satisfies |Rm| ≤ ρ−2 there. An inspection of the proof of I.8.2 shows that in order to make the argument work it suffices to check that for any barely admissible curve γ, parametrized by t ∈ [tγ , t0 ], t0 − r02 ≤ tγ ≤ t0 , such that γ(t0 ) = x, we have an estimate p 2 t0 − tγ

Z

t0

tγ

√

2 t0 − t(R(γ(t), t) + |γ(t)| ˙ )dt ≥ C(A)r02

(6.1)

for a certain function C(A) that can be made explicit. Now we would like to conclude the proof by using Lemma 5.3. However, unlike the situation in Lemma 5.2, here Lemma 5.3 provides the estimate we need only if t0 − tγ is bounded away from zero, and otherwise we only get an estimate ρ2 in place of C(A)r02 . Therefore we have to return to the proof of I.8.2. Recall that in that proof we scaled the solution to make r0 = 1 and worked on the time interval [1/2, 1]. The maximum principle for the evolution equation of the scalar curvature implies that on this time interval we have R ≥ −3. We ˆ t))L(y, ˆ τ ), where φ is a certain considered a function of the form h(y, t) = φ(d(y, ˆ ˆ τ ) = L(y, ¯ τ ) + 7, cutoff function, τ = 1 − t, d(y, t) = distt (x0 , y) − A(2t − 1), L(y, ¯ ˆ ˆ ¯ τ) + and in [I,(7.15)]. Now we redefine L, taking L(y, τ ) = L(y, √ √ L was defined 2 ˆ 2 τ . Clearly, L > 0 because R ≥ −3 and 2 τ > 4τ for 0 < τ ≤ 1/2. Then the computations and estimates of I.8.2 yield 1 2h ≥ −C(A)h − (6 + √ )φ τ

14

Now denoting by h0 (τ ) the minimum of h(y, 1 − t), we can estimate √ h0 (τ ) 6 τ +1 1 50 d √ − (log( √ )) ≤ C(A) + ≤ C(A) + √ , dτ τ 2τ − 4τ 2 τ 2τ τ whence h0 (τ ) ≤

√

√ τ exp(C(A)τ + 100 τ ),

(6.2)

(6.3)

because the left hand side of (6.2) tends to zero as τ → 0 + . Now we can return to our proof, replace the right hand side of (6.1) by the right hand side of (6.3) times r02 , with τ = r0−2 (t0 − tγ ), and apply Lemma 5.3. (b) Assume the contrary, take a sequence K1α → ∞ and consider the solu−2 tions violating the statement. Clearly, K1α (r0α )−2 < (r(tα , whence tα 0 )) 0 → ∞; When K1 is large enough, we can, arguing as in the proof of Claim 1 in ¯= [I.10.1], find a point (¯ x, t¯), x ∈ B(x0 , t¯, 2Ar0 ), t¯ ∈ [t0 − r02 /2, t0 ], such that Q R(¯ x, t¯) > K1 r0−2 , (¯ x, t¯) does not satisfy the canonical neighborhood assumption, ¯ does, where P¯ is the set of all (x, t) but each point (x, t) ∈ P¯ with R(x, t) ≥ 4Q 1 1 −1 ¯ ¯ − 12 . (Note ≤ t ≤ t¯, distt (x0 , x) ≤ distt¯(x0 , x¯) + K12 Q satisfying t¯ − 4 K1 Q that P¯ is not a parabolic neighborhood.) Clearly we can use (a) with slightly different parameters to ensure κ-noncollapsing in P¯ . Now we apply the argument from 5.4. First, by Claim 2 in 4.2, for any ¯Q ¯ in B(¯ ¯ − 12 ) when K1 is large A¯ < ∞ we have an estimate R ≤ Q(A) x, t¯, A¯Q ¯ about enough; therefore we can take a limit as α → ∞ of scalings with factor Q (¯ x, t¯), shifting the time t¯ to zero; the limit at time zero would be a smooth complete nonnegatively curved manifold. Next we observe that this limit has curvature uniformly bounded, say, by Q0 , and therefore, for each fixed A¯ and for ¯ − 12 , −ǫη −1 Q−1 Q ¯ −1 ) sufficiently large K1 , the parabolic neighborhood P (¯ x, t¯, A¯Q 0 ¯ is contained in P . (Here we use the estimate of distance change, given by Lemma I.8.3(a).) Thus we can take a limit on the interval [−ǫη −1 Q−1 0 , 0]. (The possibility of surgeries is ruled out as in 5.4) Then we repeat the procedure indefinitely, getting an ancient κ-solution in the limit, which means a contradiction. (c) If x ∈ B(x0 , t0 , Ar0 ) has very large curvature, then on the shortest geodesic γ at time t0 , that connects x0 and x, we can find a point y, such that R(y, t0 ) = K1 (A)r0−2 and the curvature is larger at all points of the segment of γ between x and y. Then our statement follows from Claim 2 in 4.2, applied to this segment. ¯ to be min(δ(t), ¯ From now on we redefine the function δ(t) δ¯2t (2t)), so that the proposition above always holds for A = t0 . 6.4 Proposition. There exist τ > 0, r¯ > 0, K < ∞ with the following property. Suppose we have a solution to the Ricci flow with δ(t)-cutoff on the time √ interval [0, t0 ], with normalized initial data. Let r0 , t0 satisfy 2C1 h ≤ r0 ≤ r¯ t0 , where h is the maximal cutoff radius for surgeries in [t0 /2, t0 ], and assume that the ball B(x0 , t0 , r0 ) has sectional curvatures at least −r0−2 at each point, and the volume of any subball B(x, t0 , r) ⊂ B(x0 , t0 , r0 ) with any radius r > 0 is at least (1 − ǫ) times the volume of the euclidean ball of the same radius. Then the solution is defined in P (x0 , t0 , r0 /4, −τ r02 ) and satisfies R < Kr0−2 there. 15

Proof. Let us first consider the case r0 ≤ r(t0 ). Then clearly R(x0 , t0 ) ≤ C12 r0−2 , since an ǫ-neck of radius r can not contain an almost euclidean ball of radius ≥ r. Thus we can take K = 2C12 , τ = ǫη −1 C1−2 in this case, and since 2 r0 ≥ 2C1 h, the surgeries do not interfere in P (x0 , t0 , r√ 0 /4, −τ r0 ). In order to handle the other case r(t0 ) < r0 ≤ r¯ t0 we need a couple of lemmas. 6.5 Lemma. There exist τ0 > 0 and K0 < ∞, such that if we have a smooth solution to the Ricci flow in P (x0 , 0, 1, −τ ), τ ≤ τ0 , having sectional curvatures at least −1, and the volume of the ball B(x0 , 0, 1) is at least (1 − ǫ) times the volume of the euclidean unit ball, then (a) R ≤ K0 τ −1 in P (x0 , 0, 1/4, −τ /2), and 1 (b) the ball B(x0 , 1/4, −τ ) has volume at least 10 times the volume of the euclidean ball of the same radius. The proof can be extracted from the proof of Lemma I.11.6. 6.6 Lemma. For any w > 0 there exists θ0 = θ0 (w) > 0, such that if B(x, 1) is a metric ball of volume at least w, compactly contained in a manifold without boundary with sectional curvatures at least −1, then there exists a ball B(y, θ0 ) ⊂ B(x, 1), such that every subball B(z, r) ⊂ B(y, θ0 ) of any radius r has volume at least (1 − ǫ) times the volume of the euclidean ball of the same radius. This is an elementary fact from the theory of Aleksandrov spaces. 6.7 Now we continue the proof of the proposition. We claim that one can take τ = min(τ0 /2, ǫη −1 C1−2 ), K = max(2K0 τ −1 , 2C12 ). Indeed, assume the contrary, and take a sequence of r¯α → 0 and solutions, violating our assertion for α α α the chosen τ, K. Let tα 0 be the first time when it is violated, and let B(x0 , t0 , r0 ) α be the counterexample with the smallest radius. Clearly r0 > r(tα 0 ) and −1 (r0α )2 (tα ) → 0 as α → ∞. 0 Consider any ball B(x1 , t0 , r) ⊂ B(x0 , t0 , r0 ), r < r0 . Clearly we can apply our proposition to this ball and get the solution in P (x1 , t0 , r/4, −τ r2 ) with the curvature bound R < Kr−2 . Now if r02 t−1 0 is small enough, then we can apply proposition 6.3(c) to get an estimate R(x, t) ≤ K ′ (A)r−2 for (x, t) satisfying t ∈ [t0 − τ r2 /2, t0 ], distt (x, x1 ) < Ar, for some function K ′ (A) that can be made explicit. Let us choose A = 100r0 r−1 ; then we get the solution with a curvature estimate in P (x0 , t0 , r0 , −△t), where △t = K ′ (A)−1 r2 . Now the pinching estimate implies Rm ≥ −r0−2 on this set, if r02 t−1 is small enough 0 while rr0−1 is bounded away from zero. Thus we can use lemma 6.5(b) to 1 estimate the volume of the ball B(x0 , t0 − △t, r0 /4) by at least 10 of the volume of the euclidean ball of the same radius, and then by lemma 6.6 we can find a 1 )r0 /4), satisfying the assumptions of our proposition. subball B(x2 , t0 −△t, θ0 ( 10 1 Therefore, if we put r = θ0 ( 10 )r0 /4, then we can repeat our procedure as many times as we like, until we reach the time t0 − τ0 r02 , when the lemma 6.5(b) stops working. But once we reach this time, we can apply lemma 6.5(a) and get the required curvature estimate, which is a contradiction. 6.8 Corollary. For any w > 0 one can find τ = τ (w) > 0, K = K(w) < ∞, r¯ = r¯(w) > 0, θ = θ(w) > 0 with the following property. Suppose we have a solution to the Ricci flow with δ(t)-cutoff on the time interval [0, t0 ], with 16

√ normalized initial data. Let t0 , r0 satisfy θ−1 (w)h ≤ r0 ≤ r¯ t0 , and assume that the ball B(x0 , t0 , r0 ) has sectional curvatures at least −r02 at each point, and volume at least wr03 . Then the solution is defined in P (x0 , t0 , r0 /4, −τ r02 ) and satisfies R < Kr0−2 there. Indeed, we can apply proposition 6.4 to a smaller ball, provided by lemma 6.6, and then use proposition 6.3(c).

7

Long time behavior II

In this section we adapt the arguments of Hamilton [H 4] to a more general setting. Hamilton considered smooth Ricci flow with bounded normalized curvature; we drop both these assumptions. In the end of [I,13.2] I claimed that the volumes of the maximal horns can be effectively bounded below, which would imply that the solution must be smooth from some time on; however, the argument I had in mind seems to be faulty. On the other hand, as we’ll see below, the presence of surgeries does not lead to any substantial problems. From now on we assume that our initial manifold does not admit a metric with nonnegative scalar curvature, and that once we get a component with nonnegative scalar curvature, it is immediately removed. 7.1 (cf. [H 4,§2,7]) Recall that for a solution to the smooth Ricci flow the scalar curvature satisfies the evolution equation 2 d R = △R + 2|Ric|2 = △R + 2|Ric◦ |2 + R2 , dt 3 where Ric◦ is the trace-free part of Ric. Then Rmin (t) satisfies whence 1 3 Rmin (t) ≥ − 2 t + 1/4

d dt Rmin

(7.1) 2 ≥ 32 Rmin ,

(7.2)

for a solution withR normalized initial data. The evolution equation for the d volume is dt V = − RdV, in particular d V ≤ −Rmin V, dt

(7.3)

3 whence by (7.2) the function V (t)(t+1/4)− 2 is non-increasing in t. Let V¯ denote its limit as t → ∞. ˆ = Rmin V 23 satisfies Now the scale invariant quantity R Z d ˆ 2 ˆ −1 R(t) ≥ RV (Rmin − R)dV, (7.4) dt 3

which is nonnegative whenever Rmin ≤ 0, which we have assumed from the ¯ denote the limit of R(t) ˆ beginning of the section. Let R as t → ∞. Assume for a moment that V¯ > 0. Then it follows from (7.2) and (7.3) that 3 ¯ V¯ − 23 = − 3 . Now the inequality Rmin (t) is asymptotic to − 2t ; in other words, R 2 (7.4) implies that whenever we have a sequence of parabolic neighborhoods 17

√ P (xα , tα , r tα , −r2 tα ), for tα → ∞ and some fixed small r > 0, such that the scalings of our solution with factor tα smoothly converge to some limit solution, defined in an abstract parabolic neighborhood P (¯ x, 1, r, −r2 ), then the scalar curvature of this limit solution is independent of the space variables and equals 3 at time t ∈ [1 − r2 , 1]; moreover, the strong maximum principle for (7.1) − 2t implies that the sectional curvature of the limit at time t is constant and equals 1 − 4t . This conclusion is also valid without the a priori assumption that V¯ > 0, since otherwise it is vacuous. Clearly the inequalities and conclusions above hold for the solutions to the Ricci flow with δ(t)-cutoff, defined in the previous sections. From now on we assume that we are given such a solution, so the estimates below may depend on it. 7.2 Lemma. (a) Given w > 0,√ r > 0, ξ > 0 one can find T = T (w, r, ξ) < ∞, such that if the ball B(x0 , t0 , r t0 ) at some time t0 ≥ T has volume at least wr3 and sectional curvature at least −r−2 t−1 0 , then curvature at x0 at time t = t0 satisfies |2tRij + gij | < ξ. (7.5) (b) Given in addition A < ∞ and √ allowing T to depend on A, we can ensure (7.5) for all points in B(x0 , t0 , Ar t0 ). √ (c) The same is true for P (x0 , t0 , Ar t0 , Ar2 t0 ). Proof. (a) If T is large enough √ then we can apply corollary 6.8 to the ball B(x0 , t0 , r0 ) for r0 = min(r, r¯(w)) t0 ; then use the conclusion of 7.1. (b) The curvature control in P (x0 , t0 , r0 /4, −τ r02 ), provided by corollary 6.8, allows us to apply proposition 6.3 (a),(b) to a controllably smaller neighborhood √ P (x0 , t0 , r0′ , −(r0′ )2 ). Thus by 6.3(b) we know that each point in B(x0 , t0 , Ar t0 ) −2 ′ with scalar curvature at least Q = K1 (A)r0 has a canonical neighborhood. This implies that for T large enough such points do not exist, since if there was a point with R larger than Q, there would be a point having a canonical neighborhood with R = Q in the same ball, and that contradicts the already proved assertion (a). Therefore we have curvature control in the ball in question, and applying 6.3(a) we also get volume control there, so our assertion has been reduced to (a). √ (c) If ξ is small enough, then the solution in the ball B(x0 , t0 , Ar t0 ) would 2 stay almost homothetic to itself on the time interval [t0 , t0 + Ar t0 ] until (7.5) is violated at some (first) time t′ in this interval. However, if T is large enough, then this violation could not happen, because we can apply the already proved assertion (b) at time t′ for somewhat larger A. 7.3 Let ρ(x, t) denote the radius ρ of the ball B(x, t, ρ) where inf Rm = −ρ−2 . It follows from corollary 6.8, proposition 6.3(c), and the pinching estimate √ (5.1) that for any w > 0 we can find ρ¯ = ρ¯(w) > 0, such that if ρ(x, t) < ρ¯ t, then V ol B(x, t, ρ(x, t)) < wρ3 (x, t), (7.6) provided that t is large enough (depending on w). Let M − (w, t) denote the thin part of M, that is the set of x ∈ M where (7.6) holds at time t, and let M + (w, t) be its complement. Then for t large enough 18

(depending on w) every point of M + satisfies the assumptions of lemma 7.2. Assume first that for some w > 0 the set M + (w, t) is not empty for a sequence of t → ∞. Then the arguments of Hamilton [H 4,§8-12] work in our situation. In particular, if we take a sequence of points xα ∈ M + (w, tα ), tα → α ∞, then the scalings of gij about xα with factors (tα )−1 converge, along a subsequence of α → ∞, to a complete hyperbolic manifold of finite volume. The limits may be different for different choices of (xα , tα ). If none of the limits is closed, and H1 is such a limit with the least number of cusps, then, by an argument in [H 4,§8-10], based on hyperbolic rigidity, for all sufficiently small w′ , 0 < w′ < w(H ¯ 1 ), there exists a standard truncation H1 (w′ ) of H1 , such that, for t large enough, M + (w′ /2, t) contains an almost isometric copy of H1 (w′ ), which in turn contains a component of M + (w′ , t); moreover, this embedded copy of H1 (w′ ) moves by isotopy as t increases to infinity. If for some w > 0 the complement M + (w, t) \ H1 (w) is not empty for a sequence of t → ∞, then we can repeat the argument and get another complete hyperbolic manifold H2 , etc., until we find a finite collection of Hj , 1 ≤ j ≤ i, such that for each sufficiently small w > 0 the embeddings of Hj (w) cover M + (w, t) for all sufficiently large t. Furthermore, the boundary tori of Hj (w) are incompressible in M. This is proved [H 4,§11,12] by a minimal surface argument, using a result of Meeks and Yau. This argument does not use the uniform bound on the normalized curvature, and goes through even in the presence of surgeries, because the area of the least area disk in question can only decrease when we make a surgery. 7.4 Let us redefine the thin part in case the thick one isn’t empty, M − (w, t) = M \(H1 (w)∪...∪Hi (w)). Then, for sufficiently small w > 0 and sufficiently large t, M − (w, t) is diffeomorphic to a graph manifold, as implied by the following general result on collapsing with local lower curvature bound, applied to the metrics t−1 gij (t). α Theorem. Suppose (M α , gij ) is a sequence of compact oriented riemannian 3-manifolds, closed or with convex boundary, and wα → 0. Assume that (1) for each point x ∈ M α there exists a radius ρ = ρα (x), 0 < ρ < 1, not exceeding the diameter of the manifold, such that the ball B(x, ρ) in the metric α gij has volume at most wα ρ3 and sectional curvatures at least −ρ−2 ; (2) each component of the boundary of M α has diameter at most wα , and has a (topologically trivial) collar of length one, where the sectional curvatures are between −1/4 − ǫ and −1/4 + ǫ; (3) For every w′ > 0 there exist r¯ = r¯(w′ ) > 0 and Km = Km (w′ ) < ∞, m = 0, 1, 2..., such that if α is large enough, 0 < r ≤ r¯, and the ball B(x, r) α in gij has volume at least w′ r3 and sectional curvatures at least −r2 , then the curvature and its m-th order covariant derivatives at x, m = 1, 2..., are bounded by K0 r−2 and Km r−m−2 respectively. Then M α for sufficiently large α are diffeomorphic to graph manifolds. Indeed, there is only one exceptional case, not covered by the theorem above, namely, when M = M − (w, t), and ρ(x, t), for some x ∈ M, is much larger than the diameter d(t) of the manifold, whereas the ratio V (t)/d3 (t) is bounded away from zero. In this case, since by the observation after formula (7.3) the

19

3

2 volume V (t) can not √ grow faster than const · t , the diameter does not grow faster than const · t, hence if we scale our metrics gij (t) to keep the diameter equal to one, the scaled metrics would satisfy the assumption (3) of the theorem above and have the minimum of sectional curvatures tending to zero. Thus we can take a limit and get a smooth solution to the Ricci flow with nonnegative sectional curvature, but not strictly positive scalar curvature. Therefore, in this exceptional case M is diffeomorphic to a flat manifold. The proof of the theorem above will be given in a separate paper; it has nothing to do with the Ricci flow; its main tool is the critical point theory for distance functions and maps, see [P,§2] and references therein. The assumption (3) is in fact redundant; however, it allows to simplify the proof quite a bit, by avoiding 3-dimensional Aleksandrov spaces, and in particular, the nonelementary Stability Theorem. Summarizing, we have shown that for large t every component of the solution is either diffeomorphic to a graph manifold, or to a closed hyperbolic manifold, or can be split by a finite collection of disjoint incompressible tori into parts, each being diffeomorphic to either a graph manifold or to a complete noncompact hyperbolic manifold of finite volume. The topology of graph manifolds is well understood [W]; in particular, every graph manifold can be decomposed in a connected sum of irreducible graph manifolds, and each irreducible one can in turn be split by a finite collection of disjoint incompressible tori into Seifert fibered manifolds.

8

On the first eigenvalue of the operator −4△+R

8.1 Recall from [I,§1,2] that Ricci flow is the gradient flow for the first eigenvalue 2 d λ of the operator −4△ + R; moreover, dt λ(t) ≥ 23 λ2 (t) and λ(t)V 3 (t) is nondecreasing whenever it is nonpositive. We would like to extend these inequalities to the case of Ricci flow with δ(t)-cutoff. Recall that we immediately remove components with nonnegative scalar curvature. Lemma. Given any positive continuous function ξ(t) one can chose δ(t) in such a way that for any solution to the Ricci flow with δ(t)-cutoff, with normalized initial data, and any surgery time T0 , after which there is at least one component, where the scalar curvature is not strictly positive, we have an estimate λ+ (T0 )−λ− (T0 ) ≥ ξ(T0 )(V + (T0 )−V − (T0 )), where V − , V + and λ− , λ+ are the volumes and the first eigenvalues of −4△+R before and after the surgery respectively. Proof. Consider the minimizer a for the functional Z (4|∇a|2 + Ra2 ) (8.1) R under normalization a2 = 1, for the metric after the surgery on a component where scalar curvature is not strictly positive. Clearly is satisfies the equation 4△a = Ra − λ− a 20

(8.2)

Observe that since the metric contains an ǫ-neck of radius about r(T0 ), we can estimate λ− (T0 ) from above by about r(T0 )−2 . Let Mcap denote the cap, added by the surgery. It is attached to a long tube, consisting of ǫ-necks of various radii. Let us restrict our attention to a maximal subtube, on which the scalar curvature at each point is at least 2λ− (T0 ). Choose any ǫ-neck in this subtube, say, with radius r0 , and consider the distance function with range [0, 2ǫ−1 r0 ], whose level sets Mz are almost round two-spheres; let Mz+ ⊃ Mcap be the part of M, chopped off by Mz . Then Z Z Z −4aaz = (4|∇a|2 + Ra2 − λ− a2 ) > r0−2 /2 a2 Mz+

Mz

On the other hand, Z |

Mz

Z 2aaz − (

Mz+

2

Mz

a )z | ≤ const ·

These two inequalities easily imply that Z Z a2 ≥ exp(ǫ−1 /10) M0+

Z

Mz

ǫr0−1 a2

a2

M +−1 ǫ

r0

Now the chosen subtube contains at least about −ǫ−1 log(λ− (T0 )h2 (T0 )) disjoint ǫ-necks, where h denotes the cutoff radius, as before. Since h tends to zero with δ, whereas r(T0 ), that occurs in the bound for λ− , is independent of δ, we that the number of necks is greater then log h, and therefore, R can 2ensure 6 a < h , say. Then standard estimates for the equation (8.2) show that Mcap 2 2 |∇a| and Ra are bounded by const · h on Mcap , which makes it possible to extend a to the metric before surgery in such a way that the functional (8.1) is preserved up to const · h4 . However, the loss of volume in the surgery is at least h3 , so it suffices to take δ so small that h is much smaller than ξ. 8.2 The arguments above lead to the following result (a) If (M, gij ) has λ > 0, then, for an appropriate choice of the cutoff parameter, the solution becomes extinct in finite time. Thus, if M admits a metric with λ > 0 then it is diffeomorphic to a connected sum of a finite collection of S2 × S1 and metric quotients of the round S3 . Conversely, every such connected sum admits a metric with R > 0, hence with λ > 0. ¯ denote the (b) Suppose M does not admit any metric with λ > 0, and let λ 2 ¯ = 0 implies that supremum of λV 3 over all metrics on this manifold. Then λ ¯ < 0. M is a graph manifold. Conversely, a graph manifold can not have λ ¯ 32 . Then V¯ is the minimum of V, such ¯ < 0 and let V¯ = (− 2 λ) (c) Suppose λ 3 that M can be decomposed in connected sum of a finite collection of S2 × S1 , metric quotients of the round S3 , and some other components, the union of which will be denoted by M ′ , and there exists a (possibly disconnected) complete hyperbolic manifold, with sectional curvature −1/4 and volume V, which can be embedded in M ′ in such a way that the complement (if not empty) is a graph

21

manifold. Moreover, if such a hyperbolic manifold has volume V¯ , then its cusps (if any) are incompressible in M ′ . For the proof one needs in addition easily verifiable statements that one can put metrics on connected sums preserving the lower bound for scalar curvature [G-L], that one can put metrics on graph manifolds with scalar curvature bounded below and volume tending to zero [C-G], and that one can close a compressible cusp, preserving the lower bound for scalar curvature and reducing the volume, cf. [A,5.2]. Notice that using these results we can avoid the hyperbolic rigidity and minimal surface arguments, quoted in 7.3, which, however, have the advantage of not requiring any a priori topological information about the complement of the hyperbolic piece. The results above are exact analogs of the conjectures for the Sigma constant, formulated by Anderson [A], at least in the nonpositive case.

References [I] G.Perelman The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 v1 [A] M.T.Anderson Scalar curvature and geometrization conjecture for threemanifolds. Comparison Geometry (Berkeley, 1993-94), MSRI Publ. 30 (1997), 49-82. [C-G] J.Cheeger, M.Gromov Collapsing Riemannian manifolds while keeping their curvature bounded I. Jour. Diff. Geom. 23 (1986), 309-346. [G-L] M.Gromov, H.B.Lawson Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math. IHES 58 (1983), 83-196. [H 1] R.S.Hamilton Three-manifolds with positive Ricci curvature. Jour. Diff. Geom. 17 (1982), 255-306. [H 2] R.S.Hamilton Formation of singularities in the Ricci flow. Surveys in Diff. Geom. 2 (1995), 7-136. [H 3] R.S.Hamilton The Harnack estimate for the Ricci flow. Jour. Diff. Geom. 37 (1993), 225-243. [H 4] R.S.Hamilton Non-singular solutions of the Ricci flow on three-manifolds. Commun. Anal. Geom. 7 (1999), 695-729. [H 5] R.S.Hamilton Four-manifolds with positive isotropic curvature. Commun. Anal. Geom. 5 (1997), 1-92. G.Perelman Spaces with curvature bounded below. Proceedings of ICM1994, 517-525. F.Waldhausen Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I,II. Invent. Math. 3 (1967), 308-333 and 4 (1967), 87-117.

22

arXiv:math/0307245v1 [math.DG] 17 Jul 2003

Finite extinction time for the solutions to the Ricci flow on certain three-manifolds Grisha Perelman∗ February 1, 2008 In our previous paper we constructed complete solutions to the Ricci flow with surgery for arbitrary initial riemannian metric on a (closed, oriented) three-manifold [P,6.1], and used the behavior of such solutions to classify threemanifolds into three types [P,8.2]. In particular, the first type consisted of those manifolds, whose prime factors are diffeomorphic copies of spherical space forms and S2 × S1 ; they were characterized by the property that they admit metrics, that give rise to solutions to the Ricci flow with surgery, which become extinct in finite time. While this classification was sufficient to answer topological questions, an analytical question of significant independent interest remained open, namely, whether the solution becomes extinct in finite time for every initial metric on a manifold of this type. In this note we prove that this is indeed the case. Our argument (in conjunction with [P,§1-5]) also gives a direct proof of the so called ”elliptization conjecture”. It turns out that it does not require any substantially new ideas: we use only a version of the least area disk argument from [H,§11] and a regularization of the curve shortening flow from [A-G].

1

Finite time extinction

1.1 Theorem. Let M be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. Then for any initial metric on M the solution to the Ricci flow with surgery becomes extinct in finite time. Proof for irreducible M . Let ΛM denote the space of all contractible loops in C 1 (S1 → M ). Given a riemannian metric g on M and c ∈ ΛM, define A(c, g) to be the infimum of the areas of all lipschitz maps from D2 to M, whose restriction to ∂D2 = S1 is c. For a family Γ ⊂ ΛM let A(Γ, g) be the supremum of A(c, g) over all c ∈ Γ. Finally, for a nontrivial homotopy class α ∈ π∗ (ΛM, M ) let A(α, g) be the infimum of A(Γ, g) over all Γ ∈ α. Since M is not aspherical, it follows from a classical (and elementary) result of Serre that such a nontrivial homotopy class exists. ∗ St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191023, Russia. Email: [email protected] or [email protected]

1

1.2 Lemma. (cf. [H,§11]) If g t is a smooth solution to the Ricci flow, then for any α the rate of change of the function At = A(α, g t ) satisfies the estimate 1 t d t A ≤ −2π − Rmin At dt 2 t (in the sense of the lim sup of the forward difference quotients), where Rmin t denotes the minimum of the scalar curvature of the metric g . A rigorous proof of this lemma will be given in §3, but the idea is simple and can be explained here. Let us assume that at time t the value At is attained by the family Γ, such that the loops c ∈ Γ where A(c, g t ) is close to At are embedded and sufficiently smooth. For each such c consider the minimal disk Dc with boundary c and with area A(c, g t ). Now let the metric evolve by the Ricci flow and let the curves c evolve by the curve shortening flow (which moves every point of the curve in the direction of its curvature vector at this point) with the same time parameter. Then the rate of change of the area of Dc can be computed as Z Z T (−Tr(Ric )) + (−kg ) Dc

c

T

where Ric is the Ricci tensor of M restricted to the tangent plane of Dc , and kg is the geodesic curvature of c with respect to Dc (cf. [A-G, Lemma 3.2]). In three dimensions the first integrand equals − 21 R − (K − det II), where K is the intrinsic curvature of Dc and det II, the determinant of the second fundamental form, is nonpositive, because Dc is minimal. Thus, the rate of change of the area of Dc can be estimated from above by Z Z Z 1 1 (− R) − 2π (− R − K) + (−kg ) = 2 2 c Dc Dc by the Gauss-Bonnet theorem, and the statement of the lemma follows. The problem with this argument is that if Γ contains curves, which are not immersed (for instance, a curve could pass an arc once in one direction and then make an about turn and pass the same arc in the opposite direction), then it is not clear how to define curve shortening flow so that it would be continuous both in the time parameter and in the family parameter. In §3 we’ll explain how to circumvent this difficulty, essentially by adding one dimension to the ambient manifold. This regularization of the curve shortening flow has been worked out by Altschuler and Grayson [A-G] (who were interested in approximating the singular curve shortening flow on the plane and obtained for that case more precise results than what we need). 1.3 Now consider the solution to the Ricci flow with surgery. Since M is assumed irreducible, the surgeries are topologically trivial, that is one of the components of the post-surgery manifold is diffeomorphic to the pre-surgery manifold, and all the others are spheres. Moreover, by the construction of the surgery [P,4.4], the diffeomorphism from the pre-surgery manifold to the post-surgery one can be chosen to be distance non-increasing ( more precisely, (1 + ξ)-lipschitz, where ξ > 0 can be made as small as we like). It follows that 2

the conclusion of the lemma above holds for the solutions to the Ricci flow with surgery as well. Now recall that the evolution equation for the scalar curvature 2 d R = △R + 2|Ric|2 = △R + R2 + 2|Ric◦ |2 dt 3 1 At t implies the estimate Rmin ≥ − 23 t+const . It follows that Aˆt = t+const satisfies 2π d ˆt A ≤ − , which implies finite extinction time since the right hand side dt t+const t ˆ is non-integrable at infinity whereas A can not become negative. 1.4 Remark. The finite time extinction result for irreducible non-aspherical manifolds already implies (in conjuction with the work in [P,§1-5] and the Kneser finiteness theorem) the so called ”elliptization conjecture”, claiming that a closed manifold with finite fundamental group is diffeomorphic to a spherical space form. The analysis of the long time behavior in [P,§6-8] is not needed in this case; moreover the argument in [P,§5] can be slightly simplified, replacing ¯ since we already have an upper the sequences rj , κj , δ¯j by single values r, κ, δ, bound on the extinction time in terms of the initial metric. In fact, we can even avoid the use of the Kneser theorem. Indeed, if we start from an initial metric on a homotopy sphere (not assumed irreducible), then at each surgery time we have (almost) distance non-increasing homotopy equivalences from the pre-surgery manifold to each of the post-surgery components, and this is enough to keep track of the nontrivial relative homotopy class of the loop space. 1.5 Proof of theorem 1.1 for general M . The Kneser theorem implies that our solution undergoes only finitely many topologically nontrivial surgeries, so from some time T on all the surgeries are trivial. Moreover, by the Milnor uniqueness theorem, each component at time T satisfies the assumption of the theorem. Since we already know from 1.4 that there can not be any simply connected prime factors, it follows that every such component is either irreducible, or has nontrivial π2 ; in either case the proof in 1.1-1.3 works.

2

Preliminaries on the curve shortening flow

In this section we rather closely follow [A-G]. 2.1 Let M be a closed n-dimensional manifold, n ≥ 3, and let g t be a smooth family of riemannian metrics on M evolving by the Ricci flow on a finite time interval [t0 , t1 ]. It is known [B] that g t for t > t0 are real analytic. Let ct be a solution to the curve shortening flow in (M, g t ), that is ct satisfies the equation d t t 1 t dt c (x) = H (x), where x is the parameter on S , and H is the curvature vector field of ct with respect to g t . It is known [G-H] that for any smoothly immersed initial curve c the solution ct exists on some time interval [t0 , t′1 ), each ct for t > t0 is an analytic immersed curve, and either t′1 = t1 , or the curvature 1 k t = g t (H t , H t ) 2 is unbounded when t → t′1 .

3

1

Denote by X t the tangent vector field to ct , and let S t = g t (X t , X t )− 2 X t be the unit tangent vector field; then H = ∇S S (from now on we drop the superscript t except where this omission can cause confusion). We compute d g(X, X) = −2Ric(X, X) − 2g(X, X)k 2 , dt

(1)

[H, S] = (k 2 + Ric(S, S))S

(2)

d 2 k = (k 2 )′′ − 2g((∇S H)⊥ , (∇S H)⊥ ) + 2k 4 + ..., dt

(3)

which implies Now we can compute

where primes denote differentiation with respect to the arclength parameter s, and where dots stand for the terms containing the curvature tensor of g, which can be estimated in absolute value by const · (k 2 + k). Thus the curvature k satisfies d k ≤ k ′′ + k 3 + const · (k + 1) (4) dt Now Rit follows from (1) and (4) that the length L and the total curvature Θ = kds satisfy Z d L ≤ (const − k 2 )ds, (5) dt Z d Θ ≤ const · (k + 1)ds (6) dt

In particular, both quantities can grow at most exponentially in t (they would be non-increasing in a flat manifold). 2.2 In general the curvature of ct may concentrate near certain points, creating singularities. However, if we know that this does not happen at some time t∗ , then we can estimate the curvature and higher derivatives at times shortly thereafter. More precisely, there exist constants ǫ, C1 , C2 , ... (which may depend on the curvatures of the ambient space and their derivatives, but are independent of ct ), such that if at time t∗ for some r > 0 the length of ct is at least r and the total curvature of each arc of length r does not exceed ǫ, then for every t ∈ (t∗ , t∗ + ǫr2 ) the curvature k and higher derivatives satisfy the estimates k 2 = g(H, H) ≤ C0 (t − t∗ )−1 , g(∇S H, ∇S H) ≤ C1 (t − t∗ )−2 , ... This can be proved by adapting the arguments of Ecker and Huisken [E-Hu]; see also [A-G,§4]. ¯ , g¯t ) × S1 , 2.3 Now suppose that our manifold (M, g t ) is a metric product (M λ where the second factor is the circle of constant length λ; let U denote the unit tangent vector field to this factor. Then u = g(S, U ) satisfies the evolution equation d u = u′′ + (k 2 + Ric(S, S))u (7) dt

4

Assume that u was strictly positive everywhere at time t0 (in this case the curve is called a ramp). Then it will remain positive and bounded away from zero as long as the solution exists. Now combining (4) and (7) we can estimate the right hand side of the evolution equation for the ratio uk and conclude that this ratio, and hence the curvature k, stays bounded (see [A-G,§2]). It follows that ct is defined on the whole interval [t0 , t1 ]. 2.4 Assume now that we have two ramp solutions ct1 , ct2 , each winding once around the S1λ factor. Let µt be the infimum of the areas of the annuli with boundary ct1 ∪ ct2 . Then d t µ ≤ (2n − 1)|Rmt |µt , dt

(8)

where |Rmt | denotes a bound on the absolute value of sectional curvatures of g t . Indeed, the curves ct1 and ct2 , being ramps, are embedded and without substantial loss of generality we may assume them to be disjoint. In this case the results of Morrey [M] and Hildebrandt [Hi] yield an analytic minimal annulus A, immersed, except at most finitely many branch points, with prescribed boundary and with area µ. The rate of change of the area of A can be computed as Z Z Z (−Tr(RicT )) + (−kg ) ≤ (−Tr(RicT ) + K) A

∂A

≤

Z

A

A

(−Tr(RicT ) + RmT ) ≤ (2n − 1)|Rm|µ,

where the first inequality comes from the Gauss-Bonnet theorem, with possible contribution of the branch points, and the second one is due to the fact that a minimal surface has nonpositive extrinsic curvature with respect to any normal vector. 2.5 The estimate (8) implies that µt can grow at most exponentially; in particular, if ct1 and ct2 were very close at time t0 , then they would be close for all t ∈ [t0 , t1 ] in the sense of minimal annulus area. In general this does not imply that the lengths of the curves are also close. However, an elementary argument shows that if ǫ > 0 is small then, given any r > 0, one can find µ ¯, depending only on r and on upper bound for sectional curvatures of the ambient space, such that if the length of ct1 is at least r, each arc of ct1 with length r has total curvature at most ǫ, and µt ≤ µ ¯ , then L(ct2 ) ≥ (1 − 100ǫ)L(ct1).

3

Proof of lemma 1.2

3.1 In this section we prove the following statement Let M be a closed three-manifold, and let (M, g t ) be a smooth solution to the Ricci flow on a finite time interval [t0 , t1 ]. Suppose that Γ ⊂ ΛM is a compact family. Then for any ξ > 0 one can construct a continuous deformation Γt , t ∈ [t0 , t1 ], Γt0 = Γ, such that for each curve c ∈ Γ either the value A(ct1 , g t1 ) is bounded from above by ξ plus the value at t = t1 of the solution to the ODE 5

t = −2π − 21 Rmin w(t) with the initial data w(t0 ) = A(ct0 , g t0 ), or L(c ) ≤ ξ; moreover, if c was a constant map, then all ct are constant maps. It is clear that our statement implies lemma 1.2, because a family consisting of very short loops can not represent a nontrivial relative homotopy class. 3.2 As a first step of the proof of the statement we can replace Γ by a family, which consists of piecewise geodesic loops with some large fixed number of vertices and with each segment reparametrized in some standard way to make the parametrizations of the whole curves twice continuously differentiable. Now consider the manifold Mλ = M × S1λ , 0 < λ < 1, and for each c ∈ Γ consider the smooth embedded closed curve cλ such that p1 cλ (x) = c(x) and p2 cλ (x) = λx mod λ, where p1 and p2 are projections of Mλ to the first and second factor respectively, and x is the parameter of the curve c on the standard circle of length one. Using 2.3 we can construct a solution ctλ , t ∈ [t0 , t1 ] to the curve shortening flow with initial data cλ . The required deformation will be obtained as Γt = p1 Γtλ (where Γtλ denotes the family consisting of ctλ ) for certain sufficiently small λ > 0. We’ll verify that an appropriate λ can be found for each individual curve c, or for any finite number of them, and then show that if our λ works for all elements of a µ-net in Γ, for sufficiently small µ > 0, then it works for all elements of Γ. 3.3 In the following estimates we shall denote by C large constants that may depend on metrics g t , family Γ and ξ, but are independent of λ, µ and a particular curve c. The first step in 3.2 implies that the lengths and total curvatures of cλ are uniformly bounded, so by 2.1 the same is true for all ctλ . It follows that the area swept by ctλ , t ∈ [t′ , t′′ ] ⊂ [t0 , t1 ] is bounded above by C(t′′ − t′ ), and therefore ′′ ′′ ′ ′ we have the estimates A(p1 ctλ , g t ) ≤ C, A(p1 ctλ , g t ) − A(p1 ctλ , g t ) ≤ C(t′′ − t′ ). R t1 R 2 3.4 It follows from (5) that t0 k dsdt ≤ C for any ctλ . Fix some large constant B, to be chosen later. ThenR there is a subset RIB (cλ ) ⊂ [t0 , t1 ] of measure at least t1 − t0 − CB −1 where k 2 ds ≤ B, hence kds ≤ ǫ on any arc of length ≤ ǫ2 B −1 . Assuming that ctλ are at least that long, we can apply 2.2 and construct another subset JB (cλ ) ⊂ [t0 , t1 ] of measure at least t1 −t0 −CB −1 , consisting of finitely many intervals of measure at least C −1 B −2 each, such that for any t ∈ JB (cλ ) we have pointwise estimates on ctλ for curvature and higher derivatives, of the form k ≤ CB, ... Now fix c, B, and consider any sequence of λ → 0. Assume again that the lengths of ctλ are bounded below by ǫ2 B −1 , at least for t ∈ [t0 , t2 ], where t2 = t1 − B −1 . Then an elementary argument shows that we can find a subsequence Λc and a subset JB (c) ⊂ [t0 , t2 ] of measure at least t1 − t0 − CB −1 , consisting of finitely many intervals, such that JB (c) ⊂ JB (cλ ) for all λ ∈ Λc . It follows that on every interval of JB (c) the curve shortening flows ctλ smoothly converge (as λ → 0 in some subsequence of Λc ) to a curve shortening flow in M. d t Let wc (t) be the solution of the ODE dt wc (t) = −2π − 12 Rmin wc (t) with t0 initial data wc (t0 ) = A(c, g ). Then for sufficiently small λ ∈ Λc we have A(p1 ctλ , g t ) ≤ wc (t) + 12 ξ provided that B > Cξ −1 . Indeed, on the intervals of JB (c) we can estimate the change of A for the limit flow using the minimal d dt w(t) t1

6

disk argument as in 1.2, and this implies the corresponding estimate for p1 ctλ if λ ∈ Λc is small enough, whereas for the intervals of the complement of JB (c) we can use the estimate in 3.3. On the other hand, if our assumption on the lower bound for lengths does not hold, then it follows from (5) that L(ctλ2 ) ≤ CB −1 ≤ 21 ξ. 3.5 Now apply the previous argument to all elements of some finite µ-net ˆ ⊂ Γ for small µ > 0 to be determined later. We get a λ > 0 such that for each Γ ˆ either A(p1 cˆt1 , g t1 ) ≤ wcˆ(t1 ) + 1 ξ or L(ˆ cˆ ∈ Γ ctλ2 ) ≤ 12 ξ. Now for any curve c ∈ Γ λ 2 ˆ µ-close to c, and apply the result of 2.4. It follows that if pick a curve cˆ ∈ Γ, t1 t1 A(p1 cˆλ , g ) ≤ wcˆ(t1 ) + 21 ξ and µ ≤ C −1 ξ, then A(p1 ctλ1 , g t1 ) ≤ wc (t1 ) + ξ. On the other hand, if L(ˆ ctλ2 ) ≤ 12 ξ, then we can conclude that L(ctλ1 ) ≤ ξ provided that µ > 0 is small enough in comparison with ξ and B −1 . Indeed, if L(ctλ1 ) > ξ, then L(ctλ ) > 34 ξ for Rall t ∈ [t2 , t1 ]; on the other hand, using (5) we can find a t ∈ [t2 , t1 ], such that k 2 ds ≤ CB for ctλ ; hence, applying 2.5, we get L(ˆ ctλ ) > 32 ξ t2 1 for this t, which is incompatible with L(ˆ cλ ) ≤ 2 ξ. The proof of the statement 3.1 is complete.

References [A-G] S.Altschuler, M.Grayson Shortening space curves and flow through singularities. Jour. Diff. Geom. 35 (1992), 283-298. [B] S.Bando Real analyticity of solutions of Hamilton’s equation. Math. Zeit. 195 (1987), 93-97. [E-Hu] K.Ecker, G.Huisken Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105 (1991), 547-569. [G-H] M.Gage, R.S.Hamilton The heat equation shrinking convex plane curves. Jour. Diff. Geom. 23 (1986), 69-96. [H] R.S.Hamilton Non-singular solutions of the Ricci flow on three-manifolds. Commun. Anal. Geom. 7 (1999), 695-729. [Hi] S.Hildebrandt Boundary behavior of minimal surfaces. Arch. Rat. Mech. Anal. 35 (1969), 47-82. [M] C.B.Morrey The problem of Plateau on a riemannian manifold. Ann. Math. 49 (1948), 807-851. [P] G.Perelman Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109 v1

7

c 2006 International Press

ASIAN J. MATH. Vol. 10, No. 2, pp. 165–492, June 2006

001

´ AND A COMPLETE PROOF OF THE POINCARE GEOMETRIZATION CONJECTURES – APPLICATION OF THE HAMILTON-PERELMAN THEORY OF THE RICCI FLOW∗ HUAI-DONG CAO† AND XI-PING ZHU‡ Abstract. In this paper, we give a complete proof of the Poincar´ e and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow. Key words. Ricci flow, Ricci flow with surgery, Hamilton-Perelman theory, Poincar´ e Conjecture, geometrization of 3-manifolds AMS subject classifications. 53C21, 53C44

CONTENTS Introduction

167

1 Evolution Equations 1.1 The Ricci Flow . . . . . . . . . . . . . . . . . 1.2 Short-time Existence and Uniqueness . . . . . 1.3 Evolution of Curvatures . . . . . . . . . . . . 1.4 Derivative Estimates . . . . . . . . . . . . . . 1.5 Variational Structure and Dynamic Property

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172 172 177 183 190 199

2 Maximum Principle and Li-Yau-Hamilton Inequalities 2.1 Preserving Positive Curvature . . . . . . . . . . . . . . . . 2.2 Strong Maximum Principle . . . . . . . . . . . . . . . . . 2.3 Advanced Maximum Principle for Tensors . . . . . . . . . 2.4 Hamilton-Ivey Curvature Pinching Estimate . . . . . . . . 2.5 Li-Yau-Hamilton Estimates . . . . . . . . . . . . . . . . . 2.6 Perelman’s Estimate for Conjugate Heat Equations . . . .

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210 210 213 217 223 226 234

3 Perelman’s Reduced Volume 3.1 Riemannian Formalism in Potentially Infinite Dimensions 3.2 Comparison Theorems for Perelman’s Reduced Volume . . 3.3 No Local Collapsing Theorem I . . . . . . . . . . . . . . . 3.4 No Local Collapsing Theorem II . . . . . . . . . . . . . .

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239 239 243 255 261

4 Formation of Singularities 4.1 Cheeger Type Compactness 4.2 Injectivity Radius Estimates 4.3 Limiting Singularity Models 4.4 Ricci Solitons . . . . . . . .

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267 267 286 291 302

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∗ Received

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December 12, 2005; accepted for publication April 16, 2006. of Mathematics, Lehigh University, Bethlehem, PA 18015, USA ([email protected]). ‡ Department of Mathematics, Zhongshan University, Guangzhou 510275, P. R. China (stszxp@ zsu.edu.cn). † Department

165

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5 Long Time Behaviors 307 5.1 The Ricci Flow on Two-manifolds . . . . . . . . . . . . . . . . . . . . 308 5.2 Differentiable Sphere Theorems in 3-D and 4-D . . . . . . . . . . . . . 321 5.3 Nonsingular Solutions on Three-manifolds . . . . . . . . . . . . . . . . 336 6 Ancient κ-solutions 6.1 Preliminaries . . . . . . . . . . . . . . . . 6.2 Asymptotic Shrinking Solitons . . . . . . 6.3 Curvature Estimates via Volume Growth . 6.4 Ancient κ-solutions on Three-manifolds .

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357 357 364 373 384

7 Ricci Flow on Three-manifolds 7.1 Canonical Neighborhood Structures . . . . . . . . . . . . 7.2 Curvature Estimates for Smooth Solutions . . . . . . . . . 7.3 Ricci Flow with Surgery . . . . . . . . . . . . . . . . . . . 7.4 Justification of the Canonical Neighborhood Assumptions 7.5 Curvature Estimates for Surgically Modified Solutions . . 7.6 Long Time Behavior . . . . . . . . . . . . . . . . . . . . . 7.7 Geometrization of Three-manifolds . . . . . . . . . . . . .

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398 398 405 413 432 452 468 481

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References

486

Index

491

THE HAMILTON-PERELMAN THEORY OF RICCI FLOW

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Introduction. In this paper, we shall present the Hamilton-Perelman theory of Ricci flow. Based on it, we shall give the first written account of a complete proof of the Poincaré conjecture and the geometrization conjecture of Thurston. While the complete work is an accumulated efforts of many geometric analysts, the major contributors are unquestionably Hamilton and Perelman. An important problem in differential geometry is to find a canonical metric on a given manifold. In turn, the existence of a canonical metric often has profound topological implications. A good example is the classical uniformization theorem in two dimensions which, on one hand, provides a complete topological classification for compact surfaces, and on the other hand shows that every compact surface has a canonical geometric structure: a metric of constant curvature. How to formulate and generalize this two-dimensional result to three and higher dimensional manifolds has been one of the most important and challenging topics in modern mathematics. In 1977, W. Thurston [122], based on ideas about Riemann surfaces, Haken’s work and Mostow’s rigidity theorem, etc, formulated a geometrization conjecture for three-manifolds which, roughly speaking, states that every compact orientable three-manifold has a canonical decomposition into pieces, each of which admits a canonical geometric structure. In particular, Thurston’s conjecture contains, as a special case, the Poincaré conjecture: A closed three-manifold with trivial fundamental group is necessarily homeomorphic to the 3-sphere S3 . In the past thirty years, many mathematicians have contributed to the understanding of this conjecture of Thurston. While Thurston’s theory is based on beautiful combination of techniques from geometry and topology, there has been a powerful development of geometric analysis in the past thirty years, lead by S.-T. Yau, R. Schoen, C. Taubes, K. Uhlenbeck, and S. Donaldson, on the construction of canonical geometric structures based on nonlinear PDEs (see, e.g., Yau’s survey papers [129, 130]). Such canonical geometric structures include Kähler-Einstein metrics, constant scalar curvature metrics, and self-dual metrics, among others. However, the most important contribution for geometric analysis on three-manifolds is due to Hamilton. In 1982, Hamilton [58] introduced the Ricci flow ∂gij = −2Rij ∂t to study compact three-manifolds with positive Ricci curvature. The Ricci flow, which evolves a Riemannian metric by its Ricci curvature, is a natural analogue of the heat equation for metrics. As a consequence, the curvature tensors evolve by a system of diffusion equations which tends to distribute the curvature uniformly over the manifold. Hence, one expects that the initial metric should be improved and evolve into a canonical metric, thereby leading to a better understanding of the topology of the underlying manifold. In the celebrated paper [58], Hamilton showed that on a compact three-manifold with an initial metric having positive Ricci curvature, the Ricci flow converges, after rescaling to keep constant volume, to a metric of positive constant sectional curvature, proving the manifold is diffeomorphic to the three-sphere S3 or a quotient of the three-sphere S3 by a linear group of isometries. Shortly after, Yau suggested that the Ricci flow should be the best way to prove the structure theorem for general three-manifolds. In the past two decades, Hamilton proved many important and remarkable theorems for the Ricci flow, and laid the foundation for the program to approach the Poincaré conjecture and Thurston’s geometrization conjecture via the Ricci flow.

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The basic idea of Hamilton’s program can be briefly described as follows. For any given compact three-manifold, one endows it with an arbitrary (but can be suitably normalized by scaling) initial Riemannian metric on the manifold and then studies the behavior of the solution to the Ricci flow. If the Ricci flow develops singularities, then one tries to find out the structures of singularities so that one can perform (geometric) surgery by cutting off the singularities, and then continue the Ricci flow after the surgery. If the Ricci flow develops singularities again, one repeats the process of performing surgery and continuing the Ricci flow. If one can prove there are only a finite number of surgeries during any finite time interval and if the long-time behavior of solutions of the Ricci flow with surgery is well understood, then one would recognize the topological structure of the initial manifold. Thus Hamilton’s program, when carried out successfully, will give a proof of the Poincaré conjecture and Thurston’s geometrization conjecture. However, there were obstacles, most notably the verification of the so called “Little Loop Lemma” conjectured by Hamilton [63] (see also [17]) which is a certain local injectivity radius estimate, and the verification of the discreteness of surgery times. In the fall of 2002 and the spring of 2003, Perelman [103, 104] brought in fresh new ideas to figure out important steps to overcome the main obstacles that remained in the program of Hamilton. (Indeed, in page 3 of [103], Perelman said “the implementation of Hamilton program would imply the geometrization conjecture for closed three-manifolds” and “In this paper we carry out some details of Hamilton program”.) Perelman’s breakthrough on the Ricci flow excited the entire mathematics community. His work has since been examined to see whether the proof of the Poincaré conjecture and geometrization program, based on the combination of Hamilton’s fundamental ideas and Perelman’s new ideas, holds together. The present paper grew out of such an effort. Now we describe the three main parts of Hamilton’s program in more detail. (i) Determine the structures of singularities Given any compact three-manifold M with an arbitrary Riemannian metric, one evolves the metric by the Ricci flow. Then, as Hamilton showed in [58], the solution g(t) to the Ricci flow exists for a short time and is unique (also see Theorem 1.2.1). In fact, Hamilton [58] showed that the solution g(t) will exist on a maximal time interval [0, T ), where either T = ∞, or 0 < T < ∞ and the curvature becomes unbounded as t tends to T . We call such a solution g(t) a maximal solution of the Ricci flow. If T < ∞ and the curvature becomes unbounded as t tends to T , we say the maximal solution develops singularities as t tends to T and T is the singular time. In the early 1990s, Hamilton systematically developed methods to understand the structure of singularities. In [61], based on suggestion by Yau, he proved the fundamental Li-Yau [82] type differential Harnack estimate (the Li-Yau-Hamilton estimate) for the Ricci flow with nonnegative curvature operator in all dimensions. With the help of Shi’s interior derivative estimate [114], he [62] established a compactness theorem for smooth solutions to the Ricci flow with uniformly bounded curvatures and uniformly bounded injectivity radii at the marked points. By imposing an injectivity radius condition, he rescaled the solution to show that each singularity is asymptotic to one of the three types of singularity models [63]. In [63] he discovered (also independently by Ivey [73]) an amazing curvature pinching estimate for the Ricci flow on three-manifolds. This pinching estimate implies that any three-dimensional singularity model must have nonnegative curvature. Thus in dimension three, one only needs to obtain a complete classification for nonnegatively curved singularity models.


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For Type I singularities in dimension three, Hamilton [63] established an isoperimetric ratio estimate to verify the injectivity radius condition and obtained spherical or necklike structures for any Type I singularity model. Based on the Li-Yau-Hamilton estimate, he showed that any Type II singularity model with nonnegative curvature is either a steady Ricci soliton with positive sectional curvature or the product of the so called cigar soliton with the real line [66]. (Characterization for nonnegatively curved Type III models was obtained in [30].) Furthermore, he developed a dimension reduction argument to understand the geometry of steady Ricci solitons [63]. In the three-dimensional case, he showed that each steady Ricci soliton with positive curvature has some necklike structure. Hence Hamilton had basically obtained a canonical neighborhood structure at points where the curvature is comparable to the maximal curvature for solutions to the three-dimensional Ricci flow. However two obstacles remained: (a) the verification of the imposed injectivity radius condition in general; and (b) the possibility of forming a singularity modelled on the product of the cigar soliton with a real line which could not be removed by surgery. The recent spectacular work of Perelman [103] removed these obstacles by establishing a local injectivity radius estimate, which is valid for the Ricci flow on compact manifolds in all dimensions. More precisely, Perelman proved two versions of “no local collapsing” property (Theorem 3.3.3 and Theorem 3.3.2), one with an entropy functional he introduced in [103], which is monotone under the Ricci flow, and the other with a space-time distance function obtained by path integral, analogous to what Li-Yau did in [82], which gives rise to a monotone volume-type (called reduced volume by Perelman) estimate. By combining Perelman’s no local collapsing theorem I′ (Theorem 3.3.3) with the injectivity radius estimate of Cheng-Li-Yau (Theorem 4.2.2), one immediately obtains the desired injectivity radius estimate, or the Little Loop Lemma (Theorem 4.2.4) conjectured by Hamilton. Furthermore, Perelman [103] developed a refined rescaling argument (by considering local limits and weak limits in Alexandrov spaces) for singularities of the Ricci flow on three-manifolds to obtain a uniform and global version of the canonical neighborhood structure theorem. We would like to point out that our proof of the singularity structure theorem (Theorem 7.1.1) is different from that of Perelman in two aspects: (1) we avoid using his crucial estimate in Claim 2 in Section 12.1 of [103]; (2) we give a new approach to extend the limit backward in time to an ancient solution. These differences are due to the difficulties in understanding Perelman’s arguments at these points. (ii) Geometric surgeries and the discreteness of surgery times After obtaining the canonical neighborhoods (consisting of spherical, necklike and caplike regions) for the singularities, one would like to perform geometric surgery and then continue the Ricci flow. In [64], Hamilton initiated such a surgery procedure for the Ricci flow on four-manifolds with positive isotropic curvature and presented a concrete method for performing the geometric surgery. His surgery procedures can be roughly described as follows: cutting the neck-like regions, gluing back caps, and removing the spherical regions. As will be seen in Section 7.3 of this paper, Hamilton’s geometric surgery method also works for the Ricci flow on compact orientable threemanifolds. Now an important challenge is to prevent surgery times from accumulating and make sure one performs only a finite number of surgeries on each finite time interval. The problem is that, when one performs the surgeries with a given accuracy at each surgery time, it is possible that the errors may add up to a certain amount which

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could cause the surgery times to accumulate. To prevent this from happening, as time goes on, successive surgeries must be performed with increasing accuracy. In [104], Perelman introduced some brilliant ideas which allow one to find “fine” necks, glue “fine” caps, and use rescaling to prove that the surgery times are discrete. When using the rescaling argument for surgically modified solutions of the Ricci flow, one encounters the difficulty of how to apply Hamilton’s compactness theorem (Theorem 4.1.5), which works only for smooth solutions. The idea to overcome this difficulty consists of two parts. The first part, due to Perelman [104], is to choose the cutoff radius in neck-like regions small enough to push the surgical regions far away in space. The second part, due to the authors and Chen-Zhu [34], is to show that the surgically modified solutions are smooth on some uniform (small) time intervals (on compact subsets) so that Hamilton’s compactness theorem can still be applied. To do so, we establish three time-extension results (see Step 2 in the proof of Proposition 7.4.1.). Perhaps, this second part is more crucial. Without it, Shi’s interior derivative estimate (Theorem 1.4.2) may not applicable, and hence one cannot be certain that Hamilton’s compactness theorem holds when only having the uniform C 0 bound on curvatures. We remark that in our proof of this second part, as can be seen in the proof of Proposition 7.4.1, we require a deep comprehension of the prolongation of the gluing “fine” caps for which we will use the recent uniqueness theorem of BingLong Chen and the second author [33] for solutions of the Ricci flow on noncompact manifolds. Once surgeries are known to be discrete in time, one can complete the classification, started by Schoen-Yau [109, 110], for compact orientable three-manifolds with positive scalar curvature. More importantly, for simply connected three-manifolds, if one can show that solutions to the Ricci flow with surgery become extinct in finite time, then the Poincaré conjecture would follow. Such a finite extinction time result was proposed by Perelman [105], and a proof also appears in Colding-Minicozzi [42]. Thus, the combination of Theorem 7.4.3 (i) and the finite extinction time result provides a complete proof to the Poincaré conjecture. (iii) The long-time behavior of surgically modified solutions. To approach the structure theorem for general three-manifolds, one still needs to analyze the long-time behavior of surgically modified solutions to the Ricci flow. In [65], Hamilton studied the long time behavior of the Ricci flow on compact threemanifolds for a special class of (smooth) solutions, the so called nonsingular solutions. These are the solutions that, after rescaling to keep constant volume, have (uniformly) bounded curvature for all time. Hamilton [65] proved that any three-dimensional nonsingular solution either collapses or subsequently converges to a metric of constant curvature on the compact manifold or, at large time, admits a thick-thin decomposition where the thick part consists of a finite number of hyperbolic pieces and the thin part collapses. Moreover, by adapting Schoen-Yau’s minimal surface arguments in [110] and using a result of Meeks-Yau [86], Hamilton showed that the boundary of hyperbolic pieces are incompressible tori. Consequently, when combined with the collapsing results of Cheeger-Gromov [24, 25], this shows that any nonsingular solution to the Ricci flow is geometrizable in the sense of Thurston [122]. Even though the nonsingular assumption seems very restrictive and there are few conditions known so far which can guarantee a solution to be nonsingular, nevertheless the ideas and arguments of Hamilton’s work [65] are extremely important. In [104], Perelman modified Hamilton’s arguments to analyze the long-time be-


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havior of arbitrary smooth solutions to the Ricci flow and solutions with surgery to the Ricci flow in dimension three. Perelman also argued that the proof of Thurston’s geometrization conjecture could be based on a thick-thin decomposition, but he could only show the thin part will only have a (local) lower bound on the sectional curvature. For the thick part, based on the Li-Yau-Hamilton estimate, Perelman [104] established a crucial elliptic type estimate, which allowed him to conclude that the thick part consists of hyperbolic pieces. For the thin part, he announced in [104] a new collapsing result which states that if a three-manifold collapses with (local) lower bound on the sectional curvature, then it is a graph manifold. Assuming this new collapsing result, Perelman [104] claimed that the solutions to the Ricci flow with surgery have the same long-time behavior as nonsingular solutions in Hamilton’s work, a conclusion which would imply a proof of Thurston’s geometrization conjecture. Although the proof of this new collapsing result promised by Perelman in [104] is still not available in literature, Shioya-Yamaguchi [118] has published a proof of the collapsing result in the special case when the manifold is closed. In the last section of this paper (see Theorem 7.7.1), we will provide a proof of Thurston’s geometrization conjecture by only using Shioya-Yamaguchi’s collapsing result. In particular, this gives another proof of the Poincaré conjecture. We would like to point out that Perelman [104] did not quite give an explicit statement of the thick-thin decomposition for surgical solutions. When we were trying to write down an explicit statement, we needed to add a restriction on the relation between the accuracy parameter ε and the collapsing parameter w. Nevertheless, we are still able to obtain a weaker version of the thick-thin decomposition (Theorem 7.6.3) that is sufficient to deduce the geometrization result. In this paper, we shall give complete and detailed proofs of what we outlined above, especially of Perelman’s work in his second paper [104] in which many key ideas of the proofs are sketched or outlined but complete details of the proofs are often missing. As we pointed out before, we have to substitute several key arguments of Perelman by new approaches based on our study, because we were unable to comprehend these original arguments of Perelman which are essential to the completion of the geometrization program. Our paper is aimed at both graduate students and researchers who want to learn Hamilton’s Ricci flow and to understand the Hamilton-Perelman theory and its application to the geometrization of three-manifolds. For this purpose, we have made the paper to be essentially self-contained so that the proof of the geometrization is accessible to those who are familiar with basics of Riemannian geometry and elliptic and parabolic partial differential equations. The reader may find some original papers, particularly those of Hamilton’s on the Ricci flow, before the appearance of Perelman’s preprints in the book “Collected Papers on Ricci Flow” [17]. For introductory materials to the Hamilton-Perelman theory of Ricci flow, we also refer the reader to the recent book by B. Chow and D. Knopf [39] and the forthcoming book by B. Chow, P. Lu and L. Ni [41]. We remark that there have also appeared several sets of notes on Perelman’s work, including the one written by B. Kleiner and J. Lott [78], which cover part of the materials that are needed for the geometrization program. There also have appeared several survey articles by Cao-Chow [16], Milnor [91], Anderson [4] and Morgan [95] for the geometrization of three-manifolds via the Ricci flow. We are very grateful to Professor S.-T. Yau, who suggested us to write this paper based on our notes, for introducing us to the wonderland of the Ricci flow. His vision and strong belief in the Ricci flow encouraged us to persevere. We also thank

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him for his many suggestions and constant encouragement. Without him, it would be impossible for us to finish this paper. We are enormously indebted to Professor Richard Hamilton for creating the Ricci flow and developing the entire program to approach the geometrization of three-manifolds. His work on the Ricci flow and other geometric flows has influenced on virtually everyone in the field. The first author especially would like to thank Professor Hamilton for teaching him so much about the subject over the past twenty years, and for his constant encouragement and friendship. We are indebted to Dr. Bing-Long Chen, who contributed a great deal in the process of writing this paper. We benefited a lot from constant discussions with him on the subjects of geometric flows and geometric analysis. He also contributed many ideas in various proofs in the paper. We would like to thank Ms. Huiling Gu, a Ph.D student of the second author, for spending many months of going through the entire paper and checking the proofs. Without both of them, it would take much longer time for us to finish this paper. The first author would like to express his gratitude to the John Simon Guggenheim Memorial Foundation, the National Science Foundation (grants DMS-0354621 and DMS-0506084), and the Outstanding Overseas Young Scholar Fund of Chinese National Science Foundation for their support for the research in this paper. He also would like to thank Tsinghua University in Beijing for its hospitality and support while he was working there. The second author wishes to thank his wife, Danlin Liu, for her understanding and support over all these years. The second author is also indebted to the National Science Foundation of China for the support in his work on geometric flows, some of which has been incorporated in this paper. The last part of the work in this paper was done and the material in Chapter 3, Chapter 6 and Chapter 7 was presented while the second author was visiting the Harvard Mathematics Department in the fall semester of 2005 and the early spring semester of 2006. He wants to especially thank Professor Shing-Tung Yau, Professor Cliff Taubes and Professor Daniel W. Stroock for the enlightening comments and encouragement during the lectures. Also he gratefully acknowledges the hospitality and the financial support of Harvard University. 1. Evolution Equations. In this chapter, we introduce Hamilton’s Ricci flow and derive evolution equations of curvatures. The short time existence and uniqueness theorem of the Ricci flow on a compact manifold is proved in Section 1.2. In Section 1.4, we prove Shi’s local derivative estimate, which plays an important role in the Ricci flow. Perelman’s two functionals and their monotonicity properties are discussed in Section 1.5. 1.1. The Ricci Flow. Let M be an n-dimensional complete Riemannian manifold with the Riemannian metric gij . The Levi-Civita connection is given by the Christoffel symbols ∂gil ∂gij 1 kl ∂gjl k + − Γij = g 2 ∂xi ∂xj ∂xl where g ij is the inverse of gij . The summation convention of summing over repeated indices is used here and throughout the book. The Riemannian curvature tensor is given by k Rijl =

∂Γkjl ∂xi

−

∂Γkil + Γkip Γpjl − Γkjp Γpil . ∂xj


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We lower the index to the third position, so that p Rijkl = gkp Rijl .

The curvature tensor Rijkl is anti-symmetric in the pairs i, j and k, l and symmetric in their interchange: Rijkl = −Rjikl = −Rijlk = Rklij . Also the first Bianchi identity holds (1.1.1)

Rijkl + Rjkil + Rkijl = 0.

The Ricci tensor is the contraction Rik = g jl Rijkl , and the scalar curvature is R = g ij Rij . ∂ We denote the covariant derivative of a vector field v = v j ∂x j by

∇i v j =

∂v j + Γjik v k ∂xi

∇i vj =

∂vj − Γkij vk . ∂xi

and of a 1-form by

These definitions extend uniquely to tensors so as to preserve the product rule and contractions. For the exchange of two covariant derivatives, we have (1.1.2) (1.1.3)

l ∇i ∇j v l − ∇j ∇i v l = Rijk vk ,

∇i ∇j vk − ∇j ∇i vk = Rijkl g lm vm ,

and similar formulas for more complicated tensors. The second Bianchi identity is given by (1.1.4)

∇m Rijkl + ∇i Rjmkl + ∇j Rmikl = 0.

i For any tensor T = Tjk we define its length by i 2 i l |Tjk | = gil g jm g kp Tjk Tmp ,

and we define its Laplacian by i i , ∆Tjk = g pq ∇p ∇q Tjk

the trace of the second iterated covariant derivatives. Similar definitions hold for more general tensors. The Ricci flow of Hamilton [58] is the evolution equation (1.1.5)

∂gij = −2Rij ∂t

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for a family of Riemannian metrics gij (t) on M . It is a nonlinear system of second order partial differential equations on metrics. In order to get a feel for the Ricci flow (1.1.5) we first present some examples of specific solutions. (1) Einstein metrics A Riemannian metric gij is called Einstein if Rij = λgij for some constant λ. A smooth manifold M with an Einstein metric is called an Einstein manifold. If the initial metric is Ricci flat, so that Rij = 0, then clearly the metric does not change under (1.1.5). Hence any Ricci flat metric is a stationary solution of the Ricci flow. This happens, for example, on a flat torus or on any K3-surface with a Calabi-Yau metric. If the initial metric is Einstein with positive scalar curvature, then the metric will shrink under the Ricci flow by a time-dependent factor. Indeed, since the initial metric is Einstein, we have Rij (x, 0) = λgij (x, 0),

∀x ∈ M

and some λ > 0. Let gij (x, t) = ρ2 (t)gij (x, 0). From the definition of the Ricci tensor, one sees that Rij (x, t) = Rij (x, 0) = λgij (x, 0). Thus the equation (1.1.5) corresponds to ∂(ρ2 (t)gij (x, 0)) = −2λgij (x, 0). ∂t This gives the ODE (1.1.6)

dρ λ =− , dt ρ

whose solution is given by ρ2 (t) = 1 − 2λt. Thus the evolving metric gij (x, t) shrinks homothetically to a point as t → T = 1/2λ. Note that as t → T , the scalar curvature becomes infinite like 1/(T − t). By contrast, if the initial metric is an Einstein metric of negative scalar curvature, the metric will expand homothetically for all times. Indeed if Rij (x, 0) = −λgij (x, 0) with λ > 0 and gij (x, t) = ρ2 (t)gij (x, 0).


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Then ρ(t) satisfies the ODE (1.1.7)

λ dρ = , dt ρ

with the solution ρ2 (t) = 1 + 2λt. Hence the evolving metric gij (x, t) = ρ2 (t)gij (x, 0) exists and expands homothetically for all times, and the curvature will fall back to zero like −1/t. Note that now the evolving metric gij (x, t) only goes back in time to −1/2λ, when the metric explodes out of a single point in a “big bang”. (2) Ricci Solitons We will call a solution to an evolution equation which moves under a oneparameter subgroup of the symmetry group of the equation a steady soliton. The symmetry group of the Ricci flow contains the full diffeomorphism group. Thus a solution to the Ricci flow (1.1.5) which moves by a one-parameter group of diffeomorphisms ϕt is called a steady Ricci soliton. If ϕt is a one-parameter group of diffeomorphisms generated by a vector field V on M , then the Ricci soliton is given by (1.1.8)

gij (x, t) = ϕ∗t gij (x, 0)

which implies that the Ricci term −2Ric on the RHS of (1.1.5) is equal to the Lie derivative LV g of the evolving metric g. In particular, the initial metric gij (x, 0) satisfies the following steady Ricci soliton equation (1.1.9)

2Rij + gik ∇j V k + gjk ∇i V k = 0.

If the vector field V is the gradient of a function f then the soliton is called a steady gradient Ricci soliton. Thus (1.1.10)

Rij + ∇i ∇j f = 0,

or Ric + ∇2 f = 0,

is the steady gradient Ricci soliton equation. Conversely, it is clear that a metric gij satisfying (1.1.10) generates a steady gradient Ricci soliton gij (t) given by (1.1.8). For this reason we also often call such a metric gij a steady gradient Ricci soliton and do not necessarily distinguish it with the solution gij (t) it generates. More generally, we can consider a solution to the Ricci flow (1.1.5) which moves by diffeomorphisms and also shrinks or expands by a (time-dependent) factor at the same time. Such a solution is called a homothetically shrinking or homothetically expanding Ricci soliton. The equation for a homothetic Ricci soliton is (1.1.11)

2Rij + gik ∇j V k + gjk ∇i V k − 2λgij = 0,

or for a homothetic gradient Ricci soliton, (1.1.12)

Rij + ∇i ∇j f − λgij = 0,

where λ is the homothetic constant. For λ > 0 the soliton is shrinking, for λ < 0 it is expanding. The case λ = 0 is a steady Ricci soliton, the case V = 0 (or f being

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a constant function) is an Einstein metric. Thus Ricci solitons can be considered as natural extensions of Einstein metrics. In fact, the following result states that there are no nontrivial gradient steady or expanding Ricci solitons on any compact manifold. We remark that if the underlying manifold M is a complex manifold and the initial metric is Kähler, then it is well known (see, e.g., [62, 11]) that the solution metric to the Ricci flow (1.1.5) remains Kähler. For this reason, the Ricci flow on a Kähler manifold is called the K¨ ahler-Ricci flow. A (steady, or shrinking, or expanding) Ricci soliton to the Kähler-Ricci flow is called a (steady, or shrinking, or expanding repectively) K¨ ahler-Ricci soliton. Proposition 1.1.1. On a compact n-dimensional manifold M , a gradient steady or expanding Ricci soliton is necessarily an Einstein metric. Proof. We shall only prove the steady case and leave the expanding case as an exercise. Our argument here follows that of Hamilton [63]. Let gij be a complete steady gradient Ricci soliton on a manifold M so that Rij + ∇i ∇j f = 0. Taking the trace, we get (1.1.13)

R + ∆f = 0.

Also, taking the covariant derivatives of the Ricci soliton equation, we have ∇i ∇j ∇k f − ∇j ∇i ∇k f = ∇j Rik − ∇i Rjk . On the other hand, by using the commutating formula (1.1.3), we otain ∇i ∇j ∇k f − ∇j ∇i ∇k f = Rijkl ∇l f. Thus ∇i Rjk − ∇j Rik + Rijkl ∇l f = 0. Taking the trace on j and k, and using the contracted second Bianchi identity (1.1.14)

∇j Rij =

1 ∇i R, 2

we get ∇i R − 2Rij ∇j f = 0. Then ∇i (|∇f |2 + R) = 2∇j f (∇i ∇j f + Rij ) = 0. Therefore (1.1.15) for some constant C.

R + |∇f |2 = C


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Taking the difference of (1.1.13) and (1.1.15), we get ∆f − |∇f |2 = −C.

(1.1.16)

We claim C = 0 when M is compact. Indeed, this follows either from Z Z (1.1.17) 0=− ∆(e−f )dV = (∆f − |∇f |2 )e−f dV, M

M

or from considering (1.1.16) at both the maximum point and minimum point of f . Then, by integrating (1.1.16) we obtain Z |∇f |2 dV = 0. M

Therefore f is a constant and gij is Ricci flat. Remark 1.1.2. By contrast, there do exist nontrivial compact gradient shrinking Ricci solitons (see Koiso [80], Cao [13] and Wang-Zhu [127] ). Also, there exist complete noncompact steady gradient Ricci solitons that are not Ricci flat. In two dimensions Hamilton [60] wrote down the first such example on R2 , called the cigar soliton, where the metric is given by (1.1.18)

ds2 =

dx2 + dy 2 , 1 + x2 + y 2

and the vector field is radial, given by V = −∂/∂r = −(x∂/∂x + y∂/∂y). This metric has positive curvature and is asymptotic to a cylinder of finite circumference 2π at ∞. Higher dimensional examples were found by Robert Bryant [10] on Rn in the Riemannian case, and by the first author [13] on Cn in the Kähler case. These examples are complete, rotationally symmetric, of positive curvature and found by solving certain nonlinear ODE (system). Noncompact expanding solitons were also constructed by the first author [13]. More recently, Feldman, Ilmanen and Knopf [46] constructed new examples of noncompact shrinking and expanding Kähler-Ricci solitons. 1.2. Short-time Existence and Uniqueness. In this section we establish the short-time existence and uniqueness result for the Ricci flow (1.1.5) on a compact ndimensional manifold M . We will see that the Ricci flow is a system of second order nonlinear weakly parabolic partial differential equations. In fact, the degeneracy of the system is caused by the diffeomorphism group of M which acts as the gauge group of the Ricci flow. For any diffeomorphism ϕ of M , we have Ric (ϕ∗ (g)) = ϕ∗ (Ric (g)). Thus, if g(t) is a solution to the Ricci flow (1.1.5), so is ϕ∗ (g(t)). Because the Ricci flow (1.1.5) is only weakly parabolic, even the existence and uniqueness result on a compact manifold does not follow from standard PDE theory. The short-time existence and uniqueness result in the compact case is first proved by Hamilton [58] using the Nash-Moser implicit function theorem. Shortly after Denis De Turck [43] gave a much simpler proof using the gauge fixing idea which we will present here. In the noncompact case, the short-time existence was established by Shi [114] in 1989, but the uniqueness result has been proved only very recently by Bing-Long Chen and the second author. These results will be presented at the end of this section.

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Let M be a compact n-dimensional Riemannian manifold. The Ricci flow equation is a second order nonlinear partial differential system ∂ gij = E(gij ), ∂t

(1.2.1)

for a family of Riemannian metrics gij (·, t) on M , where E(gij ) = −2Rij ∂ k ∂ k p k k p = −2 Γ − Γ + Γ Γ − Γ Γ kp ij ip kj ∂xk ij ∂xi kj ∂ ∂ ∂ ∂ ∂ kl ∂ kl = g g − g g + g − g kl jl il ij ∂xi ∂xj ∂xk ∂xi ∂xj ∂xl + 2Γkip Γpkj − 2Γkkp Γpij .

The linearization of this system is ∂˜ gij = DE(gij )˜ gij ∂t where g˜ij is the variation in gij and DE is the derivative of E given by DE(gij )˜ gij = g

kl

∂ 2 g˜kl ∂ 2 g˜jl ∂ 2 g˜il ∂ 2 g˜ij − − + ∂xi ∂xj ∂xi ∂xk ∂xj ∂xk ∂xk ∂xl

+ (lower order terms).

We now compute the symbol of DE. This is to take the highest order derivatives and ∂ replace ∂x i by the Fourier transform variable ζi . The symbol of the linear differential operator DE(gij ) in the direction ζ = (ζ1 , . . . , ζn ) is σDE(gij )(ζ)˜ gij = g kl (ζi ζj g˜kl + ζk ζl g˜ij − ζi ζk g˜jl − ζj ζk g˜il ). To see what the symbol does, we can always assume ζ has length 1 and choose coordinates at a point such that   gij = δij , Then



ζ = (1, 0, . . . , 0).

(σDE(gij )(ζ))(˜ gij ) = g˜ij + δi1 δj1 (˜ g11 + · · · + gñn ) − δi1 g˜1j − δj1 g˜1i ,

i.e., [σDE(gij )(ζ)(˜ gij )]11 = g˜22 + · · · + gñn ,

[σDE(gij )(ζ)(˜ gij )]1k = 0, [σDE(gij )(ζ)(˜ gij )]kl = g˜kl ,

if k 6= 1, if k = 6 1, l 6= 1.

179


In particular 

  (˜ gij ) =  

∗ ∗ .. . ∗

∗ ··· 0 ··· .. . . . . 0 ···

∗ 0 .. . 0

    

are zero eigenvectors of the symbol. The presence of the zero eigenvalue shows that the system can not be strictly parabolic. Therefore, instead of considering the system (1.2.1) (or the Ricci flow equation (1.1.5)) we will follow a trick of De Turck[43] to consider a modified evolution equation, which turns out to be strictly parabolic, so that we can apply the standard theory of parabolic equations. Suppose gîj (x, t) is a solution of the Ricci flow (1.1.5), and ϕt : M → M is a family of diffeomorphisms of M . Let gij (x, t) = ϕ∗t gîj (x, t) be the pull-back metrics. We now want to find the evolution equation for the metrics gij (x, t). Denote by y(x, t) = ϕt (x) = {y 1 (x, t), y 2 (x, t), . . . , y n (x, t)} in local coordinates. Then (1.2.2)

gij (x, t) =

∂y α ∂y β gˆαβ (y, t), ∂xi ∂xj

and ∂ ∂y α ∂y β ∂ gij (x, t) = gˆαβ (y, t) ∂t ∂t ∂xi ∂xj =

α β ∂y α ∂y β ∂ ∂ ∂y ∂y g ˆ (y, t) + gˆαβ (y, t) αβ i j i ∂x ∂x ∂t ∂x ∂t ∂xj β ∂y α ∂ ∂y + gˆαβ (y, t). ∂xi ∂xj ∂t

Let us choose a normal coordinate {xi } around a fixed point p ∈ M such that at p. Since gαβ ∂y γ ∂ ˆ αβ (y, t) + ∂ˆ gˆαβ (y, t) = −2R , ∂t ∂y γ ∂t

∂gij ∂xk

=0

180


we have in the normal coordinate, ∂ gij (x, t) ∂t ∂y α ∂y β ∂ˆ gαβ ∂y γ ∂y α ∂y β ˆ Rαβ (y, t) + = −2 i j i j ∂x ∂x ∂x ∂x ∂y γ ∂t α β β α ∂ ∂y ∂y ∂ ∂y ∂y + g ˆ (y, t) + gˆαβ (y, t) αβ ∂xi ∂t ∂xj ∂xj ∂t ∂xi α k ∂y α ∂y β ∂ˆ gαβ ∂y γ ∂ ∂y ∂x = −2Rij (x, t) + + i gjk ∂xi ∂xj ∂y γ ∂t ∂x ∂t ∂y α β k ∂y ∂x ∂ gik + ∂xj ∂t ∂y β α k ∂y α ∂y β ∂ˆ gαβ ∂y γ ∂ ∂y ∂x = −2Rij (x, t) + + i gjk ∂xi ∂xj ∂y γ ∂t ∂x ∂t ∂y α β k k k ∂ ∂y ∂x ∂y α ∂ ∂x ∂y β ∂ ∂x + g − g − gik . ik jk j β i α j ∂x ∂t ∂y ∂t ∂x ∂y ∂t ∂x ∂y β The second term on the RHS gives, in the normal coordinate, k gαβ ∂y α ∂y β ∂y γ ∂ ∂x ∂xl ∂y α ∂y β ∂y γ ∂ˆ = g kl ∂xi ∂xj ∂t ∂y γ ∂xi ∂xj ∂t ∂y γ ∂y α ∂y β k ∂y α ∂y γ ∂ ∂xk ∂y β ∂y γ ∂ ∂x = g + gik jk ∂xi ∂t ∂y γ ∂y α ∂xj ∂t ∂y γ ∂y β ∂y α ∂ 2 xk ∂y β gjk + ∂t ∂y α ∂y β ∂xi k ∂x ∂y α ∂ gjk + = ∂t ∂xi ∂y α =

∂y β ∂t ∂y β ∂t

∂ 2 xk ∂y α gik ∂y α ∂y β ∂xj k ∂ ∂x gik . ∂xj ∂y β

So we get (1.2.3)

∂ gij (x, t) ∂t = −2Rij (x, t) + ∇i

∂y α ∂xk gjk ∂t ∂y α

+ ∇j

If we define y(x, t) = ϕt (x) by the equations  ok   ∂yα = ∂yαk g jl (Γk − Γ ), jl jl ∂t ∂x (1.2.4)   α y (x, 0) = xα , ok

∂y β ∂xk gik . ∂t ∂y β

and Vi = gik g jl (Γkjl − Γjl ), we get the following evolution equation for the pull-back metric  ∂  ∂t gij (x, t) = −2Rij (x, t) + ∇i Vj + ∇j Vi , (1.2.5) o  gij (x, 0) =gij (x),

181

THE HAMILTON-PERELMAN THEORY OF RICCI FLOW ok

o

where g ij (x) is the initial metric and Γjl is the connection of the initial metric. Lemma 1.2.1. The modified evolution equation (1.2.5) is a strictly parabolic system. Proof. The RHS of the equation (1.2.5) is given by − 2Rij (x, t) + ∇i Vj + ∇j Vi ∂ ∂ ∂gjl ∂gil ∂gij kl ∂gkl kl = g − g + − ∂xi ∂xj ∂xk ∂xi ∂xj ∂xl 1 kl ∂gpl ∂gql ∂gpq ∂ g + − + gjk g pq i ∂x 2 ∂xq ∂xp ∂xl 1 kl ∂gpl ∂gql ∂gpq pq ∂ + gik g g + − ∂xj 2 ∂xq ∂xp ∂xl + (lower order terms) 2 ∂ 2 gjl ∂ 2 gil ∂ 2 gij ∂ gkl kl − i k − + k l =g ∂xi ∂xj ∂x ∂x ∂xj ∂xk ∂x ∂x 2 2 2 1 pq ∂ gpj ∂ gqj ∂ gpq + g + − i j 2 ∂xi ∂xq ∂xi ∂xp ∂x ∂x 2 ∂ gpi ∂ 2 gqi ∂ 2 gpq 1 pq + j p− i j + g 2 ∂xj ∂xq ∂x ∂x ∂x ∂x + (lower order terms)

= g kl

∂ 2 gij + (lower order terms). ∂xk ∂xl

Thus its symbol is (g kl ζk ζl )˜ gij . Hence the equation in (1.2.5) is strictly parabolic. Now since the equation (1.2.5) is strictly parabolic and the manifold M is compact, it follows from the standard theory of parabolic equations (see for example [81]) that (1.2.5) has a solution for a short time. From the solution of (1.2.5) we can obtain a solution of the Ricci flow from (1.2.4) and (1.2.2). This shows existence. Now we argue the uniqueness of the solution. Since Γkjl =

∂xk ∂ 2 y α ∂y α ∂y β ∂xk ˆ γ Γ + , ∂xj ∂xl ∂y γ αβ ∂y α ∂xj ∂xl

the initial value problem (1.2.4) can be written as  o k ∂y α α ∂ 2 yα  jl ˆ α ∂yβ  ∂y − Γ j l jl ∂xk + Γγβ ∂xj ∂t = g ∂x ∂x (1.2.6)   α y (x, 0) = xα .

∂y γ ∂xl

,

(1)

(2)

This is clearly a strictly parabolic system. For any two solutions gîj (·, t) and gîj (·, t) of the Ricci flow (1.1.5) with the same initial data, we can solve the initial value (1) (2) problem (1.2.6) (or equivalently, (1.2.4)) to get two families ϕt and ϕt of dif(1) (1) ∗ (1) feomorphisms of M . Thus we get two solutions, gij (·, t) = (ϕt ) gîj (·, t) and (2)

(2)

(2)

gij (·, t) = (ϕt )∗ gîj (·, t), to the modified evolution equation (1.2.5) with the same

182


initial metric. The uniqueness result for the strictly parabolic equation (1.2.5) implies (1) (2) that gij = gij . Then by equation (1.2.4) and the standard uniqueness result of ODE (1)

(2)

systems, the corresponding solutions ϕt and ϕt of (1.2.4) (or equivalently (1.2.6)) (1) (2) must agree. Consequently the metrics gîj and gîj must agree also. Thus we have proved the following result. Theorem 1.2.2 (Hamilton [58], De Turck [43]). Let (M, gij (x)) be a compact Riemannian manifold. Then there exists a constant T > 0 such that the initial value problem  ∂  gij (x, t) = −2Rij (x, t)  ∂t  g (x, 0) = g (x) ij ij

has a unique smooth solution gij (x, t) on M × [0, T ). The case of a noncompact manifold is much more complicated and involves a huge amount of techniques from the theory of partial differential equations. Here we will only state the existence and uniqueness results and refer the reader to the cited references for the proofs. The following existence result was obtained by Shi [114] in his thesis published in 1989. Theorem 1.2.3 (Shi [114]). Let (M, gij (x)) be a complete noncompact Riemannian manifold of dimension n with bounded curvature. Then there exists a constant T > 0 such that the initial value problem  ∂  gij (x, t) = −2Rij (x, t)  ∂t  g (x, 0) = g (x) ij ij

has a smooth solution gij (x, t) on M × [0, T ] with uniformly bounded curvature. The Ricci flow is a heat type equation. It is well-known that the uniqueness of a heat equation on a complete noncompact manifold is not always held if there are no further restrictions on the growth of the solutions. For example, the heat equation on Euclidean space with zero initial data has a nontrivial solution which grows faster than exp(a|x|2 ) for any a > 0 whenever t > 0. This implies that even for the standard linear heat equation on Euclidean space, in order to ensure the uniqueness one can only allow the solution to grow at most as exp(C|x|2 ) for some constant C > 0. Note that 2 on a Kähler manifold, the Ricci curvature is given by Rαβ¯ = − ∂zα∂∂ z¯β log det(gγ δ¯). So the reasonable growth rate for the uniqueness of the Ricci flow to hold is that the solution has bounded curvature. Thus the following uniqueness result of Bing-Long Chen and the second author [33] is essentially the best one can hope for. Theorem 1.2.4 (Chen-Zhu [33]). Let (M, gîj ) be a complete noncompact Riemannian manifold of dimension n with bounded curvature. Let gij (x, t) and g¯ij (x, t) be two solutions, defined on M × [0, T ], to the Ricci flow (1.1.5) with gîj as initial data and with bounded curvatures. Then gij (x, t) ≡ g¯ij (x, t) on M × [0, T ].


183

1.3. Evolution of Curvatures. The Ricci flow is an evolution equation on the metric. The evolution for the metric implies a nonlinear heat equation for the Riemannian curvature tensor Rijkl which we will now derive. Proposition 1.3.1 (Hamilton [58]). Under the Ricci flow (1.1.5), the curvature tensor satisfies the evolution equation ∂ Rijkl = ∆Rijkl + 2(Bijkl − Bijlk − Biljk + Bikjl ) ∂t − g pq (Rpjkl Rqi + Ripkl Rqj + Rijpl Rqk + Rijkp Rql ) where Bijkl = g pr g qs Rpiqj Rrksl and ∆ is the Laplacian with respect to the evolving metric. Proof. Choose {x1 , . . . , xm } to be a normal coordinate system at a fixed point. At this point, we compute 1 hm ∂ ∂ ∂ ∂ ∂ ∂ ∂ h Γ = g glm + l gjm − m gjl ∂t jl 2 ∂xj ∂t ∂x ∂t ∂x ∂t 1 = g hm (∇j (−2Rlm ) + ∇l (−2Rjm ) − ∇m (−2Rjl )), 2 ∂ ∂ h ∂ ∂ h ∂ h R = Γ − j Γ , ∂t ijl ∂xi ∂t jl ∂x ∂t il ∂ ∂ h ∂ghk h Rijkl = ghk Rijl + R . ∂t ∂t ∂t ijl Combining these identities we get ∂ 1 hm Rijkl = ghk ∇i [g (∇j (−2Rlm ) + ∇l (−2Rjm ) − ∇m (−2Rjl ))] ∂t 2 1 hm ∇j [g (∇i (−2Rlm ) + ∇l (−2Rim ) − ∇m (−2Ril ))] − 2 h − 2Rhk Rijl

= ∇i ∇k Rjl − ∇i ∇l Rjk − ∇j ∇k Ril + ∇j ∇l Rik − Rijlp g pq Rqk − Rijkp g pq Rql − 2Rijpl g pq Rqk

= ∇i ∇k Rjl − ∇i ∇l Rjk − ∇j ∇k Ril + ∇j ∇l Rik − g pq (Rijkp Rql + Rijpl Rqk ).

Here we have used the exchanging formula (1.1.3). Now it remains to check the following identity, which is analogous to the Simon′ s identity in extrinsic geometry, (1.3.1)

∆Rijkl + 2(Bijkl − Bijlk − Biljk + Bikjl ) = ∇i ∇k Rjl − ∇i ∇l Rjk − ∇j ∇k Ril + ∇j ∇l Rik + g pq (Rpjkl Rqi + Ripkl Rqj ).

Indeed, from the second Bianchi identity (1.1.4), we have ∆Rijkl = g pq ∇p ∇q Rijkl = g pq ∇p ∇i Rqjkl − g pq ∇p ∇j Rqikl .

184


Let us examine the first term on the RHS. By using the exchanging formula (1.1.3) and the first Bianchi identity (1.1.1), we have g pq ∇p ∇i Rqjkl − g pq ∇i ∇p Rqjkl

= g pq g mn (Rpiqm Rnjkl + Rpijm Rqnkl + Rpikm Rqjnl + Rpilm Rqjkn ) = Rim g mn Rnjkl + g pq g mn Rpimj (Rqkln + Rqlnk ) + g pq g mn Rpikm Rqjnl + g pq g mn Rpilm Rqjkn = Rim g mn Rnjkl − Bijkl + Bijlk − Bikjl + Biljk ,

while using the contracted second Bianchi identity g pq ∇p Rqjkl = ∇k Rjl − ∇l Rjk ,

(1.3.2) we have

g pq ∇i ∇p Rqjkl = ∇i ∇k Rjl − ∇i ∇l Rjk . Thus g pq ∇p ∇i Rqjkl

= ∇i ∇k Rjl − ∇i ∇l Rjk − (Bijkl − Bijlk − Biljk + Bikjl ) + g pq Rpjkl Rqi .

Therefore we obtain ∆Rijkl = g pq ∇p ∇i Rqjkl − g pq ∇p ∇j Rqikl = ∇i ∇k Rjl − ∇i ∇l Rjk − (Bijkl − Bijlk − Biljk + Bikjl ) + g pq Rpjkl Rqi

− ∇j ∇k Ril + ∇j ∇l Rik + (Bjikl − Bjilk − Bjlik + Bjkil ) − g pq Rpikl Rqj = ∇i ∇k Rjl − ∇i ∇l Rjk − ∇j ∇k Ril + ∇j ∇l Rik + g pq (Rpjkl Rqi + Ripkl Rqj ) − 2(Bijkl − Bijlk − Biljk + Bikjl )

as desired, where in the last step we used the symmetries (1.3.3)

Bijkl = Bklij = Bjilk .

Corollary 1.3.2. The Ricci curvature satisfies the evolution equation ∂ Rik = ∆Rik + 2g pr g qs Rpiqk Rrs − 2g pq Rpi Rqk . ∂t Proof. ∂ ∂ Rik = g jl Rijkl + ∂t ∂t

∂ jl g Rijkl ∂t

= g jl [∆Rijkl + 2(Bijkl − Bijlk − Biljk + Bikjl ) − g pq (Rpjkl Rqi + Ripkl Rqj + Rijpl Rqk + Rijkp Rql )] ∂ − g jp gpq g ql Rijkl ∂t

= ∆Rik + 2g jl (Bijkl − 2Bijlk ) + 2g pr g qs Rpiqk Rrs − 2g pq Rpk Rqi .


185

We claim that g jl (Bijkl − 2Bijlk ) = 0. Indeed by using the first Bianchi identity, we have g jl Bijkl = g jl g pr g qs Rpiqj Rrksl = g jl g pr g qs Rpqij Rrskl = g jl g pr g qs (Rpiqj − Rpjqi )(Rrksl − Rrlsk ) = 2g jl (Bijkl − Bijlk )

as desired. Thus we obtain ∂ Rik = ∆Rik + 2g pr g qs Rpiqk Rrs − 2g pq Rpi Rqk . ∂t

Corollary 1.3.3. The scalar curvature satisfies the evolution equation ∂R = ∆R + 2|Ric |2 . ∂t

Proof. ∂R ik ∂Rik ip ∂gpq qk =g + −g g Rik ∂t ∂t ∂t

= g ik (∆Rik + 2g pr g qs Rpiqk Rrs − 2g pq Rpi Rqk ) + 2Rpq Rik g ip g qk = ∆R + 2|Ric |2 .

To simplify the evolution equations of curvatures, we will represent the curvature tensors in an orthonormal frame and evolve the frame so that it remains orthonormal. More precisely, let us pick an abstract vector bundle V over M isomorphic to the tangent bundle T M . Locally, the frame F = {F1 , . . . , Fa , . . . , Fn } of V is ∂ i i given by Fa = Fai ∂x i with the isomorphism {Fa }. Choose {Fa } at t = 0 such that F = {F1 , . . . , Fa , . . . , Fn } is an orthonormal frame at t = 0, and evolve {Fai } by the equation ∂ i F = g ij Rjk Fak . ∂t a Then the frame F = {F1 , . . . , Fa , . . . , Fn } will remain orthonormal for all times since the pull back metric on V hab = gij Fai Fbj remains constant in time. In the following we will use indices a, b, . . . on a tensor to denote its components in the evolving orthonormal frame. In this frame we have the

186


following: Rabcd = Fai Fbj Fck Fdl Rijkl , Γajb = Fia

∂Fbi + Γijk Fia Fbk , ∂xj

((Fia ) = (Fai )−1 )

∂ a V + Γaib V b , ∂xi ∇b V a = Fbi ∇i V a , ∇i V a =

where Γajb is the metric connection of the vector bundle V with the metric hab . Indeed, by direct computations, ∂Fbj + Fbk Γjik − Fcj Γcib ∂xi k ∂Fbj k j j c ∂Fb l c k + Fb Γik − Fc Fk + Γik Fl Fb = ∂xi ∂xi

∇i Fbj =

= 0,

∇i hab = ∇i (gij Fai Fbj ) = 0. So ∇a Vb = Fai Fbj ∇i V j , and ∆Rabcd = ∇l ∇l Rabcd

= g ij ∇i ∇j Rabcd

= g ij Fak Fbl Fcm Fdn ∇i ∇j Rklmn . In an orthonormal frame F = {F1 , . . . , Fa , . . . , Fn }, the evolution equations of curvature tensors become (1.3.4) (1.3.5) (1.3.6)

∂ Rabcd = ∆Rabcd + 2(Babcd − Babdc − Badbc + Bacbd ) ∂t ∂ Rab = ∆Rab + 2Racbd Rcd ∂t ∂ R = ∆R + 2|Ric |2 ∂t

where Babcd = Raebf Rcedf . Equation (1.3.4) is a reaction-diffusion equation. We can understand the quadratic terms of this equation better if we think of the curvature tensor Rabcd as a symmetric bilinear form on the two-forms Λ2 (V ) given by the formula Rm(ϕ, ψ) = Rabcd ϕab ψcd ,

for ϕ, ψ ∈ Λ2 (V ).

A two-form ϕ ∈ Λ2 (V ) can be regarded as an element of the Lie algebra so(n) (i.e. the skew-symmetric matrix (ϕab )n×n ), where the metric on Λ2 (V ) is given by hϕ, ψi = ϕab ψab


187

and the Lie bracket is given by [ϕ, ψ]ab = ϕac ψbc − ψac ϕbc . Choose an orthonormal basis of Λ2 (V ) Φ = {ϕ1 , . . . , ϕα , . . . , ϕ

n(n−1) 2

}

where ϕα = {ϕα ab }. The Lie bracket is given by [ϕα , ϕβ ] = Cγαβ ϕγ , where C αβγ = Cσαβ δ σγ = h[ϕα , ϕβ ], ϕγ i are the Lie structure constants. β Write Rabcd = Mαβ ϕα ab ϕcd . We now claim that the first part of the quadratic terms in (1.3.4) is given by β 2(Babcd − Babdc ) = Mαγ Mβγ ϕα ab ϕcd .

(1.3.7)

Indeed, by the first Bianchi identity, Babcd − Babdc = Raebf Rcedf − Raebf Rdecf = Raebf (−Rcef d − Rcf de ) = Raebf Rcdef .

On the other hand, Raebf Rcdef = (−Rabf e − Raf eb )Rcdef

= Rabef Rcdef − Raf eb Rcdef = Rabef Rcdef − Raf be Rcdf e

which implies Raebf Rcdef = 12 Rabef Rcdef . Thus we obtain β 2(Babcd − Babdc ) = Rabef Rcdef = Mαγ Mβγ ϕα ab ϕcd .

We next consider the last part of the quadratic terms: 2(Bacbd − Badbc )

= 2(Raecf Rbedf − Raedf Rbecf )

= 2(Mγδ ϕγae ϕδcf Mηθ ϕηbe ϕθdf − Mγθ ϕγae ϕθdf Mηδ ϕηbe ϕδcf )

γ δ δ θ η θ = 2[Mγδ (ϕηae ϕγbe + Cαγη ϕα ab )ϕcf Mηθ ϕdf − Mηθ ϕae ϕdf Mγδ ϕbe ϕcf ]

= 2Mγδ ϕδcf Mηθ ϕθdf Cαγη ϕα ab . But Mγδ ϕδcf Mηθ ϕθdf Cαγη ϕα ab

θ δ δθ β = Mγδ Mηθ Cαγη ϕα ab [ϕcf ϕdf + Cβ ϕcd ] γη δθ α β δ θ = −Mηθ Mγδ Cαγη ϕα ab ϕcf ϕdf + Mγδ Mηθ Cα Cβ ϕab ϕcd

γη δθ α β δ θ = −Mηθ Mγδ Cαγη ϕα ab ϕcf ϕdf + (Cα Cβ Mγδ Mηθ )ϕab ϕcd

188


which implies Mγδ ϕδcf Mηθ ϕθdf Cαγη ϕα ab =

1 γη δθ β (C C Mγδ Mηθ )ϕα ab ϕcd . 2 α β

Then we have β 2(Bacbd − Badbc ) = (Cαγη Cβδθ Mγδ Mηθ )ϕα ab ϕcd .

(1.3.8)

Therefore, combining (1.3.7) and (1.3.8), we can reformulate the curvature evolution equation (1.3.4) as follows. β Proposition 1.3.4 (Hamilton [59]). Let Rabcd = Mαβ ϕα ab ϕcd . Then under the Ricci flow (1.1.5), Mαβ satisfies the evolution equation

∂Mαβ # 2 = ∆Mαβ + Mαβ + Mαβ ∂t

(1.3.9)

# 2 where Mαβ = Mαγ Mβγ is the operator square and Mαβ = (Cαγη Cβδθ Mγδ Mηθ ) is the Lie algebra square. # Let us now consider the operator Mαβ in dimensions 3 and 4 in more detail. In dimension 3, let ω1 , ω2 , ω3 be a positively oriented orthonormal basis for oneforms. Then √ √ √ ϕ1 = 2ω1 ∧ ω2 , ϕ2 = 2ω2 ∧ ω3 , ϕ3 = 2ω3 ∧ ω1

form an orthonormal basis for two-forms Λ2 . Write ϕα = {ϕα ab }, α = 1, 2, 3, as     √ 2 0 0 0 0 0√ √ 2    2 0√ (ϕ1ab ) = − 2 , (ϕ2ab ) = 0  0 0 2 , 2 0 − 22 0 0 0 0  √  0 0 − 22   3 0 , (ϕab ) =  √0 0 2 0 0 2

then

     √ √ 2 0 0 0√ 0 0 √0 0 − 0 0√ 22 0 2     √   [ϕ1 , ϕ2 ] =  − 2 0 0   0 √0 − 22  −  0 0√ 22   2 0 0  2 2 2 2 0 0 0 0 0 0 0 0 2 0− 2 0   0 0 − 21  = 00 0  1 2 0 0 √ 2 3 ϕ . = 2 

So C 123 = h[ϕ1 , ϕ2 ], ϕ3 i =

√ 2 2 ,

in particular ( √ ± 22 , if α 6= β 6= γ, αβγ C = 0, otherwise.


189

# Hence the matrix M # = (Mαβ ) is just the adjoint matrix of M = (Mαβ ):

M # = det M · t M −1 .

(1.3.10)

In dimension 4, we can use the Hodge star operator to decompose the space of two-forms Λ2 as Λ2 = Λ2+ ⊕Λ2− where Λ2+ (resp. Λ2− ) is the eigenspace of the star operator with eigenvalue +1 (resp. −1). Let ω1 , ω2 , ω3 , ω4 be a positively oriented orthonormal basis for one-forms. A basis for Λ2+ is then given by ϕ1 = ω1 ∧ ω2 + ω3 ∧ ω4 ,

ϕ2 = ω1 ∧ ω3 + ω4 ∧ ω2 ,

ϕ3 = ω1 ∧ ω4 + ω2 ∧ ω3 ,

while a basis for Λ2− is given by ψ 1 = ω1 ∧ ω2 − ω3 ∧ ω4 ,

ψ 2 = ω1 ∧ ω3 − ω4 ∧ ω2 ,

ψ 3 = ω1 ∧ ω4 − ω2 ∧ ω3 .

In particular, {ϕ1 , ϕ2 , ϕ3 , ψ 1 , ψ 2 , ψ 3 } forms an orthonormal basis for the space of two-forms Λ2 . By using this basis we obtain a block decomposition of the curvature operator matrix M as A B M = (Mαβ ) = . t B C Here A, B and C are 3 × 3 matrices with A and C being symmetric. Then we can write each element of the basis as a skew-symmetric 4 × 4 matrix and compute as above to get # A B# # (1.3.11) M # = (Mαβ )=2 t # , B C# where A# , B # , C # are the adjoint of 3 × 3 submatrices as before. For later applications in Chapter 5, we now give some computations for the entries of the matrices A, C and B as follows. First for the matrices A and C, we have A11 = Rm(ϕ1 , ϕ1 ) = R1212 + R3434 + 2R1234 A22 = Rm(ϕ2 , ϕ2 ) = R1313 + R4242 + 2R1342 A33 = Rm(ϕ3 , ϕ3 ) = R1414 + R2323 + 2R1423 and C11 = Rm(ψ 1 , ψ 1 ) = R1212 + R3434 − 2R1234 C22 = Rm(ψ 2 , ψ 2 ) = R1313 + R4242 − 2R1342 C33 = Rm(ψ 3 , ψ 3 ) = R1414 + R2323 − 2R1423 .

190


By the Bianchi identity R1234 + R1342 + R1423 = 0, so we have trA = trC =

1 R. 2

Next for the entries of the matrix B, we have B11 = Rm(ϕ1 , ψ 1 ) = R1212 − R3434 B22 = Rm(ϕ2 , ψ 2 ) = R1313 − R4242 B33 = Rm(ϕ3 , ψ 3 ) = R1414 − R2323 and B12 = Rm(ϕ1 , ψ 2 ) = R1213 + R3413 − R1242 − R3442

etc.

Thus the entries of B can be written as B11 =

1 (R11 + R22 − R33 − R44 ) 2

B22 =

1 (R11 + R33 − R44 − R22 ) 2

B33 =

1 (R11 + R44 − R22 − R33 ) 2

and B12 = R23 − R14

etc.

If we choose the frame {ω1 , ω2 , ω3 , ω4 } so that the Ricci tensor is diagonal, then the matrix B is also diagonal. In particular, the matrix B is identically zero when the four-manifold is Einstein. 1.4. Derivative Estimates. In the previous section we have seen that the curvatures satisfy nonlinear heat equations with quadratic growth terms. The parabolic nature will give us a bound on the derivatives of the curvatures at any time t > 0 in terms of a bound of the curvatures. We begin with the global version of the derivative estimates. Theorem 1.4.1 (Shi [114]). There exist constants Cm , m = 1, 2, . . . , such that if the curvature of a complete solution to Ricci flow is bounded by |Rijkl | ≤ M up to time t with 0 < t ≤ by

1 M,

then the covariant derivative of the curvature is bounded √ |∇Rijkl | ≤ C1 M/ t


191

and the mth covariant derivative of the curvature is bounded by |∇m Rijkl | ≤ Cm M/t 2 . m

Here the norms are taken with respect to the evolving metric. Proof. We shall only give the proof for the compact case. The noncompact case can be deduced from the next local derivative estimate theorem. Let us denote the curvature tensor by Rm and denote by A ∗ B any tensor product of two tensors A and B when we do not need the precise expression. We have from Proposition 1.3.1 that ∂ Rm = ∆Rm + Rm ∗ Rm. ∂t

(1.4.1) Since

∂ i ∂gkl 1 il ∂gjl ∂gjk Γ = g ∇j + ∇k − ∇l ∂t jk 2 ∂t ∂t ∂t = ∇Rm,

it follows that (1.4.2)

∂ (∇Rm) = ∆(∇Rm) + Rm ∗ (∇Rm). ∂t

Thus ∂ |Rm|2 ≤ ∆|Rm|2 − 2|∇Rm|2 + C|Rm|3 , ∂t ∂ |∇Rm|2 ≤ ∆|∇Rm|2 − 2|∇2 Rm|2 + C|Rm| · |∇Rm|2 , ∂t for some constant C depending only on the dimension n. Let A >0 be a constant (to be determined) and set F = t|∇Rm|2 + A|Rm|2 . We compute ∂ ∂ ∂F = |∇Rm|2 + t |∇Rm|2 + A |Rm|2 ∂t ∂t ∂t ≤ ∆(t|∇Rm|2 + A|Rm|2 ) + |∇Rm|2 (1 + tC|Rm| − 2A) + CA|Rm|3 . Taking A ≥ C + 1, we get ∂F ¯ 3 ≤ ∆F + CM ∂t for some constant C¯ depending only on the dimension n. We then obtain 2 ¯ 3 t ≤ (A + C)M ¯ F ≤ F (0) + CM ,

and then 2 ¯ |∇Rm|2 ≤ (A + C)M /t.

192


The general case follows in the same way. If we have bounds |∇k Rm| ≤ Ck M/t 2 , k

we know from (1.4.1) and (1.4.2) that ∂ k CM 3 |∇ Rm|2 ≤ ∆|∇k Rm|2 − 2|∇k+1 Rm|2 + k , ∂t t and CM 3 ∂ k+1 |∇ Rm|2 ≤ ∆|∇k+1 Rm|2 − 2|∇k+2 Rm|2 + CM |∇k+1 Rm|2 + k+1 . ∂t t Let Ak > 0 be a constant (to be determined) and set Fk = tk+2 |∇k+1 Rm|2 + Ak tk+1 |∇k Rm|2 . Then ∂ ∂ Fk = (k + 2)tk+1 |∇k+1 Rm|2 + tk+2 |∇k+1 Rm|2 ∂t ∂t ∂ + Ak (k + 1)tk |∇k Rm|2 + Ak tk+1 |∇k Rm|2 ∂t ≤ (k + 2)tk+1 |∇k+1 Rm|2 CM 3 + tk+2 ∆|∇k+1 Rm|2 − 2|∇k+2 Rm|2 + CM |∇k+1 Rm|2 + k+1 t + Ak (k + 1)tk |∇k Rm|2 CM 3 + Ak tk+1 ∆|∇k Rm|2 − 2|∇k+1 Rm|2 + k t

≤ ∆Fk + Ck+1 M 2

for some positive constant Ck+1 , by choosing Ak large enough. This implies that |∇k+1 Rm| ≤

Ck+1 M t

k+1 2

.

The above derivative estimate is a somewhat standard Bernstein estimate in PDEs. By using a cutoff argument, we will derive the following local version, which is called Shi’s derivative estimate. The following proof is adapted from Hamilton [63]. Theorem 1.4.2 (Shi [114]). There exist positive constants θ, Ck , k = 1, 2, . . . , depending only on the dimension with the following property. Suppose that the curvature of a solution to the Ricci flow is bounded θ |Rm| ≤ M, on U × 0, M where U is an open set of the manifold. Assume that the closed ball B0 (p, r), centered at p of radius r with respect to the metric at t = 0, is contained in U and the time t ≤ θ/M . Then we can estimate the covariant derivatives of the curvature at (p, t) by 1 1 + + M , |∇Rm(p, t)|2 ≤ C1 M 2 r2 t


193

and the k th covariant derivative of the curvature at (p, t) by 1 1 k 2 2 k |∇ Rm(p, t)| ≤ Ck M + k +M . r2k t √ Proof. Without loss of generality, we may assume r ≤ θ/ M and the exponential map at p at time t = 0 is injective on the ball of radius r (by passing to a local cover if necessary, and pulling back the local solution of the Ricci flow to the ball of radius r in the tangent space at p at time t = 0). Recall ∂ |Rm|2 ≤ ∆|Rm|2 − 2|∇Rm|2 + C|Rm|3 , ∂t ∂ |∇Rm|2 ≤ ∆|∇Rm|2 − 2|∇2 Rm|2 + C|Rm| · |∇Rm|2 . ∂t Define S = (BM 2 + |Rm|2 )|∇Rm|2 where B is a positive constant to be determined. By choosing B ≥ C 2 /4 and using the Cauchy inequality, we have ∂ S ≤ ∆S − 2BM 2 |∇2 Rm|2 − 2|∇Rm|4 ∂t + CM |∇Rm|2 · |∇2 Rm| + CBM 3 |∇Rm|2 ≤ ∆S − |∇Rm|4 + CB 2 M 6

≤ ∆S −

S2 + CB 2 M 6 . (B + 1)2 M 4

If we take F = b(BM 2 + |Rm|2 )|∇Rm|2 /M 4 = bS/M 4 , and b ≤ min{1/(B + 1)2 , 1/CB 2 }, we get (1.4.3)

∂F ≤ ∆F − F 2 + M 2 . ∂t

We now want to choose a cutoff function ϕ with the support in the ball B0 (p, r) such that at t = 0, ϕ(p) = r,

0 ≤ ϕ ≤ Ar,

and |∇ϕ| ≤ A,

|∇2 ϕ| ≤

A r

for some positive constant A depending only on the dimension. Indeed, let g : (−∞, +∞) → [0, +∞) be a smooth, nonnegative function satisfying ( 1, u ∈ (− 21 , 12 ), g(u) = 0, outside (−1, 1).

194


Set ϕ = rg

s2 r2

,

where s is the geodesic distance function from p with respect to the metric at t = 0. Then 2 s 1 · 2s∇s ∇ϕ = g ′ r r2 and hence |∇ϕ| ≤ 2C1 . Also, ∇2 ϕ =

1 ′′ g r

s2 r2

1 2 1 4s ∇s · ∇s + g ′ 2 r r

s2 r2

1 2∇s · ∇s + g ′ r

s2 r2

· 2s∇2 s.

Thus, by using the standard Hessian comparison, C1 C1 + |s∇2 s| r r C2 √ C1 1+s + M ≤ r s C3 . ≤ r

|∇2 ϕ| ≤

Here C1 , C2 and C3 are positive constants depending only on the dimension. θ Now extend ϕ to U × [0, M ] by letting ϕ to be zero outside B0 (p, r) and independent of time. Introduce the barrier function √ (12 + 4 n)A2 1 (1.4.4) H= + +M 2 ϕ t which is defined and smooth on the set {ϕ > 0} × (0, T ]. As the metric evolves, we will still have 0 ≤ ϕ ≤ Ar (since ϕ is independent of time t); but |∇ϕ|2 and ϕ|∇2 ϕ| may increase. By continuity it will be a while before they double. Claim 1. As long as |∇ϕ|2 ≤ 2A2 , ϕ|∇2 ϕ| ≤ 2A2 , we have ∂H > ∆H − H 2 + M 2 . ∂t Indeed, by the definition of H, we have √ 1 (12 + 4 n)2 A4 2 + 2 + M 2, H > 4 ϕ t ∂H 1 = − 2, ∂t t


195

and √ 1 2 ∆H = (12 + 4 n)A ∆ ϕ2 √ 6|∇ϕ|2 − 2ϕ∆ϕ 2 = (12 + 4 n)A ϕ4 √ 2 √ 12A + 4 nA2 2 ≤ (12 + 4 n)A ϕ4 √ 2 4 (12 + 4 n) A . = ϕ4 Therefore, H 2 > ∆H −

∂H + M 2. ∂t

Claim 2. If the constant θ > 0 is small enough √ compared to b, B and A, then we have the following property: as long as r ≤ θ/ M , t ≤ θ/M and F ≤ H, we will have |∇ϕ|2 ≤ 2A2

and

ϕ|∇2 ϕ| ≤ 2A2 .

Indeed, by considering the evolution of ∇ϕ, we have ∂ ∂ ∇a ϕ = (Fai ∇i ϕ) ∂t ∂t ∂ϕ = Fai ∇i + ∇i ϕRki Fak ∂t = Rab ∇b ϕ

which implies ∂ |∇ϕ|2 ≤ CM |∇ϕ|2 , ∂t and then |∇ϕ|2 ≤ A2 eCMt ≤ 2A2 , provided t ≤ θ/M with θ ≤ log 2/C. By considering the evolution of ∇2 ϕ, we have ∂ ∂ (∇a ∇b ϕ) = (Fai Fbj ∇i ∇j ϕ) ∂t ∂t 2 ∂ ∂ ϕ i j k ∂ϕ = Fa Fb − Γij k ∂t ∂xi ∂xj ∂x ∂ϕ + Rac ∇b ∇c ϕ + Rbc ∇a ∇c ϕ = ∇a ∇b ∂t + (∇c Rab − ∇a Rbc − ∇b Rac )∇c ϕ

196


which implies ∂ 2 |∇ ϕ| ≤ C|Rm| · |∇2 ϕ| + C|∇Rm| · |∇ϕ|. ∂t

(1.4.5)

By assumption F ≤ H, we have √ 1 2M 2 (12 + 4 n)A2 2 + , (1.4.6) |∇Rm| ≤ bB ϕ2 t

for t ≤ θ/M.

Thus by noting ϕ independent of t and ϕ ≤ Ar, we get from (1.4.5) and (1.4.6) that ∂ r 2 2 √ (ϕ|∇ ϕ|) ≤ CM ϕ|∇ ϕ| + 1 + ∂t t which implies Z t r ϕ|∇2 ϕ| ≤ eCMt (ϕ|∇2 ϕ|)|t=0 + CM 1 + √ dt t 0 h √ i CMt 2 ≤e A + CM (t + 2r t) ≤ 2A2

√ provided r ≤ θ/ M , and t ≤ θ/M with θ small enough. Therefore we have obtained Claim 2. The combination of Claim 1 and Claim 2 gives us

√ as long as r ≤ θ/ M ,

∂H > ∆H − H 2 + M 2 ∂t t ≤ θ/M and F ≤ H. And (1.4.3) tells us ∂F ≤ ∆F − F 2 + M 2 . ∂t

Then the standard maximum principle immediately gives the estimate θ i 1 1 + + M on {ϕ > 0} × 0, , |∇Rm|2 ≤ CM 2 ϕ2 t M

which implies the first order derivative estimate. The higher order derivative estimates can be obtained in the same way by induction. Suppose we have the bounds 1 1 k |∇k Rm|2 ≤ Ck M 2 + + M . r2k tk As before, by (1.4.1) and (1.4.2), we have ∂ k |∇ Rm|2 ≤ ∆|∇k Rm|2 − 2|∇k+1 Rm|2 + CM 3 ∂t

1 1 k + k +M , r2k t

and ∂ k+1 |∇ Rm|2 ≤ ∆|∇k+1 Rm|2 − 2|∇k+2 Rm|2 ∂t k+1

+ CM |∇

2

Rm| + CM

3

1

r2(k+1)

+

1 tk+1

+M

k+1

.


197

Here and in the following we denote by C various positive constants depending only on Ck and the dimension. Define

Sk = Bk M

2

1 1 + k + Mk 2k r t

k

2

+ |∇ Rm|

· |∇k+1 Rm|2

where Bk is a positive constant to be determined. By choosing Bk large enough and Cauchy inequality, we have ∂ Sk ≤ ∂t

k + ∆|∇k Rm|2 − 2|∇k+1 Rm|2 tk+1 1 1 k + + M · |∇k+1 Rm|2 + CM 3 r2k tk 1 1 k k 2 + Bk M 2 + |∇ Rm| + + M r2k tk · ∆|∇k+1 Rm|2 − 2|∇k+2 Rm|2 + CM |∇k+1 Rm|2 1 1 k+1 + + CM 3 + M tk+1 r2(k+1) k ≤ ∆Sk + 8|∇k Rm| · |∇k+1 Rm|2 · |∇k+2 Rm| − k+1 |∇k+1 Rm|2 t 1 1 k+1 4 3 k+1 2 k − 2|∇ Rm| + CM |∇ Rm| + + M r2k tk 1 1 k k 2 + + M + |∇ Rm| − 2|∇k+2 Rm|2 Bk M 2 r2k tk 1 1 k k 2 + CM |∇k+1 Rm|2 Bk M 2 + |∇ Rm| + + M r2k tk 1 1 + CM 3 + k+1 + M k+1 t r2(k+1) 1 1 k k 2 · Bk M 2 + + M + |∇ Rm| r2k tk 1 1 2k + + M ≤ ∆Sk − |∇k+1 Rm|4 + CBk2 M 6 r4k t2k 1 1 2k+1 + CBk M 5 + + M t2k+1 r2(2k+1) 1 1 2k+1 + M ≤ ∆Sk − |∇k+1 Rm|4 + CBk2 M 5 + t2k+1 r2(2k+1) Sk ≤ ∆Sk − 2 (B + 1)2 M 4 r12k + t1k + M k 1 1 2 5 2k+1 + 2k+1 + M + CBk M . t r2(2k+1) −

198


Let u = 1/r2 + 1/t + M and set Fk = bSk /uk . Then ∂Fk Fk2 ≤ ∆Fk − + bCBk2 M 5 uk+1 + kFk u ∂t b(Bk + 1)2 M 4 uk Fk2 + b(C + 2k 2 )(Bk + 1)2 M 4 uk+2 . ≤ ∆Fk − 2b(Bk + 1)2 M 4 uk By choosing b ≤ 1/(2(C + 2k 2 )(Bk + 1)2 M 4 ), we get 1 ∂Fk ≤ ∆Fk − k Fk2 + uk+2 . ∂t u Introduce Hk = 5(k + 1)(2(k + 1) + 1 +

√ n)A2 ϕ−2(k+1) + Lt−(k+1) + M k+1 ,

where L ≥ k + 2. Then by using Claim 1 and Claim 2, we have ∂Hk = −(k + 1)Lt−(k+2) , ∂t ∆Hk ≤ 20(k + 1)2 (2(k + 1) + 1 +

√ n)A4 ϕ−2(k+2)

and Hk2 > 25(k + 1)2 (2(k + 1) + 1 +

√

n)A4 ϕ−4(k+1) + L2 t−2(k+1) + M 2(k+1) .

These imply 1 ∂Hk > ∆Hk − k Hk2 + uk+2 . ∂t u Then the maximum principle immediately gives the estimate Fk ≤ Hk . In particular, b Bk M 2 uk

1 1 + k + Mk 2k r t

· |∇k+1 Rm|2 √ ≤ 5(k + 1) 2(k + 1) + 1 + n A2 ϕ−2(k+1) + Lt−(k+1) + M k+1 .

So by the definition of u and the choosing of b, we obtain the desired estimate |∇k+1 Rm|2 ≤ Ck+1 M 2

1 r2(k+1)

+

1 tk+1

+ M k+1 .

Therefore we have completed the proof of the theorem.


199

1.5. Variational Structure and Dynamic Property. In this section, we introduce two functionals of Perelman [103], F and W, and discuss their relations with the Ricci flow. It was not known whether the Ricci flow is a gradient flow until Perelman [103] showed that the Ricci flow is, in a certain sense, the gradient flow of the functional F . If we consider the Ricci flow as a dynamical system on the space of Riemannian metrics, then these two functionals are of Lyapunov type for this dynamical system. Obviously, Ricci flat metrics are fixed points of the dynamical system. When we consider the space of Riemannian metrics modulo diffeomorphism and scaling, fixed points of the Ricci flow dynamical system correspond to steady, or shrinking, or expanding Ricci solitons. The following concept corresponds to a periodic orbit.

Definition 1.5.1. A metric gij (t) evolving by the Ricci flow is called a breather if for some t1 < t2 and α > 0 the metrics αgij (t1 ) and gij (t2 ) differ only by a diffeomorphism; the case α = 1, α < 1, α > 1 correspond to steady, shrinking and expanding breathers, respectively.

Clearly, (steady, shrinking or expanding) Ricci solitons are trivial breathers for which the metrics gij (t1 ) and gij (t2 ) differ only by diffeomorphism and scaling for every pair t1 and t2 . We always assume M is a compact n-dimensional manifold in this section. Let us first consider the functional

F (gij , f ) =

(1.5.1)

Z

M

(R + |∇f |2 )e−f dV

of Perelman [103] defined on the space of Riemannian metrics, and smooth functions on M . Here R is the scalar curvature of gij .

Lemma 1.5.2 (Perelman [103]). If δgij = vij and δf = h are variations of gij and f respectively, then the first variation of F is given by Z δF (vij , h) =

h v i −vij (Rij + ∇i ∇j f ) + − h (2∆f − |∇f |2 + R) e−f dV 2 M

where v = g ij vij .

200


Proof. In any normal coordinates at a fixed point, we have ∂ ∂ (δΓh ) − (δΓhil ) ∂xi jl ∂xj ∂ 1 hm = g (∇j vlm + ∇l vjm − ∇m vjl ) ∂xi 2 ∂ 1 hm − j g (∇i vlm + ∇l vim − ∇m vil ) , ∂x 2 ∂ 1 im g (∇j vlm + ∇l vjm − ∇m vjl ) δRjl = ∂xi 2 ∂ 1 im − j g (∇i vlm + ∇l vim − ∇m vil ) ∂x 2 1 ∂ 1 ∂ [∇j vli + ∇l vji − ∇i vjl ] − [∇l v], = 2 ∂xi 2 ∂xj δR = δ(g jl Rjl )

h δRijl =

= −vjl Rjl + g jl δRjl 1 ∂ 1 ∂ = −vjl Rjl + [∇l vli + ∇l v il − ∇i v] − [∇j v] i 2 ∂x 2 ∂xj = −vjl Rjl + ∇i ∇l vil − ∆v. Thus δR(vij ) = −∆v + ∇i ∇j vij − vij Rij .

(1.5.2)

The first variation of the functional F (gij , f ) is (1.5.3)

(R + |∇f |2 )e−f dV M Z = [δR(vij ) + δ(g ij ∇i f ∇j f )]e−f dV

δ

Z

M

h v i + (R + |∇f |2 ) −he−f dV + e−f dV 2 Z = − ∆v + ∇i ∇j vij − Rij vij − vij ∇i f ∇j f M v + 2h∇f, ∇hi + (R + |∇f |2 ) − h e−f dV. 2 On the other hand, Z

M

(∇i ∇j vij − vij ∇i f ∇j f )e

Z

(∇i f ∇j vij − vij ∇i f ∇j f )e−f dV Z (∇i ∇j f )vij e−f dV, =− ZM Z Z −f 2h∇f, ∇hie dV = −2 h∆f e−f dV + 2 |∇f |2 he−f dV, M

−f

dV =

M

M

M


201

and Z

(−∆v)e

M

−f

Z

dV = − h∇f, ∇vie−f dV Z M Z = v∆f e−f dV − |∇f |2 ve−f dV. M

M

Plugging these identities into (1.5.3) the first variation formula follows. Now let us study the functional F when the metric evolves under the Ricci flow and the function evolves by a backward heat equation. Proposition 1.5.3 (Perelman [103]). Let gij (t) and f (t) evolve according to the coupled flow ( ∂gij ∂t

∂f ∂t

= −2Rij , = −∆f + |∇f |2 − R.

Then d F (gij (t), f (t)) = 2 dt

Z

M

|Rij + ∇i ∇j f |2 e−f dV

R and M e−f dV is constant. In particular F (gij (t), f (t)) is nondecreasing in time and the monotonicity is strict unless we are on a steady gradient soliton. Proof. Under the coupled flow and using the first variation formula in Lemma 1.5.2, we have d F (gij (t), f (t)) dt Z =

− (−2Rij )(Rij + ∇i ∇j f ) ∂f 1 (−2R) − (2∆f − |∇f |2 + R) e−f dV + 2 ∂t

M

=

Z

M

[2Rij (Rij + ∇i ∇j f ) + (∆f − |∇f |2 )(2∆f − |∇f |2 + R)]e−f dV.

Now Z

(∆f − |∇f |2 )(2∆f − |∇f |2 )e−f dV M Z = −∇i f ∇i (2∆f − |∇f |2 )e−f dV M Z = −∇i f (2∇j (∇i ∇j f ) − 2Rij ∇j f − 2h∇f, ∇i ∇f i)e−f dV M Z = −2 [(∇i f ∇j f −∇i ∇j f )∇i ∇j f −Rij ∇i f ∇j f −h∇f, ∇i ∇f i∇i f ]e−f dV Z M =2 [|∇i ∇j f |2 + Rij ∇i f ∇j f ]e−f dV, M

202


and Z

(∆f − |∇f |2 )Re−f dV M Z = −∇i f ∇i Re−f dV M Z Z −f =2 ∇i ∇j f Rij e dV − 2 M

M

∇i f ∇j f Rij e−f dV.

Here we have used the contracted second Bianchi identity. Therefore we obtain d F (gij (t), f (t)) dt Z =

M

=2

Z

[2Rij (Rij + ∇i ∇j f ) + 2(∇i ∇j f )(∇i ∇j f + Rij )]e−f dV

M

|Rij + ∇i ∇j f |2 e−f dV.

R It remains to show M e−f dV is a constant. Note that the volume element dV = p detgij dx evolves under the Ricci flow by ∂ ∂ p dV = ( det gij )dx ∂t ∂t 1 ∂ = log(det gij ) dV 2 ∂t 1 ij ∂ gij )dV = (g 2 ∂t = −RdV.

(1.5.4)

Hence (1.5.5)

∂f − − R dV ∂t

∂ −f e dV = e−f ∂t

= (∆f − |∇f |2 )e−f dV = −∆(e−f )dV.

It then follows that d dt

Z

M

e

−f

dV = −

Z

∆(e−f )dV = 0.

M

This finishes the proof of the proposition. Next we define the associated energy Z (1.5.6) λ(gij ) = inf F (gij , f ) | f ∈ C ∞ (M ),

M

e−f dV = 1 .

If we set u = e−f /2 , then the functional F can be expressed in terms of u as Z F= (Ru2 + 4|∇u|2 )dV, M


203

R R and the constraint M e−f dV = 1 becomes M u2 dV = 1. Therefore λ(gij ) is just the first eigenvalue of the operator −4∆ + R. Let u0 > 0 be a first eigenfunction of the operator −4∆ + R satisfying −4∆u0 + Ru0 = λ(gij )u0 . The f0 = −2 log u0 is a minimizer: λ(gij ) = F (gij , f0 ). Note that f0 satisfies the equation (1.5.7)

−2∆f0 + |∇f0 |2 − R = −λ(gij ).

Observe that the evolution equation ∂f = −∆f + |∇f |2 − R ∂t can be rewritten as the following linear equation ∂ −f (e ) = −∆(e−f ) + R(e−f ). ∂t Thus we can always solve the evolution equation for f backwards inR time. Suppose at t = t0 , the infimum λ(gij ) is achieved by some function f0 with M e−f0 dV = 1. We solve the backward heat equation ( ∂f ∂t

= −∆f + |∇f |2 − R

f |t=t0 = f0

to obtain a solution f (t) for t ≤ t0 which satisfies Proposition 1.5.3 that

R

M

e−f dV = 1. It then follows from

λ(gij (t)) ≤ F(gij (t), f (t)) ≤ F(gij (t0 ), f (t0 )) = λ(gij (t0 )). Also note λ(gij ) is invariant under diffeomorphism. Thus we have proved Corollary 1.5.4. (i) λ(gij (t)) is nondecreasing along the Ricci flow and the monotonicity is strict unless we are on a steady gradient soliton; (ii) A steady breather is necessarily a steady gradient soliton. To deal with the expanding case we consider a scale invariant version ¯ ij ) = λ(gij )V n2 (gij ). λ(g Here V = V ol(gij ) denotes the volume of M with respect to the metric gij .

204


Corollary 1.5.5. ¯ ij ) is nondecreasing along the Ricci flow whenever it is nonpositive; more(i) λ(g over, the monotonicity is strict unless we are on a gradient expanding soliton; (ii) An expanding breather is necessarily an expanding gradient soliton. Proof. Let f0 be a minimizer of λ(gij (t)) at t = t0 and solve the backward heat equation ∂f = −∆f + |∇f |2 − R ∂t

R to obtain f (t), t ≤ t0 , with M e−f (t) dV = 1. We compute the derivative (understood in the barrier sense) at t = t0 , d¯ λ(gij (t)) dt 2 d ≥ (F (gij (t), f (t)) · V n (gij (t))) dt Z 2 n

2|Rij + ∇i ∇j f |2 e−f dV M Z Z 2 2−n (−R)dV · (R + |∇f |2 )e−f dV + V n n M M 2 Z 2 Rij + ∇i ∇j f − 1 (R + ∆f )gij e−f dV = 2V n n M Z Z Z 1 1 1 (R+∆f )2 e−f dV + − (R+|∇f |2 )e−f dV RdV , + n M n V M M

=V

where we have used the formula (1.5.4) in the computation of dV /dt. Suppose λ(gij (t0 )) ≤ 0, then the last term on the RHS is given by, Z Z 1 1 2 −f − (R + |∇f | e dV ) RdV n V MZ Z M 1 2 −f 2 −f − (R + |∇f | )e dV (R + |∇f | )e dV ≥ n M M Z 2 1 =− (R + ∆f )e−f dV . n M Thus at t = t0 , (1.5.8)

d¯ λ(gij (t)) dt Z ≥ 2V +

2 n

1 n

1 (R + ∆f )gij |2 e−f dV n M Z 2 ! Z 2 −f −f (R + ∆f ) e dV − (R + ∆f )e dV ≥0 M

|Rij + ∇i ∇j f −

M

by the Cauchy-Schwarz inequality. Thus we have proved statement (i). We note that on an expanding breather on [t1 , t2 ] with αgij (t1 ) and gij (t2 ) differ only by a diffeomorphism for some α > 1, it would necessary have dV > 0, for some t ∈ [t1 , t2 ]. dt


205

On the other hand, for every t, −

d 1 log V = dt V

Z

M

RdV ≥ λ(gij (t))

by the definition of λ(gij (t)). It follows that on an expanding breather on [t1 , t2 ], ¯ ij (t)) = λ(gij (t))V n2 (gij (t)) < 0 λ(g for some t ∈ [t1 , t2 ]. Then by using statement (i), it implies ¯ ij (t1 )) < λ(g ¯ ij (t2 )) λ(g ¯ ij (t)) is invariant unless we are on an expanding gradient soliton. We also note that λ(g under diffeomorphism and scaling which implies ¯ ij (t1 )) = λ(g ¯ ij (t2 )). λ(g Therefore the breather must be an expanding gradient soliton. In particular part (ii) of Corollaries 1.5.4 and 1.5.5 imply that all compact steady or expanding Ricci solitons are gradient ones. Combining this fact with Proposition (1.1.1), we immediately get Proposition 1.5.6. On a compact manifold, a steady or expanding breather is necessarily an Einstein metric. In order to handle the shrinking case, we introduce the following important functional, also due to Perelman [103], Z n (1.5.9) W(gij , f, τ ) = [τ (R + |∇f |2 ) + f − n](4πτ )− 2 e−f dV M

where gij is a Riemannian metric, f is a smooth function on M , and τ is a positive scale parameter. Clearly the functional W is invariant under simultaneous scaling of τ and gij (or equivalently the parabolic scaling), and invariant under diffeomorphism. Namely, for any positive number a and any diffeomorphism ϕ (1.5.10)

W(aϕ∗ gij , ϕ∗ f, aτ ) = W(gij , f, τ ).

Similar to Lemma 1.5.2, we have the following first variation formula for W. Lemma 1.5.7 (Perelman [103]). If vij = δgij , h = δf, and η = δτ , then δW(vij , h, η) Z n 1 gij (4πτ )− 2 e−f dV = −τ vij Rij + ∇i ∇j f − 2τ M Z n n v −h− η [τ (R + 2∆f − |∇f |2 ) + f − n − 1](4πτ )− 2 e−f dV + 2 2τ ZM n n + η R + |∇f |2 − (4πτ )− 2 e−f dV. 2τ M Here v = g ij vij as before.

206


Proof. Arguing as in the proof of Lemma 1.5.2, the first variation of the functional W can be computed as follows, δW (vij , h, η) Z = [η(R + |∇f |2 ) + τ (−∆v + ∇i ∇j vij − Rij vij − vij ∇i f ∇j f M

n

+ 2h∇f, ∇hi) + h](4πτ )− 2 e−f dV Z h nη i v n + (τ (R + |∇f |2 ) + f − n) − + − h (4πτ )− 2 e−f dV 2τ 2 M Z n = [η(R + |∇f |2 ) + h](4πτ )− 2 e−f dV M Z n + [−τ vij (Rij + ∇i ∇j f ) + τ (v − 2h)(∆f − |∇f |2 )](4πτ )− 2 e−f dV ZM h nη i n v + (τ (R + |∇f |2 ) + f − n) − + − h (4πτ )− 2 e−f dV 2τ 2 ZM 1 n =− τ vij Rij + ∇i ∇j f − gij (4πτ )− 2 e−f dV 2τ ZM n v −h− η [τ (R + |∇f |2 ) + 2τ M 2

+ f − n + 2τ (∆f − |∇f |2 )](4πτ )− 2 e−f dV Z h n n v n i + η R + |∇f |2 − + h− + η (4πτ )− 2 e−f dV 2τ 2 2τ Z M 1 n = −τ vij Rij + ∇i ∇j f − gij (4πτ )− 2 e−f dV 2τ M Z v n n + −h− η [τ (R + 2∆f − |∇f |2 ) + f − n − 1](4πτ )− 2 e−f dV 2 2τ ZM n n + η R + |∇f |2 − (4πτ )− 2 e−f dV. 2τ M n

The following result is analogous to Proposition 1.5.3. Proposition 1.5.8. If gij (t), f (t) and τ (t) evolve according to the system  ∂gij   = −2Rij ,   ∂t     ∂f n = −∆f + |∇f |2 − R + ,  ∂t 2τ        ∂τ = −1, ∂t

then we have the identity

d W(gij (t), f (t), τ (t)) = dt

2 1 n 2τ Rij + ∇i ∇j f − gij (4πτ )− 2 e−f dV 2τ M

Z

R n and M (4πτ )− 2 e−f dV is constant. In particular W(gij (t), f (t), τ (t)) is nondecreasing in time and the monotonicity is strict unless we are on a shrinking gradient soliton.

207


Proof. Using Lemma 1.5.7, we have (1.5.11)

d W(gij (t), f (t), τ (t)) dt Z =

2τ Rij

M

+ −

Z

1 n gij (4πτ )− 2 e−f dV 2τ

(∆f − |∇f |2 )[τ (R + 2∆f − |∇f |2 ) + f ](4πτ )− 2 e−f dV

ZM M

Rij + ∇i ∇j f −

n

R + |∇f |2 −

n n (4πτ )− 2 e−f dV. 2τ

R Here we have used the fact that M (∆f − |∇f |2 )e−f dV = 0. The second term on the RHS of (1.5.11) is Z n (∆f − |∇f |2 )[τ (R + 2∆f − |∇f |2 ) + f ](4πτ )− 2 e−f dV M Z n = (∆f − |∇f |2 )(2τ ∆f − τ |∇f |2 )(4πτ )− 2 e−f dV M Z Z n −f 2 −n 2 (−∇i f )(∇i R)(4πτ )− 2 e−f dV − |∇f | (4πτ ) e dV + τ M ZM n 2 =τ (−∇i f )(∇i (2∆f − |∇f | ))(4πτ )− 2 e−f dV Z ZM n n ∇i f ∇j Rij (4πτ )− 2 e−f dV − ∆f (4πτ )− 2 e−f dV − 2τ M M Z n = −2τ (∇i f )(∇i ∆f − h∇f, ∇i ∇f i)(4πτ )− 2 e−f dV ZM n + 2τ [(∇i ∇j f )Rij − ∇i f ∇j f Rij ](4πτ )− 2 e−f dV ZM 1 n − gij (∇i ∇j f )(4πτ )− 2 e−f dV + 2τ 2τ ZM = −2τ [(∇i f ∇j f − ∇i ∇j f )∇i ∇j f − Rij ∇i f ∇j f M

n

− ∇i ∇j f ∇i f ∇j f ](4πτ )− 2 e−f dV Z n + 2τ [(∇i ∇j f )Rij − ∇i f ∇j f Rij ](4πτ )− 2 e−f dV ZM 1 n + 2τ − gij (∇i ∇j f )(4πτ )− 2 e−f dV 2τ M Z n 1 gij (4πτ )− 2 e−f dV. = 2τ (∇i ∇j f ) ∇i ∇j f + Rij − 2τ M

Also the third term on the RHS of (1.5.11) is Z n n (4πτ )− 2 e−f dV − R + |∇f |2 − 2τ M Z n n = − R + ∆f − (4πτ )− 2 e−f dV 2τ M Z −1 1 n = 2τ gij Rij + ∇i ∇j f − gij (4πτ )− 2 e−f dV. 2τ 2τ M

208


Therefore, by combining the above identities, we obtain 2 Z d Rij + ∇i ∇j f − 1 gij (4πτ )− n2 e−f dV. W(gij (t), f (t), τ (t)) = 2τ dt 2τ M

Finally, by using the computations in (1.5.5) and the evolution equations of f and τ , we have ∂ n n −f n ∂ −f (4πτ )− 2 e−f dV = (4πτ )− 2 (e dV ) + e dV ∂t ∂t 2τ n

= −(4πτ )− 2 ∆(e−f )dV.

Hence d dt

Z

n

M

n

(4πτ )− 2 e−f dV = −(4πτ )− 2

Z

∆(e−f )dV = 0.

M

Now we set µ(gij , τ ) = inf W(gij , f, τ ) | f ∈ C ∞ (M ),

(1.5.12)

1 (4πτ )n/2

Z

e−f dV = 1

M

and ν(gij ) = inf W(g, f, τ ) | f ∈ C ∞ (M ), τ > 0,

1 (4πτ )n/2

Z

e−f dV = 1 .

Note that if we let u = e−f /2 , then the functional W can be expressed as Z n W(gij , f, τ ) = [τ (Ru2 + 4|∇u|2 ) − u2 log u2 − nu2 ](4πτ )− 2 dV M

R

R n n and the constraint M (4πτ )− 2 e−f dV = 1 becomes M u2 (4πτ )− 2 dV = 1. Thus µ(gij , τ ) corresponds to the best constant of a logarithmic Sobolev inequality. Since the nonquadratic term is subcritical (in view of Sobolev exponent), it is rather straightforward to show that Z Z n 2 2 2 2 2 −n 2 − inf [τ (4|∇u| + Ru ) − u log u − nu ](4πτ ) 2 dV u (4πτ ) 2 dV = 1 M

M

is achieved by some nonnegative function u ∈ H 1 (M ) which satisfies the EulerLagrange equation τ (−4∆u + Ru) − 2u log u − nu = µ(gij , τ )u.

One can further show that u is positive (see [108]). Then the standard regularity theory of elliptic PDEs shows that u is smooth. We refer the reader to Rothaus [108] for more details. It follows that µ(gij , τ ) is achieved by a minimizer f satisfying the nonlinear equation (1.5.13)

τ (2∆f − |∇f |2 + R) + f − n = µ(gij , τ ).


209

Corollary 1.5.9. (i) µ(gij (t), τ −t) is nondecreasing along the Ricci flow; moveover, the monotonicity is strict unless we are on a shrinking gradient soliton; (ii) A shrinking breather is necessarily a shrinking gradient soliton. Proof. Fix any time t0 , let f0 be a minimizer of µ(gij (t0 ), τ − t0 ). Note that the backward heat equation ∂f n = −∆f + |∇f |2 − R + ∂t 2τ is equivalent to the linear equation n n n ∂ ((4πτ )− 2 e−f ) = −∆((4πτ )− 2 e−f ) + R((4πτ )− 2 e−f ). ∂t

Thus we can solve the backward heat equation of f with f |t=t0 = f0 to obtain R n f (t), t ≤ t0 , with M (4πτ )− 2 e−f (t) dV = 1. It then follows from Proposition 1.5.8 that µ(gij (t), τ − t) ≤ W(gij (t), f (t), τ − t) ≤ W(gij (t0 ), f (t0 ), τ − t0 ) = µ(gij (t0 ), τ − t0 )

for t ≤ t0 and the second inequality is strict unless we are on a shrinking gradient soliton. This proves statement (i). Consider a shrinking breather on [t1 , t2 ] with αgij (t1 ) and gij (t2 ) differ only by a diffeomorphism for some α < 1. Recall that the functional W is invariant under simultaneous scaling of τ and gij and invariant under diffeomorphism. Then for τ > 0 to be determined, µ(gij (t1 ), τ − t1 ) = µ(αgij (t1 ), α(τ − t1 )) = µ(gij (t2 ), α(τ − t1 )) and by the monotonicity of µ(gij (t), τ − t), µ(gij (t1 ), τ − t1 ) ≤ µ(gij (t2 ), τ − t2 ). Now take τ > 0 such that α(τ − t1 ) = τ − t2 , i.e., τ=

t2 − αt1 . 1−α

This shows the equality holds in the monotonicity of µ(gij (t), τ − t). So the shrinking breather must be a shrinking gradient soliton. Finally, we remark that Hamilton, Ilmanen and the first author [18] have obtained the second variation formulas for both λ-energy and ν-energy. We refer the reader to their paper [18] for more details and related stability questions.

210


2. Maximum Principle and Li-Yau-Hamilton Inequalities. The maximum principle is a fundamental tool in the study of parabolic equations in general. In this chapter, we present various maximum principles for tensors developed by Hamilton in the Ricci flow. As an immediate consequence, the Ricci flow preserves the nonnegativity of the curvature operator. We also present the two crucial estimates in the Ricci flow: the Hamilton-Ivey curvature pinching estimate (when dimension n = 3), and the Li-Yau-Hamilton estimate from which one obtains the Harnack inequality for the evolved scalar curvature via a Li-Yau path integral. Finally, we describe Perelman’s Li-Yau type estimate for solutions to the conjugate heat equation and show how Li-Yau type path integral leads to a space-time distance function (i.e., what Perelman called the reduced distance). 2.1. Preserving Positive Curvature. Let M be an n-dimensional complete manifold. Consider a family of smooth metrics gij (t) evolving by the Ricci flow with uniformly bounded curvature for t ∈ [0, T ] with T < +∞. Denote by dt (x, y) the distance between two points x, y ∈ M with respect to the metric gij (t). Lemma 2.1.1. There exists a smooth function f on M such that f ≥ 1 everywhere, f (x) → +∞ as d0 (x, x0 ) → +∞ (for some fixed x0 ∈ M ), |∇f |gij (t) ≤ C

and

|∇2 f |gij (t) ≤ C

on M × [0, T ] for some positive constant C. Proof. Let ϕ(v) be a smooth function on Rn which is nonnegative, rotationRally symmetric and has compact support in a small ball centered at the origin with Rn ϕ(v)dv = 1. For each x ∈ M , set Z f (x) = ϕ(v)(d0 (x0 , expx (v)) + 1)dv, Rn

where the integral is taken over the tangent space Tx M at x which we have identified with Rn . If the size of the support of ϕ(v) is small compared to the maximum curvature, then it is well known that this defines a smooth function f on M with f (x) → +∞ as d0 (x, x0 ) → +∞, while the bounds on the first and second covariant derivatives of f with respect to the metric gij (·, 0) follow from the Hessian comparison theorem. Thus it remains to show these bounds hold with respect to the evolving metric gij (t). We compute, using the frame {Fai ∇i f } introduced in Section 1.3, ∂ ∂ ∇a f = (Fai ∇i f ) = Rab ∇b f. ∂t ∂t Hence |∇f | ≤ C1 · eC2 t , where C1 , C2 are some positive constants depending only on the dimension. Also 2 ∂ ∂ ∂ f i j k ∂f (∇a ∇b f ) = Fa Fb − Γij k ∂t ∂t ∂xi ∂xj ∂x = Rac ∇b ∇c f + Rbc ∇a ∇c f + (∇c Rab − ∇a Rbc − ∇b Rac )∇c f.


211

Then by Shi’s derivative estimate (Theorem 1.4.1), we have ∂ 2 C3 |∇ f | ≤ C3 |∇2 f | + √ , ∂t t which implies Z |∇2 f |gij (t) ≤ eC3 t |∇2 f |gij (0) +

t

0

C √3 e−C3 τ dτ τ

for some positive constants C3 depending only on the dimension and the curvature bound. We now use the weak maximum principle to derive the following Proposition 2.1.2. If the scalar curvature R of the solution gij (t), 0 ≤ t ≤ T , to the Ricci flow is nonnegative at t = 0, then it remains so on 0 ≤ t ≤ T . Proof. Let f be the function constructed in Lemma 2.1.1 and recall ∂R = ∆R + 2|Ric |2 . ∂t For any small constant ε > 0 and large constant A > 0, we have ∂R ∂ (R + εeAt f ) = + εAeAt f ∂t ∂t = ∆(R + εeAt f ) + 2|Ric|2 + εeAt (Af − ∆f ) > ∆(R + εeAt f )

by choosing A large enough. We claim that R + εeAt f > 0

on M × [0, T ].

Suppose not, then there exist a first time t0 > 0 and a point x0 ∈ M such that (R + εeAt f )(x0 , t0 ) = 0, ∇(R + εeAt f )(x0 , t0 ) = 0,

and

∆(R + εeAt f )(x0 , t0 ) ≥ 0, ∂ (R + εeAt f )(x0 , t0 ) ≤ 0. ∂t

Then 0≥

∂ (R + εeAt f )(x0 , t0 ) > ∆(R + εeAt f )(x0 , t0 ) ≥ 0, ∂t

which is a contradiction. So we have proved that R + εeAt f > 0

on M × [0, T ].

Letting ε → 0, we get R≥0

on M × [0, T ].

212


This finishes the proof of the proposition. Next we derive a maximum principle of Hamilton for tensors. Let M be a complete manifold with a metric g = {gij }, V a vector bundle over M with a metric h = {hαβ } and a connection ∇ = {Γα iβ } compatible with h, and suppose h is fixed but g and ∇ may vary smoothly with time t. Let Γ(V ) be the vector space of C ∞ sections of V . The Laplacian ∆ acting on a section σ ∈ Γ(V ) is defined by ∆σ = g ij ∇i ∇j σ. Let Mαβ be a symmetric bilinear form on V . We say Mαβ ≥ 0 if Mαβ v α v β ≥ 0 for all vectors v = {v α }. Assume Nαβ = P(Mαβ , hαβ ) is a polynomial in Mαβ formed by contracting products of Mαβ with itself using the metric h = {hαβ }. Assume that the tensor Mαβ is uniformly bounded in space-time and let gij evolve by the Ricci flow with bounded curvature. Lemma 2.1.3. Suppose that on 0 ≤ t ≤ T , ∂ Mαβ = ∆Mαβ + ui ∇i Mαβ + Nαβ ∂t where ui (t) is a time-dependent vector field on M with uniform bound and Nαβ = P(Mαβ , hαβ ) satisfies Nαβ v α v β ≥ 0

whenever

Mαβ v β = 0.

If Mαβ ≥ 0 at t = 0, then it remains so on 0 ≤ t ≤ T . Proof. Set ˜ αβ = Mαβ + εeAt f hαβ , M where A > 0 is a suitably large constant (to be chosen later) and f is the function constructed in Lemma 2.1.1. ˜ αβ > 0 on M × [0, T ] for every ε > 0. If not, then for some ε > 0, We claim that M ˜ αβ acquires a null vector v α of unit length there will be a first time t0 > 0 where M at some point x0 ∈ M . At (x0 , t0 ), ˜αβ v α v β Nαβ v α v β ≥ Nαβ v α v β − N ≥ −CεeAt0 f (x0 ),

˜αβ = P(M ˜ αβ , hαβ ), and C is a positive constant (depending on the bound of where N Mαβ , but independent of A). Let us extend v α to a local vector field in a neighborhood of x0 by parallel translating v α along geodesics (with respect to the metric gij (t0 )) emanating radially out of x0 , with v α independent of t. Then, at (x0 , t0 ), we have ∂ ˜ (Mαβ v α v β ) ≤ 0, ∂t ˜ αβ v α v β ) = 0, ∇(M

˜ αβ v α v β ) ≥ 0. and ∆(M


213

But 0≥

∂ ˜ ∂ (Mαβ v α v β ) = (Mαβ v α v β + εeAt f ), ∂t ∂t ˜ αβ v α v β ) − ∆(εeAt f ) + ui ∇i (M ˜ αβ v α v β ) = ∆(M − ui ∇i (εeAt f ) + Nαβ v α v β + εAeAt0 f (x0 )

≥ −CεeAt0 f (x0 ) + εAeAt0 f (x0 ) > 0 when A is chosen sufficiently large. This is a contradiction. By applying Lemma 2.1.3 to the evolution equation ∂ # 2 Mαβ = ∆Mαβ + Mαβ + Mαβ ∂t

of the curvature operator Mαβ , we immediately obtain the following important result. Proposition 2.1.4 (Hamilton [59]). Nonnegativity of the curvature operator Mαβ is preserved by the Ricci flow. In the Kähler case, the nonnegativity of the holomorpic bisectional curvature is preserved under the Kähler-Ricci flow. This result is proved by Bando [5] for complex dimension n = 3 and by Mok [92] for general dimension n when the manifold is compact, and by Shi [115] when the manifold is noncompact. Proposition 2.1.5. Under the K¨ ahler-Ricci flow if the initial metric has positive (nonnegative) holomorphic bisectional curvature then the evolved metric also has positive (nonnegative) holomorphic bisectional curvature. 2.2. Strong Maximum Principle. Let Ω be a bounded, connected open set of a complete n-dimensional manifold M , and let gij (x, t) be a smooth solution to the Ricci flow on Ω × [0, T ]. Consider a vector bundle V over Ω with a fixed metric hαβ (independent of time), and a connection ∇ = {Γα iβ } which is compatible with hαβ and may vary with time t. Let Γ(V ) be the vector space of C ∞ sections of V over Ω. The Laplacian ∆ acting on a section σ ∈ Γ(V ) is defined by ∆σ = g ij (x, t)∇i ∇j σ. Consider a family of smooth symmetric bilinear forms Mαβ evolving by (2.2.1)

∂ Mαβ = ∆Mαβ + Nαβ , ∂t

on Ω × [0, T ],

where Nαβ = P (Mαβ , hαβ ) is a polynomial in Mαβ formed by contracting products of Mαβ with itself using the metric hαβ and satisfies Nαβ ≥ 0, whenever Mαβ ≥ 0. The following result, due to Hamilton [59], shows that the solution of (2.2.1) satisfies a strong maximum principle. Theorem 2.2.1 (Hamilton’s strong maximum principle). Let Mαβ be a smooth solution of the equation (2.2.1). Suppose Mαβ ≥ 0 on Ω × [0, T ]. Then there exists a positive constant 0 < δ ≤ T such that on Ω × (0, δ), the rank of Mαβ is constant, and

214


the null space of Mαβ is invariant under parallel translation and invariant in time and also lies in the null space of Nαβ . Proof. Set l = max{rank of Mαβ (x, 0)}. x∈Ω

Then we can find a nonnegative smooth function ρ(x), which is positive somewhere and has compact support in Ω, so that at every point x ∈ Ω, n−l+1 X i=1

Mαβ (x, 0)viα viβ ≥ ρ(x)

for any (n − l + 1) orthogonal unit vectors {v1 , . . . , vn−l+1 } at x. Let us evolve ρ(x) by the heat equation ∂ ρ = ∆ρ ∂t with the Dirichlet condition ρ|∂Ω = 0 to get a smooth function ρ(x, t) defined on Ω×[0, T ]. By the standard strong maximum principle, we know that ρ(x, t) is positive everywhere in Ω for all t ∈ (0, T ]. For every ε > 0, we claim that at every point (x, t) ∈ Ω × [0, T ], there holds n−l+1 X

Mαβ (x, t)viα viβ + εet > ρ(x, t)

i=1

for any (n − l + 1) orthogonal unit vectors {v1 , . . . , vn−l+1 } at x. We argue by contradiction. Suppose not, then for some ε > 0, there will be a first time t0 > 0 and some (n − l + 1) orthogonal unit vectors {v1 , . . . , vn−l+1 } at some point x0 ∈ Ω so that n−l+1 X

Mαβ (x0 , t0 )viα viβ + εet0 = ρ(x0 , t0 )

i=1

Let us extend each vi (i = 1, . . . , n − l + 1) to a local vector field, independent of t, in a neighborhood of x0 by parallel translation along geodesics (with respect to the metric gij (t0 )) emanating radially out of x0 . Clearly {v1 , . . . , vn−l+1 } remain orthogonal unit vectors in the neighborhood. Then, at (x0 , t0 ), we have ∂ ∂t and ∆

n−l+1 X

Mαβ viα viβ

+ εe − ρ

Mαβ viα viβ

t

i=1

n−l+1 X i=1

t

!

+ εe − ρ

!

≤ 0, ≥ 0.


But, since Nαβ ≥ 0 by our assumption, we have n−l+1 X

∂ 0≥ ∂t =

i=1

=

t

+ εe − ρ

!

n−l+1 X i=1

≥

Mαβ viα viβ

215

(∆Mαβ + Nαβ )viα viβ + εet − ∆ρ

n−l+1 X i=1

n−l+1 X i=1

∆(Mαβ viα viβ ) + εet − ∆ρ ∆(Mαβ viα viβ + εet − ρ) + εet

≥ εet > 0. This is a contradiction. Thus by letting ε → 0, we prove that n−l+1 X i=1

Mαβ (x, t)viα viβ ≥ ρ(x, t)

for any (n − l + 1) orthogonal unit vectors {v1 , . . . , vn−l+1 } at x ∈ Ω and t ∈ [0, T ]. Hence Mαβ has at least rank l everywhere in the open set Ω for all t ∈ (0, T ]. Therefore we can find a positive constant δ(≤ T ) such that the rank Mαβ is constant over Ω × (0, δ). Next we proceed to analyze the null space of Mαβ . Let v be any smooth section of V in the null of Mαβ on 0 < t < δ. Then ∂ (Mαβ v α v β ) ∂t ∂ ∂v β = Mαβ v α v β + 2Mαβ v α ∂t ∂t ∂ Mαβ v α v β , = ∂t

0=

and 0 = ∆(Mαβ v α v β ) = (∆Mαβ )v α v β + 4g kl ∇k Mαβ · v α ∇l v β

+ 2Mαβ g kl ∇k v α · ∇l v β + 2Mαβ v α ∆v β

= (∆Mαβ )v α v β + 4g kl ∇k Mαβ · v α ∇l v β + 2Mαβ g kl ∇k v α · ∇l v β . By noting that 0 = ∇k (Mαβ v β ) = (∇k Mαβ )v α + Mαβ ∇k v α and using the evolution equation (2.2.1), we get Nαβ v α v β + 2Mαβ g kl ∇k v α · ∇l v β = 0.

216


Since Mαβ ≥ 0 and Nαβ ≥ 0, we must have v ∈ null (Nαβ )

and ∇i v ∈ null (Mαβ ),

for all i.

The first inclusion shows that null (Mαβ ) ⊂ null (Nαβ ), and the second inclusion shows that null (Mαβ ) is invariant under parallel translation. To see null (Mαβ ) is also invariant in time, we first note that ∆v = ∇i (∇i v) ∈ null (Mαβ ) and then g kl ∇k Mαβ · ∇l v α = g kl ∇k (Mαβ ∇l v α ) − Mαβ ∆v α = 0. Thus we have 0 = ∆(Mαβ v α ) = (∆Mαβ )v α + 2g kl ∇k Mαβ · ∇l v α + Mαβ ∆v α

= (∆Mαβ )v α , and hence 0=

∂ (Mαβ v α ) ∂t

= (∆Mαβ + Nαβ )v α + Mαβ = Mαβ

∂v α . ∂t

∂v α ∂t

This shows that ∂v ∈ null (Mαβ ), ∂t so the null space of Mαβ is invariant in time. We now apply Hamilton’s strong maximum principle to the evolution equation of the curvature operator Mαβ . Recall ∂Mαβ # 2 = ∆Mαβ + Mαβ + Mαβ ∂t # where Mαβ = Cαξγ Cβηθ Mξη Mγθ . Suppose we have a solution to the Ricci flow with nonnegative curvature operator. Then by Theorem 2.2.1, the null space of the curvature operator Mαβ of the solution has constant rank and is invariant in time and under parallel translation over some time interval 0 < t < δ . Moreover the null space # of Mαβ must also lie in the null space of Mαβ . Denote by (n − k) the rank of Mαβ on 0 < t < δ. Let us diagonalize Mαβ so that # Mαα = 0 if α ≤ k and Mαα > 0 if α > k. Then we have Mαα = 0 also for α ≤ k from the evolution equation of Mαα . Since # 0 = Mαα = Cαξγ Cαηθ Mξη Mγθ ,


217

it follows that Cαξγ = hv α , [v ξ , v γ ]i = 0, if α ≤ k and ξ, γ > k. This says that the image of Mαβ is a Lie subalgebra (in fact it is the subalgebra of the restricted holonomy group by using the Ambrose-Singer holonomy theorem [3]). This proves the following result. Theorem 2.2.2 (Hamilton [59]). Suppose the curvature operator Mαβ of the initial metric is nonnegative. Then, under the Ricci flow, for some interval 0 < t < δ the image of Mαβ is a Lie subalgebra of so(n) which has constant rank and is invariant under parallel translation and invariant in time. 2.3. Advanced Maximum Principle for Tensors. In this section we present Hamilton’s advanced maximum principle for tensors which generalizes Lemma 2.1.3 and shows how a tensor evolving by a nonlinear heat equation may be controlled by a system of ODEs. An important application of the advanced maximum principle is the Hamilton-Ivey curvature pinching estimate for the Ricci flow on three-manifolds given in the next section. More applications will be given in Chapter 5. Let M be a complete manifold equipped with a one-parameter family of Riemannian metrics gij (t), 0 ≤ t ≤ T , with T < +∞. Let V → M be a vector bundle with a time-independent bundle metric hab and Γ(V ) be the vector space of C ∞ sections of V . Let ∇t :

Γ(V ) → Γ(V ⊗ T ∗ M ),

t ∈ [0, T ]

be a smooth family of time-dependent connections compatible with hab , i.e. ∆

(∇t )i hab = (∇t )

∂ ∂xi

hab = 0,

for any local coordinate { ∂x∂ 1 , . . . , ∂x∂ n }. The Laplacian ∆t acting on a section σ ∈ Γ(V ) is defined by ∆t σ = g ij (x, t)(∇t )i (∇t )j σ. For the application to the Ricci flow, we will always assume that the metrics gij (·, t) evolve by the Ricci flow. Since M may be noncompact, we assume that, for the sake of simplicity, the curvature of gij (t) is uniformly bounded on M × [0, T ]. Let N : V × [0, T ] → V be a fiber preserving map, i.e., N (x, σ, t) is a timedependent vector field defined on the bundle V and tangent to the fibers. We assume that N (x, σ, t) is continuous in x, t and satisfies |N (x, σ1 , t) − N (x, σ2 , t)| ≤ CB |σ1 − σ2 | for all x ∈ M , t ∈ [0, T ] and |σ1 | ≤ B, |σ2 | ≤ B, where CB is a positive constant depending only on B. Then we can form the nonlinear heat equation (PDE)

∂ σ(x, t) = ∆t σ(x, t) + ui (∇t )i σ(x, t) + N (x, σ(x, t), t) ∂t

where ui = ui (t) is a time-dependent vector field on M which is uniformly bounded on M × [0, T ]. Let K be a closed subset of V . One important question is under what conditions will solutions of the PDE which start in K remain in K. To answer this question, Hamilton [59] imposed the following two conditions on K:

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(H1) K is invariant under parallel translation defined by the connection ∇t for each t ∈ [0, T ]; ∆ (H2) in each fiber Vx , the set Kx = Vx ∩ K is closed and convex. Then one can judge the behavior of the PDE by comparing to that of the following ODE dσx = N (x, σx , t) dt

(ODE) for σx = σx (t) in each fiber Vx .

Theorem 2.3.1 (Hamilton’s advanced maximum principle [59]). Let K be a closed subset of V satisfying the hypothesis (H1) and (H2). Suppose that for any x ∈ M and any initial time t0 ∈ [0, T ), any solution σx (t) of the (ODE) which starts in Kx at t0 will remain in Kx for all later times. Then for any initial time t0 ∈ [0, T ) the solution σ(x, t) of the (PDE) will remain in K for all later times provided σ(x, t) starts in K at time t0 and σ(x, t) is uniformly bounded with respect to the bundle metric hab on M × [t0 , T ]. We remark that Lemma 2.1.3 is a special case of the above theorem where V is given by a symmetric tensor product of a vector bundle and K corresponds to the convex set consisting of all nonnegative symmetric bilinear forms. We also remark that Hamilton [59] established the above theorem for a general evolving metric gij (x, t) which does not necessarily satisfy the Ricci flow. Before proving Theorem 2.3.1, we need to establish three lemmas. Let ϕ : [a, b] → R be a Lipschitz function. We consider dϕ dt (t) at t ∈ [a, b) in the sense of limsup of the forward difference quotients, i.e., dϕ ϕ(t + h) − ϕ(t) (t) = lim sup . dt h h→0+ Lemma 2.3.2. Suppose ϕ : [a, b] → R is Lipschitz continuous and suppose for some constant C < +∞, d ϕ(t) ≤ Cϕ(t), dt and

whenever ϕ(t) ≥ 0 on [a, b),

ϕ(a) ≤ 0.

Then ϕ(t) ≤ 0 on [a, b]. Proof. By replacing ϕ by e−Ct ϕ, we may assume

and

d ϕ(t) ≤ 0, dt ϕ(a) ≤ 0.

whenever ϕ(t) ≥ 0 on [a, b),

For arbitrary ε > 0, we shall show ϕ(t) ≤ ε(t − a) on [a, b]. Clearly we may assume ϕ(a) = 0. Since lim sup h→0+

ϕ(a + h) − ϕ(a) ≤ 0, h

there must be some interval a ≤ t < δ on which ϕ(t) ≤ ε(t − a).


219

Let a ≤ t < c be the largest interval with c ≤ b such that ϕ(t) ≤ ε(t − a) on [a, c). Then by continuity ϕ(t) ≤ ε(t − a) on the closed interval [a, c]. We claim that c = b. Suppose not, then we can find δ > 0 such that ϕ(t) ≤ ε(t − a) on [a, c + δ] since lim sup h→0+

ϕ(c + h) − ϕ(c) ≤ 0. h

This contradicts the choice of the largest interval [a, c). Therefore, since ε > 0 can be arbitrary small, we have proved ϕ(t) ≤ 0 on [a, b].

The second lemma below is a general principle on the derivative of a sup-function which will bridge solutions between ODEs and PDEs. Let X be a complete smooth manifold and Y be a compact subset of X. Let ψ(x, t) be a smooth function on X × [a, b] and let ϕ(t) = sup{ψ(y, t) | y ∈ Y }. Then it is clear that ϕ(t) is Lipschitz continuous. We have the following useful estimate on its derivative. Lemma 2.3.3. d ϕ(t) ≤ sup dt

∂ψ (y, t) | y ∈ Y satisfies ψ(y, t) = ϕ(t) . ∂t

Proof. Choose a sequence of times {tj } decreasing to t for which lim

tj →t

dϕ(t) ϕ(tj ) − ϕ(t) = . tj − t dt

Since Y is compact, we can choose yj ∈ Y with ϕ(tj ) = ψ(yj , tj ). By passing to a subsequence, we can assume yj → y for some y ∈ Y . By continuity, we have ϕ(t) = ψ(y, t). It follows that ψ(yj , t) ≤ ψ(y, t), and then ϕ(tj ) − ϕ(t) ≤ ψ(yj , tj ) − ψ(yj , t) =

∂ ψ(yj , t˜j ) · (tj − t) ∂t

for some t˜j ∈ [t, tj ] by the mean value theorem. Thus we have lim

tj →t

ϕ(tj ) − ϕ(t) ∂ ≤ ψ(y, t). tj − t ∂t

This proves the result. We remark that the above two lemmas are somewhat standard facts in the theory of PDEs and we have implicitly used them in the previous sections when we apply the maximum principle. The third lemma gives a characterization of when a system of ODEs preserve closed convex sets in Euclidean space. Let Z ⊂ Rn be a closed convex subset. We define the tangent cone Tϕ Z to the closed convex set Z at a point ϕ ∈ ∂Z as the smallest closed convex cone with vertex at ϕ which contains Z. Lemma 2.3.4. Let U ⊂ Rn be an open set and Z ⊂ U be a closed convex subset. Consider the ODE (2.3.1)

dϕ = N (ϕ, t) dt

where N : U × [0, T ] → Rn is continuous and Lipschitz in ϕ. Then the following two statements are equivalent.

220


(i) For any initial time t0 ∈ [0, T ], any solution of the ODE (2.3.1) which starts in Z at t0 will remain in Z for all later times; (ii) ϕ + N (ϕ, t) ∈ Tϕ Z for all ϕ ∈ ∂Z and t ∈ [0, T ). Proof. We say that a linear function l on Rn is a support function for Z at ϕ ∈ ∂Z and write l ∈ Sϕ Z if |l| = 1 and l(ϕ) ≥ l(η) for all η ∈ Z. Then ϕ+N (ϕ, t) ∈ Tϕ Z if and only if l(N (ϕ, t)) ≤ 0 for all l ∈ Sϕ Z. Suppose l(N (ϕ, t)) > 0 for some ϕ ∈ ∂Z and some l ∈ Sϕ Z. Then d dϕ l(ϕ) = l = l(N (ϕ, t)) > 0, dt dt so l(ϕ) is strictly increasing and the solution ϕ(t) of the ODE (2.3.1) cannot remain in Z. To see the converse, first note that we may assume Z is compact. This is because we can modify the vector field N (ϕ, t) by multiplying a cutoff function which is everywhere nonnegative, equals one on a large ball and equals zero on the complement of a larger ball. The paths of solutions of the ODE are unchanged inside the first large ball, so we can intersect Z with the second ball to make Z convex and compact. If there were a counterexample before the modification there would still be one after as we chose the first ball large enough. Let s(ϕ) be the distance from ϕ to Z in Rn . Clearly s(ϕ) = 0 if ϕ ∈ Z. Then s(ϕ) = sup{l(ϕ − η) | η ∈ ∂Z and l ∈ Sη Z}. The sup is taken over a compact subset of Rn × Rn . Hence by Lemma 2.3.3 d s(ϕ) ≤ sup{l(N (ϕ, t)) | η ∈ ∂Z, l ∈ Sη Z and s(ϕ) = l(ϕ − η)}. dt It is clear that the sup on the RHS of the above inequality can be takeen only when η is the unique closest point in Z to ϕ and l is the linear function of length one with gradient in the direction of ϕ − η. Since N (ϕ, t) is Lipschitz in ϕ and continuous in t, we have |N (ϕ, t) − N (η, t)| ≤ C|ϕ − η| for some constant C and all ϕ and η in the compact set Z. By hypothesis (ii), l(N (η, t)) ≤ 0, and for the unique η, the closest point in Z to ϕ, |ϕ − η| = s(ϕ). Thus d s(ϕ) ≤ sup dt

(

≤ Cs(ϕ).

l(N (η, t)) + |l(N (ϕ, t)) − l(N (η, t))| l ∈ Sη Z,

| η ∈ ∂Z,

and s(ϕ) = l(ϕ − η)

)


221

Since s(ϕ) = 0 to start at t0 , it follows from Lemma 2.3.2 that s(ϕ) = 0 for t ∈ [t0 , T ]. This proves the lemma. We are now ready to prove Theorem 2.3.1. Proof of Theorem 2.3.1. Since the solution σ(x, t) of the (PDE) is uniformly bounded with respect to the bundle metric hab on M × [t0 , T ] by hypothesis, we may assume that K is contained in a tubular neighborhood V (r) of the zero section in V whose intersection with each fiber Vx is a ball of radius r around the origin measured by the bundle metric hab for some large r > 0. Recall that gij (·, t), t ∈ [0, T ], is a smooth solution to the Ricci flow with uniformly bounded curvature on M × [0, T ]. From Lemma 2.1.1, we have a smooth function f such that f ≥ 1 everywhere, f (x) → +∞ as d0 (x, x0 ) → +∞ for some fixed point x0 ∈ M , and the first and second covariant derivatives with respect to the metrics gij (·, t) are uniformly bounded on M × [0, T ]. Using the metric hab in each fiber Vx and writing |ϕ − η| for the distance between ϕ ∈ Vx and η ∈ Vx , we set ∆

s(t) = sup {inf{|σ(x, t) − η| | η ∈ Kx = K ∩ Vx } − ǫeAt f (x)} x∈M

where ǫ is an arbitrarily small positive number and A is a positive constant to be determined. We rewrite the function s(t) as s(t) = sup{l(σ(x, t) − η) − ǫeAt f (x) | x ∈ M, η ∈ ∂Kx and l ∈ Sη Kx }. By the construction of the function f , we see that the sup is taken in a compact subset of M × V × V ∗ for all t. Then by Lemma 2.3.3, ∂ ds(t) ≤ sup l(σ(x, t) − η) − ǫAeAt f (x) (2.3.2) dt ∂t where the sup is over all x ∈ M, η ∈ ∂Kx and l ∈ Sη Kx such that l(σ(x, t) − η) − ǫeAt f (x) = s(t); in particular we have |σ(x, t) − η| = l(σ(x, t) − η), where η is the unique closest point in Kx to σ(x, t), and l is the linear function of length one on the fiber Vx with gradient in the direction of η to σ(x, t). We compute at these (x, η, l), (2.3.3)

∂ l(σ(x, t) − η) − ǫAeAt f (x) ∂t ∂σ(x, t) − ǫAeAt f (x) =l ∂t

= l(∆t σ(x, t)) + l(ui (x, t)(∇t )i σ(x, t)) + l(N (x, σ(x, t), t)) − ǫAeAt f (x).

By the assumption and Lemma 2.3.4 we have η + N (x, η, t) ∈ Tη Kx . Hence, for those (x, η, l), l(N (x, η, t)) ≤ 0 and then (2.3.4)

l(N (x, σ(x, t), t)) ≤ l(N (x, η, t)) + |N (x, σ(x, t), t) − N (x, η, t)| ≤ C|σ(x, t) − η| = C(s(t) + ǫeAt f (x))

222


for some positive constant C by the assumption that N (x, σ, t) is Lipschitz in σ and the fact that the sup is taken on a compact set. Thus the combination of (2.3.2)–(2.3.4) gives (2.3.5)

ds(t) ≤ l(∆t σ(x, t)) + l(ui (x, t)(∇t )i σ(x, t)) + Cs(t) + ǫ(C − A)eAt f (x) dt

for those x ∈ M, η ∈ ∂Kx and l ∈ Sη Kx such that l(σ(x, t) − η) − ǫeAt f (x) = s(t). Next we estimate the first two terms of (2.3.5). As we extend a vector in a bundle from a point x by parallel translation along geodesics emanating radially out of x, we will get a smooth section of the bundle in some small neighborhood of x such that all the symmetrized covariant derivatives at x are zero. Now let us extend η ∈ Vx and l ∈ Vx∗ in this manner. Clearly, we continue to have |l|(·) = 1. Since K is invariant under parallel translations, we continue to have η(·) ∈ ∂K and l(·) as a support function for K at η(·). Therefore l(σ(·, t) − η(·)) − ǫeAt f (·) ≤ s(t) in the neighborhood. It follows that the function l(σ(·, t) − η(·)) − ǫeAt f (·) has a local maximum at x, so at x (∇t )i (l(σ(x, t) − η) − ǫeAt f (x)) = 0,

and ∆t (l(σ(x, t) − η) − ǫeAt f (x)) ≤ 0. Hence at x l((∇t )i σ(x, t)) − ǫeAt (∇t )i f (x) = 0,

and l(∆t σ(x, t)) − ǫeAt ∆t f (x) ≤ 0. Therefore by combining with (2.3.5), we have

d s(t) ≤ Cs(t) + ǫ(∆t f (x) + ui (∇t )i f (x) + (C − A)f (x))eAt dt ≤ Cs(t) for A > 0 large enough, since f (x) ≥ 1 and the first and second covariant derivatives of f are uniformly bounded on M × [0, T ]. So by applying Lemma 2.3.2 and the arbitrariness of ǫ, we have completed the proof of Theorem 2.3.1. Finally, we would like to state a useful generalization of Theorem 2.3.1 by Chow and Lu in [40] which allows the set K to depend on time. One can consult the paper [40] for the proof. Theorem 2.3.5 (Chow and Lu [40]). Let K(t) ⊂ V , t ∈ [0, T ] be closed subsets which satisfy the following hypotheses (H3) K(t) is invariant under parallel translation defined by the connection ∇t for each t ∈ [0, T ]; ∆ (H4) in each fiber Vx , the set Kx (t) = K(t) ∩ Vx is nonempty, closed and convex for each t ∈ [0, T ]; S (H5) the space-time track (∂K(t) × {t}) is a closed subset of V × [0, T ]. t∈[0,T ]

223


Suppose that, for any x ∈ M and any initial time t0 ∈ [0, T ), and for any solution σx (t) of the (ODE) which starts in Kx (t0 ), the solution σx (t) will remain in Kx (t) for all later times. Then for any initial time t0 ∈ [0, T ) the solution σ(x, t) of the (PDE) will remain in K(t) for all later times if σ(x, t) starts in K(t0 ) at time t0 and the solution σ(x, t) is uniformly bounded with respect to the bundle metric hab on M × [t0 , T ]. 2.4. Hamilton-Ivey Curvature Pinching Estimate. The Hamilton-Ivey curvature pinching estimate roughly says that if a solution to the Ricci flow on a three-manifold becomes singular (i.e., the curvature goes to infinity) as time t approaches the maximal time T , then the most negative sectional curvature will be small compared to the most positive sectional curvature. This pinching estimate plays a crucial role in analyzing the formation of singularities in the Ricci flow on three-manifolds. Consider a complete solution to the Ricci flow ∂ gij = −2Rij ∂t on a complete three-manifold with bounded curvature in space for each time t ≥ 0. Recall from Section 1.3 that the evolution equation of the curvature operator Mαβ is given by (2.4.1)

∂ # 2 Mαβ = ∆Mαβ + Mαβ + Mαβ ∂t

2 where Mαβ is the operator square 2 Mαβ = Mαγ Mβγ # and Mαβ is the Lie algebra so(n) square # Mαβ = Cαγζ Cβηθ Mγη Mζθ . # In dimension n = 3, we know that Mαβ is the adjoint matrix of Mαβ . If we diagonalize Mαβ with eigenvalues λ ≥ µ ≥ ν so that   λ , µ (Mαβ ) =  ν # 2 then Mαβ and Mαβ are also diagonal, with  2   λ µν 2  and (M # ) =  µ2 (Mαβ )= αβ ν2

λν λµ



.

Thus the ODE corresponding to PDE (2.4.1) for Mαβ (in the space of 3 × 3 matrices) is given by the following system  d 2   dt λ = λ + µν,    d 2 (2.4.2) dt µ = µ + λν,      d 2 dt ν = ν + λµ.

224


Let P be the principal bundle of the manifold and form the associated bundle V = P ×G E, where G = O(3) and E is the vector space of symmetric bilinear forms on so(3). The curvature operator Mαβ is a smooth section of V = P ×G E. According to Theorem 2.3.1, any closed convex set of curvature operator matrices Mαβ which is O(3)-invariant (and hence invariant under parallel translation) and preserved by ODE (2.4.2) is also preserved by the Ricci flow. We are now ready to state and prove the Hamilton-Ivey pinching estimate . Theorem 2.4.1 (Hamilton [63], Ivey [73]). Suppose we have a solution to the Ricci flow on a three-manifold which is complete with bounded curvature for each t ≥ 0. Assume at t = 0 the eigenvalues λ ≥ µ ≥ ν of the curvature operator at each point are bounded below by ν ≥ −1. The scalar curvature R = λ + µ + ν is their sum. Then at all points and all times t ≥ 0 we have the pinching estimate R ≥ (−ν)[log(−ν) − 3], whenever ν < 0. Proof. Consider the function y = f (x) = x(log x − 3) defined on e2 ≤ x < +∞. It is easy to check that f is increasing and convex with range −e2 ≤ y < +∞. Let f −1 (y) = x be the inverse function, which is also increasing but concave and satisfies (2.4.3)

f −1 (y) =0 y→∞ y lim

Consider also the set K of matrices Mαβ defined by the inequalities   λ + µ + ν ≥ −3, (2.4.4) K:  ν + f −1 (λ + µ + ν) ≥ 0.

By Theorem 2.3.1 and the assumptions in Theorem 2.4.1 at t = 0, we only need to check that the set K defined above is closed, convex and preserved by the ODE (2.4.2). Clearly K is closed because f −1 is continuous. λ + µ + ν is just the trace function of 3 × 3 matrices which is a linear function. Hence the first inequality in (2.4.4) defines a linear half-space, which is convex. The function ν is the least eigenvalue function, which is concave. Also note that f −1 is concave. Thus the second inequality in (2.4.4) defines a convex set as well. Therefore we proved K is closed and convex. Under the ODE (2.4.2) d (λ + µ + ν) = λ2 + µ2 + ν 2 + λµ + λν + µν dt 1 = [(λ + µ)2 + (λ + ν)2 + (µ + ν)2 ] 2 ≥ 0. Thus the first inequality in (2.4.4) is preserved by the ODE.


225

The second inequality in (2.4.4) can be written as whenever ν ≤ −e2 ,

λ + µ + ν ≥ f (−ν), which becomes (2.4.5)

whenever ν ≤ −e2 .

λ + µ ≥ (−ν)[log(−ν) − 2],

To show the inequality is preserved we only need to look at points on the boundary of the set. If ν + f −1 (λ + µ + ν) = 0 then ν = −f −1 (λ + µ + ν) ≤ −e2 since f −1 (y) ≥ e2 . Hence the RHS of (2.4.5) is nonnegative. We thus have λ ≥ 0 because λ ≥ µ. But µ may have either sign. We split our consideration into two cases: Case (i): µ ≥ 0. We need to verify dλ dµ d(−ν) + ≥ (log(−ν) − 1) dt dt dt when λ + µ = (−ν)[log(−ν) − 2]. Solving for log(−ν) − 2 =

λ+µ (−ν)

and substituting above, we must show 2

2

λ + µν + µ + λν ≥

λ+µ + 1 (−ν 2 − λµ) (−ν)

which is equivalent to (λ2 + µ2 )(−ν) + λµ(λ + µ + (−ν)) + (−ν)3 ≥ 0. Since λ, µ and (−ν) are all nonnegative we are done in the first case. Case (ii): µ < 0. We need to verify dλ d(−µ) d(−ν) ≥ + (log(−ν) − 1) dt dt dt when λ = (−µ) + (−ν)[log(−ν) − 2]. Solving for log(−ν) − 2 =

λ − (−µ) (−ν)

and substituting above, we need to show λ − (−µ) 2 2 λ + µν ≥ −µ − λν + + 1 (−ν 2 − λµ) (−ν) or λ2 + (−µ)(−ν) ≥ λ(−ν) − (−µ)2 +

λ − (−µ) + 1 (λ(−µ) − (−ν)2 ) (−ν)

226


which reduces to λ2 (−ν) + λ(−µ)2 + (−µ)2 (−ν) + (−ν)3 ≥ λ2 (−µ) + λ(−µ)(−ν) or equivalently (λ2 − λ(−µ) + (−µ)2 )((−ν) − (−µ)) + (−µ)3 + (−ν)3 ≥ 0. Since λ2 − λ(−µ) + (−µ)2 ≥ 0 and (−ν) − (−µ) ≥ 0 we are also done in the second case. Therefore the proof is completed. 2.5. Li-Yau-Hamilton Estimates. In [82], Li-Yau developed a fundamental gradient estimate, now called Li-Yau estimate, for positive solutions to the heat equation on a complete Riemannian manifold with nonnegative Ricci curvature. They used it to derive the Harnack inequality for such solutions by path integration. Then based on the suggestion of Yau, Hamilton [60] developed a similar estimate for the scalar curvature of solutions to the Ricci flow on a Riemann surface with positive curvature, and later obtained a matrix version of the Li-Yau estimate for solutions to the Ricci flow with positive curvature operator in all dimensions. This matrix version of the Li-Yau estimate is the Li-Yau-Hamilton estimate, which we will present in this section. The Li-Yau-Hamilton estimate plays a central role in the analysis of formation of singularities and the application of the Ricci flow to three-manifold topology. We have seen that in the Ricci flow the curvature tensor satisfies a nonlinear heat equation, and the nonnegativity of the curvature operator is preserved by the Ricci flow. Roughly speaking the Li-Yau-Hamilton estimate says the nonnegativity of a certain combination of the derivatives of the curvature up to second order is also preserved by the Ricci flow. Let us begin by describing the Li-Yau estimate for positive solutions to the heat equation on a complete Riemannian manifold with nonnegative Ricci curvature. Theorem 2.5.1 (Li-Yau [82]). Let (M, gij ) be an n-dimensional complete Riemannian manifold with nonnegative Ricci curvature. Let u(x, t) be any positive solution to the heat equation ∂u = ∆u ∂t

on M × [0, ∞).

Then we have (2.5.1)

∂u |∇u|2 n − + u≥0 ∂t u 2t

on

M × (0, ∞).

We remark that one can in fact prove the following quadratic version that for any vector field V i on M , (2.5.2)

n ∂u + 2∇u · V + u|V |2 + u ≥ 0. ∂t 2t

If we take the optimal vector field V = −∇u/u, we recover the inequality (2.5.1). Now we consider the Ricci flow on a Riemann surface. Since in dimension two the Ricci curvature is given by Rij =

1 Rgij , 2


227

the Ricci flow (1.1.5) becomes ∂gij = −Rgij . ∂t

(2.5.3)

Now let gij (x, t) be a complete solution of the Ricci flow (2.5.3) on a Riemann surface M and 0 ≤ t < T . Then the scalar curvature R(x, t) evolves by the semilinear equation ∂R = △R + R2 ∂t on M × [0, T ). Suppose the scalar curvature of the initial metric is bounded, nonnegative everywhere and positive somewhere. Then it follows from Proposition 2.1.2 that the scalar curvature R(x, t) of the evolving metric remains nonnegative. Moreover, from the standard strong maximum principle (which works in each local coordinate neighborhood), the scalar curvature is positive everywhere for t > 0. In [60], Hamilton obtained the following Li-Yau estimate for the scalar curvature R(x, t). Theorem 2.5.2 (Hamilton [60]). Let gij (x, t) be a complete solution of the Ricci flow on a surface M . Assume the scalar curvature of the initial metric is bounded, nonnegative everywhere and positive somewhere. Then the scalar curvature R(x, t) satisfies the Li-Yau estimate R ∂R |∇R|2 − + ≥ 0. ∂t R t

(2.5.4)

Proof. By the above discussion, we know R(x, t) > 0 for t > 0. If we set L = log R(x, t)

for t > 0,

then 1 ∂ L = (△R + R2 ) ∂t R = △L + |∇L|2 + R and (2.5.4) is equivalent to ∂L 1 1 − |∇L|2 + = △L + R + ≥ 0. ∂t t t Following Li-Yau [82] in the linear heat equation case, we consider the quantity (2.5.5)

Q=

∂L − |∇L|2 = △L + R. ∂t

Then by a direct computation, ∂Q ∂ = (△L + R) ∂t ∂t ∂L ∂R =△ + R△L + ∂t ∂t

= △Q + 2∇L · ∇Q + 2|∇2 L|2 + 2R(△L) + R2 ≥ △Q + 2∇L · ∇Q + Q2 .

228


So we get ∂ 1 1 1 1 1 Q+ ≥△ Q+ + 2∇L · ∇ Q + + Q− Q+ . ∂t t t t t t Hence by a similar maximum principle argument as in the proof of Lemma 2.1.3, we obtain Q+

1 ≥ 0. t

This proves the theorem. As an immediate consequence, we obtain the following Harnack inequality for the scalar curvature R by taking the Li-Yau type path integral as in [82]. Corollary 2.5.3 (Hamilton [60]). Let gij (x, t) be a complete solution of the Ricci flow on a surface with bounded and nonnegative scalar curvature. Then for any points x1 , x2 ∈ M , and 0 < t1 < t2 , we have R(x2 , t2 ) ≥

t1 −dt (x1 ,x2 )2 /4(t2 −t1 ) e 1 R(x1 , t1 ). t2

Proof. Take the geodesic path γ(τ ), τ ∈ [t1 , t2 ], from x1 to x2 at time t1 with constant velocity dt1 (x1 , x2 )/(t2 − t1 ). Consider the space-time path η(τ ) = (γ(τ ), τ ), τ ∈ [t1 , t2 ]. We compute log

R(x2 , t2 ) = R(x1 , t1 ) =

Z

Z

t2

t1 t2

t1

≥

Z

t2

t1

d L(γ(τ ), τ )dτ dτ dγ 1 ∂R + ∇R · dτ R ∂τ dτ ! 2 ∂L 1 dγ dτ. − |∇L|2gij (τ ) − ∂τ 4 dτ gij (τ )

Then by Theorem 2.5.2 and the fact that the metric is shrinking (since the scalar curvature is nonnegative), we have ! 2 Z t2 1 1 dγ R(x2 , t2 ) dτ ≥ − − log R(x1 , t1 ) τ 4 dτ gij (τ ) t1 = log

dt (x1 , x2 )2 t1 − 1 t2 4(t2 − t1 )

After exponentiating above, we obtain the desired Harnack inequality. To prove a similar inequality as (2.5.4) for the scalar curvature of solutions to the Ricci flow in higher dimensions is not so simple. First of all, we will need to require nonnegativity of the curvature operator (which we know is preserved under the Ricci flow). Secondly, one does not get inequality (2.5.4) directly, but rather indirectly as the trace of certain matrix estimate. The key ingredient in formulating this matrix version is to derive some identities from the soliton solutions and prove an elliptic


229

inequality based on these quantities. Hamilton found such a general principle which was based on the idea of Li-Yau [82] when an identity is checked on the heat kernel before an inequality was found. To illustrate this point, let us first examine the heat equation case. Consider the heat kernel u(x, t) = (4πt)−n/2 e−|x|

2

/4t

for the standard heat equation on Rn which can be considered as an expanding soliton solution. Differentiating the function u, we get xj or ∇j u + uVj = 0, (2.5.6) ∇j u = −u 2t where xj ∇j u Vj = =− . 2t u Differentiating (2.5.6), we have (2.5.7)

∇i ∇j u + ∇i uVj +

u δij = 0. 2t

To make the expression in (2.5.7) symmetric in i, j, we multiply Vi to (2.5.6) and add to (2.5.7) and obtain u (2.5.8) ∇i ∇j u + ∇i uVj + ∇j uVi + uVi Vj + δij = 0. 2t Taking the trace in (2.5.8) and using the equation ∂u/∂t = ∆u, we arrive at ∂u n + 2∇u · V + u|V |2 + u = 0, ∂t 2t which shows that the Li-Yau inequality (2.5.1) becomes an equality on our expanding soliton solution u! Moreover, we even have the matrix identity (2.5.8). Based on the above observation and using a similar process, Hamilton found a matrix quantity, which vanishes on expanding gradient Ricci solitons and is nonnegative for any solution to the Ricci flow with nonnegative curvature operator. Now we describe the process of finding the Li-Yau-Hamilton quadratic for the Ricci flow in arbitrary dimension. Consider a homothetically expanding gradient soliton g, we have 1 gab = ∇a Vb 2t in the orthonormal frame coordinate chosen as in Section 1.3. Here Vb = ∇b f for some function f . Differentiating (2.5.9) and commuting give the first order relations

(2.5.9)

(2.5.10)

Rab +

∇a Rbc − ∇b Rac = ∇a ∇b Vc − ∇b ∇a Vc = Rabcd Vd ,

and differentiating again, we get ∇a ∇b Rcd − ∇a ∇c Rbd = ∇a (Rbcde Ve ) = ∇a Rbcde Ve + Rbcde ∇a Ve = ∇a Rbcde Ve + Rae Rbcde +

1 Rbcda . 2t

230


We further take the trace of this on a and b to get ∆Rcd − ∇a ∇c Rad − Rae Racde +

1 Rcd − ∇a Racde Ve = 0, 2t

and then by commuting the derivatives and second Bianchi identity, 1 1 ∆Rcd − ∇c ∇d R + 2Rcade Rae − Rce Rde + Rcd + (∇e Rcd − ∇d Rce )Ve = 0. 2 2t Let us define 1 1 Mab = ∆Rab − ∇a ∇b R + 2Racbd Rcd − Rac Rbc + Rab , 2 2t Pabc = ∇a Rbc − ∇b Rac . Then (2.5.11)

Mab + Pcba Vc = 0,

We rewrite (2.5.10) as Pabc = Rabcd Vd and then (2.5.12)

Pcab Vc + Racbd Vc Vd = 0.

Adding (2.5.11) and (2.5.12) we have Mab + (Pcab + Pcba )Vc + Racbd Vc Vd = 0 and then Mab Wa Wb + (Pcab + Pcba )Wa Wb Vc + Racbd Wa Vc Wb Vd = 0. If we write Uab =

1 (Va Wb − Vb Wa ) = V ∧ W, 2

then the above identity can be rearranged as (2.5.13)

∆

Q = Mab Wa Wb + 2Pabc Uab Wc + Rabcd Uab Ucd = 0.

This is the Li-Yau-Hamilton quadratic we look for. Note that the proof of the LiYau-Hamilton estimate below does not depend on the existence of such an expanding gradient Ricci soliton. It is only used as inspiration. Now we are ready to state the remarkable Li-Yau-Hamilton estimate for the Ricci flow. Theorem 2.5.4 (Hamilton [61]). Let gij (x, t) be a complete solution with bounded curvature to the Ricci flow on a manifold M for t in some time interval (0, T ) and suppose the curvature operator of gij (x, t) is nonnegative. Then for any one-form Wa and any two-form Uab we have Mab Wa Wb + 2Pabc Uab Wc + Rabcd Uab Ucd ≥ 0

231


on M × (0, T ). The proof of this theorem requires some rather intense calculations. Here we only give a sketch of the proof. For more details, we refer the reader to Hamilton’s original paper [61]. Sketch of the Proof. Let gij (x, t) be the complete solution with bounded and nonnegative curvature operator. Recall that in the orthonormal frame coordinate system, the curvatures evolve by  ∂   ∂t Rabcd = ∆Rabcd + 2(Babcd − Babdc − Badbc + Bacbd ),    ∂ ∂t Rab = ∆Rab + 2Racbd Rcd ,      ∂ 2 ∂t R = ∆R + 2|Ric| , where Babcd = Raebf Rcedf . By a long but straightforward computation from these evolution equations, one can get ∂ − ∆ Pabc = 2Radbe Pdec + 2Radce Pdbe + 2RbdcePade − 2Rde ∇d Rabce ∂t and

∂ − ∆ Mab = 2Racbd Mcd + 2Rcd (∇c Pdab + ∇c Pdba ) ∂t + 2Pacd Pbcd − 4Pacd Pbdc + 2Rcd Rce Radbe −

1 Rab . 2t2

Now consider ∆

Q = Mab Wa Wb + 2Pabc Uab Wc + Rabcd Uab Ucd . At a point where

(2.5.14)

1 ∂ ∂ − ∆ Wa = Wa , − ∆ Uab = 0, ∂t t ∂t

and (2.5.15)

∇a Wb = 0, ∇a Ubc =

1 1 (Rab Wc − Rac Wb ) + (gab Wc − gac Wb ), 2 4t

we have (2.5.16)

∂ − ∆ Q = 2Racbd Mcd Wa Wb − 2Pacd Pbdc Wa Wb ∂t

+ 8RadcePdbe Uab Wc + 4Raecf Rbedf Uab Ucd

+ (Pabc Wc + Rabcd Ucd )(Pabe We + Rabef Uef ). For simplicity we assume the manifold is compact and the curvature operator is strictly positive. (For the general case we shall mess the formula up a bit to sneak in the term ǫeAt f , as done in Lemma 2.1.3). Suppose not; then there will be a first time

232


when the quantity Q is zero, and a point where this happens, and a choice of U and W giving the null eigenvectors. We can extend U and W any way we like in space and time and still have Q ≥0, up to the critical time. In particular we can make the first derivatives in space and time to be anything we like, so we can extend first in space to make (2.5.15) hold at that point. And then, knowing ∆Wa and ∆Uab , we can extend in time to make (2.5.14) hold at that point and that moment. Thus we have (2.5.16) at the point. In the RHS of (2.5.16) the quadratic term (Pabc Wc + Rabcd Ucd )(Pabe We + Rabef Uef ) is clearly nonnegative. By similar argument as in the proof of Lemma 2.1.3, to get a contradiction we only need to show the remaining part in the RHS of (2.5.16) is also nonnegative. A nonnegative quadratic form can always be written as a sum of squares of linear forms. This is equivalent to diagonalizing a symmetric matrix and writing each nonnegative eigenvalue as a square. Write X n(n − 1) k Q= (Xak Wa + Yab Uab )2 , 1≤k ≤n+ . 2 k

This makes Mab =

X

Xak Xbk ,

Pabc =

k

X

k Yab Xck

k

and Rabcd =

X

k k Yab Ycd .

k

It is then easy to compute 2Racbd Mcd Wa Wb − 2Pacd Pbdc Wa Wb + 8Radce Pdbe Uab We + 4Raecf Rbedf Uab Ucd ! ! X X k k l l =2 Yac Ybd Xa Yc Wa Wb k

−2 +8

X

k Xdk Yac

k

X

k k Yad Yce

k

+4 =

X k,l

X k

k k Yae Ycf

!

!

!

l

X

l Ybd Xcl

l

X l

X l

l Ydb Xel

!

!

l Ybe Ydfl

Wa Wb

!

Uab Wc Uab Ucd

k l k l (Yac Xcl Wa − Yac Xck Wa − 2Yac Ybc Uab )2

≥ 0.

This says that the remaining part in the RHS of (2.5.16) is also nonnegative. Therefore we have completed the sketch of the proof.


233

By taking Uab = 12 (Va Wb − Vb Wa ) and tracing over Wa , we immediately get Corollary 2.5.5 (Hamilton [61]). For any one-form Va we have ∂R R + + 2∇a R · Va + 2Rab Va Vb ≥ 0. ∂t t In particular by taking V ≡ 0, we see that the function tR(x, t) is pointwise nondecreasing in time. By combining this property with the local derivative estimate of curvature, we have the following elliptic type estimate. Corollary 2.5.6. Suppose we have a solution to the Ricci flow for t > 0 which is complete with bounded curvature, and has nonnegative curvature operator. Suppose also that at some time t > 0 we have the scalar curvature R ≤ M for some constant M in the ball of radius r around some point p. Then for k = 1, 2, . . ., the k th order derivatives of the curvature at p at the time t satisfy a bound 1 1 k k 2 2 + k +M |∇ Rm(p, t)| ≤ Ck M r2k t for some constant Ck depending only on the dimension and k. Proof. Since tR is nondecreasing in time, we get a bound R ≤ 2M in the given region for times between t/2 and t. The nonnegative curvature hypothesis tells us the metric is shrinking. So we can apply the local derivative estimate in Theorem 1.4.2 to deduce the result. By a similar argument as in Corollary 2.5.3, one readily has the following Harnack inequality. Corollary 2.5.7. Let gij (x, t) be a complete solution of the Ricci flow on a manifold with bounded and nonnegative curvature operator, and let x1 , x2 ∈ M, 0 < t1 < t2 . Then the following inequality holds R(x2 , t2 ) ≥

t1 −dt (x1 ,x2 )2 /2(t2 −t1 ) e 1 · R(x1 , t1 ). t2

In the above discussion, we assumed that the solution to the Ricci flow exists on 0 ≤ t < T , and we derived the Li-Yau-Hamilton estimate with terms 1/t in it. When the solution happens to be ancient, i.e., defined on −∞ < t < T , Hamilton [61] found an interesting and simple procedure for getting rid of them. Suppose we have a solution on α < t < T we can replace t by t − α in the Li-Yau-Hamilton estimate. If we let α → −∞, then the expression 1/(t − α) → 0 and disappears! In particular the trace Li-Yau-Hamilton estimate in Corollary 2.5.5 becomes (2.5.17)

∂R + 2∇a R · Va + 2Rab Va Vb ≥ 0. ∂t

By taking V = 0, we see that

∂R ∂t

≥ 0. Thus, we have the following

Corollary 2.5.8 (Hamilton [61]). Let gij (x, t) be a complete ancient solution of the Ricci flow on M × (−∞, T ) with bounded and nonnegative curvature operator, then the scalar curvature R(x, t) is pointwise nondecreasing in time t.

234


Corollary 2.5.8 will be very useful later on when we study ancient κ-solutions in Chapter 6, especially combined with Shi’s derivative estimate. We end this section by stating the Li-Yau-Hamilton estimate for the Kähler-Ricci flow, due to the first author [12], under the weaker curvature assumption of nonnegative holomorphic bisectional curvature. Note that the following Li-Yau-Hamilton estimate in the Kähler case is really a Li-Yau-Hamilton estimate for the Ricci tensor of the evolving metric, so not only can we derive an estimate on the scalar curvature, which is the trace of the Ricci curvature, similar to Corollary 2.5.5 but also an estimate on the determinant of the Ricci curvature as well. Theorem 2.5.9 (Cao [12]). Let gαβ¯(x, t) be a complete solution to the K¨ ahlerRicci flow on a complex manifold M with bounded curvature and nonnegtive bisectional curvature and 0 ≤ t < T . For any point x ∈ M and any vector V in the holomorphic tangent space Tx1,0 M , let Qαβ¯ =

∂ 1 γ δ¯ R ¯ + Rα¯γ Rγ β¯ + ∇γ Rαβ¯ V γ + ∇γ¯ Rαβ¯ V γ¯ + Rαβγ ¯ δ¯V V + Rαβ¯ . ∂t αβ t

Then we have ¯

Qαβ¯W α W β ≥ 0 for all x ∈ M , V, W ∈ Tx1,0 M , and t > 0. Corollary 2.5.10 (Cao [12]). Under the assumptions of Theorem 2.5.9, we have (i) the scalar curvature R satisfies the estimate R ∂R |∇R|2 − + ≥ 0, ∂t R t and (ii) assuming Rαβ¯ > 0, the determinant φ = det(Rαβ¯ )/ det(gαβ¯) of the Ricci curvature satisfies the estimate ∂φ |∇φ|2 nφ − + ≥0 ∂t nφ t for all x ∈ M and t > 0. 2.6. Perelman’s Estimate for Conjugate Heat Equations. In [103] Perelman obtained a Li-Yau type estimate for fundamental solutions of the conjugate heat equation, which is a backward heat equation, when the metric evolves by the Ricci flow. In this section we shall describe how to get this estimate along the same line as in the previous section. More importantly, we shall show how the Li-Yau path integral, when applied to Perelman’s Li-Yau type estimate, leads to an important space-time distance function introduced by Perelman [103]. We learned from Hamilton [67] this idea of looking at Perelman’s Li-Yau estimate. We saw in the previous section that the Li-Yau quantity and the Li-Yau-Hamilton quantity vanish on expanding solutions. Note that when we consider a backward heat equation, shrinking solitons can be viewed as expanding backward in time. So we start by looking at shrinking gradient Ricci solitons.


235

Suppose we have a shrinking gradient Ricci soliton gij with potential function f on manifold M and −∞ < t < 0 so that, for τ = −t, (2.6.1)

Rij + ∇j ∇i f −

1 gij = 0. 2τ

Then, by taking the trace, we have (2.6.2)

R + ∆f −

n = 0. 2τ

Also, by similar calculations as in deriving (1.1.15), we get (2.6.3)

R + |∇f |2 −

f =C τ

where C is a constant which we can set to be zero. Moreover, observe ∂f = |∇f |2 ∂t

(2.6.4)

because f evolves in time with the rate of change given by the Lie derivative in the direction of ∇f generating the one-parameter family of diffeomorphisms. Combining (2.6.2) with (2.6.4), we see f satisfies the backward heat equation (2.6.5)

∂f n = −∆f + |∇f |2 − R + , ∂t 2τ

or equivalently (2.6.6)

n ∂f = ∆f − |∇f |2 + R − . ∂τ 2τ

Recall the Li-Yau-Hamilton quadratic is a certain combination of the second order space derivative (or first order time derivative), first order space derivatives and zero orders. Multiplying (2.6.2) by a factor of 2 and subtracting (2.6.3) yields 2∆f − |∇f |2 + R +

n 1 f− =0 τ τ

valid for our potential function f of the shrinking gradient Ricci soliton. The quantity on the LHS of the above identity is precisely the Li-Yau-Hamilton type quadratic found by Perelman [103]. Note that a function f satisfies the backward heat equation (2.6.6) if and only if the function n

u = (4πτ )− 2 e−f satisfies the so called conjugate heat equation (2.6.7)

∗ u ,

∂u − ∆u + Ru = 0. ∂τ

Lemma 2.6.1 (Perelman [103]). Let gij (x, t), 0 ≤ t < T , be a complete solution n to the Ricci flow on an n-dimensional manifold M and let u = (4πτ )− 2 e−f be a solution to the conjugate equation (2.6.7) with τ = T − t. Set H = 2∆f − |∇f |2 + R +

f −n τ

236


and v = τ Hu = τ (R + 2∆f − |∇f |2 ) + f − n u. Then we have

and

2 1 1 ∂H = ∆H − 2∇f · ∇H − H − 2 Rij + ∇i ∇j f − gij , ∂τ τ 2τ 2 ∂v 1 = ∆v − Rv − 2τ u Rij + ∇i ∇j f − gij . ∂τ 2τ Proof. By direct computations, we have f −n 2 2△f − |∇f | + R + τ ∂f ∂f − 2h2Rij , fij i − 2 ∇f, ∇ + 2Ric (∇f, ∇f ) = 2△ ∂τ ∂τ 1 ∂ f −n ∂ R+ f− + ∂τ τ ∂τ τ2 n = 2△ △f − |∇f |2 + R − − 4hRij , fij i + 2Ric (∇f, ∇f ) 2τ D n E − 2 ∇f, ∇ △f − |∇f |2 + R − − △R − 2|Rij |2 2τ n f −n 1 △f − |∇f |2 + R − − , + τ 2τ τ 2 f −n ∇H = ∇ 2△f − |∇f |2 + R + τ 1 = 2∇(△f ) − 2h∇∇i f, ∇i f i + ∇R + ∇f, τ f −n △H = △ 2△f − |∇f |2 + R + τ 1 = 2△(△f ) − △(|∇f |2 ) + △R + △f, τ

∂ ∂ H= ∂τ ∂τ

and 1 ∇f, ∇f i τ 2 = 2 [h2∇(△f ), ∇f i − 2hfij , fi fj i + h∇R, ∇f i] + |∇f |2 . τ

2∇H · ∇f = 2h2∇(△f ) − 2h∇∇i f, ∇i f i + ∇R +


237

Thus we get 1 ∂ H − △H + 2∇f · ∇H + H ∂τ τ = −4hRij , fij i + 2Ric (∇f, ∇f ) − 2|Rij |2 − △(|∇f |2 ) + 2h∇(△f ), ∇f i 2 2 n + △f + R − 2 τ τ 2τ n R 1 2 2 = −2 |Rij | + |fij | + 2 + 2hRij , fij i − − △f 4τ τ τ 2 1 gij , = −2 Rij + ∇i ∇j f − 2τ

and

∂ ∂ −△+R v = − △ + R (τ Hu) ∂τ ∂τ ∂ − △ (τ H) · u − 2h∇(τ H), ∇ui = ∂τ ∂ = − △ (τ H) − 2h∇(τ H), ∇f i u ∂τ 1 ∂H − △H + 2∇f · ∇H + H u =τ ∂τ τ 2 1 = −2τ u Rij + ∇i ∇j f − gij . 2τ

Note that, since f satisfies the equation (2.6.6), we can rewrite H as (2.6.8)

H=2

1 ∂f + |∇f |2 − R + f. ∂τ τ

Then, by Lemma 2.6.1, we have 2 ∂ 1 (τ H) = ∆(τ H) − 2∇f · ∇(τ H) − 2τ Ric + ∇2 f − g . ∂τ 2τ

So by the maximum principle, we find max(τ H) is nonincreasing as τ increasing. When u is chosen to be a fundamental solution to (2.6.7), one can show that limτ →0 τ H ≤ 0 and hence H ≤ 0 on M for all τ ∈ (0, T ] (see, for example, [99]). Since this fact is not used in later chapters and will be only used in the rest of the section to introduce a space-time distance via Li-Yau path integral, we omit the details of the proof. Once we have Perelman’s Li-Yau type estimate H ≤ 0, we can apply the Li-Yau path integral as in [82] to estimate the above solution u (i.e., a heat kernel estimate for the conjugate heat equation, see also the earlier work of Cheeger-Yau [28]). Let p, q ∈ M be two points and γ(τ ), τ ∈ [0, τ¯], be a curve joining p and q, with γ(0) = p

238


and γ(¯ τ ) = q. Then along the space-time path (γ(τ ), τ ), τ ∈ [0, τ¯], we have √ ∂f d √ 1 2 τ f (γ(τ ), τ ) = 2 τ + ∇f · γ(τ ˙ ) +√ f dτ ∂τ τ √ √ 2 ˙ ) + τR ≤ τ − |∇f |gij (τ ) + 2∇f · γ(τ √ √ = − τ |∇f − γ(τ ˙ )|2gij (τ ) + τ (R + |γ(τ ˙ )|2gij (τ ) ) √ ≤ τ (R + |γ(τ ˙ )|2gij (τ ) ) where we have used the fact that H ≤ 0 and the expression for H in (2.6.8). Integrating the above inequality from τ = 0 to τ = τ¯, we obtain Z τ¯ √ √ τ (R + |γ(τ ˙ )|2gij (τ ) )dτ, 2 τ¯f (q, τ¯) ≤ 0

or

1 f (q, τ¯) ≤ √ L(γ), 2 τ¯ where L(γ) ,

(2.6.9) Denote by

Z

τ¯

0

√

τ (R + |γ(τ ˙ )|2gij (τ ) )dτ.

1 l(q, τ¯) , inf √ L(γ), γ 2 τ ¯

(2.6.10)

where the inf is taken over all space curves γ(τ ), 0 ≤ τ ≤ τ¯, joining p and q. The space-time distance function l(q, τ¯) obtained by the above Li-Yau path integral argument is first introduced by Perelman in [103] and is what Perelman calls reduced distance. Since Perelman pointed out in page 19 of [103] that “an even closer reference is [82], where they use ‘length’, associated to a linear parabolic equation, which is pretty much the same as in our case”, it is natural to call l(q, τ¯) the Li-YauPerelman distance. See Chapter 3 for much more detailed discussions. Finally, we conclude this section by relating the quantity H (or v) and the Wfunctional of Perelman defined in (1.5.9). Observe that v happens to be the integrand of the W-functional, Z W(gij (t), f, τ ) = vdV. M

Hence, when M is compact,

∂ v + Rv dV ∂τ M 2 Z Ric + ∇2 f − 1 g udV = −2τ 2τ M

d W= dτ

Z

≤ 0,

or equivalently,

d W(gij (t), f (t), τ (t)) = dt

2 n 1 2τ Rij + ∇i ∇j f − gij (4πτ )− 2 e−f dV, 2τ M

Z

which is the same as stated in Proposition 1.5.8.


239

3. Perelman’s Reduced Volume. In Section 1.5 we introduced the F functional and the W-functional of Perelman and proved their monotonicity properties under the Ricci flow. In the last section of the previous chapter we have defined the Li-Yau-Perelman distance. The main purpose of this chapter is to use the Li-YauPerelman distance to define the Perelman’s reduced volume, which was introduced by Perelman in [103], and prove the monotonicity property of the reduced volume under the Ricci flow. This new monotonicity formula is more useful for local considerations, especially when we consider the formation of singularities in Chapter 6 and work on the Ricci flow with surgery in Chapter 7. As first applications we will present two no local collapsing theorems of Perelman [103] in this chapter. More applications can be found in Chapter 6 and 7. 3.1. Riemannian Formalism in Potentially Infinite Dimensions. In Section 2.6, from an analytic view point, we saw how the Li-Yau path integral of Perelman’s estimate for fundamental solutions to the conjugate heat equation leads to the Li-Yau-Perelman distance. In this section we present, from a geometric view point, another motivation why one is lead to the consideration of the Li-Yau-Perelman distance function, as well as a reduced volume concept. Interestingly enough, the LiYau-Hamilton quadratic introduced in Section 2.5 appears again in this geometric consideration. We consider the Ricci flow ∂ gij = −2Rij ∂t on a manifold M where we assume that gij (·, t) are complete and have uniformly bounded curvatures. Recall from Section 2.5 that the Li-Yau-Hamilton quadratic introduced in [61] is Q = Mij Wi Wj + 2Pijk Uij Wk + Rijkl Uij Ukl where 1 1 Mij = ∆Rij − ∇i ∇j R + 2Rikjl Rkl − Rik Rjk + Rij , 2 2t Pijk = ∇i Rjk − ∇j Rik and Uij is any two-form and Wi is any 1-form. Here and throughout this chapter we do not always bother to raise indices; repeated indices is short hand for contraction with respect to the metric. In [63], Hamilton predicted that the Li-Yau-Hamilton quadratic is some sort of jet extension of positive curvature operator on some larger space. Such an interpretation of the Li-Yau-Hamilton quadratic as a curvature operator on the space M × R+ was found by Chow and Chu [38] where a potentially degenerate Riemannian metric on M × R+ was constructed. The degenerate Riemannian metric on M × R+ is the limit of the following two-parameter family of Riemannian metrics gN,δ (x, t) = g(x, t) + (R(x, t) +

N )dt2 2(t + δ)

as N tends to infinity and δ tends to zero, where g(x, t) is the solution of the Ricci flow on M and t ∈ R+ .

240


˜ = M ×SN ×R+ To avoid the degeneracy, Perelman [103] considers the manifold M with the following metric: g˜ij = gij , g˜αβ = τ gαβ , N + R, gõo = 2τ g˜iα = g˜io = g˜αo = 0, where i, j are coordinate indices on M ; α, β are coordinate indices on SN ; and the coordinate τ on R+ has index o. Let τ = T − t for some fixed constant T . Then gij ∂ will evolve with τ by the backward Ricci flow ∂τ gij = 2Rij . The metric gαβ on SN is 1 a metric with constant sectional curvature 2N . We remark that the metric g˜αβ on SN is chosen so that the product metric (˜ gij , g˜αβ ) on M × SN evolves by the Ricci flow, while the component gõo is just the ˜ = M × SN × R+ is scalar curvature of (˜ gij , g˜αβ ). Thus the metric g˜ defined on M exactly a “regularization” of Chow-Chu’s degenerate metric on M × R+ . Proposition 3.1.1. The components of the curvature tensor of the metric g˜ coincide (modulo N −1 ) with the components of the Li-Yau-Hamilton quadratic.

Proof. By definition, the Christoffel symbols of the metric g˜ are given by the following list: ˜ kij = Γkij , Γ ˜ γ = 0, ˜ k = 0 and Γ Γ iβ ij k ˜ ˜ Γαβ = 0 and Γγiβ = 0, ˜ o = −˜ ˜ k = g kl Rli and Γ g oo Rij , Γ io ij ˜ o = 1 gõo ∂ R, ˜ k = − 1 g kl ∂ R and Γ Γ io oo 2 ∂xl 2 ∂xi γ o k ˜ iβ = 0, Γ ˜ oβ = 0 and Γ ˜ = 0, Γ oj γ γ ˜ Γ =Γ , αβ

˜γ Γ αo ˜ γoo Γ õ Γ oo

αβ

1 γ ˜ o = − 1 gõo gαβ , = δα and Γ αβ 2τ 2 o ˜ = 0 and Γoβ = 0, 1 oo N ∂ = g˜ − 2+ R . 2 2τ ∂τ

Fix a point (p, s, τ ) ∈ M × SN × R+ and choose normal coordinates around p ∈ M and normal coordinates around s ∈ SN such that Γkij (p) = 0 and Γγαβ (s) = 0 for all ˜ of the metric g˜ at the point i, j, k and α, β, γ. We compute the curvature tensor Rm

241


as follows:

˜ oil = Rijkl + O ˜ ijkl = Rijkl + Γ ˜ kio Γ ˜ ojl − Γ ˜ kjo Γ R ˜ ijkδ = 0, R ˜ ijγδ = 0 R

1 N

,

˜ iβkδ = Γ ˜ kio Γ ˜ oβδ − Γ ˜ kβo Γ ˜ oiδ = − 1 gõo gβδ g kl Rli = O and R 2

˜ iβγδ = 0, R

1 N

,

˜ kio Γ ˜ ojo − Γ ˜ kjo Γ ˜ oio = Pijk + O 1 , ˜ ijko = ∂ Rjk − ∂ Rik + Γ R ∂xi ∂xj N 2 1 ∂ ∂ ˜ ioko = − ˜ kio Γ ˜ ooo − Γ ˜ koj Γ ˜j − Γ ˜ koo Γ ˜ oio R R− (Ril g lk ) + Γ io 2 ∂xi ∂xk ∂τ ∂ 1 1 1 ∇ ∇ R − R + 2R R − R − R R + O =− i k ik ik lk ik ij jk 2 ∂τ 2τ N 1 = Mik + O( ), N ˜ ijγo = 0 and R ˜ iγjo = 0, R 1 γ õ ˜ ˜ ˜ ioγδ = 0, Riβγo = −τ Γβo Γio = O and R N ˜ ioγo = 0, R

˜ αβγo = 0, R 1 1 γ ˜γ õ γ ˜β ˜ ˜ δα + Γαo Γoo − Γoβ Γαo τ = O , Rαoγo = 2 2τ N ˜ αβγδ = O 1 . R N

Thus the components of the curvature tensor of the metric g˜ coincide (modulo N −1 ) with the components of the Li-Yau-Hamilton quadratic.

The following observation due to Perelman [103] gives an important motivation to define Perelman’s reduced volume.

Corollary 3.1.2. All components of the Ricci tensor of g˜ are zero (modulo N −1 ).

Proof. From the list of the components of the curvature tensor of g˜ given above,

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we have ˜ ij = g˜kl R ˜ ijkl + g˜αβ R ˜ iαjβ + gõo R ˜ iojo R 1 1 1 αβ oo αβ oo g g˜ g Rij + g˜ Mij − Rij + O = Rij − 2τ 2τ N N oo 1 = Rij − g˜ Rij + O 2τ N 1 , =O N ˜ iγ = g˜kl R ˜ ikγl + g˜αβ R ˜ iαγβ + gõo R ˜ ioγo = 0, R ˜ io = g˜kl R ˜ ikol + g˜αβ R ˜ iαoβ + gõo R ˜ iooo R 1 , = −g kl Pikl + O N ˜ αβ = g˜kl R ˜ αkβl + g˜γδ R ˜ αγβδ + gõo R ˜ αoβo R 1 =O , N ˜ αo = g˜kl R ˜ αkol + g˜βγ R ˜ αβoγ + gõo R ˜ αooo = 0, R ˜ oo = g˜kl R ˜ okol + g˜αβ R ˜ oαoβ + gõo R ˜ oooo R 1 1 +O . = g kl Mkl + O N N Since gõo is of order N −1 , we see that the norm of the Ricci tensor is given by 1 ˜ |Ric|g˜ = O . N This proves the result. We now use the Ricci-flatness of the metric g˜ to interpret the Bishop-Gromov relative volume comparison theorem which will motivate another monotonicity formula for the Ricci flow. The argument in the following will not be rigorous. However it ˜ , g˜) gives an intuitive picture of what one may expect. Consider a metric ball in (M N ˜ centered at some point (p, s, 0) ∈ M . Note that the metric of the sphere S at τ = 0 degenerates and it shrinks to a point. Then the shortest geodesic γ(τ ) between (p, s, 0) ˜ is always orthogonal to the SN fibre. The length and an arbitrary point (q, s¯, τ¯) ∈ M of γ(τ ) can be computed as s

N + R + |γ(τ ˙ )|2gij (τ ) dτ 2τ 0 Z τ¯ √ √ 3 1 τ (R + |γ(τ ˙ )|2gij )dτ + O(N − 2 ). = 2N τ¯ + √ 2N 0

Z

τ¯

Thus a shortest geodesic should minimize L(γ) =

Z

0

τ¯

√

τ (R + |γ(τ ˙ )|2gij )dτ.


243

Let minimum. We claim that a metric sphere √ L(q, τ¯) denote the corresponding √ ˜ of radius 2N τ¯ centered at (p, s, 0) is O(N −1 )-close to the hyperSM˜ ( 2N τ¯) in M √ surface {τ = τ¯}. Indeed, if (x, s′ , τ (x)) lies on the metric sphere SM˜ ( 2N τ¯), then the distance between (x, s′ , τ (x)) and (p, s, 0) is p √ 3 1 L(x, τ (x)) + O N − 2 2N τ¯ = 2N τ (x) + √ 2N which can be written as p √ 1 L(x, τ (x)) + O(N −2 ) = O(N −1 ). τ (x) − τ¯ = − 2N √ This shows that the metric sphere SM˜ ( 2N τ¯) is O(N −1 )-close to the hypersurface 1 {τ = τ¯}. Note that the metric gαβ on SN has constant sectional curvature 2N . Thus √ 2N τ¯ Vol SM˜ Z Z ≈ dVτ (x)gαβ dVgij (x) SN ZM N = (τ (x)) 2 Vol (S N )dVM M

N √ 1 −2 ≈ (2N ) ωN L(x, τ (x)) + O(N ) dVM τ¯ − 2N M N Z √ N 1 ≈ (2N ) 2 ωN τ¯ − L(x, τ¯) + o(N −1 ) dVM , 2N M N 2

Z

where ωN is the volume of √ the standard N -dimensional sphere. Now the volume of Euclidean sphere of radius 2N τ¯ in Rn+N +1 is √ N +n Vol (SRn+N +1 ( 2N τ¯)) = (2N τ¯) 2 ωn+N . Thus we have √ Z Vol (SM˜ ( 2N τ¯)) 1 −n −n 2 2 √ · (¯ τ) exp − √ L(x, τ¯) dVM . ≈ const · N 2 τ¯ Vol (SRn+N +1 ( 2N τ¯)) M ˜ is zero (modulo N −1 ), the Bishop-Gromov volume Since the Ricci curvature of M comparison theorem then suggests that the integral Z 1 ∆ −n ˜ 2 exp − √ L(x, τ¯) dVM , V (¯ τ) = (4π¯ τ) 2 τ¯ M which we will call Perelman’s reduced volume, should be nonincreasing in τ¯. A rigorous proof of this monotonicity property will be given in the next section. One should note the analog of reduced volume with the heat kernel and there is a parallel calculation for the heat kernel of the Shrödinger equation in the paper of Li-Yau [82]. 3.2. Comparison Theorems for Perelman’s Reduced Volume. In this section we will write the Ricci flow in the backward version ∂ gij = 2Rij ∂τ

244


on a manifold M with τ = τ (t) satisfying dτ /dt = −1 (in practice we often take τ = t0 − t for some fixed time t0 ). We always assume that either M is compact or gij (τ ) are complete and have uniformly bounded curvature. To each (smooth) space curve γ(τ ), 0 < τ1 ≤ τ ≤ τ2 , in M , we define its L-length as Z τ2 √ L(γ) = τ (R(γ(τ ), τ ) + |γ(τ ˙ )|2gij (τ ) )dτ. τ1

Let X(τ ) = γ(τ ˙ ), and let Y (τ ) be any (smooth) vector field along γ(τ ). First of all, we compute the first variation formula for L-length. Lemma 3.2.1 (First variation formula). Z τ2 √ √ 1 δY (L) = 2 τ hX, Y i|ττ21 + τ Y, ∇R − 2∇X X − 4Ric (·, X) − X dτ τ τ1 where h·, ·i denotes the inner product with respect to the metric gij (τ ). Proof. By direct computations, Z τ2 √ δY (L) = τ (h∇R, Y i + 2hX, ∇Y Xi)dτ τ1 Z τ2 √ = τ (h∇R, Y i + 2hX, ∇X Y i)dτ τ1 Z τ2 √ d τ h∇R, Y i + 2 hX, Y i − 2h∇X X, Y i − 4Ric (X, Y ) dτ = dτ τ1 Z τ2 √ √ 1 = 2 τ hX, Y i|ττ21 + τ Y, ∇R − 2∇X X − 4Ric (·, X) − X dτ. τ τ1 A smooth curve γ(τ ) in M is called an L-geodesic if it satisfies the following L-geodesic equation (3.2.1)

1 1 ∇X X − ∇R + X + 2Ric (X, ·) = 0. 2 2τ

Given any two points p, q ∈ M and τ2 > τ1 > 0, there always exists an L-shortest curve (or shortest L-geodesic) γ(τ ): [τ1 , τ2 ] → M connecting p to q which satisfies √ the above L-geodesic equation. Multiplying the L-geodesic equation (3.2.1) by τ , we get √ √ √ τ ∇X ( τ X) = ∇R − 2 τ Ric (X, ·) on [τ1 , τ2 ], 2 or equivalently √ √ d √ τ ( τ X) = ∇R − 2Ric ( τ X, ·) on [τ1 , τ2 ]. dτ 2 Thus if a continuous curve, defined on [0, τ2 ], satisfies the L-geodesic equation on √ every subinterval 0 < τ1 ≤ τ ≤ τ2 , then τ1 X(τ1 ) has a limit as τ1 → 0+ . This allows us to extend the definition of the L-length to include the case τ1 = 0 for


245

all those (continuous) curves γ : [0, τ2 ] → M which are smooth on (0, τ2 ] and have √ limits lim+ τ γ(τ ˙ ). Clearly, there still exists an L-shortest curve γ(τ ) : [0, τ2 ] → M τ →0

connecting arbitrary two points p, q ∈ M and satisfying the L-geodesic equation (3.2.1) on (0, τ2 ]. Moreover, √ for any vector v ∈ Tp M , we can find an L-geodesic γ(τ ) ˙ ) = v. starting at p with lim τ γ(τ τ →0+

From now on, we fix a point p ∈ M and set τ1 = 0. The L-distance function on the space-time M × R+ is denoted by L(q, τ¯) and defined to be the L-length of the L-shortest curve γ(τ ) connecting p and q with 0 ≤ τ ≤ τ¯. Consider a shortest L-geodesic γ : [0, τ¯] → M connecting p to q. In the computations below we pretend that L-shortest geodesics between p and q are unique for all pairs (q, τ¯); if this is not the case, the inequalities that we obtain are still valid, by a standard barrier argument, when understood in the sense of distributions (see, for example, [112]). The first variation formula in Lemma 3.2.1 implies that E D √ τ ), Y (¯ τ) . ∇Y L(q, τ¯) = 2 τ¯X(¯ Thus

√ τ ), ∇L(q, τ¯) = 2 τ¯X(¯ and |∇L|2 = 4¯ τ |X|2 = −4¯ τ R + 4¯ τ (R + |X|2 ).

(3.2.2) We also compute (3.2.3)

d L(γ(τ ), τ )|τ =¯τ − h∇L, Xi dτ √ √ = τ¯(R + |X|2 ) − 2 τ¯|X|2 √ √ = 2 τ¯R − τ¯(R + |X|2 ).

Lτ¯ (γ(¯ τ ), τ¯) =

To evaluate R + |X|2 , we compute by using (3.2.1), d (R(γ(τ ), τ ) + |X(τ )|2gij (τ ) ) dτ = Rτ + h∇R, Xi + 2h∇X X, Xi + 2Ric (X, X) 1 1 = Rτ + R + 2h∇R, Xi − 2Ric (X, X) − (R + |X|2 ) τ τ 1 2 = −Q(X) − (R + |X| ), τ where Q(X) = −Rτ −

R − 2h∇R, Xi + 2Ric (X, X) τ

is the trace Li-Yau-Hamilton quadratic in Corollary 2.5.5. Hence 3 1√ d 3 (τ 2 (R + |X|2 ))|τ =¯τ = τ¯(R + |X|2 ) − τ¯ 2 Q(X) dτ 2 3 1 d = L(γ(τ ), τ )|τ =¯τ − τ¯ 2 Q(X). 2 dτ

246


Therefore, 3

τ¯ 2 (R + |X|2 ) =

(3.2.4)

1 L(q, τ¯) − K, 2

where (3.2.5)

K=

Z

τ¯

3

τ 2 Q(X)dτ. 0

Combining (3.2.2) with (3.2.3), we obtain 2 4 |∇L|2 = −4¯ τR + √ L − √ K τ¯ τ¯

(3.2.6) and

√ 1 1 Lτ¯ = 2 τ¯R − L + K. 2¯ τ τ¯

(3.2.7)

Next we compute the second variation of an L-geodesic. Lemma 3.2.2 (Second variation formula). For any L-geodesic γ, we have Z τ¯ √ √ τ [2|∇X Y |2 + 2hR(Y, X)Y, Xi δY2 (L) = 2 τ h∇Y Y, Xi|τ0¯ + 0

+ ∇Y ∇Y R + 2∇X Ric (Y, Y ) − 4∇Y Ric (Y, X)]dτ.

Proof. We compute δY2

Z

τ¯

√

τ (Y (R) + 2h∇Y X, Xi)dτ (L) = Y 0 Z τ¯ √ = τ (Y (Y (R)) + 2h∇Y ∇Y X, Xi + 2|∇Y X|2 )dτ 0 Z τ¯ √ = τ (Y (Y (R)) + 2h∇Y ∇X Y, Xi + 2|∇X Y |2 )dτ 0

and 2h∇Y ∇X Y, Xi = 2h∇X ∇Y Y, Xi + 2hR(Y, X)Y, Xi d = 2 h∇Y Y, Xi − 4Ric (∇Y Y, X) − 2h∇Y Y, ∇X Xi dτ d ∇Y Y, X − 2h∇X ∇Y Y, Xi + 2hR(Y, X)Y, Xi − 2 dτ d = 2 h∇Y Y, Xi − 4Ric (∇Y Y, X) − 2h∇Y Y, ∇X Xi dτ ∂ i j kl − 2 Y Y (g (∇i Rlj + ∇j Rli − ∇l Rij )) k , X + 2hR(Y, X)Y, Xi ∂x d = 2 h∇Y Y, Xi − 4Ric (∇Y Y, X) − 2h∇Y Y, ∇X Xi − 4∇Y Ric (X, Y ) dτ + 2∇X Ric (Y, Y ) + 2hR(Y, X)Y, Xi,


247

where we have used the computation ∂ k Γ = g kl (∇i Rlj + ∇j Rli − ∇l Rij ). ∂τ ij Thus by using the L-geodesic equation (3.2.1), we get Z τ¯ √ d δY2 (L) = τ Y (Y (R)) + 2 h∇Y Y, Xi − 4Ric (∇Y Y, X) dτ 0

− 2h∇Y Y, ∇X Xi − 4∇Y Ric (X, Y ) + 2∇X Ric (Y, Y ) 2 + 2hR(Y, X)Y, Xi + 2|∇X Y | dτ Z τ¯ √ d 1 = 2 τ h∇Y Y, Xi + √ h∇Y Y, Xi dτ dτ τ 0 Z τ¯ √ + τ [Y (Y (R)) − h∇Y Y, ∇Ri − 4∇Y Ric (X, Y ) 0

+ 2∇X Ric (Y, Y ) + 2hR(Y, X)Y, Xi + 2|∇X Y |2 ]dτ Z τ¯ √ √ = 2 τ h∇Y Y, Xi|τ0¯ + τ [2|∇X Y |2 + 2hR(Y, X)Y, Xi 0

+ ∇Y ∇Y R − 4∇Y Ric (X, Y ) + 2∇X Ric (Y, Y )]dτ.

We now use the above second variation formula to estimate the Hessian of the L-distance function. Let γ(τ ) : [0, τ¯] → M be an L-shortest curve connecting p and q so that the L-distance function L = L(q, τ¯) is given by the L-length of γ. We fix a vector Y at τ = τ¯ with |Y |gij (¯τ ) = 1, and extend Y along the L-shortest geodesic γ on [0, τ¯] by solving the following ODE ∇X Y = −Ric (Y, ·) +

(3.2.8)

1 Y. 2τ

This is similar to the usual parallel translation and multiplication with proportional parameter. Indeed, suppose {Y1 , . . . , Yn } is an orthonormal basis at τ = τ¯ (with respect to the metric gij (¯ τ )) and extend this basis along the L-shortest geodesic γ by solving the above ODE (3.2.8). Then d hYi , Yj i = 2Ric (Yi , Yj ) + h∇X Yi , Yj i + hYi , ∇X Yj i dτ 1 = hYi , Yj i τ for all i, j. Hence, (3.2.9)

hYi (τ ), Yj (τ )i =

τ δij τ¯

and {Y1 (τ ), . . . , Yn (τ )} remains orthogonal on [0, τ¯] with Yi (0) = 0, i = 1, . . . , n. Proposition 3.2.3. Given any unit vector Y at any point q ∈ M with τ = τ¯, consider an L-shortest geodesic γ connecting p to q and extend Y along γ by solving

248


the ODE (3.2.8). Then the Hessian of the L-distance function L on M with τ = τ¯ satisfies √ 1 HessL (Y, Y ) ≤ √ − 2 τ¯Ric (Y, Y ) − τ¯

Z

τ¯

√ τ Q(X, Y )dτ

0

in the sense of distributions, where Q(X, Y ) = −∇Y ∇Y R − 2hR(Y, X)Y, Xi − 4∇X Ric (Y, Y ) + 4∇Y Ric (Y, X) 1 − 2Ric τ (Y, Y ) + 2|Ric (Y, ·)|2 − Ric (Y, Y ) τ is the Li-Yau-Hamilton quadratic. Moreover the equality holds if and only if the vector field Y (τ ), τ ∈ [0, τ¯], is an L-Jacobian field (i.e., Y is the derivative of a variation of γ by L-geodesics). Proof. As said before, we pretend that the shortest L-geodesics between p and q are unique so that L(q, τ¯) is smooth. Otherwise, the inequality is still valid, by a standard barrier argument, when understood in the sense of distributions (see, for example, [112]). √ √ Recall that ∇L(q, τ¯) = 2 τ¯X. Then h∇Y Y, ∇Li = 2 τ¯h∇Y Y, Xi. We compute by using Lemma 3.2.2, (3.2.8) and (3.2.9), HessL (Y, Y ) = Y (Y (L))(¯ τ ) − h∇Y Y, ∇Li(¯ τ) √ 2 τ) ≤ δY (L) − 2 τ¯h∇Y Y, Xi(¯ Z τ¯ √ = τ [2|∇X Y |2 + 2hR(Y, X)Y, Xi + ∇Y ∇Y R 0

+ 2∇X Ric (Y, Y ) − 4∇Y Ric (Y, X)]dτ 2 Z τ¯ √ 1 Y + 2hR(Y, X)Y, Xi + ∇Y ∇Y R = τ 2 − Ric (Y, ·) + 2τ 0 + 2∇X Ric (Y, Y ) − 4∇Y Ric (Y, X) dτ Z τ¯ √ 1 2 = + 2hR(Y, X)Y, Xi τ 2|Ric (Y, ·)|2 − Ric (Y, Y ) + τ 2τ τ¯ 0 + ∇Y ∇Y R + 2∇X Ric (Y, Y ) − 4∇Y Ric (Y, X) dτ.

Since d Ric (Y, Y ) = Ric τ (Y, Y ) + ∇X Ric (Y, Y ) + 2Ric (∇X Y, Y ) dτ = Ric τ (Y, Y ) + ∇X Ric (Y, Y ) − 2|Ric (Y, ·)|2 +

1 Ric (Y, Y ), τ


249

we have HessL (Y, Y ) Z τ¯ √ 1 2 + 2hR(Y, X)Y, Xi ≤ τ 2|Ric (Y, ·)|2 − Ric (Y, Y ) + τ 2τ τ¯ 0 d + ∇Y ∇Y R − 4(∇Y Ric )(X, Y ) − 2 Ric (Y, Y ) − 2Ric τ (Y, Y ) dτ 2 2 + 4|Ric (Y, ·)| − Ric (Y, Y ) + 4∇X Ric (Y, Y ) dτ τ Z τ¯ Z τ¯ √ d 1 1 1 √ dτ =− 2 τ Ric (Y, Y ) + √ Ric (Y, Y ) dτ + dτ 2¯ τ 0 τ τ 0 Z τ¯ √ 1 τ 2hR(Y, X)Y, Xi + ∇Y ∇Y R + Ric (Y, Y ) + τ 0

+ 4(∇X Ric (Y, Y ) − ∇Y Ric (X, Y )) + 2Ric τ (Y, Y ) − 2|Ric (Y, ·)|2 dτ

√ 1 = √ − 2 τ¯Ric (Y, Y ) − τ¯

Z

τ¯

√

τ Q(X, Y )dτ.

0

This proves the inequality. As usual, the quadratic form I(V, V ) =

Z

0

τ¯

√

τ [2|∇X V |2 + 2hR(V, X)V, Xi + ∇V ∇V R

+2∇X Ric (V, V ) − 4∇V Ric (V, X)]dτ, for any vector field V along γ, is called the index form. Since γ is shortest, the standard Dirichlet principle for I(V, V ) implies that the equality holds if and only if the vector field Y is the derivative of a variation of γ by L-geodesics. Corollary 3.2.4. We have √ 1 n ∆L ≤ √ − 2 τ¯R − K τ¯ τ¯ in the sense of distribution. Moreover, the equality holds if and only if we are on a gradient shrinking soliton with 1 1 gij . Rij + √ ∇i ∇j L = 2¯ τ 2 τ¯

Proof. Choose an orthonormal basis {Y1 , . . . , Yn } at τ = τ¯ and extend them along the shortest L-geodesic γ to get vector fields Yi (τ ), i = 1, . . . , n, by solving the ODE (3.2.8), with hYi (τ ), Yj (τ )i = ττ¯ δij on [0, τ¯]. Taking Y = Yi in Proposition 3.2.3 and

250


summing over i, we get n Z τ¯ X √ √ n τ Q(X, Yi )dτ ∆L ≤ √ − 2 τ¯R − τ¯ i=1 0 Z τ¯ √ √ τ n = √ − 2 τ¯R − τ Q(X)dτ τ¯ τ¯ 0 √ 1 n = √ − 2 τ¯R − K. τ¯ τ¯

(3.2.10)

Moreover, by Proposition 3.2.3, the equality in (3.2.10) holds everywhere if and only if for each (q, τ¯) and any shortest L-geodesic γ on [0, τ¯] connecting p and q, and for any unit vector Y at τ = τ¯, the extended vector field Y (τ ) along γ by the ODE (3.2.8) must be an L-Jacobian field. When Yi (τ ), i = 1, . . . , n are L-Jacobian fields along γ, we have d hYi (τ ), Yj (τ )i dτ = 2Ric (Yi , Yj ) + h∇X Yi , Yj i + hYi , ∇X Yj i 1 1 √ ∇L , Yj + Yi , ∇Yj √ ∇L = 2Ric (Yi , Yj ) + ∇Yi 2 τ¯ 2 τ¯ 1 = 2Ric (Yi , Yj ) + √ HessL (Yi , Yj ) τ¯ and then by (3.2.9), 1 1 2Ric (Yi , Yj ) + √ HessL (Yi , Yj ) = δij , τ¯ τ¯

at τ = τ¯.

Therefore the equality in (3.2.10) holds everywhere if and only if we are on a gradient shrinking soliton with 1 1 Rij + √ ∇i ∇j L = gij . 2¯ τ 2 τ¯

In summary, from (3.2.6), (3.2.7) and Corollary 3.2.4, we have      

∂L ∂ τ¯

√ = 2 τ¯R −

L 2¯ τ

+

K τ¯ ,

|∇L|2 = −4¯ τ R + √2τ¯ L − √4τ¯ K,      ∆L ≤ −2√τ¯R + √n − K , τ¯ τ¯

in the sense of distributions. Now the Li-Yau-Perelman distance l = l(q, τ¯) is defined by √ l(q, τ¯) = L(q, τ¯)/2 τ¯. We thus have the following


251

Lemma 3.2.5. For the Li-Yau-Perelman distance l(q, τ¯) defined above, we have l 1 ∂l = − + R + 3/2 K, ∂ τ¯ τ¯ 2¯ τ 1 l 2 |∇l| = −R + − 3/2 K, τ¯ τ¯ n 1 ∆l ≤ −R + − 3/2 K, 2¯ τ 2¯ τ

(3.2.11) (3.2.12) (3.2.13)

in the sense of distributions. Moreover, the equality in (3.2.13) holds if and only if we are on a gradient shrinking soliton. As the first consequence, we derive the following upper bound on the minimum of l(·, τ ) for every τ which will be useful in proving the no local collapsing theorem in the next section. Corollary 3.2.6. Let gij (τ ), τ ≥ 0, be a family of metrics evolving by the Ricci ∂ gij = 2Rij on a compact n-dimensional manifold M . Fix a point p in M and flow ∂τ let l(q, τ ) be the Li-Yau-Perelman distance from (p, 0). Then for all τ , min{l(q, τ ) | q ∈ M } ≤

n . 2

Proof. Let ¯ τ ) = 4τ l(q, τ ). L(q, Then, it follows from (3.2.11) and (3.2.13) that ¯ 2K ∂L = 4τ R + √ , ∂τ τ and ¯ ≤ −4τ R + 2n − 2K √ . ∆L τ Hence ¯ ∂L ¯ ≤ 2n. + ∆L ∂τ ¯ τ ) − 2nτ | q ∈ M } is Thus, by a standard maximum principle argument, min{L(q, ¯ nonincreasing and therefore min{L(q, τ )| q ∈ M } ≤ 2nτ . As another consequence of Lemma 3.2.5, we obtain ∂l n − ∆l + |∇l|2 − R + ≥ 0. ∂ τ¯ 2¯ τ or equivalently

∂ −∆+R ∂ τ¯

n (4π¯ τ )− 2 exp(−l) ≤ 0.

252


If M is compact, we define Perelman’s reduced volume by V˜ (τ ) =

Z

n

(4πτ )− 2 exp(−l(q, τ ))dVτ (q),

M

where dVτ denotes the volume element with respect to the metric gij (τ ). Note that Perelman’s reduced volume resembles the expression in Huisken’s monotonicity formula for the mean curvature flow [72]. It follows, from the above computation, that d d¯ τ

Z

n

(4π¯ τ )− 2 exp(−l(q, τ¯))dVτ¯ (q) Z n n ∂ ((4π¯ τ )− 2 exp(−l(q, τ¯))) + R(4π¯ = τ )− 2 exp(−l(q, τ¯)) dVτ¯ (q) ∂ τ¯ ZM n ≤ ∆((4π¯ τ )− 2 exp(−l(q, τ¯)))dVτ¯ (q) M

M

= 0.

This says that if M is compact, then Perelman’s reduced volume V˜ (τ ) is nonincreasing in τ ; moreover, the monotonicity is strict unless we are on a gradient shrinking soliton. In order to define and to obtain the monotonicity of Perelman’s reduced volume for a complete noncompact manifold, we need to formulate the monotonicity of Perelman’s reduced volume in a local version. This local version is very important and will play a crucial role in the analysis of the Ricci flow with surgery in Chapter 7. We define the L-exponential map (with parameter τ¯) L exp(¯ τ ) : Tp M → M as follows: for any X ∈ Tp M , we set L expX (¯ τ ) = γ(¯ τ)

√ where γ(τ ) is the L-geodesic, starting at p and having X as the limit of τ γ(τ ˙ ) as τ → 0+ . The associated Jacobian of the L-exponential map is called L-Jacobian. We denote by J (τ ) the L-Jacobian of L exp(τ ) : Tp M → M . We can now deduce an estimate for the L-Jacobian as follows. Let q =√L expX (¯ τ ) and γ(τ ), τ ∈ [0, τ¯], be the shortest L-geodesic connecting p ˙ ) → X as τ → 0+ . For any vector v ∈ Tp M , we consider the family and q with τ γ(τ of L-geodesics: γs (τ ) = L exp(X+sv) (τ ),

0 ≤ τ ≤ τ¯, s ∈ (−ǫ, ǫ).

The associated variation vector field V (τ ), 0 ≤ τ ≤ τ¯, is an L-Jacobian field with V (0) = 0 and V (τ ) = (L expX (τ ))∗ (v). Let v1 , . . . , vn be n linearly independent vectors in Tp M . Then Vi (τ ) = (L expX (τ ))∗ (vi ),

i = 1, 2, . . . , n,

are n L-Jacobian fields along γ(τ ), τ ∈ [0, τ¯]. The L-Jacobian J (τ ) is given by J (τ ) = |V1 (τ ) ∧ · · · ∧ Vn (τ )|gij (τ ) /|v1 ∧ · · · ∧ vn |.


253

Now for fixed b ∈ (0, τ¯), we can choose linearly independent vectors v1 , . . . , vn ∈ Tp M such that hVi (b), Vj (b)igij (b) = δij . We compute d 2 J dτ =

n X 2 hV1 ∧ · · · ∧ ∇X Vj ∧ · · · ∧ Vn , V1 ∧ · · · ∧ Vn igij (τ ) |v1 ∧ · · · ∧ vn |2 j=1

+

n X 2 hV1 ∧ · · · ∧ Ric (Vj , ·) ∧ · · · ∧ Vn , V1 ∧ · · · ∧ Vn igij (τ ) . |v1 ∧ · · · ∧ vn |2 j=1

At τ = b, n X d 2 2 J (b) = (h∇X Vj , Vj igij (b) + Ric (Vj , Vj )). dτ |v1 ∧ · · · ∧ vn |2 j=1

Thus, n

X d log J (b) = (h∇X Vj , Vj igij (b) + Ric (Vj , Vj )) dτ j=1 ! n X 1 √ ∇L , Vj = ∇Vj + Ric (Vj , Vj ) 2 b gij (b) j=1   n X 1 HessL (Vj , Vj ) + R = √  2 b j=1 1 = √ ∆L + R. 2 b

Therefore, in view of Corollary 3.2.4, we obtain the following estimate for L-Jacobian: (3.2.14)

d n 1 log J (τ ) ≤ − 3/2 K dτ 2τ 2τ

on [0, τ¯].

On the other hand, by the definition of the Li-Yau-Perelman distance and (3.2.4), we have (3.2.15)

d 1 1 d l(τ ) = − l + √ L dτ 2τ 2 τ dτ 1 √ 1 = − l + √ ( τ (R + |X|2 )) 2τ 2 τ 1 = − 3/2 K. 2τ

Here and in the following we denote by l(τ ) = l(γ(τ ), τ ). Now the combination of (3.2.14) and (3.2.15) implies the following important Jacobian comparison theorem of Perelman [103]. Theorem 3.2.7 (Perelman’s Jacobian comparison theorem). Let gij (τ ) be a ∂ family of complete solutions to the Ricci flow ∂τ gij = 2Rij on a manifold M with

254


bounded curvature. Let γ : [0, τ¯] → M be a shortest L-geodesic starting from a fixed point p. Then Perelman’s reduced volume element n

(4πτ )− 2 exp(−l(τ ))J (τ ) is nonincreasing in τ along γ. We now show how to integrate Perelman’s reduced volume element over Tp M to deduce the following monotonicity result of Perelman [103]. Theorem 3.2.8 (Monotonicity of Perelman’s reduced volume). Let gij be a ∂ family of complete metrics evolving by the Ricci flow ∂τ gij = 2Rij on a manifold M with bounded curvature. Fix a point p in M and let l(q, τ ) be the reduced distance from (p, 0). Then (i) Perelman’s reduced volume Z n ˜ V (τ ) = (4πτ )− 2 exp(−l(q, τ ))dVτ (q) M

is finite and nonincreasing in τ ; (ii) the monotonicity is strict unless we are on a gradient shrinking soliton. Proof. For any v ∈ Tp M we can find an L-geodesic γ(τ ), starting at p, with √ lim+ τ γ(τ ˙ ) = v. Recall that γ(τ ) satisfies the L-geodesic equation

τ →0

1 1 ∇γ(τ ˙ ) − ∇R + γ(τ ˙ ) + 2Ric (γ(τ ˙ ), ·) = 0. ˙ ) γ(τ 2 2τ √ Multiplying this equation by τ , we get (3.2.16)

√ d √ 1√ ˙ − τ ∇R + 2Ric ( τ γ(τ ˙ ), ·) = 0. ( τ γ) dτ 2

Since the curvature of the metric gij (τ ) is bounded, it follows from Shi’s derivative estimate (Theorem 1.4.1) that |∇R| is also bounded for small τ > 0. Thus by integrating (3.2.16), we have √ (3.2.17) | τ γ(τ ˙ ) − v| ≤ Cτ (|v| + 1) for τ small enough and for some positive constant C depending only the curvature bound. Let v1 , . . . , vn be n linearly independent vectors in Tp M and let Vi (τ ) = (L expv (τ ))∗ (vi ) =

d |s=0 L exp(v+svi ) (τ ), i = 1, . . . , n. ds

The L-Jacobian J (τ ) is given by J (τ ) = |V1 (τ ) ∧ · · · ∧ Vn (τ )|gij (τ ) /|v1 ∧ · · · ∧ vn | By (3.2.17), we see that √ d τ L exp(v+sv ) (τ ) − (v + svi ) ≤ Cτ (|v| + |vi | + 1) i dτ


255

for τ small enough and all s ∈ (−ǫ, ǫ) (for some ǫ > 0 small) and i = 1, . . . , n. This implies that √ lim+ τ V˙ i (τ ) = vi , i = 1, . . . , n, τ →0

so we deduce that (3.2.18)

n

lim τ − 2 J (τ ) = 1.

τ →0+

Meanwhile, by using (3.2.17), we have Z τ √ 1 l(τ ) = √ τ (R + |γ(τ ˙ )|2 )dτ 2 τ 0 → |v|2 as τ → 0+ .

Thus l(0) = |v|2 .

(3.2.19)

Combining (3.2.18) and (3.2.19) with Theorem 3.2.7, we get Z n V˜ (τ ) = (4πτ )− 2 exp(−l(q, τ ))dVτ (q) ZM n ≤ (4πτ )− 2 exp(−l(τ ))J (τ )|τ =0 dv Tp M Z n exp(−|v|2 )dv = (4π)− 2 Rn

< +∞.

This proves that Perelman’s reduced volume is always finite and hence well defined. Now the monotonicity assertion in (i) follows directly from Theorem 3.2.7. For the assertion (ii), we note that the equality in (3.2.13) holds everywhere when the monotonicity of Perelman’s reduced volume is not strict. Therefore we have completed the proof of the theorem. 3.3. No Local Collapsing Theorem I. In this section we apply the monotonicity of Perelman’s reduced volume in Theorem 3.2.8 to prove Perelman’s no local collapsing theorem I, which is extremely important not only because it gives a local injectivity radius estimate in terms of local curvature bound but also it will survive the surgeries in Chapter 7. Definition 3.3.1. Let κ, r be two positive constants and let gij (t), 0 ≤ t < T, be a solution to the Ricci flow on an n-dimensional manifold M . We call the solution gij (t) κ-noncollapsed at (x0 , t0 ) ∈ M ×[0, T ) on the scale r if it satisfies the following property: whenever |Rm|(x, t) ≤ r−2 for all x ∈ Bt0 (x0 , r) and t ∈ [t0 − r2 , t0 ], we have V olt0 (Bt0 (x0 , r)) ≥ κrn .

256


Here Bt0 (x0 , r) is the geodesic ball centered at x0 ∈ M and of radius r with respect to the metric gij (t0 ). Now we are ready to state the no local collapsing theorem I of Perelman [103]. Theorem 3.3.2 (No local collapsing theorem I). Given any metric gij on an n-dimensional compact manifold M . Let gij (t) be the solution to the Ricci flow on [0, T ), with T < +∞, starting at gij . Then there exist positive constants κ and ρ0 such that for any t0 ∈ [0, T ) and any point x0 ∈ M , the solution gij (t) is κ-noncollapsed at (x0 , t0 ) on all scales less than ρ0 . Proof. We argue by contradiction. Suppose that there are sequences pk ∈ M , tk ∈ [0, T ) and rk → 0 such that |Rm|(x, t) ≤ rk−2

(3.3.1)

for x ∈ Bk = Btk (pk , rk ) and tk − rk2 ≤ t ≤ tk , but (3.3.2)

1

ǫk = rk−1 V oltk (Bk ) n → 0 as k → ∞.

Without loss of generality, we may assume that tk → T as k → +∞.

Let τ¯(t) = tk − t, p = pk and Z n ˜ Vk (¯ τ) = (4π¯ τ )− 2 exp(−l(q, τ¯))dVtk −¯τ (q), M

where l(q, τ¯) is the Li-Yau-Perelman distance with respect to p = pk . Step 1. We first want to show that for k large enough, n

V˜k (ǫk rk2 ) ≤ 2ǫk2 . For any v ∈ Tp M we can find an L-geodesic γ(τ ) starting at p with lim

τ →0

√

τ γ(τ ˙ )

= v. Recall that γ(τ ) satisfies the equation (3.2.16). It follows from assumption (3.3.1) and Shi’s local derivative estimate (Theorem 1.4.2) that |∇R| has a bound in the order of 1/rk3 for t ∈ [tk − ǫk rk2 , tk ]. Thus by integrating (3.2.16) we see that for τ ≤ ǫk rk2 satisfying the property that γ(σ) ∈ Bk as long as σ < τ , there holds √ ˙ ) − v| ≤ Cǫk (|v| + 1) (3.3.3) | τ γ(τ where C is some positive constant depending only on the dimension. Here we have implicitly used the fact that the metric gij (t) is equivalent for x ∈ Bk and t ∈ [tk − ∂g ǫk rk2 , tk ]. In fact since ∂tij = −2Rij and |Rm| ≤ rk−2 on Bk × [tk − rk2 , tk ], we have (3.3.4)

e−2ǫk gij (x, tk ) ≤ gij (x, t) ≤ e2ǫk gij (x, tk ),

for x ∈ Bk and t ∈ [tk − ǫk rk2 , tk ]. −1

Suppose v ∈ Tp M with |v| ≤ 14 ǫk 2 . Let τ ≤ ǫk rk2 such √ that γ(σ) ∈ Bk as long as ˙ ) = v. Then, by (3.3.3) σ < τ , where γ is the L-geodesic starting at p with lim τ γ(τ τ →0


257

and (3.3.4), for k large enough, dtk (pk , γ(τ )) ≤

Z

τ

|γ(σ)| ˙ gij (tk ) dσ Z 1 − 1 τ dσ √ < ǫk 2 2 σ 0 − 12 √ = ǫk τ 0

≤ rk . This shows that for k large enough, L exp{|v|≤ 1 ǫ−1/2 } (ǫk rk2 ) ⊂ Bk = Btk (pk , rk ).

(3.3.5)

4 k

We now estimate the integral of V˜k (ǫk rk2 ) as follows, (3.3.6) V˜k (ǫk rk2 ) =

Z

(4πǫk rk2 )− 2 exp(−l(q, ǫk rk2 ))dVtk −ǫk rk2 (q) M Z n (4πǫk rk2 )− 2 exp(−l(q, ǫk rk2 ))dVtk −ǫk rk2 (q) = n

2 −1/2 (ǫk rk ) } {|v|≤ 1 ǫ 4 k

L exp

Z

+

(4πǫk rk2 )− 2 exp(−l(q, ǫk rk2 ))dVtk −ǫk rk2 (q). n

2 −1/2 (ǫk rk ) } {|v|≤ 1 ǫ 4 k

M\L exp

We observe that for each q ∈ Bk , Z

L(q, ǫk rk2 ) =

2 ǫ k rk

0

3 √ 3 τ (R + |γ| ˙ 2 )dτ ≥ −C(n)rk−2 (ǫk rk2 ) 2 = −C(n)ǫk2 rk ,

hence l(q, ǫk rk2 ) ≥ −C(n)ǫk . Thus, the first term on the RHS of (3.3.6) can be estimated by Z

(3.3.7)

(4πǫk rk2 )− 2 exp(−l(q, ǫk rk2 ))dVtk −ǫk rk2 (q) n

2 −1/2 (ǫk rk ) } {|v|≤ 1 ǫ 4 k

L exp

≤e

nǫk

Z

(4πǫk rk2 )− 2 exp(−l(q, ǫk rk2 ))dVtk (q) n

Bk −n 2

≤ enǫk (4π)− 2 · eC(n)ǫk · ǫk n

n

= e(n+C(n))ǫk (4π)− 2 · ǫk2 , n

· (rk−n Vol tk (Bk ))

where we have also used (3.3.5) and (3.3.4). Meanwhile, by using (3.2.18), (3.2.19) and the Jacobian Comparison Theorem

258


3.2.7, the second term on the RHS of (3.3.6) can be estimated as follows Z n (4πǫk rk2 )− 2 exp(−l(q, ǫk rk2 ))dVtk −ǫk rk2 (q) (3.3.8) M\L exp

−1 {|v|≤ 1 ǫ 2 } 4 k

Z

≤

2) (ǫk rk

n

(4πτ )− 2 exp(−l(τ ))J (τ )|τ =0 dv −1 2

{|v|> 41 ǫk

−n 2

= (4π)

}

Z

exp(−|v|2 )dv −1 2

{|v|> 41 ǫk

}

n 2

≤ ǫk , for k sufficiently large. Combining (3.3.6)-(3.3.8), we finish the proof of Step 1. Step 2. We next want to show n V˜k (tk ) = (4πtk )− 2

Z

exp(−l(q, tk ))dV0 (q) > C ′ M

for all k, where C ′ is some positive constant independent of k. It suffices to show the Li-Yau-Perelman distance l(·, tk ) is uniformly bounded from above on M . By Corollary 3.2.6 we know that the minimum of l(·, τ ) does not exceed T n 2 for each τ > 0. Choose qk ∈ M such that the minimum of l(·, tk − 2 ) is attained at qk . We now construct a path γ : [0, tk ] → M connecting pk to any given point q ∈ M as follows: the first half path γ|[0,tk − T ] connects pk to qk so that 2

1 T = q l qk , tk − 2 2 tk −

T 2

Z

tk − T2 0

√

τ (R + |γ(τ ˙ )|2 )dτ ≤

n 2

and the second half path γ|[tk − T ,tk ] is a shortest geodesic connecting qk to q with 2 respect to the initial metric gij (0). Then, for any q ∈ M n , 1 l(q, tk ) = √ L(q, tk ) 2 tk Z tk − T2 Z tk ! √ 1 ≤ √ + τ (R + |γ(τ ˙ )|2 )dτ 2 tk 0 tk − T2 ! r Z tk √ T 1 τ (R + |γ(τ ˙ )|2 )dτ n tk − + ≤ √ 2 2 tk tk − T2 ≤C for some constant C > 0, since all geometric quantities in gij are uniformly bounded when t ∈ [0, T2 ] (or equivalently, τ ∈ [tk − T2 , tk ]). Combining Step 1 with Step 2, and using the monotonicity of V˜k (τ ), we get n

C ′ < V˜k (tk ) ≤ V˜k (ǫk rk2 ) ≤ 2ǫk2 → 0


259

as k → ∞. This gives the desired contradiction. Therefore we have proved the theorem. The above no local collapsing theorem I says that if |Rm| ≤ r−2 on the parabolic ball {(x, t) | dt0 (x, x0 ) ≤ r, t0 − r2 ≤ t ≤ t0 }, then the volume of the geodesic ball Bt0 (x0 , r) (with respect to the metric gij (t0 )) is bounded from below by κrn . In [103], Perelman used the monotonicity of the W-functional (defined by (1.5.9)) to obtain a stronger version of the no local collapsing theorem, where the curvature bound assumption on the parabolic ball is replaced by that on the geodesic ball Bt0 (x0 , r). The following result, called no local collapsing theorem I′ , gives a further extension where the bound on the curvature tensor is replaced by the bound on the scalar curvature only. We now follow a clever argument by Bing-Long Chen. Theorem 3.3.3 (No local collapsing theorem I′ ). Suppose M is a compact Riemannian manifold, and gij (t), 0 ≤ t < T < +∞, is a solution to the Ricci flow. Then there exists a positive constant κ depending only the initial metric and T such that for any (x0 , t0 ) ∈ M × (0, T ) if R(x, t0 ) ≤ r−2 ,

√ with 0 < r ≤ T , then we have

∀x ∈ Bt0 (x0 , r)

Volt0 (Bt0 (x0 , r)) ≥ κrn . Proof. We will prove the assertion Volt0 (Bt0 (x0 , a)) ≥ κan

(∗)a

for all 0 < a ≤ r. Recall that Z −n −f 2 (4πτ ) e dV = 1 . µ(gij , τ ) = inf W(gij , f, τ ) M

Set

µ0 =

inf

0≤τ ≤2T

µ(gij (0), τ ) > −∞.

By Corollary 1.5.9, we have (3.3.9)

µ(gij (t0 ), b) ≥ µ(gij (0), t0 + b) ≥ µ0

for 0 < b ≤ r2 . Let 0 < ζ ≤ 1 be a positive smooth function on R where ζ(s) = 1 for |s| ≤ 21 , |ζ ′ |2 /ζ ≤ 20 everywhere, and ζ(s) is very close to zero for |s| ≥ 1. Define a function f on M by n dt0 (x, x0 ) n , (4πr2 )− 2 e−f (x) = e−c (4πr2 )− 2 ζ r R n where the constant c is chosen so that M (4πr2 )− 2 e−f dVt0 = 1. Then it follows from (3.3.9) that Z n (3.3.10) W(gij (t0 ), f, r2 ) = [r2 (|∇f |2 + R) + f − n](4πr2 )− 2 e−f dVt0 M

≥ µ0 .

260


Note that 1= ≥

Z

2 −n 2 −c

(4πr )

ZM

e

ζ

dt0 (x, x0 ) r

dVt0

(4πr2 )− 2 e−c dVt0 n

Bt0 (x0 , r2 )

n r . = (4πr2 )− 2 e−c Vol t0 Bt0 x0 , 2 By combining with (3.3.10) and the scalar curvature bound, we have Z ′ 2 (ζ ) n c≥− − log ζ · ζ e−c (4πr2 )− 2 dVt0 + (n − 1) + µ0 ζ M

≥ −2(20 + e−1 )e−c (4πr2 )− 2 Vol t0 (Bt0 (x0 , r)) + (n − 1) + µ0 n

≥ −2(20 + e−1 )

Volt0 (Bt0 (x0 , r)) + (n − 1) + µ0 , Vol t0 (Bt0 (x0 , 2r ))

where we used the fact that ζ(s) is very close to zero for |s| ≥ 1. Note also that Z Z n n 2 e−c (4πr2 )− 2 dVt0 ≥ (4πr2 )− 2 e−f dVt0 = 1. Bt0 (x0 ,r)

Let us set κ = min

M

1 1 exp(−2(20 + e−1 )3−n + (n − 1) + µ0 ), αn 2 2

where αn is the volume of the unit ball in Rn . Then we obtain n 1 Volt0 (Bt0 (x0 , r)) ≥ ec (4πr2 ) 2 2 n 1 ≥ (4π) 2 exp(−2(20 + e−1 )3−n + (n − 1) + µ0 ) · rn 2 ≥ κrn

provided Volt0 (Bt0 (x0 , r2 )) ≥ 3−n V olt0 (Bt0 (x0 , r)). Note that the above argument also works for any smaller radius a ≤ r. Thus we have proved the following assertion: (3.3.11)

Volt0 (Bt0 (x0 , a)) ≥ κan

whenever a ∈ (0, r] and Volt0 (Bt0 (x0 , 2a )) ≥ 3−n Volt0 (Bt0 (x0 , a)). Now we argue by contradiction to prove the assertion (∗)a for any a ∈ (0, r]. Suppose (∗)a fails for some a ∈ (0, r]. Then by (3.3.11) we have a Vol t0 (Bt0 (x0 , )) < 3−n Vol t0 (Bt0 (x0 , a)) 2 < 3−n κan a n . 0 with the following property: if gij (t) is a complete solution to the Ricci flow on 0 ≤ t ≤ r02 with bounded curvature and satifying |Rm|(x, t) ≤ r0−2

on B0 (x0 , r0 ) × [0, r02 ]

and Vol 0 (B0 (x0 , r0 )) ≥ A−1 r0n , then gij (t) is κ-noncollapsed on all scales less than r0 at every point (x, r02 ) with dr02 (x, x0 ) ≤ Ar0 .

264


Proof. From the evolution equation of the Ricci flow, we know that the metrics gij (·, t) are equivalent to each other on B0 (x0 , r0 ) × [0, r02 ]. Thus, without loss of generality, we may assume that the curvature of the solution is uniformly bounded for all t ∈ [0, r02 ] and all points in Bt (x0 , r0 ). Fix a point (x, r02 ) ∈ M × {r02 }. By scaling we may assume r0 = 1. We may also assume d1 (x, x0 ) = A. Let p = x, τ¯ = 1 − t, and consider Perelman’s reduced volume Z n ˜ V (¯ τ) = (4π¯ τ )− 2 exp(−l(q, τ¯))dV1−¯τ (q), M

where l(q, τ¯) = inf

1 √ 2 τ¯

Z

0

τ¯

√ τ (R + |γ| ˙ 2 )dτ | γ : [0, τ¯] → M with γ(0) = p, γ(¯ τ) = q

is the Li-Yau-Perelman distance. We argue by contradiction. Suppose for some 0 < r < 1 we have |Rm|(y, t) ≤ r−2 1

whenever y ∈ B1 (x, r) and 1 − r2 ≤ t ≤ 1, but ǫ = r−1 Vol 1 (B1 (x, r)) n is very small. Then arguing as in the proof of the no local collapsing theorem I (Theorem 3.3.2), we see that Perelman’s reduced volume n V˜ (ǫr2 ) ≤ 2ǫ 2 .

On the other hand, from the monotonicity of Perelman’s reduced volume we have Z −n 2 exp(−l(q, 1))dV0 (q) = V˜ (1) ≤ V˜ (ǫr2 ). (4π) M

Thus once we bound the function l(q, 1) over B0 (x0 , 1) from above, we will get the desired contradiction and will prove the theorem. For any q ∈ B0 (x0 , 1), exactly as in the proof of the no local collapsing theorem 1 I, we choose a path γ : [0, 1] → M with γ(0) = x, γ(1) = q, γ( 21 ) = y ∈ B 12 (x0 , 10 ) 1 and γ(τ ) ∈ B1−τ (x0 , 1) for τ ∈ [ 2 , 1] such that L(γ|[0, 21 ] ) = 2

r

1 1 1 l y, = L y, . 2 2 2

R1√ Now L(γ|[ 12 ,1] ) = 1 τ (R(γ(τ ), 1 − τ ) + |γ(τ ˙ )|2gij (1−τ ) )dτ is bounded from above by 2 a uniform constant since all geometric quantities in gij are uniformly bounded on {(y, t) | t ∈ [0, 1/2], y ∈ Bt (x0 , 1)} (where t ∈ [0, 1/2] is equivalent to τ ∈ [1/2, 1]). ¯ 1) = Thus all we need is to estimate the minimum of l(·, 12 ), or equivalently L(·, 2 1 1 1 4 2 l(·, 2 ), in the ball B 12 (x0 , 10 ). ¯ satisfies the differential inequality Recall that L (3.4.1)

¯ ∂L ¯ ≤ 2n. + ∆L ∂τ


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We will use this in a maximum principle argument. Let us define ¯ 1 − t) + 2n + 1) h(y, t) = φ(d(y, t) − A(2t − 1)) · (L(y, 1 ), where d(y, t) = dt (y, x0 ), and φ is a function of one variable, equal to 1 on (−∞, 20 1 1 and rapidly increasing to infinity on ( 20 , 10 ) in such a way that:

(3.4.2)

2

(φ′ )2 − φ′′ ≥ (2A + 100n)φ′ − C(A)φ φ

for some constant C(A) < +∞. The existence of such a function φ can be justified ′ as follows: put v = φφ , then the condition (3.4.2) for φ can be written as 3v 2 − v ′ ≥ (2A + 100n)v − C(A) which can be solved for v. Since the scalar curvature R evolves by ∂R 2 = ∆R + 2|Rc|2 ≥ ∆R + R2 , ∂t n we can apply the maximum principle as in Chapter 2 to deduce R(x, t) ≥ −

n for t ∈ (0, 1] and x ∈ M. 2t

Thus for τ¯ = 1 − t ∈ [0, 21 ], √ ¯ τ¯) = 2 τ¯ L(·,

Z

τ¯

√

τ (R + |γ| ˙ 2 )dτ Z τ¯ √ √ n dτ ≥ 2 τ¯ τ − 2(1 − τ ) 0 Z τ¯ √ √ ≥ 2 τ¯ τ (−n)dτ 0

0

> −2n.

That is 1 ¯ L(·, 1 − t) + 2n + 1 ≥ 1, for t ∈ ,1 . 2

(3.4.3)

1 ) and Clearly min h(y, 12 ) is achieved by some y ∈ B 21 (x0 , 10 y∈M

(3.4.4)

min h(y, 1) ≤ h(x, 1) = 2n + 1.

y∈M

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We compute ∂ ∂ ¯ 1 − t) + 2n + 1) −∆ h= − ∆ φ · (L(y, ∂t ∂t ∂ ¯ 1 − t) − 2h∇φ, ∇L(y, ¯ 1 − t)i +φ· − ∆ L(y, ∂t ∂ ′′ 2 ′ ¯ + 2n + 1) − ∆ d − 2A − φ |∇d| · (L = φ ∂t ∂ ¯ − 2h∇φ, ∇Li ¯ +φ· − −∆ L ∂τ ∂ ′′ ′ ¯ + 2n + 1) − ∆ d − 2A − φ · (L ≥ φ ∂t ¯ − 2nφ − 2h∇φ, ∇Li by using (3.4.1). At a minimizing point of h we have ¯ ∇L ∇φ =− ¯ . φ (L + 2n + 1) Hence ¯ =2 −2h∇φ, ∇Li

(φ′ )2 ¯ |∇φ|2 ¯ (L + 2n + 1) = 2 (L + 2n + 1). φ φ

Then at the minimizing point of h, we compute ∂ ∂ ¯ + 2n + 1) − ∆ h ≥ φ′ − ∆ d − 2A − φ′′ · (L ∂t ∂t (φ′ )2 ¯ (L + 2n + 1) − 2nφ + 2 φ ∂ ¯ + 2n + 1) − ∆ d − 2A − φ′′ · (L ≥ φ′ ∂t (φ′ )2 ¯ (L + 2n + 1) − 2nh + 2 φ for t ∈ [ 21 , 1] and ∆h ≥ 0. Let us denote by hmin (t) = min h(y, t). By applying Lemma 3.4.1(i) to the set where y∈M

φ′ 6= 0, we further obtain

(φ′ )2 d ′ ′′ ¯ hmin ≥ (L + 2n + 1) · φ (−100n − 2A) − φ + 2 − 2nhmin dt φ 1 ≥ −(2n + C(A))hmin , for t ∈ [ , 1]. 2

This implies that hmin (t) cannot decrease too fast. By combining (3.4.3) and (3.4.4) ¯ 1 ) in the ball B 1 (x0 , 1 ). we get the required estimate for the minimum L(·, 2 10 2 Therefore we have completed the proof of the theorem.


267

4. Formation of Singularities. Let gij (x, t) be a solution to the Ricci flow on M × [0, T ) and suppose [0, T ), T ≤ ∞, is the maximal time interval. If T < +∞, then the short time existence theorem tells us the curvature of the solution becomes unbounded as t → T . We then say the solution develops a singularity as t → T . As in the minimal surface theory and harmonic map theory, one usually tries to understand the structure of a singularity of the Ricci flow by rescaling the solution (or blow up) to obtain a sequence of solutions to the Ricci flow with uniformly bounded curvature on compact subsets and looking at its limit. The main purpose of this chapter is to establish a convergence theorem for a sequence of solutions to the Ricci flow with uniform bounded curvature on compact subsets and to use the convergence theorem to give a rough classification for singularities of solutions to the Ricci flow. Further studies on the structures of singularities of the Ricci flow will be given in Chapter 6 and 7. ∞ 4.1. Cheeger Type Compactness. We begin with the concept of Cloc convergence of tensors on a given manifold M . Let Ti be a sequence of tensors on M . We ∞ say that Ti converges to a tensor T in the Cloc topology if we can find a covering {(Us , ϕs )}, ϕs : Us → Rn , of C ∞ coordinate charts so that for every compact set K ⊂ M , the components of Ti converge in the C ∞ topology to the components of T in the intersections of K with these coordinate charts, considered as functions on ϕs (Us ) ⊂ Rn . Consider a Riemannian manifold (M, g). A marking on M is a choice of a point p ∈ M which we call the origin. We will refer to such a triple (M, g, p) as a marked Riemannian manifold.

Definition 4.1.1. Let (Mk , gk , pk ) be a sequence of marked complete Riemannian manifolds, with metrics gk and marked points pk ∈ Mk . Let B(pk , sk ) ⊂ Mk denote the geodesic ball centered at pk ∈ Mk and of radius sk (0 < sk ≤ +∞). We say a sequence of marked geodesic balls (B(pk , sk ), gk , pk ) with sk → s∞ (≤ +∞) ∞ converges in the Cloc topology to a marked (maybe noncomplete) manifold (B∞ , g∞ , p∞ ), which is an open geodesic ball centered at p∞ ∈ B∞ and of radius s∞ with respect to the metric g∞ , if we can find a sequence of exhausting open sets Uk in B∞ containing p∞ and a sequence of diffeomorphisms fk of the sets Uk in B∞ to open sets Vk in B(pk , sk ) ⊂ Mk mapping p∞ to pk such that the pull-back metrics g˜k = (fk )∗ gk converge in C ∞ topology to g∞ on every compact subset of B∞ . ∞ We remark that this concept of Cloc -convergence of a sequence of marked mani∞ folds (Mk , gk , pk ) is not the same as that of Cloc -convergence of metric tensors on a given manifold, even when we are considering the sequence of Riemannian metric gk on the same space M . This is because one can have a sequence of diffeomorphisms ∞ fk : M → M such that (fk )∗ gk converges in Cloc topology while gk itself does not converge. There have been a lot of work in Riemannian geometry on the convergence of a sequence of compact manifolds with bounded curvature, diameter and injectivity radius (see for example Gromov [53], Peters [106], and Greene and Wu [51]). The following theorem, which is a slight generalization of Hamilton’s convergence theorem [62], modifies these results in three aspects: the first one is to allow noncompact limits and then to avoid any diameter bound; the second one is to avoid having to assume a uniform lower bound for the injectivity radius over the whole manifold, a hypothesis which is much harder to satisfy in applications; the last one is to avoid a uniform curvature bound over the whole manifold so that we can take a local limit.

Theorem 4.1.2 (Hamilton [62]). Let (Mk , gk , pk ) be a sequence of marked

268


complete Riemannian manifolds of dimension n. Consider a sequence of geodesic balls B(pk , sk ) ⊂ Mk of radius sk (0 < sk ≤ ∞), with sk → s∞ (≤ ∞), around the base point pk of Mk in the metric gk . Suppose (a) for every radius r < s∞ and every integer l ≥ 0 there exists a constant Bl,r , independent of k, and positive integer k(r, l) < +∞ such that as k ≥ k(r, l), the curvature tensors Rm(gk ) of the metrics gk and their lth -covariant derivatives satisfy the bounds |∇l Rm(gk )| ≤ Bl,r on the balls B(pk , r) of radius r around pk in the metrics gk ; and (b) there exists a constant δ > 0 independent of k such that the injectivity radii inj (Mk , pk ) of Mk at pk in the metric gk satisfy the bound inj (Mk , pk ) ≥ δ. Then there exists a subsequence of the marked geodesic balls (B(pk , sk ), gk , pk ) which converges to a marked geodesic ball ∞ (B(p∞ , s∞ ), g∞ , p∞ ) in Cloc topology. Moreover the limit is complete if s∞ = +∞. Proof. In [62] and Theorem 16.1 of [63], Hamilton proved this convergence theorem for the case s∞ = +∞. In the following we only need to modify Hamilton’s argument to prove the remaining case of s∞ < +∞. Suppose we are given a sequence of geodesic balls (B(pk , sk ), gk , pk ) ⊂ (Mk , gk , pk ), with sk → s∞ (< +∞), satisfying the assumptions of Theorem 4.1.2. We will split the proof into three steps. Step 1: Picking the subsequence. By the local injectivity radius estimate (4.2.2) in Corollary 4.2.3 of the next section, we can find a positive decreasing C 1 function ρ(r), 0 ≤ r < s∞ , independent of k such that (4.1.1) (4.1.2)

1 (s∞ − r), 100 1 0 ≥ ρ′ (r) ≥ − , 1000

ρ(r)
0 is bound, each ball B(xα k,ρ k k some constant depending on r but independent of k. Now these balls are all disjoint and contained in the ball B(pk , (r + s∞ )/2). On the other hand, for large enough k, we can estimate the volume of this ball from above, again using the curvature bound, by a positive function of r that is independent of k. Thus there is a k ′ (r) > 0 such that for each k ≥ k ′ (r), there holds #{α | rkα ≤ r} ≤ λ(r)

(4.1.4)

˜˜α ) for some positive constant λ(r) depending only on r, and the geodesic balls B(xα k , 2ρ k for α ≤ λ(r) cover the ball B(pk , r). By the way, since rkα ≤ rkα−1 + ρ˜˜α−1 + ρ˜˜α k k ≤ rα−1 + 2ρ˜˜α−1 , k

k

and by (4.1.1) 1 (s∞ − rkα−1 ), ρ˜˜kα−1 ≤ 100 we get by induction 1 49 α−1 r + s∞ 50 k 50 α α−1 ! 49 1 49 49 0 ≤ rk + 1+ + ···+ s∞ 50 50 50 50 α 49 = 1− s∞ . 50

rkα ≤

So for each α, with α ≤ λ(r) (r < s∞ ), there holds λ(r) ! 49 α s∞ (4.1.5) rk ≤ 1 − 50 for all k. And by passing to a subsequence (using a diagonalization argument) we α may assume that rkα converges to some rα for each α. Then ρ˜˜α ˜α k (respectively ρ k , ρk ) α α α α α α converges to ρ˜˜ = ρ˜˜(r ) (respectively ρ˜ = ρ˜(r ), ρ = ρ(r )).

270


Hence for all α we can find k(α) such that 1 ˜α ˜α ρ˜ ≤ ρ˜k ≤ 2ρ˜˜α 2 1 α ρ˜ ≤ ρ˜α ˜α k ≤ 2ρ 2

and

1 α α ρ ≤ ρα k ≤ 2ρ 2

˜˜α are comparable when k is large enough whenever k ≥ k(α). Thus for all α, ρ˜˜α k and ρ so we can work with balls of a uniform size, and the same is true for ρ˜α ˜α , and k and ρ α α α α α α α α α α α ˆ ˆα. ˜ ˜ ˜ ˜ ˜ ρk and ρ . Let Bk = B(xk , 4ρ˜ ), then ρ˜k ≤ 2ρ˜ and B(xk , 2ρ˜k ) ⊂ B(xk , 4ρ˜ ) = B k So for every r if we let k(r) = max{k(α) | α ≤ λ(r)} then when k ≥ k(r), the balls ˆ α for α ≤ λ(r) cover the ball B(pk , r) as well. Suppose that B ˆ α and B ˆ β meet for B k k k β k ≥ k(α) and k ≥ k(β), and suppose rk ≤ rkα . Then, by the triangle inequality, we must have rkα ≤ rkβ + 4ρ˜˜α + 4ρ˜˜β ≤ rkβ + 8ρ˜˜β < rkβ + 16ρ˜βk . This then implies ρ˜˜βk = ρ˜˜(rkβ ) = ρ˜(rkβ + 20ρ˜(rkβ )) < ρ˜(rkα ) = ρ˜α k and hence ρ˜˜β ≤ 4ρ˜α . ˆ β ⊂ B(xα , 36ρ˜α ) whenever B ˆ α and B ˆ β meet and k ≥ max{k(α), k(β)}. Therefore B k k k k ˜ α = B(xα , ρ˜˜α /2). Note that B ˜α ˜˜α ) and B Next we define the balls Bkα = B(xα , 5 ρ k k k k α α α α α α ˜ ⊂ B(x , ρ˜˜ ). Since B ˆ ⊂ B , the balls B cover B(pk , r) for are disjoint since B k k k k k k α ≤ λ(r) as before. If Bkα and Bkβ meet for k ≥ k(α) and k ≥ k(β) and rkβ ≤ rkα , then by the triangle inequality we get rkα ≤ rkβ + 10ρ˜˜β < rkβ + 20ρ˜βk , and hence ρ˜˜β ≤ 4ρ˜α again. Similarly, ρ˜βk = ρ˜(rkβ ) = ρ(rkβ + 20ρ(rkβ )) < ρ(rkα ) = ρα k. This makes ρ˜β ≤ 4ρα . Now any point in Bkβ has distance at most 5ρ˜˜α + 5ρ˜˜β + 5ρ˜˜β ≤ 45ρ˜α β α from xα ˜α ). Likewise, whenever Bkα and Bkβ meet for k ≥ k(α) k , so Bk ⊂ B(xk , 45ρ and k ≥ k(β), any point in the larger ball B(xβk , 45ρ˜β ) has distance at most

5ρ˜˜α + 5ρ˜˜β + 45ρ˜β ≤ 205ρα


271

β α α ¯α from xα ˜β ) ⊂ B(xα ˜α ) k and hence B(xk , 45ρ k , 205ρ ). Now we define Bk = B(xk , 45ρ α ¯ = B(xα , 205ρα). Then the above discussion says that whenever B α and B β and B k k k k meet for k ≥ k(α) and k ≥ k(β), we have

(4.1.6)

¯β ⊂ B ¯α and B k. k

¯kα Bkβ ⊂ B

¯ α is still a nice embedded ball since, by (4.1.3), 205ρα ≤ 410ρα < Note that B k k inj(Mk , xα k ). We claim there exist positive numbers N (r) and k ′′ (r) such that for any given α with rα < r, as k ≥ k ′′ (r), there holds (4.1.7)

#{β | Bkα ∩ Bkβ 6= φ} ≤ N (r).

Indeed, if Bkα meets Bkβ then there is a positive k ′′ (α) such that as k ≥ k ′′ (α), rkβ ≤ rkα + 10ρ˜˜α k ≤ r + 20ρ(r) 1 ≤ r + (s∞ − r), 5 where we used (4.1.2) in the third inequality. Set k ′′ (r) = max{k ′′ (α), k ′ (r) | α ≤ λ(r)} and

1 N (r) = λ r + (s∞ − r) . 5 Then by combining with (4.1.4), these give the desired estimate (4.1.7) Next we observe that by passing to another subsequence we can guarantee that for any pair α and β we can find a number k(α, β) such that if k ≥ k(α, β) then either Bkα always meets Bkβ or it never does. Hence by setting ¯ k(r) = max k(α, β), k(α), k(β), k ′′ (r) | α ≤ λ(r) and

1 , β ≤ λ r + (s∞ − r) 5

¯ we have shown the following results: for every r < s∞ , if k ≥ k(r), we have α (i) the ball B(pk , r) in Mk is covered by the balls Bk for α ≤ λ(r), (ii) whenever Bkα and Bkβ meet for α ≤ λ(r), we have ¯kα Bkβ ⊂ B

¯ α, ¯β ⊂ B and B k k

(iii) for each α ≤ λ(r), there no more than N (r) balls ever meet Bkα , and ¯ (iv) for any α ≤ λ(r) and any β, either Bkα meets Bkβ for all k ≥ k(r) or none for ¯ all k ≥ k(r).

272


ˆ α, Eα, E ¯ α , and E¯ α be the balls of radii 4ρ˜˜α , 5ρ˜˜α , 45ρ˜α , and 205ρα Now we let E around the origin in Euclidean space Rn . At each point xα k ∈ B(pk , sk ) we define coordinate charts Hkα : E α → Bkα as the composition of a linear isometry of Rn to the tangent space Txαk Mk with the exponential map expxαk at xα k . We also get maps α α α ¯ ¯ ¯ α α α ¯ : E¯ → B ¯ and H k : E → B k in the same way. Note that (4.1.3) implies that H k k these maps are all well defined. We denote by gkα (and g¯kα and g¯α k ) the pull-backs of α ¯ α α ¯ the metric gk by Hk (and Hk and H k ). We also consider the coordinate transition ¯ α defined by ¯ α and J¯αβ : E ¯β → E functions Jkαβ : E β → E k ¯ kα )−1 H β Jkαβ = (H k

−1 ¯ β ¯α and J¯kαβ = (H Hk . k)

Clearly J¯kαβ Jkβα = I. Moreover Jkαβ is an isometry from gkβ to g¯kα and J¯kαβ from g¯kβ to g¯α k. Now for each fixed α, the metrics gkα are in geodesic coordinates and have their curvatures and their covariant derivatives uniformly bounded. Claim 1. By passing to another subsequence we can guarantee that for each α (and indeed all α by diagonalization) the metrics gkα (or g¯kα or g¯α k ) converge uniformly α ¯ α ) which α α α ¯ with their derivatives to a smooth metric g (or g¯ or g ) on E (or E¯ α or E is also in geodesic coordinates. Look now at any pair α, β for which the balls Bkα and Bkβ always meet for large k, and thus the maps Jkαβ (and J¯kαβ and Jkβα and J¯kβα ) are always defined for large k. Claim 2. The isometries Jkαβ (and J¯kαβ and Jkβα and J¯kβα ) always have a convergent subsequence. So by passing to another subsequence we may assume Jkαβ → J αβ (and J¯kαβ → ¯ α and J and Jkβα → J βα and J¯kβα → J¯βα ). The limit maps J αβ : E β → E α ¯ are isometries in the limit metrics g β and g α . Moreover ¯β → E J¯αβ : E ¯αβ

J αβ J¯βα = I. We are now done picking subsequences, except we still owe the reader the proofs of Claim 1 and Claim 2. Step 2: Finding local diffeomorphisms which are approximate isometries. Take the subsequence (B(pk , sk ), gk , pk ) chosen in Step 1 above. We claim that for every r < s∞ and every (ǫ1 , ǫ2 , . . . , ǫp ), and for all k and l sufficiently large in comparison, we can find a diffeomorphism Fkl of a neighborhood of the ball B(pk , r) ⊂ B(pk , sk ) into an open set in B(pl , sl ) which is an (ǫ1 , ǫ2 , . . . , ǫp ) approximate isometry in the sense that |t ∇Fkl ∇Fkl − I| < ǫ1 and |∇2 Fkl | < ǫ2 , . . . , |∇p Fkl | < ǫp where ∇p Fkl is the pth covariant derivative of Fkl . The idea (following Peters [106] or Greene and Wu [51]) of proving the claim is α α ¯ α ◦ (H ¯ α )−1 , resp.) from B α to B α to define the map Fkℓ = Hlα ◦ (Hkα )−1 (or F¯kℓ =H l k k ℓ


273

¯ α to B ¯ α , resp.) for k and ℓ large compared to α so as to be the identity map (or B k ℓ α ¯ α and H ¯ α , resp.), on E (or E¯ α , resp.) in the coordinate charts Hkα and Hℓα (or H k ℓ ¯ and then to define Fkℓ on a neighborhood of B(pk , r) for k, ℓ ≥ k(r) be averaging β the maps F¯kℓ for β ≤ λ(r + 51 (s∞ − r)). To describe the averaging process on Bkα with α ≤ λ(r) we only need to consider those Bkβ which meet Bkα ; there are never more than N (r) of them and each β ≤ λ(r + 15 (s∞ − r)), and they are the same for k ¯ and ℓ when k, ℓ ≥ k(r). The averaging process is defined by taking Fkℓ (x) to be the β center of mass of the F¯kℓ (x) for x ∈ Bkα averaging over those β where Bkβ meets Bkα β using weights µk (x) defined by a partition of unity. The center of mass of the points β y β = Fkℓ (x) with weights µβ is defined to be the point y such that X expy V β = y β and µβ V β = 0. When the points y β are all close and the weights µβ satisfy 0 ≤ µβ ≤ 1 then there will be a unique solution y close to y β which depends smoothly on the y β and the µβ (see [51] for the details). The point y is found by the inverse function theorem, which also provides bounds on all the derivatives of y as a function of the y β and the µβ . ¯ α , the map F¯ β = H ¯ β and B ¯β ⊆ B ¯ β ◦ (H ¯ β )−1 can be represented Since Bkα ⊆ B ℓ k ℓ kℓ l k in local coordinates by the map αβ ¯α Pkℓ : Eα → E

defined by αβ Pkℓ = J¯ℓαβ ◦ Jkβα .

Since Jkβα → J βα as k → ∞ and J¯ℓαβ → J¯αβ as ℓ → ∞ and J¯αβ ◦ J βα = I, we see αβ that the maps Pkℓ → I as k, ℓ → ∞ for each choice of α and β. The weights µβk are defined in the following way. We pick for each β a smooth function ψ β which equals ˆ β and equals 0 outside E β . We then transfer ψ β to a function ψ β on Mk by 1 on E k ¯ β (i.e. ψ β = ψ β ◦ (H ¯ β )−1 ). Then let the coordinate map H k k k .X µβk = ψkβ ψkγ γ

as usual. In the coordinate chart E α the function ψkβ looks like the composition of Jkβα with ψ β . Call this function ψkαβ = ψ β ◦ Jkβα . Then as k → ∞, ψkαβ → ψ αβ where ψ αβ = ψ β ◦ J βα .

In the coordinate chart E α the function µβk looks like .X αβ µαβ ψkαγ k = ψk γ

αβ and µαβ as k → ∞ where k → µ

µαβ = ψ αβ

.X γ

ψ αγ .

274


ˆ α cover B(pk , r), it follows that P ψ γ ≥ 1 on this set and by Since the sets B k γ k combining with (4.1.5) and (4.1.7) there is no problem bounding all these functions and their derivatives. There is a small problem in that we want to guarantee that the 0 averaged map still takes pk to pℓ . This is true at least for the map Fkℓ . Therefore it α will suffice to guarantee that µk = 0 in a neighborhood of pk if α 6= 0. This happens if the same is true for ψkα . If not, we can always replace ψkα by ψ˜kα = (1 − ψk0 )ψkα which P still leaves ψ˜kα ≥ 21 ψkα or ψk0 ≥ 12 everywhere, and this is sufficient to make γ ψ˜kγ ≥ 12 everywhere. αβ Now in the local coordinate E α we are averaging maps Pkℓ which converge to αβ the identity with respect to weights µk which converge. It follows that the averaged map converges to the identity in these coordinates. Thus Fkℓ can be made to be an (ǫ1 , ǫ2 , . . . , ǫp ) approximate isometry on B(pk , r) when k and ℓ are suitably large. At least the estimates |t ∇Fkℓ · ∇Fkℓ − I| < ǫ1 and |∇2 Fkℓ | < ǫ2 , . . . , |∇p Fkℓ | < ǫp on B(pk , r) follow from the local coordinates. We still need to check that Fkℓ is a diffeomorphism on a neighborhood of B(pk , r). This, however, follows quickly enough from the fact that we also get a map Fℓk on a slightly larger ball B(pℓ , r′ ) which contains the image of Fkℓ on B(pk , r) if we take r′ = (1 + ǫ1 )r, and Fℓk also satisfies the above estimates. Also Fkℓ and Fℓk fix the markings, so the composition Fℓk ◦ Fkℓ satisfies the same sort of estimates and fixes the origin pk . αβ αβ Since the maps Pkℓ and Pℓk converge to the identity as k, ℓ tend to infinity, Fℓk ◦Fkℓ must be very close to the identity on B(pk , r). It follows that Fkℓ is invertible. This finishes the proof of the claim and the Step 2. Step 3: Constructing the limit geodesic ball (B∞ , g∞ , p∞ ). We now know the geodesic balls (B(pk , sk ), gk , pk ) are nearly isometric for large k. We are now going to construct the limit B∞ . For a sequence of positive numbers rj ր s∞ with each rj < sj , we choose the numbers (ǫ1 (rj ), . . . , ǫj (rj )) so small that when we choose k(rj ) large in comparison and find the maps Fk(rj ),k(rj+1 ) constructed above on neighborhoods of B(pk(rj ) , rj ), in Mk(rj ) into Mk(rj+1 ) the image always lies in B(pk(rj+1 ) , rj+1 ) and the composition of Fk(rj ),k(rj+1 ) with Fk(rj+1 ),k(rj+2 ) and · · · and Fk(rs−1 ),k(rs ) for any s > j is still an (η1 (rj ), . . . , ηj (rj )) isometry for any choice of ηi (rj ), say ηi (rj ) = 1/j for 1 ≤ i ≤ j. Now we simplify the notation by writing Mj in place of Mk(rj ) and Fj in place of Fk(rj ),k(rj+1 ) . Then Fj : B(pj , rj ) → B(pj+1 , rj+1 ) is a diffeomorphism map from B(pj , rj ) into B(pj+1 , rj+1 ), and the composition Fs−1 ◦ · · · ◦ Fj : B(pj , j) → B(ps , s) is always an (η1 (rj ), . . . , ηj (rj )) approximate isometry. We now construct the limit B∞ as a topological space by identifying the balls B(pj , rj ) with each other using the homeomorphisms Fj . Given any two points x and y in B∞ , we have x ∈ B(pj , rj ) and y ∈ B(ps , rs ) for some j and s. If j ≤ s then x ∈ B(ps , rs ) also, by identification. A set in B∞ is open if and only if it intersects each B(pj , rj ) in an open set. Then choosing disjoint neighborhoods of x and y in


275

B(ps , rs ) gives disjoint neighborhoods of x and y in B∞ . Thus B∞ is a Hausdorff space. Any smooth chart on B(pj , rj ) also gives a smooth chart on B(ps , rs ) for all s > j. The union of all such charts gives a smooth atlas on B∞ . It is fairly easy to see the metrics gj on B(pj , rj ), converge to a smooth metric g∞ on B∞ uniformly together with all derivatives on compact sets. For since the Fs−1 ◦ · · · ◦ Fj are very good approximate isometries, the gj are very close to each other, and hence form a Cauchy sequence (together with their derivatives, in the sense that the covariant derivatives of gj with respect to gs are very small when j and s are both large). One checks in the usual way that such a Cauchy sequence converges. The origins pj are identified with each other, and hence with an origin p∞ in B∞ . Now it is the inverses of the maps identifying B(pj , rj ) with open subsets of B∞ that provide the diffeomorphisms of (relatively compact) open sets in B∞ into the geodesic balls B(pj , sj ) ⊂ Mj such that the pull-backs of the metrics gj converge to g∞ . This completes the proof of Step 3. Now it remains to prove both Claim 1 and Claim 2 in Step 1. Proof of Claim 1. It suffices to show the following general result: There exists a constant c > 0 depending only on the dimension, and constants Cq depending only on the dimension and q and bounds Bj on the curvature and its derivatives for j ≤ q where |Dj Rm| √ ≤ Bj , so that for any metric gkℓ in geodesic coordinates in the ball |x| ≤ r ≤ c/ B0 , we have 1 Ikℓ ≤ gkℓ ≤ 2Ikℓ 2 and ∂ ∂ j1 · · · jq gkℓ ≤ Cq , ∂x ∂x

where Ikℓ is the Euclidean metric.

Suppose√we are given a metric gij (x)dxi dxj in geodesic coordinates in the ball |x| ≤ r ≤ c/ B0 as in Claim 1. Then by definition every line through the origin is a geodesic (parametrized proportional to arc length) and gij = Iij at the origin. Also, the Gauss Lemma says that the metric gij is in geodesic coordinates if and only if gij xi = Iij xi . Note in particular that in geodesic coordinates |x|2 = gij xi xj = Iij xi xj is unambiguously defined. Also, in geodesic coordinates we have Γkij (0) = 0, and all the first derivatives for gjk vanish at the origin. Introduce the symmetric tensor Aij =

1 k ∂ x gij . 2 ∂xk

Since we have gjk xk = Ijk xk , we get xk

∂ ∂ gjk = Iij − gij = xk j gik i ∂x ∂x

276


and hence from the formula for Γijk xj Γijk = g iℓ Akℓ . Hence Akℓ xk = 0. Let Di be the covariant derivative with respect to the metric gij . Then Di xk = Iik + Γkij xj = Iik + g kℓ Aiℓ . Introduce the potential function P = |x|2 /2 =

1 gij xi xj . 2

We can use the formulas above to compute Di P = gij xj . Also we get Di Dj P = gij + Aij . The defining equation for P gives g ij Di P Dj P = 2P. If we take the covariant derivative of this equation we get g kℓ Dj Dk P Dℓ P = Dj P which is equivalent to Ajk xk = 0. But if we take the covariant derivative again we get g kℓ Di Dj Dk P Dℓ P + g kℓ Dj Dk P Di Dℓ P = Di Dj P. Now switching derivatives Di Dj Dk P = Di Dk Dj P = Dk Di Dj P + Rikjℓ g ℓm Dm P and if we use this and Di Dj P = gij + Aij and g kℓ Dℓ P = xk we find that xk Dk Aij + Aij + g kℓ Aik Ajℓ + Rikjℓ xk xℓ = 0. From our assumed curvature bounds we can take |Rijkℓ | ≤ B0 . Then we get the following estimate: |xk Dk Aij + Aij | ≤ C|Aij |2 + CB0 r2 on the ball |x| ≤ r for some constant C depending only on the dimension. We now show how to use the maximum principle on such equations. First of all, by a maximum principle argument, it is easy to show that if f is a function on a ball |x| ≤ r and λ > 0 is a constant, then ∂f λ sup |f | ≤ sup xk k + λf . ∂x


277

For any tensor T = {Ti···j } and any constant λ > 0, setting f = |T |2 in the above inequality, we have (4.1.8)

λ sup |T | ≤ sup |xk Dk T + λT |.

Applying this to the tensor Aij we get sup |Aij | ≤ C sup |Aij |2 + CB0 r2

|x|≤r

|x|≤r

for some constant depending only on the dimension. It is fairly elementary to see that there exist constants c > 0 and C0 < ∞ such that if√ the metric gij is in geodesic coordinates with |Rijkℓ | ≤ B0 in the ball of radius r ≤ c/ B0 then |Aij | ≤ C0 B0 r2 . Indeed, since the derivatives of gij vanish at the origin, so does Aij . Hence the estimate holds near the origin. But the inequality sup |Aij | ≤ C sup |Aij |2 + CB0 r2

|x|≤r

|x|≤r

says that |Aij | avoids an interval when c is chosen small. In fact the inequality X ≤ CX 2 + D is equivalent to |2CX − 1| ≥

√ 1 − 4CD

which makes X avoid an interval if 4CD < 1. (Hence in our case we need to choose c with 4C 2 c2 < 1.) Then if X is on the side containing 0 we get √ 1 − 1 − 4CD X≤ ≤ 2D. 2C This gives |Aij | ≤ C0 B0 r2 with C0 = 2C. We can also derive bounds on all the covariant derivatives of P in terms of bounds on the covariant derivatives of the curvature. To simplify the notation, we let Dq P = {Dj1 Dj2 · · · Djq P } denote the q th covariant derivative, and in estimating Dq P we will lump all the lower order terms into a general slush term Φq which will be a polynomial in D1 P, D2 P, . . . , Dq−1 P and Rm, D1 Rm, . . . , Dq−2 Rm. We already have estimates on a ball of radius r P ≤ r2 /2 |D1 P | ≤ r |Aij | ≤ C0 B0 r2

278


√ and since Di Dj P = gij + Aij and r ≤ c/ B0 if we choose c small we can make |Aij | ≤ 1/2, and we get |D2 P | ≤ C2 for some constant C2 depending only on the dimension. Start with the equation g ij Di P Dj P = 2P and apply repeated covariant derivatives. Observe that we get an equation which starts out g ij Di P Dq Dj P + · · · = 0 where the omitted terms only contain derivatives Dq P and lower. If we switch two derivatives in a term Dq+1 P or lower, we get a term which is a product of a covariant derivative of Rm of order at most q − 2 (since the two closest to P commute) and a covariant derivative of P of order at most q − 1; such a term can be lumped in with the slush term Φq . Therefore up to terms in Φq we can regard the derivatives as commuting. Then paying attention to the derivatives in D1 P we get an equation g ij Di P Dj Dk1 · · · Dkq P + g ij Di Dk1 P Dj Dk2 · · · Dkq P +g ij Di Dk2 P Dj Dk1 Dk3 · · · Dkq P + · · · + g ij Di Dkq P Dj Dk1 · · · Dkq−1 P = Dk1 · · · Dkq P + Φq . Recalling that Di Dj P = gij + Aij we can rewrite this as Φq = g ij Di P Dj Dk1 · · · Dkq + (q − 1)Dk1 · · · Dkq P

+ g ij Aik1 Dj Dk2 · · · Dkq P + · · · + g ij Aikq Dj Dk1 · · · Dkq−1 P.

Estimating the product of tensors in the usual way gives |xi Di Dq P + (q − 1)Dq P | ≤ q|A||Dq P | + |Φq |. Applying the inequality λ sup |T | ≤ sup |xk Dk T + λT | with T = Dq P gives (q − 1) sup |Dq P | ≤ sup(q|A||Dq P | + |Φq |). √ Now we can make |A| ≤ 1/2 by making r ≤ c/ B0 with c small; it is important here that c is independent of q! Then we get (q − 2) sup |Dq P | ≤ 2 sup |Φq | which is a good estimate for q ≥ 3. The term Φq is estimated inductively from the terms Dq−1 P and Dq−2 Rm and lower. This proves that there exist constants Cq for q ≥ 3 depending only on q and the dimension and on |Dj Rm| for j ≤ q − 2 such that |Dq P | ≤ Cq

√ on the ball r ≤ c/ B0 . Now we turn our attention to estimating the Euclidean metric Ijk and its covariant derivatives with respect to gjk . We will need the following elementary fact: suppose that f is a function on a ball |x| ≤ r with f (0) = 0 and ∂f i ≤ C|x|2 x i ∂x


279

for some constant C. Then |f | ≤ C|x|2

(4.1.9)

for the same constant C. As a consequence, if T = {Tj···k } is a tensor which vanishes at the origin and if |xi Di T | ≤ C|x|2 on a ball |x| ≤ r then |T | ≤ C|x|2 with the same constant C. (Simply apply the inequality (4.1.9) to the function f = |T |. In case this is not smooth, we can use p f = |T |2 + ǫ2 − ǫ and then let ǫ → 0.) Our application will be to the tensor Ijk which gives the Euclidean metric as a tensor in geodesic coordinates. We have Di Ijk = −Γpij Ipk − Γpik Ipj and since xi Γpij = g pq Ajq we get the equation xi Di Ijk = −g pq Ajp Ikq − g pq Akp Ijq . √ We already have |Ajk | ≤ C0 B0 |x|2 for |x| ≤ r ≤ c/ B0 . The tensor Ijk doesn’t vanish at the origin, but the tensor hjk = Ijk − gjk does. We can then use xi Di hjk = −g pq Ajp hkq − g pq Akq hjq − 2Ajk . Suppose M (s) = sup|x|≤s |hjk |. Then |xi Di hjk | ≤ 2[1 + M (s)]C0 B0 |x|2 and we get |hjk | ≤ 2[1 + M (s)]C0 B0 |x|2 on |x| ≤ s. This makes M (s) ≤ 2[1 + M (s)]C0 B0 s2 .

√ Then for s ≤ r ≤ c/ B0 with c small compared to C0 we get 2C0 B0 s2 ≤ 1/2 and M (s) ≤ 4C0 B0 s2 . Thus |Ijk − gjk | = |hjk | ≤ 4C0 B0 |x|2

√ for |x| ≤ r ≤ c/ B0 , and hence for c small enough

1 gjk ≤ Ijk ≤ 2gjk . 2

280


Thus the metrics are comparable. Note that this estimate only needs r small compared to B0 and does not need any bounds on the derivatives of the curvature. Now to obtain bounds on the covariant derivative of the Eucliden metric Ikℓ with respect to the Riemannian metric gkℓ we want to start with the equation xi Di Ikℓ + g mn Akm Iℓn + g mn Aℓm Ikn = 0 and apply q covariant derivatives Dj1 · · · Djq . Each time we do this we must interchange Dj and xi Di , and since this produces a term which helps we should look at it closely. If we write Rji = [Dj , Di ] for the commutator, this operator on tensors involves the curvature but no derivatives. Since Dj xi = Iji + g im Ajm we can compute [Dj , xi Di ] = Dj + g im Ajm Di + xi Rji and the term Dj in the commutator helps, while Ajm can be kept small and Rji is zero order. It follows that we get an equation of the form 0 = xi Di Dj1 · · · Djq Ikℓ + qDj1 · · · Djq Ikℓ +

q X

h=1

g im Ajh m Dj1 · · · Djh−1 Di Djh+1 · · · Djq Ikℓ

+ g mn Akm Dj1 · · · Djq Iℓn + g mn Aℓm Dj1 · · · Djq Ikn + Ψq , where the slush term Ψq is a polynomial in derivatives of Ikℓ of degree no more than q − 1 and derivatives of P of degree no more than q + 2 (remember xi = g ij Dj P and Aij = Di Dj P − gij ) and derivatives of the curvature Rm of degree no more than q − 1. We now estimate Dq Ikℓ = {Dj1 · · · Djq Ikℓ } by induction on q using (4.1.8) with λ = q. Noticing a total of q + 2 terms contracting Aij with a derivative of Ikℓ of degree q, we get the estimate q sup |Dq Ikℓ | ≤ (q + 2) sup |A| sup |Dq Ikℓ | + sup |Ψq |. and everything works. This proves that there exists a constant c > 0 depending only on the dimension, and constants Cq depending only on the dimension and q and bounds Bj on the curvature and its derivatives for j ≤ q where |Dj√Rm| ≤ Bj , so that for any metric gkℓ in geodesic coordinates in the ball |x| ≤ r ≤ c/ B0 the Euclidean metric Ikℓ satisfies 1 gkℓ ≤ Ikℓ ≤ 2gkℓ 2 and the covariant derivatives of Ikℓ with respect to gkℓ satisfy |Dj1 · · · Djq Ikℓ | ≤ Cq . The difference between a covariant derivative and an ordinary derivative is given by the connection −Γpij Ipk − Γpik Ipj


281

to get Γkij =

1 kℓ I (Dℓ Iij − Di Ijℓ − Dj Iiℓ ). 2

This gives us bounds on Γkij . We then obtain bounds on the first derivatives of gij from ∂ gjk = gkℓ Γℓij + gjℓ Γℓik . ∂xi Always proceeding inductively on the order of the derivative, we now get bounds on covariant derivatives of Γkij from the covariant derivatives of Ipk and bounds of the ordinary derivatives of Γkij by relating the to the covariant derivatives using the Γkij , and bounds on the ordinary derivatives of the gjk from bounds on the ordinary derivatives of the Γℓij . Consequently, we have estimates 1 Ikℓ ≤ gkℓ ≤ 2Ikℓ 2 and ∂ ∂ j1 · · · jq gkℓ ≤ C˜q ∂x ∂x

for similar constants C˜q . Therefore we have finished the proof of Claim 1.

Proof of Claim 2. We need to show how to estimate the derivatives of an isometry. We will prove that if y = F (x) is an isometry from a ball in Euclidean space with a metric gij dxi dxj to a ball in Euclidean space with a metric hkl dy k dy l . Then we can bound all of the derivatives of y with respect to x in terms of bounds on gij and its derivatives with respect to x and bound on hkl and its derivatives with respect to y. This would imply Claim 2. Since y = F (x) is an isometry we have the equation hpq

∂y p ∂y q = gjk . ∂xj ∂xk

Using bounds gjk ≤ CIjk and hpq ≥ cIpq comparing to the Euclidean metric, we easily get estimates ∂y p j ≤ C. ∂x

Now if we differentiate the equation with respect to xi we get hpq

∂ 2 y p ∂y q ∂y p ∂ 2 y q ∂gjk ∂hpq ∂y r ∂y p ∂y q + hpq j i k = − . i j k i ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂y r ∂xi ∂xj ∂xk

Now let Tijk = hpq

∂y p ∂ 2 y q ∂xi ∂xj ∂xk

282


and let Uijk =

∂hpq ∂y r ∂y p ∂y q ∂gjk − . i ∂x ∂y r ∂xi ∂xj ∂xk

Then the above equation says Tkij + Tjik = Uijk . Using the obvious symmetries Tijk = Tikj and Uijk = Uikj we can solve this in the usual way to obtain Tijk =

1 (Ujik + Ukij − Uijk ). 2

We can recover the second derivatives of y with respect to x from the formula ∂ 2yp ∂y p kℓ = g T . kij ∂xi ∂xj ∂xℓ Combining these gives an explicit formula giving ∂ 2 y p /∂xi ∂xj as a function of g ij , hpq , ∂gjk /∂xi , ∂hpq /∂y r , and ∂y p /∂y i . This gives bounds ∂ 2yp i j ≤ C ∂y ∂y

and bounds on all higher derivatives follow by differentiating the formula and using induction. This completes the proof of Claim 2 and hence the proof of Theorem 4.1.2. We now want to show how to use this convergence result on solutions to the Ricci flow. Let us first state the definition for the convergence of evolving manifolds. Definition 4.1.3. Let (Mk , gk (t), pk ) be a sequence of evolving marked complete Riemannian manifolds, with the evolving metrics gk (t) over a fixed time interval t ∈ (A, Ω], A < 0 ≤ Ω, and with the marked points pk ∈ Mk . We say a sequence of evolving marked (B0 (pk , sk ), gk (t), pk ) over t ∈ (A, Ω], where B0 (pk , sk ) are geodesic balls of (Mk , gk (0)) centered at pk with the radii sk → s∞ (≤ +∞), con∞ verges in the Cloc topology to an evolving marked (maybe noncomplete) manifold (B∞ , g∞ (t), p∞ ) over t ∈ (A, Ω], where, at the time t = 0, B∞ is a geodesic open ball centered at p∞ ∈ B∞ with the radius s∞ , if we can find a sequence of exhausting open sets Uk in B∞ containing p∞ and a sequence of diffeomorphisms fk of the sets Uk in B∞ to open sets Vk in B(pk , sk ) ⊂ Mk mapping p∞ to pk such that the pull-back metrics g˜k (t) = (fk )∗ gk (t) converge in C ∞ topology to g∞ (t) on every compact subset of B∞ × (A, Ω]. Now we fix a time interval A < t ≤ Ω with −∞ < A < 0 and 0 ≤ Ω < +∞. Consider a sequence of marked evolving complete manifolds (Mk , gk (t), pk ), t ∈ (A, Ω], with each gk (t), k = 1, 2, . . . , being a solution of the Ricci flow ∂ gk (t) = −2Ric k (t) ∂t on B0 (pk , sk ) × (A, Ω], where Rick is the Ricci curvature tensor of gk , and B0 (pk , sk ) is the geodesic ball of (Mk , gk (0)) centered at pk with the radii sk → s∞ (≤ +∞).


283

Assume that for each r < s∞ there are positive constants C(r) and k(r) such that the curvatures of gk (t) satisfy the bound |Rm(gk )| ≤ C(r) on B0 (pk , r)×(A, Ω] for all k ≥ k(r). We also assume that (Mk , gk (t), pk ), k = 1, 2, . . . , have a uniform injectivity radius bound at the origins pk at t = 0. By Shi’s derivatives estimate (Theorem 1.4.1), the above assumption of uniform bound of the curvatures on the geodesic balls B0 (pk , r) (r < s∞ ) implies the uniform bounds on all the derivatives of the curvatures at t = 0 on the geodesic balls B0 (pk , r) (r < s∞ ). Then by Theorem 4.1.2 we can find a subsequence of marked evolving manifolds, still denoted by (Mk , gk (t), pk ) with t ∈ (A, Ω], so that the geodesic balls (B0 (pk , sk ), gk (0), pk ) ∞ converge in the Cloc topology to a geodesic ball (B∞ (p∞ , s∞ ), g∞ (0), p∞ ). From now on, we consider this subsequence of marked evolving manifolds. By Definition 4.1.1, we have a sequence of (relatively compact) exhausting covering {Uk } of B∞ (p∞ , s∞ ) containing p∞ and a sequence of diffeomorphisms fk of the sets Uk in B∞ (p∞ , s∞ ) to open sets Vk in B0 (pk , sk ) mapping p∞ to pk such that the pull-back metrics at t = 0 C∞

loc g˜k (0) = (fk )∗ gk (0) −→ g∞ (0),

as k → +∞, on B∞ (p∞ , s∞ ).

However, the pull-back metrics g˜k (t) = (fk )∗ gk (t) are also defined at all times A < t ≤ Ω (although g∞ (t) is not yet). We also have uniform bounds on the curvature of the pull-back metrics g˜k (t) and all their derivatives, by Shi’s derivative estimates (Theorem 1.4.1), on every compact subset of B∞ (p∞ , s∞ ) × (A, Ω]. What we claim next is that we can find uniform bounds on all the covariant derivatives of the g˜k taken with respect to the fixed metric g∞ (0). Lemma 4.1.4. Let (M, g) be a Riemannian manifold, K a compact subset of M , and g˜k (t) a collection of solutions to Ricci flow defined on neighborhoods of K × [α, β] with [α, β] containing 0. Suppose that for each l ≥ 0, (a) C0−1 g ≤ g˜k (0) ≤ C0 g, on K, f or all k, (b) |∇l g˜k (0)| ≤ Cl , on K, f or all k, ˜ l Rm(˜ (c) |∇ gk )|k ≤ Cl′ , on K × [α, β], f or all k, k for some positive constants Cl , Cl′ , l = 0, 1, . . . , independent of k, where Rm(˜ gk ) are ˜ k denote covariant derivative with respect the curvature tensors of the metrics g˜k (t), ∇ to g˜k (t), | · |k are the length of a tensor with respect to g˜k (t), and | · | is the length with respect to g. Then the metrics g˜k (t) satisfy −1 C˜0 g ≤ g˜k (t) ≤ C˜0 g, on K × [α, β]

and |∇l g˜k | ≤ C˜l , on K × [α, β], l = 1, 2, . . . , for all k, where C˜l , l = 0, 1, . . . , are positive constants independent of k. Proof. First by using the equation ∂ ˜ k g˜k = −2Ric ∂t and the assumption (c) we immediately get (4.1.10)

−1 C˜0 g ≤ g˜k (t) ≤ C˜0 g, on K × [α, β]

284


for some positive constant C˜0 independent of k. ˜ k of g˜k and the Next we want to bound ∇˜ gk . The difference of the connection Γ connection Γ of g is a tensor. Taking Γ to be fixed in time, we get ∂ ˜ ∂ 1 ∂ ∂ ∂ γδ (Γk − Γ) = (˜ gk ) (˜ gk )δβ + (˜ gk )δα − δ (˜ gk )αβ ∂t ∂t 2 ∂xα ∂xβ ∂x h 1 ˜ k )α (−2(Ric ˜ k )βδ ) + (∇ ˜ k )β (−2(Ric ˜ k )αδ ) gk )γδ (∇ = (˜ 2 i ˜ k )δ (−2(Ric ˜ k )αβ ) − (∇ and then by the assumption (c) and (4.1.10), ∂ (Γ ˜ k − Γ) ≤ C, ∂t

for all k.

Note also that at a normal coordinate of the metric g at a fixed point and at the time t = 0, ∂ ∂ ∂ γδ ˜ k )γ − Γγ = 1 (˜ (4.1.11) g ) (˜ g ) + (˜ g ) − (˜ g ) (Γ k k δβ k δα k αβ αβ αβ 2 ∂xα ∂xβ ∂xδ 1 = (˜ gk )γδ (∇α (˜ gk )δβ + ∇β (˜ gk )δα − ∇δ (˜ gk )αβ ), 2 thus by the assumption (b) and (4.1.10), ˜ k (0) − Γ| ≤ C, for all k. |Γ Integrating over time we deduce that ˜ k − Γ| ≤ C, on K × [α, β], for all k. (4.1.12) |Γ By using the assumption (c) and (4.1.10) again, we have ∂ ˜ (∇˜ g ) k = | − 2∇Ric k | ∂t

˜ k Ric ˜ k + (Γ ˜ k − Γ) ∗ Ric ˜ k| = | − 2∇ ≤ C, for all k.

Hence by combining with the assumption (b) we get bounds (4.1.13) |∇˜ gk | ≤ C˜1 , on K × [α, β],

for some positive constant C˜1 independent of k. Further we want to bound ∇2 g˜k . Again regarding ∇ as fixed in time, we see ∂ ˜ k ). (∇2 g˜k ) = −2∇2 (Ric ∂t

Write ˜ k = (∇ − ∇ ˜ k )(∇Ric ˜ k) + ∇ ˜ k (∇ − ∇ ˜ k )Ric ˜ k +∇ ˜ 2 Ric ˜ k ∇2 Ric k ˜ k ) ∗ ∇Ric ˜ k+∇ ˜ k ((Γ − Γ ˜ k ) ∗ Ric ˜ k) + ∇ ˜ 2 Ric ˜ k = (Γ − Γ k ˜ k ) ∗ [(∇ − ∇ ˜ k )Ric ˜ k+∇ ˜ k Ric ˜ k] = (Γ − Γ ˜ k (˜ ˜ k) + ∇ ˜ 2 Ric ˜ k +∇ gk−1 ∗ ∇˜ gk ∗ Ric k ˜ k ) ∗ [(Γ − Γ ˜ k ) ∗ Ric ˜ k+∇ ˜ k Ric ˜ k] = (Γ − Γ ˜ k ˜ k (˜ ˜ k) + ∇ ˜ 2 Ric +∇ gk−1 ∗ ∇˜ gk ∗ Ric k


285

where we have used (4.1.11). Then by the assumption (c), (4.1.10), (4.1.12) and (4.1.13) we have |

∂ 2 ˜ k ∇˜ ∇ g˜k | ≤ C + C · |∇ gk | ∂t ˜ k − Γ) ∗ ∇˜ = C + C · |∇2 g˜k + (Γ gk | ≤ C + C|∇2 g˜k |.

Hence by combining with the assumption (b) we get |∇2 g˜k | ≤ C˜2 , on K × [α, β],

for some positive constant C˜2 independent of k. The bounds on the higher derivatives can be derived by the same argument. Therefore we have completed the proof of the lemma. We now apply the lemma to the pull-back metrics g˜k (t) = (fk )∗ gk (t) on B∞ (p∞ , s∞ ) × (A, Ω]. Since the metrics g˜k (0) have uniform bounds on their curvature and all derivatives of their curvature on every compact set of B∞ (p∞ , s∞ ) ∞ and converge to the metric g∞ (0) in Cloc topology, the assumptions (a) and (b) are certainly held for every compact subset K ⊂ B∞ (p∞ , s∞ ) with g = g∞ (0). For every compact subinterval [α, β] ⊂ (A, Ω], we have already seen from Shi’s derivative estimates (Theorem 1.4.1) that the assumption (c) is also held on K × [α, β]. Then all of the ∇l g˜k are uniformly bounded with respect to the fixed metric g = g∞ (0) on every compact set of B∞ (p∞ , s∞ ) × (A, Ω]. By using the classical Arzela-Ascoli theorem, we can find a subsequence which converges uniformly together with all its derivatives on every compact subset of B∞ (p∞ , s∞ ) × (A, Ω]. The limit metric will agree with that obtained previously at t = 0, where we know its convergence already. The limit g∞ (t), t ∈ (A, Ω], is now clearly itself a solution of the Ricci flow. Thus we obtain the following Cheeger type compactness theorem to the Ricci flow, which is essentially obtained by Hamilton in [62] and is called Hamilton’s compactness theorem. Theorem 4.1.5 (Hamilton’s compactness theorem). Let (Mk , gk (t), pk ), t ∈ (A, Ω] with A < 0 ≤ Ω, be a sequence of evolving marked complete Riemannian manifolds. Consider a sequence of geodesic balls B0 (pk , sk ) ⊂ Mk of radii sk (0 < sk ≤ +∞), with sk → s∞ (≤ +∞), around the base points pk in the metrics gk (0). Suppose each gk (t) is a solution to the Ricci flow on B0 (pk , sk ) × (A, Ω]. Suppose also (i) for every radius r < s∞ there exist positive constants C(r) and k(r) independent of k such that the curvature tensors Rm(gk ) of the evolving metrics gk (t) satisfy the bound |Rm(gk )| ≤ C(r), on B0 (pk , r) × (A, Ω] for all k ≥ k(r), and (ii) there exists a constant δ > 0 such that the injectivity radii of Mk at pk in the metric gk (0) satisfy the bound inj (Mk , pk , gk (0)) ≥ δ > 0, for all k = 1, 2, . . .. Then there exists a subsequence of evolving marked (B0 (pk , sk ), gk (t), pk ) over t ∈ ∞ (A, Ω] which converge in Cloc topology to a solution (B∞ , g∞ (t), p∞ ) over t ∈ (A, Ω] to the Ricci flow, where, at the time t = 0, B∞ is a geodesic open ball centered at p∞ ∈ B∞ with the radius s∞ . Moreover the limiting solution is complete if s∞ = +∞.

286


4.2. Injectivity Radius Estimates. We will use rescaling arguments to understand the formation of singularities and long-time behaviors of the Ricci flow. In view of the compactness property obtained in the previous section, on one hand one needs to control the bounds on the curvature, and on the other hand one needs to control the lower bounds of the injectivity radius. In applications we usually rescale the solution so that the (rescaled) curvatures become uniformly bounded on compact subsets and leave the injectivity radii of the (rescaled) solutions to be estimated in terms of curvatures. In this section we will review a number of such injectivity radius estimates in Riemannian geometry. In the end we will combine these injectivity estimates with Perelman’s no local collapsing theorem I′ to give the well-known little loop lemma to the Ricci flow which was conjectured by Hamilton in [63]. Let M be a Riemannian manifold. Recall that the injectivity radius at a point p ∈ M is defined by inj (M, p) = sup{r > 0 | expp : B(O, r)(⊂ Tp M ) → M is injective}, and the injectivity radius of M is inj (M ) = inf{inj (M, p) | p ∈ M }. We begin with a basic lemma due to Klingenberg (see for example, Corollary 5.7 in Cheeger & Ebin [22]). Klingenberg’s Lemma. Let M be a complete Riemannian manifold and let p ∈ M . Let lM (p) denote the minimal length of a nontrivial geodesic loop starting and ending at p (maybe not smooth at p). Then the injectivity radius of M at p satisfies the inequality 1 π √ , lM (p) inj (M, p) ≥ min Kmax 2 where √ Kmax denotes the supermum of the sectional curvature on M and we understand π/ Kmax to be positive infinity if Kmax ≤ 0.

Based on this lemma and a second variation argument, Klingenberg proved that the injectivity radius of an even-dimensional, compact, simply connected √ Riemannian manifold of positive sectional curvature is bounded from below by π/ Kmax . For odddimensional, compact, simply connected Riemannian manifold of positive sectional curvature, the same injectivity radius estimates was also proved by Klingenberg under an additional assumption that the sectional curvature is strictly 41 -pinched (see for example Theorem 5.9 and 5.10 in Cheeger & Ebin [22]). We also remark that in dimension 7, there exists a sequence of simply connected, homogeneous Einstein spaces whose sectional curvatures are positive and uniformly bounded from above but their injectivity radii converge to zero. (See [2].) The next result due to Gromoll and Meyer [52] shows that for complete, noncompact Riemannian manifold with positive sectional curvature, the above injectivity radius estimate actually holds without any restriction on dimension. Since the result and proof were not explicitly given in [52], we include a proof here. Theorem 4.2.1 (The Gromoll-Meyer injectivity radius estimate). Let M be a complete, noncompact Riemannian manifold with positive sectional curvature. Then the injectivity radius of M satisfies the following estimate π . inj (M ) ≥ √ Kmax


287

Proof. Let O be an arbitrary fixed √ point in M . We need to show that the injectivity radius at O is not less than π/ Kmax . We argue by contradiction. Suppose not, then by Klingenberg’s lemma there exists a closed geodesic loop γ on M starting and ending at O (may be not smooth at O). Since M has positive sectional curvature, we know from the work of GromollMeyer [52] (see also Proposition 8.5 in Cheeger & Ebin [22]) that there exists a compact totally convex subset C of M containing the geodesic loop γ. Among all geodesic loops starting and ending at the same point and lying entirely in the compact totally convex set C there will be a shortest one. Call it γ0 , and suppose γ0 starts and ends at a point we call p0 . First we claim that γ0 must be also smooth at the point p0 . Indeed by the curvature bound and implicit function theorem, there will be a geodesic loop γ˜ close to γ0 starting and ending at any point p˜ close to p0 . Let p˜ be along γ0 . Then by total convexity of the set C, γ˜ also lies entirely in C. If γ0 makes an angle different from π at p0 , the first variation formula will imply that γ˜ is shorter than γ0 . This contradicts with the choice of the geodesic loop γ0 being the shortest. Now let L : [0, +∞) → M be a ray emanating from p0 . Choose r > 0 large enough and set q = L(r). Consider the distance between q and the geodesic loop γ0 . It is clear that the distance can be realized by a geodesic β connecting the point q to a point p on γ0 . Let X be the unit tangent vector of the geodesic loop γ0 at p. Clearly X is orthogonal to the tangent vector of β at p. We then translate the vector X along the geodesic β to get a parallel vector field X(t), 0 ≤ t ≤ r. By using this vector field we can form a variation fixing one endpoint q and the other on γ0 such that the variational vector field is (1 − rt )X(t). The second variation of the arclength of this family of curves is given by t t X(t), 1 − X(t) I 1− r r 2 Z r t ∇ ∂ 1 − = X(t) ∂t r 0 t ∂ t ∂ , 1− X(t), , 1 − X(t) dt −R ∂t r ∂t r 2 Z r 1 t ∂ ∂ = − 1− R , X(t), , X(t) dt r r ∂t ∂t 0 0. Then, we have inj (M ) ≥ Cn (λ, D, v)

288


for some positive constant Cn (λ, D, v) depending only on λ, D, v and the dimension n. For general complete manifolds, it is possible to relate a lower injectivity radius bound to some lower volume bound provided one localizes the relevant geometric quantities appropriately. The following injectivity radius estimate, which was first obtained by Cheng-Li-Yau [35] for heat kernel estimates and later by Cheeger-Gromov-Taylor [27] with a wave equation argument, is a localized version of the above Cheeger’s Lemma. We now present an argument adapted from Abresch and Meyer [1]. Theorem 4.2.2 (Cheng-Li-Yau [35]). Let B(x0 , 4r0 ), 0 < r0 < ∞, be a geodesic ball in an n-dimensional complete Riemannian manifold (M, g) such that the sectional curvature K of the metric g on B(x0 , 4r0 ) satisfies the bounds λ≤K≤Λ for some constants √ λ and Λ. Then for any positive constant r ≤ r0 (we will also require r ≤ π/(4 Λ) if Λ > 0) the injectivity radius of M at x0 can be bounded from below by inj(M, x0 ) ≥ r ·

Vol (B(x0 , r)) , Vol (B(x0 , r)) + Vλn (2r)

where Vλn (2r) denotes the volume of a geodesic ball of radius 2r in the n-dimensional simply connected space form Mλ with constant sectional curvature λ. Proof. It is well known (cf. Lemma 5.6 in Cheeger and Ebin [22]) that 1 inj(M, x0 ) = min conjugate radius of x0 , lM (x0 ) 2 where lM (x0 ) denotes the length of the shortest (nontrivial) closed geodesic starting √ and ending at x0 . Since by assumption r ≤ π/(4 Λ) if Λ > 0, the conjugate radius of x0 is at least 4r. Thus it suffices to show (4.2.1)

lM (x0 ) ≥ 2r ·

Vol (B(x0 , r)) . Vol (B(x0 , r)) + Vλn (2r)

Now we follow the argument presented in [1]. The idea for proving this inequality, as indicated in [1], is to compare the geometry of the ball B(x0 , 4r) ⊆ B(x0 , 4r0 ) ⊂ ˜4r ⊂ Tx0 (M ), via the exponential map expx , M with the geometry of its lifting B 0 ˜4r → B(x0 , 4r) is a equipped with the pull-back metric g˜ = exp∗x0 g. Thus expx0 : B length-preserving local diffeomorphism. ˜r ⊂ B ˜4r with x Let x ˜0 , x ˜1 , . . . , x Ñ be the preimages of x0 in B ˜0 = 0. Clearly they one-to-one correspond to the geodesic loops γ0 , γ1 , . . . , γN at x0 of length less than r, where γ0 is the trivial loop. Now for each point x ˜i there exists exactly one isometric ˜r → B ˜4r mapping 0 to x˜i and such that exp ϕi = exp . immersion ϕi : B x0 x0 Without loss of generality, we may assume γ1 is the shortest nontrivial geodesic ˜r loop at x0 . By analyzing short homotopies, one finds that ϕi (˜ x) 6= ϕj (˜ x) for all x˜ ∈ B and 0 ≤ i < j ≤ N . This fact has two consequences: (a) N ≥ 2m, where m = [r/lM (x0 )]. To see this, we first observe that the points ˜r because ϕ1 is an isometric immersion ϕk1 (0), −m ≤ k ≤ m, are preimages of x0 in B

289


satisfying expx0 ϕ1 = expx0 . Moreover we claim they are distinct. For otherwise ϕ1 would act as a permutation on the set {ϕk1 (0) | − m ≤ k ≤ m}. Since the induced ˜r has the injectivity radius at least 2r, it follows from metric g˜ at each point in B ˜r is geodesically convex. Then the Whitehead theorem (see for example [22]) that B there would exist the unique center of mass y˜ ∈ B˜r . But then y˜ = ϕ0 (˜ y ) = ϕ1 (˜ y ), a contradiction. ˜2r . (b) Each point in B(x0 , r) has at least N +1 preimages in Ω = ∪N xi , r) ⊂ B i=0 B(˜ Hence by the Bishop volume comparison, ˜2r ) ≤ V n (2r). (N + 1)Vol (B(x0 , r)) ≤ Volg˜ (Ω) ≤ Volg˜ (B λ Now the inequality (4.2.1) follows by combining the fact N ≥ 2[r/lM (x0 )] with the above volume estimate. For our purpose of application, we now consider in a complete Riemannian manifold M a geodesic ball B(p0 , s0 ) (0 < s0 ≤ ∞) with the property that there exists a positive increasing function Λ : [0, s0 ) → [0, ∞) such that for any 0 < s < s0 the sectional curvature K on the ball B(p0 , s) of radius s around p0 satisfies the bound |K| ≤ Λ(s). Using Theorem 4.2.2, we can control the injectivity radius at any point p ∈ B(p0 , s0 ) in terms a positive constant that depends only on the dimension n, the injectivity radius at the base point p0 , the function Λ and the distance d(p0 , p) from p to p0 . We now proceed to derive such an estimate. The geometric insight of the following argument belongs to Yau [128] where he obtained a lower bound estimate for volume by comparing various geodesic balls. Indeed, it is a finite version of Yau’s Busemann function argument which gives the information on comparing geodesic balls with centers far apart. For any point p ∈ B(p0 , s0 ) with d(p0 , p) = s, set r0 = (s0 − s)/4 (we define r0 =1 if s0 = ∞). Define the set S to be the union of minimal geodesic segments that connect p to each point in B(p0 , r0 ). Now any point q ∈ S has distance at most r0 + r0 + s = s + 2r0

p from p0 and hence S ⊆ B(p0 , s + 2r0 ). For any 0 < r ≤ min{π/4 Λ(s + 2r0 ), r0 }, we denote by α(p, r) the sector S ∩B(p, r) of radius r and by α(p, s+r0 ) = S ∩B(p, s+r0 ). Let α−Λ(s+2r0 ) (r0 ) (resp. α−Λ(s+2r0 ) (s + r0 )) be a corresponding sector of the same “angles” with radius r0 (resp. s + r0 ) in the n-dimensional simply connected space form with constant sectional curvature −Λ(s+2r0 ). Since B(p0 , r0 ) ⊂ S ⊂ α(p, s+r0 ) and α(p, r) ⊂ B(p, r), the Bishop-Gromov volume comparison theorem implies that Vol (α(p, s + r0 )) Vol (B(p0 , r0 )) ≤ Vol (B(p, r)) Vol (α(p, r)) ≤

n V−Λ(s+2r (s + r0 ) Vol (α−Λ(s+2r0 ) (s + r0 )) 0) = . n Vol (α−Λ(s+2r0 ) (r)) V−Λ(s+2r0 ) (r)

Combining this inequality with the local injectivity radius estimate in Theorem 4.2.2, we get inj (M, p) ≥r

n V−Λ(s+2r (r) · Vol (B(p0 , r0 )) 0)

n n n V−Λ(s+2r (r)Vol (B(p0 , r0 )) + V−Λ(s+2r (2r)V−Λ(s+2r (s + 2r0 ) 0) 0) 0)

.

290


Thus, we have proved the following Corollary 4.2.3. Suppose B(p0 , s0 ) (0 < s0 ≤ ∞) is a geodesic ball in an n-dimensional complete Riemannian manifold M having the property that for any 0 < s < s0 the sectional curvature K on B(p0 , s) satisfies the bound |K| ≤ Λ(s) for some positive increasing function Λ defined on [0, s0 ). Then for any point p p ∈ B(p0 , s0 ) with d(p0 , p) = s and any positive number r ≤ min{π/4 Λ(s + 2r0 ), r0 } with r0 = (s0 − s)/4, the injectivity radius of M at p is bounded below by inj (M, p) ≥r

n V−Λ(s+2r (r) · Vol (B(p0 , r0 )) 0) n n n V−Λ(s+2r0 ) (r)Vol (B(p0 , r0 )) + V−Λ(s+2r (2r)V−Λ(s+2r (s 0) 0)

+ 2r0 )

.

In particular, we have inj (M, p) ≥ ρn,δ,Λ (s)

(4.2.2)

where δ > 0 is a lower bound of the injectivity radius inj (M, p0 ) at the origin p0 and ρn,δ,Λ : [0, s0 ) → R+ is a positive decreasing function that depends only on the dimension n, the lower bound δ of the injectivity radius inj (M, p0 ), and the function Λ. We remark that in the above discussion if s0 = ∞ then we can apply the standard Bishop relative volume comparison theorem to geodesic balls directly. Indeed, for any ∆ p ∈ M and any positive constants r and r0 , we have B(p0 , r0 ) ⊆ B(p, rˆ) with rˆ = max{r, r0 +d(p0 , p)}. Suppose in addition the curvature K on M is uniformly bounded by λ ≤ K ≤ Λ for some constants λ and Λ, then the Bishop volume comparison theorem implies that Vol (B(p, rˆ)) Vλ (ˆ r) Vol (B(p0 , r0 )) ≤ ≤ . Vol (B(p, r)) Vol (B(p, r)) Vλ (r) Hence (4.2.3)

inj (M, p) ≥ r

Vλn (r) · Vol (B(p0 , r0 )) . Vλn (r)Vol (B(p0 , r0 )) + Vλn (2r)Vλn (ˆ r)

So we see that the injectivity radius inj(M, p) at p falls off at worst exponentially as the distance d(p0 , p) goes to infinity. In other words, (4.2.4)

√ √ c inj (M, p) ≥ √ (δ B)n e−C Bd(p,p0 ) B

where B is an upper bound on the absolute value of the √ sectional curvature, δ is a lower bound on the injectivity radius at p0 with δ < c/ B, and c > 0 and C < +∞ are positive constants depending only on the dimension n. Finally, by combining Theorem 4.2.2 with Perelman’s no local collapsing Theorem I′ (Theorem 3.3.3) we immediately obtain the following important Little Loop Lemma conjectured by Hamilton [63].


291

Theorem 4.2.4 (Little Loop Lemma). Let gij (t), 0 ≤ t < T < +∞, be a solution of the Ricci flow on a compact manifold M . Then there exists a constant ρ > 0 having the following property: if at a point x0 ∈ M and a time t0 ∈ [0, T ), |Rm|(·, t0 ) ≤ r−2

√

on Bt0 (x0 , r)

for some r ≤ T , then the injectivity radius of M with respect to the metric gij (t0 ) at x0 is bounded from below by inj (M, x0 , gij (t0 )) ≥ ρr. 4.3. Limiting Singularity Models. Consider a solution gij (x, t) of the Ricci flow on M × [0, T ), T ≤ +∞, where either M is compact or at each time t the metric gij (·, t) is complete and has bounded curvature. We say that gij (x, t) is a maximal solution if either T = +∞ or T < +∞ and |Rm| is unbounded as t → T . Denote by Kmax (t) = sup |Rm(x, t)|gij (t) . x∈M

Definition 4.3.1. We say that {(xk , tk ) ∈ M ×[0, T )}, k = 1, 2, . . ., is a sequence of (almost) maximum points if there exist positive constants c1 and α ∈ (0, 1] such that α |Rm(xk , tk )| ≥ c1 Kmax (t), t ∈ [tk − , tk ] Kmax (tk ) for all k. Definition 4.3.2. We say that the solution satisfies injectivity radius condition if for any sequence of (almost) maximum points {(xk , tk )}, there exists a constant c2 > 0 independent of k such that c2 inj (M, xk , gij (tk )) ≥ p Kmax (tk )

for all k.

Clearly, by the Little Loop Lemma, a maximal solution on a compact manifold with the maximal time T < +∞ always satisfies the injectivity radius condition. Also by the Gromoll-Meyer injectivity radius estimate, a solution on a complete noncompact manifold with positive sectional curvature also satisfies the injectivity radius condition. According to Hamilton [63], we classify maximal solutions into three types; every maximal solution is clearly of one and only one of the following three types: Type I:

T < +∞ and

sup (T − t)Kmax (t) < +∞;

t∈[0,T )

Type II: (a) T < +∞ but

sup (T − t)Kmax (t) = +∞;

t∈[0,T )

(b) T = +∞ but

sup tKmax (t) = +∞; t∈[0,T )

292


Type III:

(a) T = +∞,

sup tKmax (t) < +∞, and t∈[0,T )

lim sup tKmax (t) > 0; t→+∞

(b) T = +∞,

sup tKmax (t) < +∞, and t∈[0,T )

lim sup tKmax (t) = 0; t→+∞

It seems that Type III (b) is not compatible with the injectivity radius condition unless it is a trivial flat solution. Indeed under the Ricci flow the length of a curve γ connecting two points x0 , x1 ∈ M evolves by Z d Lt (γ) = −Ric (γ, ˙ γ)ds ˙ dt γ ≤ C(n)Kmax (t) · Lt (γ) ǫ ≤ Lt (γ), as t large enough, t

for arbitrarily fixed ǫ > 0. Thus when we are considering the Ricci flow on a compact manifold, the diameter of the evolving manifold grows at most as tǫ . But the curvature of the evolving manifold decays faster than t−1 . This says, as choosing ǫ > 0 small enough, diamt (M )2 · |Rm(·, t)| → 0, as t → +∞. Then it is well-known from Cheeger-Gromov [54] that the manifold is a nilmanifold and the injectivity radius condition can not be satisfied as t large enough. When we are considering the Ricci flow on a complete noncompact manifold with nonnegative curvature operator or on a complete noncompact Kähler manifold with nonnegative holomorphic bisectional curvature, Li-Yau-Hamilton inequalities imply that tR(x, t) is increasing in time t. Then Type III(b) occurs only when the solution is a trivial flat metric. For each type of solution we define a corresponding type of limiting singularity model. Definition 4.3.3. A solution gij (x, t) to the Ricci flow on the manifold M , where either M is compact or at each time t the metric gij (·, t) is complete and has bounded curvature, is called a singularity model if it is not flat and of one of the following three types: Type I: The solution exists for t ∈ (−∞, Ω) for some constant Ω with 0 < Ω < +∞ and |Rm| ≤ Ω/(Ω − t) everywhere with equality somewhere at t = 0;


293

Type II: The solution exists for t ∈ (−∞, +∞) and |Rm| ≤ 1 everywhere with equality somewhere at t = 0; Type III: The solution exists for t ∈ (−A, +∞) for some constant A with 0 < A < +∞ and |Rm| ≤ A/(A + t) everywhere with equality somewhere at t = 0. Theorem 4.3.4. For any maximal solution to the Ricci flow which satisfies the injectivity radius condition and is of Type I, II(a), (b), or III(a), there exists a sequence of dilations of the solution along (almost) maximum points which converges ∞ in the Cloc topology to a singularity model of the corresponding type. Proof. and

Type I: We consider a maximal solution gij (x, t) on M × [0, T ) with T < +∞ ∆

Ω = lim sup(T − t)Kmax (t) < +∞. t→T

First we note that Ω > 0. Indeed by the evolution equation of curvature, d 2 Kmax (t) ≤ Const · Kmax (t). dt This implies that Kmax (t) · (T − t) ≥ Const > 0, because lim sup Kmax (t) = +∞. t→T

Thus Ω must be positive. Choose a sequence of points xk and times tk such that tk → T and lim (T − tk )|Rm(xk , tk )| = Ω.

k→∞

Denote by ǫk = p

1 . |Rm(xk , tk )|

We translate in time so that tk becomes 0, dilate in space by the factor ǫk and dilate in time by ǫ2k to get (k)

2˜ g˜ij (·, t˜) = ǫ−2 k gij (·, tk + ǫk t),

t˜ ∈ [−tk /ǫ2k , (T − tk )/ǫ2k ).

294


Then ∂ (k) ˜ ∂ g˜ (·, t) = ǫ−2 gij (·, t) · ǫ2k k ∂t ∂ t˜ ij = −2Rij (·, tk + ǫ2k t˜) ˜ (k) (·, t˜), = −2R ij

˜ (k) is the Ricci curvature of the metric g˜(k) . So g˜(k) (·, t˜) is still a solution to where R ij ij ij the Ricci flow which exists on the time interval [−tk /ǫ2k , (T − tk )/ǫ2k ), where tk /ǫ2k = tk |Rm(xk , tk )| → +∞ and (T − tk )/ǫ2k = (T − tk )|Rm(xk , tk )| → Ω. For any ǫ > 0 we can find a time τ < T such that for t ∈ [τ, T ), |Rm| ≤ (Ω + ǫ)/(T − t) (k) by the assumption. Then for t˜ ∈ [(τ − tk )/ǫ2k , (T − tk )/ǫ2k ), the curvature of g˜ij (·, t˜) is bounded by

˜ (k) | = ǫ2 |Rm| |Rm k

≤ (Ω + ǫ)/((T − t)|Rm(xk , tk )|) = (Ω + ǫ)/((T − tk )|Rm(xk , tk )| + (tk − t)|Rm(xk , tk )|) → (Ω + ǫ)/(Ω − t˜), as k → +∞.

This implies that {(xk , tk )} is a sequence of (almost) maximum points. And then by the injectivity radius condition and Hamilton’s compactness theorem 4.1.5, there (k) ∞ exists a subsequence of the metrics g˜ij (t˜) which converges in the Cloc topology to a (∞) ˜ (∞) ˜ limit metric g˜ij (t) on a limiting manifold M with t˜ ∈ (−∞, Ω) such that g˜ij (t˜) is a complete solution of the Ricci flow and its curvature satisfies the bound ˜ (∞) | ≤ Ω/(Ω − t˜) |Rm ˜ × (−∞, Ω) with the equality somewhere at t˜ = 0. everywhere on M Type II(a): We consider a maximal solution gij (x, t) on M × [0, T ) with T < +∞ and

lim sup(T − t)Kmax (t) = +∞. t→T

Let Tk < T < +∞ with Tk → T , and γk ր 1, as k → +∞. Pick points xk and times tk such that, as k → +∞, (Tk − tk )|Rm(xk , tk )| ≥ γk

sup x∈M,t≤Tk

(Tk − t)|Rm(x, t)| → +∞.

Again denote by ǫk = p

1 |Rm(xk , tk )|


295

and dilate the solution as before to get (k) 2˜ g˜ij (·, t˜) = ǫ−2 k gij (·, tk + ǫk t),

t˜ ∈ [−tk /ǫ2k , (Tk − tk )/ǫ2k ),

which is still a solution to the Ricci flow and satisfies the curvature bound ˜ (k) | = ǫ2k |Rm| |Rm

1 (Tk − tk ) · γk (Tk − t) tk (Tk − tk ) 1 (Tk − tk )|Rm(xk , tk )| ˜ for t ∈ − 2 , = , γk [(Tk − tk )|Rm(xk , tk )| − t˜] ǫk ǫ2k p since t = tk +ǫ2k t˜ and ǫk = 1/ |Rm(xk , tk )|. Hence {(xk , tk )} is a sequence of (almost) maximum points. And then as before, by applying Hamilton’s compactness theorem (k) ∞ 4.1.5, there exists a subsequence of the metrics g˜ij (t˜) which converges in the Cloc (∞) ˜ and t˜ ∈ (−∞, +∞) such that topology to a limit g˜ij (t˜) on a limiting manifold M ≤

(∞)

g˜ij (t˜) is a complete solution of the Ricci flow and its curvature satisfies ˜ (∞) | ≤ 1 |Rm

˜ × (−∞, +∞) and the equality holds somewhere at t˜ = 0. everywhere on M

Type II(b): We consider a maximal solution gij (x, t) on M × [0, T ) with T = +∞ and

lim sup tKmax (t) = +∞. t→T

Again let Tk → T = +∞, and γk ր 1, as k → +∞. Pick xk and tk such that tk (Tk − tk )|Rm(xk , tk )| ≥ γk

sup x∈M,t≤Tk

t(Tk − t)|Rm(x, t)|.

Define (k) 2˜ g˜ij (·, t˜) = ǫ−2 k gij (·, tk + ǫk t), p where ǫk = 1/ |Rm(xk , tk )|. Since

tk (Tk − tk )|Rm(xk , tk )| ≥ γk ≥ γk ≥ we have

sup x∈M,t≤Tk

t(Tk − t)|Rm(x, t)|

sup x∈M,t≤Tk /2

t(Tk − t)|Rm(x, t)|

Tk γk sup t|Rm(x, t)|, 2 x∈M,t≤Tk /2

Tk T k − tk

γk (Tk − tk ) = (Tk − tk )|Rm(xk , tk )| ≥ ǫ2k 2

tk γk = tk |Rm(xk , tk )| ≥ 2 ǫk 2

t˜ ∈ [−tk /ǫ2k , (Tk − tk )/ǫ2k ),

and

sup x∈M,t≤Tk /2

Tk tk

t|Rm(x, t)| → +∞,

sup x∈M,t≤Tk /2

t|Rm(x, t)| → +∞,

296


as k → +∞. As before, we also have ∂ (k) ˜ ˜ (k) (·, t˜) g˜ij (·, t) = −2R ij ˜ ∂t and ˜ (k) | |Rm = ǫ2k |Rm| 1 tk (Tk − tk ) · γk t(Tk − t) 1 tk (Tk − tk )|Rm(xk , tk )| = · γk (tk + ǫ2k t˜)[(Tk − tk ) − ǫ2k t˜] · |Rm(xk , tk )| tk (Tk − tk )|Rm(xk , tk )| 1 · = γk (tk + ǫ2k t˜)[(Tk − tk )|Rm(xk , tk )| − t˜] tk (Tk − tk )|Rm(xk , tk )| = γk (1+ t˜/(tk |Rm(xk , tk )|))[tk (Tk −tk )|Rm(xk , tk )|](1− t˜/((Tk −tk )|Rm(xk , tk )|)) ≤

→ 1,

as k → +∞.

Hence {(xk , tk )} is again a sequence of (almost) maximum points. As before, there (k) ∞ exists a subsequence of the metrics g˜ij (t˜) which converges in the Cloc topology to (∞) ˜ (∞) ˜ ˜ a limit g˜ij (t) on a limiting manifold M and t ∈ (−∞, +∞) such that g˜ij (t˜) is a complete solution of the Ricci flow and its curvature satisfies ˜ (∞) | ≤ 1 |Rm ˜ × (−∞, +∞) with the equality somewhere at t˜ = 0. everywhere on M Type III(a): T = +∞ and

We consider a maximal solution gij (x, t) on M × [0, T ) with lim sup tKmax (t) = A ∈ (0, +∞). t→T

Choose a sequence of xk and tk such that tk → +∞ and lim tk |Rm(xk , tk )| = A.

k→∞

p Set ǫk = 1/ |Rm(xk , tk )| and dilate the solution as before to get (k) 2˜ g˜ij (·, t˜) = ǫ−2 k gij (·, tk + ǫk t),

t˜ ∈ [−tk /ǫ2k , +∞)

which is still a solution to the Ricci flow. Also, for arbitrarily fixed ǫ > 0, there exists a sufficiently large positive constant τ such that for t ∈ [τ, +∞), ˜ (k) | = ǫ2k |Rm| |Rm A+ǫ 2 ≤ ǫk t A+ǫ 2 = ǫk tk + ǫ2k t˜

= (A + ǫ)/(tk |Rm(xk , tk )| + t˜),

for t˜ ∈ [(τ − tk )/ǫ2k , +∞).


297

Note that (A + ǫ)/(tk |Rm(xk , tk )| + t˜) → (A + ǫ)/(A + t˜),

as k → +∞

and (τ − tk )/ǫ2k → −A,

as k → +∞.

Hence {(xk , tk )} is a sequence of (almost) maximum points. And then as before, there (k) ∞ exists a subsequence of the metrics g˜ij (t˜) which converges in the Cloc topology to (∞) ˜ (∞) ˜ ˜ a limit g˜ij (t) on a limiting manifold M and t ∈ (−A, +∞) such that g˜ij (t˜) is a complete solution of the Ricci flow and its curvature satisfies ˜ (∞) | ≤ A/(A + t˜) |Rm ˜ × (−A, +∞) with the equality somewhere at t˜ = 0. everywhere on M In the case of manifolds with nonnegative curvature operator, or Kähler metrics with nonnegative holomorphic bisectional curvature, we can bound the Riemannian curvature by the scalar curvature R upto a constant factor depending only on the dimension. Then we can slightly modify the statements in the previous theorem as follows Corollary 4.3.5. For any complete maximal solution to the Ricci flow with bounded and nonnegative curvature operator on a Riemannian manifold, or on a K¨ ahler manifold with bounded and nonnegative holomorphic bisectional curvature, there exists a sequence of dilations of the solution along (almost ) maximum points which converges to a singular model. For Type I solutions: the limit model exists for t ∈ (−∞, Ω) with 0 < Ω < +∞ and has R ≤ Ω/(Ω − t) everywhere with equality somewhere at t = 0. For Type II solutions: the limit model exists for t ∈ (−∞, +∞) and has R≤1 everywhere with equality somewhere at t = 0. For Type III solutions: the limit model exists for t ∈ (−A, +∞) with 0 < A < +∞ and has R ≤ A/(A + t) everywhere with equality somewhere at t = 0. A natural and important question is to understand each of the three types of singularity models. The following results obtained by Hamilton [66] and Chen-Zhu [30] characterize the Type II and Type III singularity models with nonnegative curvature operator and positive Ricci curvature respectively. The corresponding results in the Kähler case with nonnegative holomorphic bisectional curvature were obtained by the first author [14].

298


Theorem 4.3.6. (i) (Hamilton [66]) Any Type II singularity model with nonnegative curvature operator and positive Ricci curvature to the Ricci flow on a manifold M must be a (steady) Ricci soliton. (ii) (Chen-Zhu [30]) Any Type III singularity model with nonnegative curvature operator and positive Ricci curvature on a manifold M must be a homothetically expanding Ricci soliton. Proof. We only give the proof of (ii), since the proof of (i) is similar and easier. After a shift of the time variable, we may assume the Type III singularity model is defined on 0 < t < +∞ and tR assumes its maximum in space-time. Recall from the Li-Yau-Hamilton inequality (Theorem 2.5.4) that for any vectors V i and W i , (4.3.1)

Mij W i W j + (Pkij + Pkji )V k W i W j + Rikjl W i W j V k V l ≥ 0,

where 1 1 Mij = ∆Rij − ∇i ∇j R + 2Ripjq Rpq − g pq Rip Rjq + Rij 2 2t and Pijk = ∇i Rjk − ∇j Rik . Take the trace on W to get (4.3.2)

∆

Q=

∂R R + + 2∇i R · V i + 2Rij V i V j ≥ 0 ∂t t

for any vector V i . Let us choose V to be the vector field minimizing Q, i.e., 1 V i = − (Ric −1 )ik ∇k R, 2

(4.3.3)

where (Ric−1 )ik is the inverse of the Ricci tensor Rij . Substitute this vector field ˜ By a direct computation from the evolution V into Q to get a smooth function Q. equations of curvatures (see [61] for details), ∂ ˜ ˜ − 2 Q. ˜ Q ≥ ∆Q ∂t t

(4.3.4)

Suppose tR assumes its maximum at (x0 , t0 ) with t0 > 0, then ∂R R + = 0, ∂t t

at (x0 , t0 ).

This implies that the quantity Q=

∂R R + + 2∇i R · V i + 2Rij V i V j ∂t t

vanishes in the direction V = 0 at (x0 , t0 ). We claim that for any earlier time t < t0 and any point x ∈ M , there is a vector V ∈ Tx M such that Q = 0.


299

We argue by contradiction. Suppose not, then there is x¯ ∈ M and 0 < t¯ < t0 such ˜ is positive at x = x¯ and t = t¯. We can find a nonnegative smooth function ρ that Q on M with support in a neighborhood of x ¯ so that ρ(¯ x) > 0 and ˜ ≥ ρ, Q t¯2 at t = t¯. Let ρ evolve by the heat equation ∂ρ = ∆ρ. ∂t It then follows from the standard strong maximum principle that ρ > 0 everywhere for any t > t¯. From (4.3.4) we see that ρ ∂ ˜ ˜− ρ −2 Q ˜− ρ Q− 2 ≥∆ Q ∂t t t2 t t2 Then by the maximum principle as in Chapter 2, we get ˜ ≥ ρ > 0, Q for all t ≥ t¯. t2 This gives a contradiction with the fact Q = 0 for V = 0 at (x0 , t0 ). We thus prove the claim. Consider each time t < t0 . The null vector field of Q satisfies the equation (4.3.5)

∇i R + 2Rij V j = 0,

by the first variation of Q in V . Since Rij is positive, we see that such a null vector field is unique and varies smoothly in space-time. Substituting (4.3.5) into the expression of Q, we have (4.3.6)

∂R R + + ∇i R · V i = 0 ∂t t

Denote by Qij = Mij + (Pkij + Pkji )V k + Rikjl V k V l . From (4.3.1) we see that Qij is nonnegative definite with its trace Q = 0 for such a null vector V . It follows that Qij = Mij + (Pkij + Pkji )V k + Rikjl V k V l = 0. Again from the first variation of Qij in V , we see that (4.3.7)

(Pkij + Pkji ) + (Rikjl + Rjkil )V l = 0,

and hence (4.3.8)

Mij − Rikjl V k V l = 0.

Applying the heat operator to (4.3.5) and (4.3.6) we get ∂ (4.3.9) − ∆ (∇i R + 2Rij V j ) 0= ∂t ∂ ∂ j −∆ V + − ∆ (∇i R) = 2Rij ∂t ∂t ∂ + 2V j − ∆ Rij − 4∇k Rij ∇k V j , ∂t

300


and (4.3.10)

∂ ∂R R i 0= −∆ + + ∇i R · V ∂t ∂t t ∂ ∂ i i −∆ V +V − ∆ (∇i R) − 2∇k ∇i R · ∇k V i = ∇i R ∂t ∂t ∂ ∂R R + −∆ + . ∂t ∂t t

Multiplying (4.3.9) by V i , summing over i and adding (4.3.10), as well as using the evolution equations on curvature, we get (4.3.11)

0 = 2V i (2∇i (|Rc|2 ) − Ril ∇l R) + 2V i V j (2Rpiqj Rpq − 2g pq Rpi Rqj ) − 4∇k Rij · ∇k V j · V i − 2∇k ∇i R · ∇k V i + 4Rij ∇i ∇j R R + 4g kl g mn g pq Rkm Rnp Rql + 4Rijkl Rik V j V l − 2 . t

From (4.3.5), we have the following equalities  i l i j pq  −2V Ril ∇ R − 4V V g Rpi Rqj = 0, (4.3.12) −4∇k Rij · ∇k V j · V i − 2∇k ∇i R · ∇k V i = 4Rij ∇k V i · ∇k V j ,   ∇i ∇j R = −2∇i Rjl · V l − 2Rjl ∇i V l . Substituting (4.3.12) into (4.3.11), we obtain

8Rij (∇k Rij · V k + Rikjl V k V l − ∇i Rjl · V l − Rjl ∇i V l ) R + 4Rij ∇k V i · ∇k V j + 4g kl g mn g pq Rkm Rnp Rql − 2 = 0. t By using (4.3.7), we know Rij (∇k Rij · V k + Rikjl V k V l − ∇i Rjl · V l ) = 0. Then we have (4.3.13)

−8Rij Rjl ∇i V l + 4Rij ∇k V i · ∇k V j + 4g kl g mn g pq Rkm Rnp Rql −

R = 0. t2

By taking the trace in the last equality in (4.3.12) and using (4.3.6) and the evolution equation of the scalar curvature, we can get (4.3.14)

Rij (Rij +

gij − ∇i Vj ) = 0. 2t

Finally by combining (4.3.13) and (4.3.14), we deduce gjk gik − ∇k Vi Rjk + − ∇k Vj = 0. 4Rij g kl Rik + 2t 2t Since Rij is positive definite, we get (4.3.15)

∇i Vj = Rij +

gij , 2t

for all i, j.


301

This means that gij (t) is a homothetically expanding Ricci soliton. Remark 4.3.7. Recall from Section 1.5 that any compact steady Ricci soliton or expanding Ricci soliton must be Einstein. If the manifold M in Theorem 4.3.6 is noncompact and simply connected, then the steady (or expanding) Ricci soliton must be a steady (or expanding) gradient Ricci soliton. For example, we know that ∇i Vj is symmetric from (4.3.15). Also, by the simply connectedness of M there exists a function F such that ∇i ∇j F = ∇i Vj ,

on M.

So Rij = ∇i ∇j F −

gij , 2t

on M

This means that gij is an expanding gradient Ricci soliton. In the Kähler case, we have the following results for Type II and Type III singularity models with nonnegative holomorphic bisectional curvature obtained by the first author in [14]. Theorem 4.3.8 (Cao [14]). (i) Any Type II singularity model on a K¨ ahler manifold with nonnegative holomorphic bisectional curvature and positive Ricci curvature must be a steady K¨ ahler-Ricci soliton. (ii) Any Type III singularity model on a K¨ ahler manifold with nonnegative holomorphic bisectional curvature and positive Ricci curvature must be an expanding K¨ ahler-Ricci soliton. To conclude this section, we state a result of Sesum [113] on compact Type I singularity models. Recall that Perelman’s functional W, introduced in Section 1.5, is given by Z n W(g, f, τ ) = (4πτ )− 2 [τ (|∇f |2 + R) + f − n]e−f dVg M

with the function f satisfying the constraint Z n (4πτ )− 2 e−f dVg = 1. M

And recall from Corollary 1.5.9 that Z n µ(g(t)) = inf W(g(t), f, T − t)| (4π(T − t))− 2 e−f dVg(t) = 1 M

is strictly increasing along the Ricci flow unless we are on a gradient shrinking soliton. If one can show that µ(g(t)) is uniformly bounded from above and the minimizing functions f = f (·, t) have a limit as t → T , then the rescaling limit model will be a shrinking gradient soliton. As shown by Natasa Sesum in [113], Type I assumption guarantees the boundedness of µ(g(t)), while the compactness assumption of the rescaling limit guarantees the existence of the limit for the minimizing functions f (·, t). Therefore we have

302


Theorem 4.3.9 (Sesum [113]). Let (M, gij (t)) be a Type I singularity model obtained as a rescaling limit of a Type I maximal solution. Suppose M is compact. Then (M, gij (t)) must be a gradient shrinking Ricci soliton. It seems that the assumption on the compactness of the rescaling limit is superfluous. We conjecture that any noncompact Type I limit is also a gradient shrinking soliton. 4.4. Ricci Solitons. We will now examine the structure of a steady Ricci soliton of the sort we get as a Type II limit. Lemma 4.4.1. Suppose we have a complete gradient steady Ricci soliton gij with bounded curvature so that Rij = ∇i ∇j F for some function F on M . Assume the Ricci curvature is positive and the scalar curvature R attains its maximum Rmax at a point x0 ∈ M . Then (4.4.1)

|∇F |2 + R = Rmax

everywhere on M , and furthermore F is convex and attains its minimum at x0 . Proof. Recall that, from (1.1.15) and noting our F here is −f there, the steady gradient Ricci soliton has the property |∇F |2 + R = C0 for some constant C0 . Clearly, C0 ≥ Rmax . If C0 = Rmax , then ∇F = 0 at the point x0 . Since ∇i ∇j F = Rij > 0, we see that F is convex and F attains its minimum at x0 . If C0 > Rmax , consider a gradient path of F in a local coordinate neighborhood through x0 = (x10 , . . . , xn0 ) : ( i x = xi (u), u ∈ (−ε, ε), i = 1, . . . , n xi0 = xi (0),

and dxi = g ij ∇j F, du

u ∈ (−ε, ε).

Now |∇F |2 = C0 − R ≥ C0 − Rmax > 0 everywhere, while |∇F |2 is smallest at x = x0 since R is largest there. But we compute d d |∇F |2 = 2g jl ∇j F ∇l F du du = 2g ik g jl ∇i ∇j F · ∇k F ∇l F = 2g ik g jl Rij ∇k F ∇l F >0

since Rij > 0 and |∇F |2 > 0. Then |∇F |2 is not smallest at x0 , and we have a contradiction.


303

We remark that when we are considering a complete expanding gradient Ricci soliton on M with positive Ricci curvature and Rij + ρgij = ∇i ∇j F for some constant ρ > 0 and some function F , the above argument gives |∇F |2 + R − 2ρF = C for some positive constant C. Moreover the function F is an exhausting and convex function. In particular, such an expanding gradient Ricci soliton is diffeomorphic to the Euclidean space Rn . Let us introduce a geometric invariant as follows. Let O be a fixed point in a Riemannian manifold M , s the distance to the fixed point O, and R the scalar curvature. We define the asymptotic scalar curvature ratio A = lim sup Rs2 . s→+∞

Clearly the definition is independent of the choice of the fixed point O and invariant under dilation. This concept is particular useful on manifolds with positive sectional curvature. The first type of gap theorem was obtained by Mok-Siu-Yau [93] in understanding the hypothesis of the paper of Siu-Yau [120]. Yau (see [49]) suggested that this should be a general phenomenon. This was later conformed by Greene-Wu [49, 50], Eschenberg-Shrader-Strake [45] and Drees [44] where they show that any complete noncompact n-dimensional (except n = 4 or 8) Riemannian manifold of positive sectional curvature must have A > 0. Similar results on complete noncompact Kähler manifolds of positive holomorphic bisectional curvature were obtained by Chen-Zhu [31] and Ni-Tam [100]. Theorem 4.4.2 (Hamilton [63]). For a complete noncompact steady gradient Ricci soliton with bounded curvature and positive sectional curvature of dimension n ≥ 3 where the scalar curvature assume its maximum at a point O ∈ M , the asymptotic scalar curvature ratio is infinite, i.e., A = lim sup Rs2 = +∞ s→+∞

where s is the distance to the point O. Proof. The solution to the Ricci flow corresponding to the soliton exists for −∞ < t < +∞ and is obtained by flowing along the gradient of a potential function F of the soliton. We argue by contradiction. Suppose Rs2 ≤ C. We will show that the limit g¯ij (x) = lim gij (x, t) t→−∞

exists for x 6= O on the manifold M and is a complete flat metric on M \ {O}. Since the sectional curvature of M is positive everywhere, it follows from Cheeger-Gromoll [23] that M is diffeomorphic to Rn . Thus M \ {O} is diffeomorphic to Sn−1 × R. But for n ≥ 3 there is no flat metric on Sn−1 × R, and this will finish the proof.

304


To see the limit metric exists, we note that R → 0 as s → +∞, so |∇F |2 → Rmax as s → +∞ by (4.4.1). The function F itself can be taken to evolve with time, using the definition ∂F ∂xi = ∇i F · = −|∇F |2 = ∆F − Rmax ∂t ∂t which pulls F back by the flow along the gradient of F . Then we continue to have ∇i ∇j F = Rij for all time, and |∇F |2 → Rmax as s → +∞ for each time. When we go backward in time, this is√equivalent to flowing outwards along the gradient of F , and our speed approaches Rmax . So, starting outside of any neighborhood of O we have p s dt (·, O) = → Rmax , |t| |t|

and (4.4.2)

R(·, t) ≤

C , Rmax · |t|2

as t → −∞

as |t| large enough.

Hence for |t| sufficiently large, 0 ≥ −2Rij ∂ = gij ∂t ≥ −2Rgij 2C gij ≥− Rmax · |t|2 which implies that for any tangent vector V , 0≤

d 2C (log(gij (t)V i V j )) ≤ . d|t| Rmax · |t|2

These two inequalities show that gij (t)V i V j has a limit g¯ij V i V j as t → −∞. Since the metrics are all essentially the same, it always takes an infinite length to get out to the infinity. This shows the limit g¯ij is complete at the infinity. One the other hand, any point P other than O will eventually be arbitrarily far from O, so the limit metric g¯ij is also complete away from O in M \ {O}. Using Shi’s derivative ∞ estimates in Chapter 1, it follows that gij (·, t) converges in the Cloc topology to a complete smooth limit metric g¯ij as t → −∞, and the limit metric is flat by (4.4.2). The above argument actually shows that (4.4.3)

lim sup Rs1+ε = +∞ s→+∞

for arbitrarily small ε > 0 and for any complete gradient Ricci soliton with bounded and positive sectional curvature of dimension n ≥ 3 where the scalar curvature assumes its maximum at a fixed point O. Finally we conclude this section with the important uniqueness of complete Ricci soliton on two-dimensional Riemannian manifolds.


305

Theorem 4.4.3 (Hamilton [60]). The only complete steady Ricci soliton on a two-dimensional manifold with bounded curvature which assumes its maximum 1 at an origin is the “cigar” soliton on the plane R2 with the metric ds2 =

dx2 + dy 2 . 1 + x2 + y 2

Proof. Recall that the scalar curvature evolves by ∂R = ∆R + R2 ∂t on a two-dimensional manifold M . Denote by Rmin (t) = inf{R(x, t) | x ∈ M }. We see from the maximum principle (see for example Chapter 2) that Rmin (t) is strictly increasing whenever Rmin (t) 6= 0, for −∞ < t < +∞. This shows that the curvature of a steady Ricci soliton on a two-dimensional manifold M must be nonnegative and Rmin (t) = 0 for all t ∈ (−∞, +∞). Further by the strong maximum principle we see that the curvature is actually positive everywhere. In particular, the manifold must be noncompact. So the manifold M is diffiomorphic to R2 and the Ricci soliton must be a gradient soliton. Let F be a potential function of the gradient Ricci soliton. Then, by definition, we have ∇i Vj + ∇j Vi = Rgij with Vi = ∇i F . This says that the vector field V must be conformal. In complex ∂ coordinate a conformal vector field is holomorphic. Hence V is locally given by V (z) ∂z for a holomorphic function V (z). At a zero of V there will be a power series expansion V (z) = az p + · · · , (a 6= 0) and if p > 1 the vector field will have closed orbits in any neighborhood of the zero. Now the vector field is gradient and a gradient flow cannot have a closed orbit. Hence V (z) has only simple zeros. By Lemma 4.4.1, we know that F is strictly convex with the only critical point being the minima, chosen to be the origin of R2 . So the holomorphic vector field V must be V (z)

∂ ∂ = cz , ∂z ∂z

for z ∈ C,

for some complex number c. We now claim that c is real. Let us write the metric as ds2 = g(x, y)(dx2 + dy 2 ) with z = x +

√ √ ∂ means that if c = a + −1b, then −1y. Then ∇F = cz ∂z ∂F = (ax − by)g, ∂x

Taking the mixed partial derivatives gives b = 0, so c is real.

∂2F ∂x∂y

∂F = (bx + ay)g. ∂y and equating them at the origin x = y = 0

306


Let (

x = eu cos v,

−∞ < u < +∞,

u

0 ≤ v ≤ 2π.

y = e sin v,

Write ds2 = g(x, y)(dx2 + dy 2 ) = g(eu cos v, eu sin v)e2u (du2 + dv 2 ) ∆

= g(u, v)(du2 + dv 2 ). Then we get the equations ∂F = ag, ∂u

∂F =0 ∂v

∂ since the gradient of F is just a ∂u for a real constant a. The second equation shows that F = F (u) is a function of u only, then the first equation shows that g = g(u) is also a function of u only. Then we can write the metric as

(4.4.4)

ds2 = g(u)(du2 + dv 2 ) = g(u)e−2u (dx2 + dy 2 ).

This implies that e−2u g(u) must be a smooth function of x2 + y 2 = e2u . So as u → −∞, (4.4.5)

g(u) = b1 e2u + b2 (e2u )2 + · · · ,

with b1 > 0. The curvature of the metric is given by R=−

1 g

g′ g

′

where (·)′ is the derivative with respect to u. Note that the soliton is by translation in u with velocity c. Hence g = g(u + ct) satisfies ∂g = −Rg ∂t which becomes cg = ′

g′ g

′

.

Thus by (4.4.5), g′ = cg + 2 g and then by integrating e2u

1 c = − e2u + b1 g 2


307

i.e., e2u . b1 − 2c e2u

g(u) =

In particular, we have c < 0 since the Ricci soliton is not flat. Therefore ds2 = g(u)e−2u (dx2 + dy 2 ) =

dx2 + dy 2 α1 + α2 (x2 + y 2 )

for some constants α1 , α2 > 0. By the normalization condition that the curvature attains its maximum 1 at the origin, we conclude that ds2 =

dx2 + dy 2 . 1 + (x2 + y 2 )

5. Long Time Behaviors. Let M be a complete manifold of dimension n. Consider a solution of the Ricci flow gij (x, t) on M and on a maximal time interval [0, T ). When M is compact, we usually consider the normalized Ricci flow 2 ∂gij = rgij − 2Rij , ∂t n

R R where r = M RdV / M dV is the average scalar curvature. The factor r serves to normalize p the Ricci flow so that the volume is constant. To see this we observe that dV = det gij dx and then p ∂ 1 ∂ log det gij = g ij gij = r − R, ∂t 2 ∂t d dt

Z

dV = M

Z

M

(r − R)dV = 0.

The Ricci flow and the normalized Ricci flow differ only by a change of scale in space and a change of parametrization in time. Indeed, we first assume that gij (t) evolves by the (unnormalized) R Ricci flow and choose the normalization factor R ψ = ψ(t) so that g˜ij = ψgij , and M d˜ µ = 1. Next we choose a new time scale t˜ = ψ(t)dt. Then for the normalized metric g˜ij we have ˜ ij = Rij , R ˜ = 1 R, r˜ = 1 r. R ψ ψ R R n Because M dV˜ = 1, we see that M dV = ψ − 2 . Then Z 2 d d log ψ = − log dV dt n dt M R ∂p det gij dx 2 M ∂tR = − n M dV 2 = r, n

308


since

∂ ∂t gij

= −2Rij for the Ricci flow. Hence it follows that ∂ ∂ d g˜ij = gij + log ψ gij ∂t dt ∂ t˜ 2 ˜ ij . = r˜g˜ij − 2R n

Thus studying the behavior of the Ricci flow near the maximal time is equivalent to studying the long-time behavior of the normalized Ricci flow. In this chapter we will obtain long-time behavior of the normalized Ricci flow for the following special cases: (1) compact two-manifolds; (2) compact three-manifolds with nonnegative Ricci curvature; (3) compact four-manifolds with nonnegative curvature operator; and (4) compact three-manifolds with uniformly bounded normalized curvature. 5.1. The Ricci Flow on Two-manifolds. Let M be a compact surface, we will discuss in this section the evolution of a Riemannian metric gij under the normalized Ricci flow. On a surface, the Ricci curvature is given by Rij =

1 Rgij 2

so the normalized Ricci flow equation becomes ∂ gij = (r − R)gij . ∂t

(5.1.1)

Recall the Gauss-Bonnet formula says Z RdV = 4πχ(M ), M

where χ(M ) is the REuler characteristic number of M . Thus the average scalar curvature r = 4πχ(M )/ M dV is constant in time. To obtain the evolution equation of the normalized curvature, we recall a simple principle in [58] for converting from the unnormalized to the normalized evolution equation on an n-dimensional manifold. Let P and Q be two expressions formed from ˜ be the corresponding expressions the metric and curvature tensors, and let P˜ and Q for the normalized Ricci flow. Since they differ by dilations, they differ by a power of the normalized factor ψ = ψ(t). We say P has degree k if P˜ = ψ k P . Thus gij has degree 1, Rij has degree 0, R has degree −1. Lemma 5.1.1. Suppose P satisfies

∂P = ∆P + Q ∂t for the unnormalized Ricci flow, and P has degree k. Then Q has degree k − 1, and for the normalized Ricci flow, ∂ P˜ ˜ P˜ + Q ˜ + 2 k˜ r P˜ . =∆ ˜ n ∂t ˜ Then Proof. We first see Q has degree k − 1 since ∂ t˜/∂t = ψ and ∆ = ψ ∆. ψ

∂ −k ˜ ˜ −k P˜ ) + ψ −k+1 Q ˜ (ψ P ) = ψ ∆(ψ ∂ t˜


309

which implies ∂ P˜ ˜ P˜ + Q ˜ + k ∂ψ P˜ =∆ ˜ ψ ∂ t˜ ∂t ˜ P˜ + Q ˜ + 2 k˜ rP˜ =∆ n since

∂ log ψ ∂ t˜

∂ = ( ∂t log ψ)ψ −1 =

2 ˜. nr

We now come back to the normalized Ricci flow (5.1.1) on a compact surface. By applying the above lemma to the evolution equation of unnormalized scalar curvature, we have (5.1.2)

∂R = ∆R + R2 − rR ∂t

for the normalized scalar curvature R. As a direct consequence, by using the maximum principle, both nonnegative scalar curvature and nonpositive scalar curvature are preserved for the normalized Ricci flow on surfaces. Let us introduce a potential function ϕ as in the Kähler-Ricci flow (see for example [11]). Since R − r has mean value zero on a compact surface, there exists a unique function ϕ, with mean value zero, such that ∆ϕ = R − r.

(5.1.3)

Differentiating (5.1.3) in time, we have ∂ ∂ R = (∆ϕ) ∂t ∂t = (R − r)∆ϕ + g ij = (R − r)∆ϕ + ∆

∂ ∂t

∂ϕ ∂t

∂2ϕ ∂ϕ − Γkij k i j ∂x ∂x ∂x

.

Combining with the equation (5.1.2), we get ∂ϕ ∆ = ∆(∆ϕ) + r∆ϕ ∂t which implies that ∂ϕ = ∆ϕ + rϕ − b(t) ∂t R for some function b(t) of time only. Since M ϕdV = 0 for all t, we have Z Z Z d 0= ϕdµ = (∆ϕ + rϕ − b(t))dµ + ϕ(r − R)dµ dt M M M Z Z = −b(t) dµ + |∇ϕ|2 dµ. (5.1.4)

M

M

Thus the function b(t) is given by b(t) =

R

|∇ϕ|2 dµ . dµ M

MR

310


Define a function h by h = ∆ϕ + |∇ϕ|2 = (R − r) + |∇ϕ|2 , and set 1 Mij = ∇i ∇j ϕ − ∆ ϕgij 2 to be the traceless part of ∇i ∇j ϕ. Lemma 5.1.2. The function h satisfies the evolution equation (5.1.5)

∂h = ∆h − 2|Mij |2 + rh. ∂t

Proof. Under the normalized Ricci flow, ∂ ∂ ij ∂ |∇ϕ|2 = g ∇i ϕ∇j ϕ + 2g ij ∇i ϕ (∇j ϕ) ∂t ∂t ∂t

= (R − r)|∇ϕ|2 + 2g ij ∇i (∆ϕ + rϕ − b(t))∇j ϕ = (R + r)|∇ϕ|2 + 2g ij (∆∇i ϕ − Rik ∇k ϕ)∇j ϕ

= (R + r)|∇ϕ|2 + ∆|∇ϕ|2 − 2|∇2 ϕ|2 − 2g ij Rik ∇k ϕ∇j ϕ

= ∆|∇ϕ|2 − 2|∇2 ϕ|2 + r|∇ϕ|2 ,

where Rik = 12 Rgik on a surface. On the other hand we may rewrite the evolution equation (5.1.2) as ∂ (R − r) = ∆(R − r) + (∆ϕ)2 + r(R − r). ∂t Then the combination of above two equations yields ∂ 1 h = ∆h − 2(|∇2 ϕ|2 − (∆ϕ)2 ) + rh ∂t 2 = ∆h − 2|Mij |2 + rh as desired. As a direct consequence of the evolution equation (5.1.5) and the maximum principle, we have R ≤ C1 ert + r

(5.1.6)

for some positive constant C1 depending only on the initial metric. On the other hand, it follows from (5.1.2) that Rmin (t) = minx∈M R(x, t) satisfies d Rmin ≥ Rmin (Rmin − r) ≥ 0 dt whenever Rmin ≤ 0. This says that (5.1.7)

Rmin (t) ≥ −C2 ,

for all t > 0


311

for some positive constant C2 depending only on the initial metric. Thus the combination of (5.1.6) and (5.1.7) implies the following long time existence result. Proposition 5.1.3. For any initial metric on a compact surface, the normalized Ricci flow (5.1.1) has a solution for all time. To investigate the long-time behavior of the solution, let us now divide the discussion into three cases: χ(M ) < 0; χ(M ) = 0; and χ(M ) > 0. Case (1): χ(M ) < 0 (i.e., r < 0). From the evolution equation (5.1.2), we have d Rmin ≥ Rmin (Rmin − r) dt ≥ r(Rmin − r), on M × [0, +∞) which implies that R − r ≥ −C˜1 ert , on M × [0, +∞) for some positive constant C˜1 depending only on the initial metric. Thus by combining with (5.1.6) we have (5.1.8)

−C˜1 ert ≤ R − r ≤ C1 ert , on M × [0, +∞).

Theorem 5.1.4 (Hamilton [60]). On a compact surface with χ(M ) < 0, for any initial metric the solution of the normalized Ricci flow (5.1.1) exists for all time and converges in the C ∞ topology to a metric with negative constant curvature. Proof. The estimate (5.1.8) shows that the scalar curvature R(x, t) converges exponentially to the negative constant r as t → +∞. Fix a tangent vector v ∈ Tx M at a point x ∈ M and let |v|2t = gij (x, t)v i v j . Then we have d 2 ∂ |v|t = gij (x, t) v i v j dt ∂t = (r − R)|v|2t

which implies d log |v|2t ≤ Cert , for all t > 0 dt

for some positive constant C depending only on the initial metric (by using (5.1.8)). Thus |v|2t converges uniformly to a continuous function |v|2∞ as t → +∞ and |v|2∞ 6= 0 if v 6= 0. Since the parallelogram law continues to hold to the limit, the limiting norm |v|2∞ comes from an inner product gij (∞). This says, the metrics gij (t) are all equivalent and as t → +∞, the metric gij (t) converges uniformly to a positive-definite metric tensor gij (∞) which is continuous and equivalent to the initial metric. By the virtue of Shi’s derivative estimates of the unnormalized Ricci flow in Section 1.4, we see that all derivatives and higher order derivatives of the curvature of the solution gij of the normalized flow are uniformly bounded on M × [0, +∞).

312


This shows that the limiting metric gij (∞) is a smooth metric with negative constant curvature and the solution gij (t) converges to the limiting metric gij (∞) in the C ∞ topology as t → +∞. Case (2): χ(M ) = 0, i.e., r = 0. From (5.1.6) and (5.1.7) we know that the curvature remains bounded above and below. To get the convergence, we consider the potential function ϕ of (5.1.3) again. The evolution of ϕ is given by (5.1.4). We renormalize the function ϕ by Z ϕ(x, ˜ t) = ϕ(x, t) + b(t)dt, on M × [0, +∞). Then, since r = 0, ϕ˜ evolves by ∂ ϕ˜ = ∆ϕ, ˜ ∂t

(5.1.9)

on M × [0, +∞).

From the proof of Lemma 5.1.2, we get ∂ |∇ϕ| ˜ 2 = ∆|∇ϕ| ˜ 2 − 2|∇2 ϕ| ˜ 2. ∂t

(5.1.10) Clearly, we have

∂ 2 ϕ˜ = ∆ϕ˜2 − 2|∇ϕ| ˜ 2. ∂t

(5.1.11) Thus it follows that

∂ (t|∇ϕ| ˜ 2 + ϕ˜2 ) ≤ ∆(t|∇ϕ| ˜ 2 + ϕ˜2 ). ∂t Hence by applying the maximum principle, there exists a positive constant C3 depending only on the initial metric such that (5.1.12)

|∇ϕ| ˜ 2 (x, t) ≤

C3 , 1+t

on M × [0, +∞).

In the following we will use this decay estimate to obtain a decay estimate for the scalar curvature. By the evolution equations (5.1.2) and (5.1.10), we have ∂ (R + 2|∇ϕ| ˜ 2 ) = ∆(R + 2|∇ϕ| ˜ 2 ) + R2 − 4|∇2 ϕ| ˜2 ∂t ≤ ∆(R + 2|∇ϕ| ˜ 2 ) − R2 since R2 = (∆ϕ) ˜ 2 ≤ 2|∇2 ϕ| ˜ 2 . Thus by using (5.1.12), we have ∂ [t(R + 2|∇ϕ| ˜ 2 )] ∂t ≤ ∆[t(R + 2|∇ϕ| ˜ 2 )] − tR2 + R + 2|∇ϕ| ˜2

≤ ∆[t(R + 2|∇ϕ| ˜ 2 )] − t(R + 2|∇ϕ| ˜ 2 )2 + (1 + 4t|∇ϕ| ˜ 2 )(R + 2|∇ϕ| ˜ 2) ≤ ∆[t(R + 2|∇ϕ| ˜ 2 )] − [t(R + 2|∇ϕ| ˜ 2 ) − (1 + 4C3 )](R + 2|∇ϕ| ˜ 2) ≤ ∆[t(R + 2|∇ϕ| ˜ 2 )]


313

wherever t(R + 2|∇ϕ| ˜ 2 ) ≥ (1 + 4C3 ). Hence by the maximum principle, there holds R + 2|∇ϕ| ˜2≤

(5.1.13)

C4 , 1+t

on M × [0, +∞)

for some positive constant C4 depending only on the initial metric. On the other hand, the scalar curvature satisfies ∂R = ∆R + R2 , ∂t

on M × [0, +∞).

It is not hard to see that R≥

(5.1.14)

Rmin (0) , 1 − Rmin (0)t

on M × [0, +∞),

by using the maximum principle. So we obtain the decay estimate for the scalar curvature |R(x, t)| ≤

(5.1.15)

C5 , 1+t

on M × [0, +∞),

for some positive constant C5 depending only on the initial metric. Theorem 5.1.5 (Hamilton [60]). On a compact surface with χ(M ) = 0, for any initial metric the solution of the normalized Ricci flow (5.1.1) exists for all time and converges in C ∞ topology to a flat metric. Proof. Since

∂ϕ ˜ ∂t

= ∆ϕ, ˜ it follows from the maximum principle that |ϕ(x, ˜ t)| ≤ C6 ,

on M × [0, +∞)

for some positive constant C6 depending only on the initial metric. Recall that ∆ϕ˜ = R. We thus obtain for any tangent vector v ∈ Tx M at a point x ∈ M , d 2 ∂ |v|t = gij (x, t) v i v j dt ∂t = −R(x, t)|v|2t

and then Z t 2 d log |v|t = log vt |2 dt| |v|20 0 dt Z t = R(x, t)dt 0

= |ϕ(x, ˜ t) − ϕ(x, ˜ 0)| ≤ 2C6 ,

for all x ∈ M and t ∈ [0, +∞). This shows that the solution gij (t) of the normalized Ricci flow are all equivalent. This gives us control of the diameter and injectivity radius. As before, by Shi’s derivative estimates of the unnormalized Ricci flow, all derivatives and higher order derivatives of the curvature of the solution gij of the normalized Ricci flow (5.1.1) are uniformly bounded on M × [0, +∞). By the virtue of Hamilton’s

314


compactness theorem (Theorem 4.1.5) we see that the solution gij (t) subsequentially converges in C ∞ topology. The decay estimate (5.1.15) implies that each limit must be a flat metric on M . Clearly, we will finish the proof if we can show that limit is unique. Note that the solution gij (t) is changing conformally under the Ricci flow (5.1.1) on surfaces. Thus each limit must be conformal to the initial metric, denoted by g¯ij . Let us denote gij (∞) = eu g¯ij to be a limiting metric. Since gij (∞) is flat, it is easy to compute ¯ − ∆u), ¯ 0 = e−u (R

on M,

¯ is the curvature of g¯ij and ∆ ¯ is the Laplacian in the metric g¯ij . The solution where R of Poission equation ¯ = R, ¯ ∆u

on M

is unique up to constant. Moreover the constant must be also uniquely determined since the area of the solution of the normalized Ricci flow (5.1.1) is constant in time. So the limit is unique and we complete the proof of Theorem 5.1.5. Case (3): χ(M ) > 0, i.e., r > 0. This is the most difficult case. There exist several proofs by now but, in contrast to the previous two cases, none of them depend only on the maximum principle type of argument. In fact, all the proofs rely on some combination of the maximum principle argument and certain integral estimate of the curvature. In the pioneer work [60], Hamilton introduced an integral quantity Z E= R log R dV, M

which he calls entropy, for the (normalized) Ricci flow on a surface M with positive curvature, and showed that the entropy is monotone decreasing under the flow. By combining this entropy estimate with the Harnack inequality for the curvature (Corollary 2.5.3), Hamilton obtained the uniform bound on the curvature of the normalized Ricci flow on M with positive curvature. Furthermore, he showed that the evolving metric converges to a shrinking Ricci soliton on M and that the shrinking Ricci soliton must be a round metric on the 2-sphere S2 . Subsequently, Chow [36] extended Hamilton’s work to the general case when the curvature may change signs. More precisely, he proved that given any initial metric on a compact surface M with χ(M ) > 0, the evolving metric under the (normalized) Ricci flow will have positive curvature after a finite time. Hence, when combined with Hamilton’s result, we know the evolving metric on M converges to the round metric on S2 . In the following we present a new argument by combining the Li-Yau-Hamilton inequality of the curvature with Perelman’s no local collapsing theorem I′ , as was done in the recent joint work of Bing-Long Chen and the authors [15] where they considered the Kähler-Ricci flow on higher dimensional Kähler manifolds of nonnegative holomorphic bisectional curvature (see [15] for more details). (There are also other proofs for Case (3) by Bartz-Struwe-Ye [6] and Struwe [121].) Given any initial metric on M with χ(M ) > 0, we consider the solution gij (t) of the normalized Ricci flow (5.1.1). Recall that the (scalar) curvature R satisfies the evolution equation ∂ R = ∆R + R2 − rR. ∂t


315

The corresponding ODE is ds = s2 − rs. dt

(5.1.16)

Let us choose c > 1 and close to 1 so that r/(1 − c) < minx∈M R(x, 0). It is clear that the function s(t) = r/(1 − cert ) < 0 is a solution of the ODE (5.1.16) with s(0) < min R(x, 0). Then the difference of R and s evolves by x∈M

∂ (R − s) = ∆(R − s) + (R − r + s)(R − s). ∂t

(5.1.17)

Since minx∈M R(x, 0) − s(0) > 0, the maximum principle implies that R − s > 0 for all times. We first extend the Li-Yau-Hamilton inequality (Theorem 2.5.2) to the normalized Ricci flow whose curvature may change signs. As in the proof of Theorem 2.5.2, we consider the quantity L = log(R − s). It is easy to compute ∂L = ∆L + |∇L|2 + R − r + s. ∂t Then we set Q=

∂L − |∇L|2 − s = ∆L + R − r. ∂t

By a direct computation and using the estimate (5.1.8), we have ∂ Q=∆ ∂t

∂L ∂t

+ (R − r)∆L +

∂R ∂t

= ∆Q + 2|∇2 L|2 + 2h∇L, ∇(∆L)i + R|∇L|2 + (R − r)∆L + ∆R + R(R − r)

= ∆Q + 2|∇2 L|2 + 2h∇L, ∇Qi + 2(R − r)∆L + (R − r)2 + (r − s)∆L + s|∇L|2 + r(R − r)

= ∆Q + 2h∇L, ∇Qi + 2|∇2 L|2 + 2(R − r)∆L + (R − r)2 + (r − s)Q + s|∇L|2 + s(R − r)

≥ ∆Q + 2h∇L, ∇Qi + Q2 + (r − s)Q + s|∇L|2 − C. Here and below C is denoted by various positive constants depending only on the initial metric. In order to control the bad term s|∇L|2 , we consider ∂ (sL) = ∆(sL) + s|∇L|2 + s(R − r + s) + s(s − r)L ∂t ≥ ∆(sL) + 2h∇L, ∇(sL)i − s|∇L|2 − C

316


by using the estimate (5.1.8) again. Thus ∂ (Q + sL) ≥ ∆(Q + sL) + 2h∇L, ∇(Q + sL)i + Q2 + (r − s)Q − C ∂t 1 ≥ ∆(Q + sL) + 2h∇L, ∇(Q + sL)i + [(Q + sL)2 − C 2 ], 2 since sL is bounded by (5.1.8). This, by the maximum principle, implies that Q ≥ −C,

for all t ∈ [0, +∞).

Then for any two points x1 , x2 ∈ M and two times t2 > t1 ≥ 0, and a path γ : [t1 , t2 ] → M connecting x1 to x2 , we have Z

t2

d L(γ(t), t)dt dt t1 Z t2 ∂L + h∇L, γi ˙ dt = ∂t t1 1 ≥ − ∆ − C(t2 − t1 ) 4

L(x2 , t2 ) − L(x1 , t1 ) =

where ∆ = ∆(x1 , t1 ; x2 , t2 ) Z t2 2 = inf |γ(t)| ˙ gij (t) dt | γ : [t1 , t2 ] → M with γ(t1 ) = x1 , γ(t2 ) = x2 . t1

Thus we have proved the following Harnack inequality. Lemma 5.1.6 (Chow [36]). There exists a positive constant C depending only on the initial metric such that for any x1 , x2 ∈ M and t2 > t1 ≥ 0, ∆

R(x1 , t1 ) − s(t1 ) ≤ e 4 +C(t2 −t1 ) (R(x2 , t2 ) − s(t2 )) where ∆ = inf

Z

t2

t1

2 |γ(t)| ˙ t dt

| γ : [t1 , t2 ] → M with γ(t1 ) = x1 , γ(t2 ) = x2 .

We now state and prove the uniform bound estimate for the curvature. Proposition 5.1.7. Let (M, gij (t)) be a solution of the normalized Ricci flow on a compact surface with χ(M ) > 0. Then there exist a time t0 > 0 and a positive constant C such that the estimate C −1 ≤ R(x, t) ≤ C holds for all x ∈ M and t ∈ [t0 , +∞). Proof. Recall that R(x, t) ≥ s(t) =

r , 1 − cert

on M × [0, +∞).


317

For any ε ∈ (0, r), there exists a large enough t0 > 0 such that R(x, t) ≥ −ε2 ,

(5.1.18)

on M × [t0 , +∞).

Let t be any fixed time with t ≥ t0 + 1. Obviously there is some point x0 ∈ M such that R(x0 , t + 1) = r. Consider the geodesic ball Bt (x0 , 1), centered at x0 and radius 1 with respect to the metric at the fixed time t. For any point x ∈ Bt (x0 , 1), we choose a geodesic γ: [t, t + 1] → M connecting x and x0 with respect to the metric at the fixed time t. Since ∂ gij = (r − R)gij ≤ 2rgij ∂t we have

Z

t+1 t

|γ(τ ˙ )|2τ dτ

≤e

2r

Z

on M × [t0 , +∞), t+1

t

|γ(τ ˙ )|2t dτ ≤ e2r .

Then by Lemma 5.1.6, we have (5.1.19)

R(x, t) ≤ s(t) + exp ≤ C1 ,

1 2r e +C 4

· (R(x0 , t + 1) − s(t + 1))

as x ∈ Bt (x0 , 1),

for some positive constant C1 depending only on the initial metric. Note the the corresponding unnormalized Ricci flow in this case has finite maximal time since its volume decreases at a fixed rate −4πχ(M ) < 0. Hence the no local collapsing theorem I′ (Theorem 3.3.3) implies that the volume of Bt (x0 , 1) with respect to the metric at the fixed time t is bounded from below by Vol t (Bt (x0 , 1)) ≥ C2

(5.1.20)

for some positive constant C2 depending only on the initial metric. We now want to bound the diameter of (M, gij (t)) from above. The following argument is analogous to Yau in [128] where he got a lower bound for the volume of geodesic balls of a complete Riemannian manifold with nonnegative Ricci curvature. Without loss of generality, we may assume that the diameter of (M, gij (t)) is at least 3. Choose a point x1 ∈ M such that the distance dt (x0 , x1 ) between x1 and x0 with respect to the metric at the fixed time t is at least a half of the diameter of (M, gij (t)). By (5.1.18), the standard Laplacian comparison theorem (c.f. [112]) implies ∆ρ2 = 2ρ∆ρ + 2 ≤ 2(1 + ερ) + 2 in the sense of distribution, where ρ is the distance function from x1 (with respect to the metric gij (t)). That is, for any ϕ ∈ C0∞ (M ), ϕ ≥ 0, we have Z Z (5.1.21) − ∇ρ2 · ∇ϕ ≤ [2(1 + ερ) + 2]ϕ. M

C0∞ (M )

M

Since functions can be approximated by Lipschitz functions in the above inequality, we can set ϕ(x) = ψ(ρ(x)), x ∈ M , where ψ(s) is given by   0 ≤ s ≤ dt (x0 , x1 ) − 1, 1, 1 ′ ψ(s) = ψ (s) = − 2 , dt (x0 , x1 ) − 1 ≤ s ≤ dt (x0 , x1 ) + 1,   0, s ≥ dt (x0 , x1 ) + 1.

318


Thus, by using (5.1.20), the left hand side of (5.1.21) is Z − ∇ρ2 · ∇ϕ M Z = ρ Bt (x1 ,dt (x0 ,x1 )+1)\Bt (x1 ,dt (x0 ,x1 )−1)

≥ (dt (x0 , x1 ) − 1)Vol t (Bt (x1 , dt (x0 , x1 ) + 1) \ Bt (x1 , dt (x0 , x1 ) − 1))

≥ (dt (x0 , x1 ) − 1)Vol t (Bt (x0 , 1)) ≥ (dt (x0 , x1 ) − 1)C2 , and the right hand side of (5.1.21) is Z Z [2(1 + ερ) + 2]ϕ ≤ M

[2(1 + ερ) + 2]

Bt (x1 ,dt (x0 ,x1 )+1)

≤ [2(1 + εdt (x0 , x1 )) + 4]Vol t (Bt (x1 , dt (x0 , x1 ) + 1)) ≤ [2(1 + εdt (x0 , x1 )) + 4]A

where A is the area of M with respect to the initial metric. Here we have used the fact that the area of solution of the normalized Ricci flow is constant in time. Hence C2 (dt (x0 , x1 ) − 1) ≤ [2(1 + εdt (x0 , x1 )) + 4]A, which implies, by choosing ε > 0 small enough, dt (x0 , x1 ) ≤ C3 for some positive constant C3 depending only on the initial metric. Therefore, the diameter of (M, gij (t)) is uniformly bounded above by (5.1.22)

diam (M, gij (t)) ≤ 2C3

for all t ∈ [t0 , +∞). We then argue, as in deriving (5.1.19), by applying Lemma 5.1.6 again to obtain R(x, t) ≤ C4 ,

on M × [t0 , +∞)

for some positive constant C4 depending only on the initial metric. It remains to prove a positive lower bound estimate of the curvature. First, we note that the function s(t) → 0 as t → +∞, and the average scalar curvature of the solution equals to r, a positive constant. Thus the Harnack inequality in Lemma 5.1.6 and the diameter estimate (5.1.22) imply a positive lower bound for the curvature. Therefore we have completed the proof of Proposition 5.1.7. Next we consider long-time convergence of the normalized flow. Recall that the trace-free part of the Hessian of the potential ϕ of the curvature is the tensor Mij defined by 1 Mij = ∇i ∇j ϕ − ∆ϕ · gij , 2 where by (5.1.3), ∆ϕ = R − r.


Lemma 5.1.8. We have ∂ |Mij |2 = ∆|Mij |2 − 2|∇k Mij |2 − 2R|Mij |2 , on M × [0, +∞). ∂t

(5.1.23)

Proof. First we note the time-derivative of the Levi-Civita connection is ∂ 1 ∂ ∂ ∂ k Γij = g kl ∇i gjl + ∇j gil − ∇l gij ∂t 2 ∂t ∂t ∂t 1 −∇i R · δjk − ∇j R · δik + ∇k R · gij . = 2

By using this and (5.1.4), we have ∂ ∂ϕ ∂ k 1 ∂ Mij = ∇i ∇j − Γij ∇k ϕ − [(R − r)gij ] ∂t ∂t ∂t 2 ∂t 1 = ∇i ∇j ∆ϕ + (∇i R · ∇j ϕ + ∇j R · ∇i ϕ − h∇R, ∇ϕigij ) 2 1 − ∆R · gij + rMij . 2 Since on a surface, Rijkl =

1 R(gil gjk − gik gjl ), 2

we have ∇i ∇j ∆ϕ

= ∇i ∇k ∇j ∇k ϕ − ∇i (Rjl ∇l ϕ)

l = ∇k ∇i ∇j ∇k ϕ − Rikj ∇l ∇k ϕ − Ril ∇j ∇l ϕ − Rjl ∇i ∇l ϕ − ∇i Rjl ∇l ϕ

l l = ∆∇i ∇j ϕ − ∇k (Rikj ∇l ϕ) − Rikj ∇l ∇k ϕ

− Ril ∇j ∇l ϕ − Rjl ∇i ∇l ϕ − ∇i Rjl ∇l ϕ 1 = ∆∇i ∇j ϕ − (∇i R · ∇j ϕ + ∇j R · ∇i ϕ − h∇R, ∇ϕigij ) 2 1 − 2R ∇i ∇j ϕ − ∆ϕ · gij . 2

Combining these identities, we get 1 ∂ Mij = ∆∇i ∇j ϕ − ∆R · gij + (r − 2R)Mij ∂t 2 1 = ∆ ∇i ∇j ϕ − (R − r)gij + (r − 2R)Mij . 2 Thus the evolution Mij is given by (5.1.24)

∂Mij = ∆Mij + (r − 2R)Mij . ∂t

Now the lemma follows from (5.1.24) and a straightforward computation.

319

320


Proposition 5.1.7 tells us that the curvature R of the solution to the normalized Ricci flow is uniformly bounded from below by a positive constant for t large. Thus we can apply the maximum principle to the equation (5.1.23) in Lemma 5.1.8 to obtain the following estimate. Proposition 5.1.9. Let (M, gij (t)) be a solution of the normalized Ricci flow on a compact surface with χ(M ) > 0. Then there exist positive constants c and C depending only on the initial metric such that |Mij |2 ≤ Ce−ct ,

on M × [0, +∞).

Now we consider a modification of the normalized Ricci flow. Consider the equation ∂ gij = 2Mij = (r − R)gij + 2∇i ∇j ϕ. ∂t

(5.1.25)

As we saw in Section 1.3, the solution of this modified flow differs from that of the normalized Ricci flow only by a one parameter family of diffeomorphisms generated by the gradient vector field of the potential function ϕ. Since the quantity |Mij |2 is invariant under diffeomorphisms, the estimate |Mij |2 ≤ Ce−ct also holds for the solution of the modified flow (5.1.25). This exponential decay estimate then implies the solution gij (x, t) of the modified flow (5.1.25) converges exponentially to a continuous metric gij (∞) as t → +∞. Furthermore, by the virtue of Hamilton’s compactness theorem (Theorem 4.1.5) we see that the solution gij (x, t) of the modified flow actually converges exponentially in C ∞ topology to gij (∞). Moreover the limiting metric gij (∞) satisfies Mij = (r − R)gij + 2∇i ∇j ϕ = 0,

on M.

That is, the limiting metric is a shrinking gradient Ricci soliton on the surface M . The next result was first obtained by Hamilton in [60]. The following simplified proof by using the Kazdan-Warner identity was widely known to experts in the field. Proposition 5.1.10. On a compact surface there are no shrinking Ricci solitons other than constant curvature. Proof. By definition, a shrinking Ricci soliton on a compact surface M is given by ∇i Xj + ∇j Xi = (R − r)gij

(5.1.26)

for some vector field X = Xj . By contracting the above equation by Rg −1 , we have 2R(R − r) = 2R div X, and hence

Z

M

(R − r)2 dV =

Z

M

R(R − r)dV =

Z

R div XdV.

M

Since X is a conformal vector field (by the Ricci soliton equation (5.1.26)), by integrating by parts and applying the Kazdan-Warner identity [77], we obtain Z Z (R − r)2 dV = − h∇R, XidV = 0. M

M


321

Hence R ≡ r, and the lemma is proved. Now back to the solution of the modified flow (5.1.25). We have seen the curvature converges exponentially to its limiting value in the C ∞ topology. But since there are no nontrivial soliton on M , we must have R converging exponentially to the constant value r in the C ∞ topology. This then implies that the unmodified flow (5.1.1) will converge to a metric of positive constant curvature in the C ∞ topology. In conclusion, we have proved the following main theorem of this section. Theorem 5.1.11 (Hamilton [60], Chow [36]). On a compact surface with χ(M ) > 0, for any initial metric, the solution of the normalized Ricci flow (5.1.1) exists for all time, and converges in the C ∞ topology to a metric with positive constant curvature. 5.2. Differentiable Sphere Theorems in 3-D and 4-D. An important problem in Riemannian geometry is to understand the influence of curvatures, in particular the sign of curvatures, on the topology of underlying manifolds. Classical results of this type include sphere theorem and its refinements stated below (see, for example, Theorem 6.1 and Theorem 7.16 in Cheeger-Ebin [22], and Theorem 6.6 in CheegerEbin [22]). In this section we shall use the long-time behavior of the Ricci flow on positively curved manifolds to establish Hamilton’s differentiable sphere theorems in dimensions three and four. Let us first recall the classical sphere theorems. Given a Remannian manifold M , we denote by KM the sectional curvature of M . Classical Sphere Theorems. Let M be a complete, simply connected ndimensional manifold. (i) If 14 < KM ≤ 1, then M is homeomorphic to the n-sphere Sn . (ii) There exists a positive constant δ ∈ ( 14 , 1) such that if δ < KM ≤ 1, then M is diffeomorphic to the n-sphere Sn . Result (ii) is called the differentiable sphere theorem. If we relax the assumptions on the strict lower bound in (i), then we have the following rigidity result. Berger’s Rigidity Theorem. Let M be a complete, simply connected ndimensional manifold with 41 ≤ KM ≤ 1. Then either M is homeomorphic to Sn or M is isometric to a symmetric space. We remark that it follows from the classification of symmetric spaces (see for example [68]) thatn the only simply connected symmetric spaces with positive curvature n are Sn , CP 2 , QP 4 , and the Cayley plane. In early and mid 80’s respectively, Hamilton [58], [59] used the Ricci flow to prove the following differential sphere theorems. Theorem 5.2.1 (Hamilton [58]). A compact three-manifold with positive Ricci curvature must be diffeomorphic to the three-sphere S3 or a quotient of it by a finite group of fixed point free isometries in the standard metric. Theorem 5.2.2 (Hamilton [59]). A compact four-manifold with positive curvature operator is diffeomorphic to the four-sphere S4 or the real projective space RP4 . Note that in above two theorems, we only assume curvatures to be strictly positive, but not any strong pinching conditions as in the classical sphere theorems. In fact, one of the important special features discovered by Hamilton is that if the initial metric has positive curvature, then the metric will get rounder and rounder as it

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evolves under the Ricci flow, at least in dimension three and four, so any small initial pinching will get improved. Indeed, the pinching estimate is a key step in proving both Theorem 5.2.1 and 5.2.2. The following results are concerned with compact three-manifolds or fourmanifolds with weakly positive curvatures. Theorem 5.2.3 (Hamilton [59]). (i) A compact three-manifold with nonnegative Ricci curvature is diffeomorphic to S3 , or a quotient of one of the spaces S3 or S2 × R1 or R3 by a group of fixed point free isometries in the standard metrics. (ii) A compact four-manifold with nonnegative curvature operator is diffeomorphic to S4 or CP2 or S2 × S2 , or a quotient of one of the spaces S4 or CP2 or S3 × R1 or S2 × S2 or S2 × R2 or R4 by a group of fixed point free isometries in the standard metrics. The rest of the section will be devoted to prove Theorems 5.2.1-5.2.3. Recall that the curvature operator Mαβ evolves by (5.2.1)

∂ # 2 Mαβ = ∆Mαβ + Mαβ + Mαβ . ∂t

2 where (see Section 1.3 and Section 2.4) Mαβ is the operator square 2 Mαβ = Mαγ Mβγ # and Mαβ is the Lie algebra so(n) square # Mαβ = Cαγζ Cβηθ Mγη Mζθ .

We begin with the curvature pinching estimates of the Ricci flow in three dimen# sions. In dimension n = 3, we know that Mαβ is the adjoint matrix of Mαβ . If we diagonalize Mαβ with eigenvalues λ ≥ µ ≥ ν so that   λ , µ (Mαβ ) =  ν # 2 then Mαβ and Mαβ are also diagonal, with  2   λ µν 2  and (M # ) =  µ2 (Mαβ )= αβ ν2

λν λµ



,

and the ODE corresponding to PDE (5.2.1) is then given by the system  d 2   dt λ = λ + µν,    d 2 (5.2.2) dt µ = µ + λν,      d 2 dt ν = ν + λµ. Lemma 5.2.4. For any ε ∈ [0, 13 ], the pinching condition Rij ≥ 0

and

Rij ≥ εRgij


323

is preserved by the Ricci flow. Proof. If we diagonalize the 3 × 3 curvature operator matrix Mαβ with eigenvalues λ ≥ µ ≥ ν, then nonnegative sectional curvature corresponds to ν ≥ 0 and nonnegative Ricci curvature corresponds to the inequality µ + ν ≥ 0. Also, the scalar curvature R = λ + µ + ν. So we need to show µ+ν ≥ 0

and µ + ν ≥ δλ, with δ = 2ε/(1 − 2ε),

are preserved by the Ricci flow. By Hamilton’s advanced maximum principle (Theorem 2.3.1), it suffices to show that the closed convex set K = {Mαβ | µ + ν ≥ 0 and µ + ν ≥ δλ} is preserved by the ODE system (5.2.2). Now suppose we have diagonalized Mαβ with eigenvalues λ ≥ µ ≥ ν at t = 0, # 2 then both Mαβ and Mαβ are diagonal, so the matrix Mαβ remains diagonal for t > 0. Moreover, since d (µ − ν) = (µ − ν)(µ + ν − λ), dt it is clear that µ ≥ ν for t > 0 also. Similarly, we have λ ≥ µ for t > 0. Hence the inequalities λ ≥ µ ≥ ν persist. This says that the solutions of the ODE system (5.2.2) agree with the original choice for the eigenvalues of the curvature operator. The condition µ + ν ≥ 0 is clearly preserved by the ODE, because d (µ + ν) = µ2 + ν 2 + λ(µ + ν) ≥ 0. dt It remains to check d d (µ + ν) ≥ δ λ dt dt or µ2 + λν + ν 2 + λµ ≥ δ(λ2 + µν) on the boundary where µ + ν = δλ ≥ 0. In fact, since (λ − ν)µ2 + (λ − µ)ν 2 ≥ 0, we have λ(µ2 + ν 2 ) ≥ (µ + ν)µν. Hence 2

2

µ + µλ + ν + νλ ≥

µ+ν λ

(λ2 + µν)

= δ(λ2 + µν).

324


So we get the desired pinching estimate. Proposition 5.2.5. Suppose that the initial metric of the solution to the Ricci flow on M 3 × [0, T ) has positive Ricci curvature. Then for any ε > 0 we can find Cε < +∞ such that Rij − 1 Rgij ≤ εR + Cε 3 for all subsequent t ∈ [0, T ).

Proof. Again we consider the ODE system (5.2.2). Let Mαβ be diagonalized with eigenvalues λ ≥ µ ≥ ν at t = 0. We saw in the proof of Lemma 5.2.4 the inequalities λ ≥ µ ≥ ν persist for t > 0. We only need to show that there are positive constants δ and C such that the closed convex set K = { Mαβ | λ − ν ≤ C(λ + µ + ν)1−δ } is preserved by the ODE. We compute d (λ − ν) = (λ − ν)(λ + ν − µ) dt and d (λ + µ + ν) = (λ + µ + ν)(λ + ν − µ) + µ2 dt + µ(µ + ν) + λ(µ − ν) ≥ (λ + µ + ν)(λ + ν − µ) + µ2 . Thus, without loss of generality, we may assume λ − ν > 0 and get d log(λ − ν) = λ + ν − µ dt and µ2 d log(λ + µ + ν) ≥ λ + ν − µ + . dt λ+µ+ν By Lemma 5.2.4, there exists a positive constant C depending only on the initial metric such that λ ≤ λ + µ ≤ C(µ + ν) ≤ 2Cµ, λ + ν − µ ≤ λ + µ + ν ≤ 6Cµ, and hence with ǫ = 1/36C 2, d log(λ + µ + ν) ≥ (1 + ǫ)(λ + ν − µ). dt


325

Therefore with (1 − δ) = 1/(1 + ǫ), d log((λ − ν)/(λ + µ + ν)1−δ ) ≤ 0. dt This proves the proposition. We now are ready to prove Theorem 5.2.1. Proof of Theorem 5.2.1. Let M be a compact three-manifold with positive Ricci curvature and let the metric evolve by the Ricci flow. By Lemma 5.2.4 we know that there exists a positive constant β > 0 such that Rij ≥ βRgij for all t ≥ 0 as long as the solution exists. The scalar curvature evolves by ∂R = ∆R + 2|Rij |2 ∂t 2 ≥ ∆R + R2 , 3 which implies, by the maximum principle, that the scalar curvature remains positive and tends to +∞ in finite time. We now use a blow up argument as in Section 4.3 to get the following gradient estimate. Claim. For any ε > 0, there exists a positive constant Cε < +∞ such that for any time τ ≥ 0, we have 3

max max |∇Rm(x, t)| ≤ ε max max |Rm(x, t)| 2 + Cε . t≤τ x∈M

t≤τ x∈M

We argue by contradiction. Suppose the above gradient estimate fails for some fixed ε0 > 0. Pick a sequence Cj → +∞, and pick points xj ∈ M and times τj such that 3

|∇Rm(xj , τj )| ≥ ε0 max max |Rm(x, t)| 2 + Cj , t≤τj x∈M

j = 1, 2, . . . .

Choose xj to be the origin, and pull the metric back to a small ball on the tangent space Txj M of radius rj proportional to the reciprocal of the square root of the maximum curvature up to time τj (i.e., maxt≤τj maxx∈M |Rm(x, t)|). Clearly the maximum curvatures go to infinity by Shi’s derivative estimate of curvature (Theorem 1.4.1). Dilate the metrics so that the maximum curvature max max |Rm(x, t)| t≤τj x∈M

becomes 1 and translate time so that τj becomes the time 0. By Theorem 4.1.5, we can take a (local) limit. The limit metric satisfies |∇Rm(0, 0)| ≥ ε0 > 0. However the pinching estimate in Proposition 5.2.5 tells us the limit metric has 1 Rij − Rgij ≡ 0. 3

326


By using the contracted second Bianchi identity 1 1 1 ∇i R = ∇j Rij = ∇j Rij − Rgij + ∇i R, 2 3 3 we get ∇i R ≡ 0

and then

∇i Rjk ≡ 0.

For a three-manifold, this in turn implies ∇Rm = 0 which is a contradiction. Hence we have proved the gradient estimate claimed. We can now show that the solution to the Ricci flow becomes round as the time t tends to the maximal time T . We have seen that the scalar curvature goes to infinity in finite time. Pick a sequence of points xj ∈ M and times τj where the curvature at xj is as large as it has been anywhere for 0 ≤ t ≤ τj and τj tends to the maximal time. Since |∇Rm| is very small compared to |Rm(xj , τj )| by the above gradient estimate and |Rij − 31 Rgij | is also very small compared to |Rm(xj , τj )| by Proposition 5.2.5, the curvature is nearly constant and positive in a large ball around xj at the time τj . But then the Bonnet-Myers’ theorem tells us this is the whole manifold. For j large enough, the sectional curvature of the solution at the time τj is sufficiently pinched. Then it follows from the Klingenberg injectivity radius estimate (see Section p 4.2) that the injectivity radius of the metric at time τj is bounded from below by c/ |Rm(xj , τj )| for some positive constant c independent of j. Dilate the metrics so that the maximum curvature |Rm(xj , τj )| = maxt≤τj maxx∈M |Rm(x, t)| becomes 1 and shift the time τj to the new time 0. Then we can apply Hamilton’s compactness theorem (Theorem 4.1.5) to take a limit. By the pinching estimate in Proposition 5.2.5, we know that the limit has positive constant curvature which is either the round S3 or a metric quotient of the round S3 . Consequently, the compact three-manifold M is diffeomorphic to the round S3 or a metric quotient of the round S3 . Next we consider the pinching estimates of the Ricci flow on a compact fourmanifold M with positive curvature operator. In dimension 4, we saw in Section 1.3 when we decompose orthogonally Λ2 = 2 Λ+ ⊕ Λ2− into the eigenspaces of Hodge star with eigenvalue ±1, we have a block decomposition of Mαβ as A B Mαβ = t B C and then # Mαβ =2

A# t # B

B# C#

where A# , B # , C # are the adjoints of 3 × 3 submatrices as before. Thus the ODE d # 2 Mαβ = Mαβ + Mαβ dt


327

corresponding to the PDE (5.2.1) breaks up into the system of three equations  d 2 t #   dt A = A + B B + 2A ,   d # (5.2.3) dt B = AB + BC + 2B ,     d 2 t # dt C = C + BB + 2C .

As shown in Section 1.3, by the Bianchi identity, we know that tr A = tr C. For the symmetric matrices A and C, we can choose an orthonormal basis x1 , x2 , x3 of Λ2+ such that   a1 0 0     A =  0 a2 0  ,   0

0

a3

and an orthonormal basis z1 , z2 , z3 of Λ2− such that 

c1

  C= 0  0

0 c2 0

0



  0 . 

c3

For matrix B, we can choose orthonormal basis y1+ , y2+ , y3+ of Λ2+ and y1− , y2− , y3− of Λ2− such that 

b1

  B= 0  0

0 b2 0

0



  0 . 

b3

with 0 ≤ b1 ≤ b2 ≤ b3 . We may also arrange the eigenvalues of A and C as a1 ≤ a2 ≤ a3 and c1 ≤ c2 ≤ c3 . In view of the advanced maximum principle Theorem 2.3.1, we only need to establish the pinching estimates for the ODE (5.2.3). Note that a1 = inf{A(x, x) | x ∈ Λ2+ and |x| = 1},

a3 = sup{A(x, x) | x ∈ Λ2+ and |x| = 1}, c1 = inf{C(z, z) | z ∈ Λ2− and |z| = 1},

c3 = sup{C(z, z) | z ∈ Λ2− and |z| = 1}.

We can compute their derivatives by Lemma 2.3.3 as follows:  d 2 2  dt a1 ≥ a1 + b1 + 2a2 a3 ,      d  dt a3 ≤ a23 + b23 + 2a1 a2 , (5.2.4)  d 2 2   dt c1 ≥ c1 + b1 + 2c2 c3 ,     d 2 2 dt c3 ≤ c3 + b3 + 2c1 c2 .

328


We shall make the pinching estimates by using the functions b2 + b3 and a − 2b + c, where a = a1 + a2 + a3 = c = c1 + c2 + c3 and b = b1 + b2 + b3 . Since b2 + b3 = B(y2+ , y2− ) + B(y3+ , y3− ) = sup{B(y + , y − ) + B(˜ y + , y˜− ) | y + , y˜+ ∈ Λ2+ with |y + | = |˜ y + | = 1, y + ⊥˜ y + , and y − , y˜− ∈ Λ2− with |y − | = |˜ y − | = 1, y − ⊥˜ y − },

We compute by Lemma 2.3.3, (5.2.5)

d d d (b2 + b3 ) ≤ B(y2+ , y2− ) + B(y3+ , y3− ) dt dt dt = AB(y2+ , y2− ) + BC(y2+ , y2− ) + 2B # (y2+ , y2− ) + AB(y3+ , y3− ) + BC(y3+ , y3− ) + 2B # (y3+ , y3− ) = b2 A(y2+ , y2+ ) + b2 C(y2− , y2− ) + 2b1 b3 + b3 A(y3+ , y3+ ) + b3 C(y3− , y3− ) + 2b1 b2 ≤ a2 b2 + a3 b3 + b2 c2 + b3 c3 + 2b1 b2 + 2b1 b3 ,

where we used the facts that A(y2+ , y2+ ) + A(y3+ , y3+ ) ≤ a2 + a3 and C(y2− , y2− ) + C(y3− , y3− ) ≤ c2 + c3 . Note also that the function a = trA = c = trC is linear, and the function b is given by

Indeed,

b = B(y1+ , y1− ) + B(y2+ , y2− ) + B(y3+ , y3− ) n = sup B(T y1+ , T˜y1− ) + B(T y2+ , T˜y2− ) + B(T y3+ , T˜y3− ) | T, T˜ are o othogonal transformations of Λ2+ and Λ2− respectively . B(T y1+ , T˜y1− ) + B(T y2+ , T˜y2− ) + B(T y3+ , T˜y3− ) = B(y + , T −1 T˜(y − )) + B(y + , T −1 T˜(y − )) + B(y + , T −1 T˜(y − )) 1

1

2

2

3

3

= b1 t11 + b2 t22 + b3 t33 where t11 , t22 , t33 are diagonal elements of the orthogonal matrix T −1 T˜ with t11 , t22 , t33 ≤ 1. Thus by using Lemma 2.3.3 again, we compute d d d d (a − 2b + c) ≥ tr A−2 B+ C dt dt dt dt = tr ((A − B)2 + (C − B)2 + 2(A# − 2B # + C # ))

evaluated in those coordinates where B is diagonal as above. Recalling the definition of Lie algebra product P #Q =

1 εαβγ εζηθ Pβη Qγθ 2

with εαβγ being the permutation tensor, we see that the Lie algebra product # gives


329

a symmetric bilinear operation on matrices, and then tr (2(A# − 2B # + C # ))

= tr ((A − C)# + (A + 2B + C)#(A − 2B + C)) 1 1 = − tr (A − C)2 + (tr (A − C))2 2 2 + tr ((A + 2B + C)#(A − 2B + C)) 1 = − tr (A − C)2 + tr ((A + 2B + C)#(A − 2B + C)) 2

by the Bianchi identity. It is easy to check that 1 1 tr (A − B)2 + tr (C − B)2 − tr (A − C)2 = tr (A − 2B + C)2 ≥ 0. 2 2 Thus we obtain d (a − 2b + c) ≥ tr ((A + 2B + C)#(A − 2B + C)) dt Since Mαβ ≥ 0 and



Mαβ = 

A

B

t

C

B



,

we see that A + 2B + C ≥ 0 and A − 2B + C ≥ 0, by applying Mαβ to the vectors (x, x) and (x, −x). It is then not hard to see tr ((A + 2B + C)#(A − 2B + C)) ≥ (a1 + 2b1 + c1 )(a − 2b + c). Hence we obtain (5.2.6)

d (a − 2b + c) ≥ (a1 + 2b1 + c1 )(a − 2b + c). dt

We now state and prove the following pinching estimates of Hamilton for the associated ODE (5.2.3). Proposition 5.2.6 (Hamilton [59]). If we choose successively positive constants G large enough, H large enough, δ small enough, J large enough, ε small enough, K large enough, θ small enough, and L large enough, with each depending on those chosen before, then the closed convex subset X of {Mαβ ≥ 0} defined by the inequalities (1) (b2 + b3 )2 ≤ Ga1 c1 , (2) a3 ≤ Ha1 and c3 ≤ Hc1 , (3) (b2 + b3 )2+δ ≤ Ja1 c1 (a − 2b + c)δ , (4) (b2 + b3 )2+ε ≤ Ka1 c1 , (5) a3 ≤ a1 + La1−θ and c3 ≤ c1 + Lc1−θ , 1 1 is preserved by ODE (5.2.3). Moreover every compact subset of {Mαβ > 0} lies in some such set X. Proof. Clearly the subset X is closed and convex. We first note that we may assume b2 + b3 > 0 because if b2 + b3 = 0, then from (5.2.5), b2 + b3 will remain

330


zero and then the inequalities (1), (3) and (4) concerning b2 + b3 are automatically satisfied. Likewise we may assume a3 > 0 and c3 > 0 from (5.2.4). Let G be a fixed positive constant. To prove the inequality (1) we only need to check a 1 c1 d log ≥0 dt (b2 + b3 )2

(5.2.7)

whenever (b2 + b3 )2 = Ga1 c1 and b2 + b3 > 0. Indeed, it follows from (5.2.4) and (5.2.5) that d (a1 − b1 )2 a3 log a1 ≥ 2b1 + 2a3 + + 2 (a2 − a1 ), dt a1 a1 (c1 − b1 )2 c3 d log c1 ≥ 2b1 + 2c3 + + 2 (c2 − c1 ), dt c1 c1

(5.2.8) (5.2.9) and (5.2.10)

b2 d log(b2 + b3 ) ≤ 2b1 + a3 + c3 − [(a3 − a2 ) + (c3 − c2 )], dt b2 + b3

which immediately give the desired inequality (5.2.7). By (5.2.4), we have d b2 2a1 log a3 ≤ a3 + 2a1 + 3 − (a3 − a2 ). dt a3 a3

(5.2.11)

From the inequality (1) there holds b23 ≤ Ga1 c1 . Since tr A = tr C, c1 ≤ c1 + c2 + c3 = a1 + a2 + a3 ≤ 3a3 which shows b23 ≤ 3Ga1 . a3

Thus by (5.2.8) and (5.2.11), d a3 log ≤ (3G + 2)a1 − a3 . dt a1 So if H ≥ (3G + 2), then the inequalities a3 ≤ Ha1 and likewise c3 ≤ Hc1 are preserved. For the inequality (3), we compute from (5.2.8)-(5.2.10) a 1 c1 (a1 − b1 )2 (c1 − b1 )2 a3 c3 d log ≥ + + 2 (a2 − a1 ) + 2 (c2 − c1 ) 2 dt (b2 + b3 ) a1 c1 a1 c1 2b2 + [(a3 − a2 ) + (c3 − c2 )]. b2 + b3 If b1 ≤ a1 /2, then (a1 − b1 )2 a1 1 ≥ ≥ a3 , a1 4 4H and if b1 ≥ a1 /2, then 2b2 1 2b2 2b2 2b2 ≥ √ . ≥√ ≥ √ ≥√ b2 + b3 Ga1 c1 3Ga1 a3 3GH · a1 3GH


331

Thus by combining with 3a3 ≥ c3 , we have a 1 c1 d log ≥ δ(a3 − a1 ) + δ(c3 − c1 ) dt (b2 + b3 )2 1 1 provided δ ≤ min( 24H , √3GH ). On the other hand, it follows from (5.2.6) and (5.2.10) that

d b2 + b3 log ≤ (a3 − a1 ) + (c3 − c1 ). dt a − 2b + c Therefore the inequality (3) (b2 + b3 )2+δ ≤ Ja1 c1 (a − 2b + c)δ will be preserved by any positive constant J. To verify the inequality (4), we first note that there is a small η > 0 such that b ≤ (1 − η)a, on the set defined by the inequality (3). Indeed, if b ≤ a2 , this is trivial and if b ≥ a2 , then a 2+δ ≤ (b2 + b3 )2+δ ≤ 2δ Ja2 (a − b)δ 3 which makes b ≤ (1 − η)a for some η > 0 small enough. Consequently, ηa ≤ a − b ≤ 3(a3 − b1 ) which implies either a3 − a1 ≥

1 ηa, 6

a1 − b 1 ≥

1 ηa. 6

or

Thus as in the proof of the inequality (3), we have d a 1 c1 log ≥ δ(a3 − a1 ) dt (b2 + b3 )2 and a 1 c1 (a1 − b1 )2 d log ≥ , dt (b2 + b3 )2 a1 which in turn implies d a 1 c1 log ≥ dt (b2 + b3 )2

1 1 2 max ηδ, η · a. 6 36

On the other hand, it follows from (5.2.10) that d log(b2 + b3 ) ≤ 2b1 + a3 + c3 ≤ 4a dt

332


since Mαβ ≥ 0. Then if ε > 0 is small enough a 1 c1 d log ≥0 dt (b2 + b3 )2+ε and it follows that the inequality (4) is preserved by any positive K. Finally we consider the inequality (5). From (5.2.8) we have d log a1 ≥ a1 + 2a3 dt and then for θ ∈ (0, 1), a1 + (1 − θ)La1−θ d 1 log(a1 + La11−θ ) ≥ (a1 + 2a3 ). dt a1 + La1−θ 1 On the other hand, the inequality (4) tells us ˜ 1−θ a3 b23 ≤ Ka 1 ˜ large enough with θ to be fixed small enough. And then for some positive constant K d ˜ 1−θ , log a3 ≤ a3 + 2a1 + Ka 1 dt by combining with (5.2.11). Thus by choosing θ ≤

1 6H

˜ and L ≥ 2K,

a1 + La1−θ La1−θ d 1 1 ˜ 1−θ log ≥ (a3 − a1 ) − θ (a1 + 2a3 ) − Ka 1 dt a3 a1 + La1−θ 1 La1−θ 1 ˜ 1−θ · 3Ha1 − Ka 1 a1 + La1−θ 1 ˜ 1−θ ≥ (a3 − a1 ) − (3θHL + K)a ≥ (a3 − a1 ) − θ

1

˜ 1−θ = [L − (3θHL + K)]a 1 ≥0

whenever a1 + La11−θ = a3 . Consequently the set {a1 + La1−θ ≥ a3 } is preserved. A 1 similar argument works for the inequality in C. This completes the proof of Proposition 5.2.6. The combination of the advanced maximum principle Theorem 2.3.1 and the pinching estimates of the ODE (5.2.3) in Proposition 5.2.6 immediately gives the following pinching estimate for the Ricci flow on a compact four-manifold. Corollary 5.2.7. Suppose that the initial metric of the solution to the Ricci flow on a compact four-manifold has positive curvature operator. Then for any ε > 0 we can find positive constant Cε < +∞ such that ◦

|Rm| ≤ εR + Cε ◦

for all t ≥ 0 as long as the solution exists, where Rm is the traceless part of the curvature operator.


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Proof of Theorem 5.2.2. Let M be a compact four-manifold with positive curvature operator and let us evolve the metric by the Ricci flow. Again the evolution equation of the scalar curvature tells us that the scalar curvature remains positive and becomes unbounded in finite time. Pick a sequence of points xj ∈ M and times τj where the curvature at xj is as large as it has been anywhere for 0 ≤ t ≤ τj . Dilate the metrics so that the maximum curvature |Rm(xj , τj )| = maxt≤τj maxx∈M |Rm(x, t)| becomes 1 and shift the time so that the time τj becomes the new time 0. The Klingenberg injectivity radius estimate in Section 4.2 tells us that the injectivity radii of the rescaled metrics at the origins xj and at the new time 0 are uniformly bounded from below. Then we can apply the Hamilton’s compactness theorem (Theorem 4.1.5) to take a limit. By the pinching estimate in Corollary 5.2.7, we know that the limit metric has positive constant curvature which is either S4 or RP4 . Therefore the compact four-manifold M is diffeomorphic to the sphere S4 or the real projective space RP4 . Remark 5.2.8. The proofs of Theorem 5.2.1 and Theorem 5.2.2 also show that the Ricci flow on a compact three-manifold with positive Ricci curvature or a compact four-manifold with positive curvature operator is subsequentially converging (up to scalings) in the C ∞ topology to the same underlying compact manifold with a metric of positive constant curvature. Of course, this subsequential convergence is in the sense of Hamilton’s compactness theorem (Theorem 4.1.5) which is also up to the pullbacks of diffeomorphisms. Actually in [58] and [59], Hamilton obtained the convergence in the stronger sense that the (rescaled) metrics converge (in the C ∞ topology) to a constant (positive) curvature metric. In the following we use the Hamilton’s strong maximum principle in Section 2.4 to prove Theorem 5.2.3. Proof of Theorem 5.2.3. In views of Theorem 5.2.1 and Theorem 5.2.2, we may assume the Ricci curvature (in dimension 3) and the curvature operator (in dimension 4) always have nontrivial kernels somewhere along the Ricci flow. (i) In the case of dimension 3, we consider the evolution equation (1.3.5) of the Ricci curvature ∂Rab = △Rab + 2Racbd Rcd ∂t in an orthonormal frame coordinate. At each point, we diagonalize Rab with eigenvectors e1 , e2 , e3 and eigenvalues λ1 ≤ λ2 ≤ λ3 . Since R1c1d Rcd = R1212 R22 + R1313 R33 1 = ((λ3 − λ2 )2 + λ1 (λ2 + λ3 )), 2 we know that if Rab ≥ 0, then Racbd Rcd ≥ 0. By Hamilton’s strong maximum principle (Theorem 2.2.1), there exists an interval 0 < t < δ on which the rank of Rab is constant and the null space of Rab is invariant under parallel translation and invariant in time and also lies in the null space of Racbd Rcd . If the null space of Rab has rank one, then λ1 = 0 and λ2 = λ3 > 0. In this case, by De Rham decomposition ˜ of the compact M splits isometrically as R × Σ2 theorem, the universal cover M 2 and the curvature of Σ has a positive lower bound. Hence Σ2 is diffeomorphic to S2 . Assume M = R × Σ2 /Γ, for some isometric subgroup Γ of R × Σ2 . Note that Γ

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remains to be an isometric subgroup of R× Σ2 during the Ricci flow by the uniqueness (Theorem 1.2.4). Since the Ricci flow on R × Σ2 /Γ converges to the standard metric by Theorem 5.1.11, Γ must be an isometric subgroup of R× S2 in the standard metric. If the null space of Rab has rank greater than one, then Rab = 0 and the manifold is flat. This proves Theorem 5.2.3 part (i). (ii) In the case of dimension 4, we classify the manifolds according to the (restricted) holonomy algebra G. Note that the curvature operator has nontrivial kernel and G is the image of the the curvature operator, we see that G is a proper subalgebra of so(4). We divide the argument into two cases. Case 1. G is reducible.

˜ splits isometrically as M ˜1 × M ˜2 . By the above In this case the universal cover M results on two and three dimensional Ricci flow, we see that M is diffeomorphic to a quotient of one of the spaces R4 , R × S3 , R2 × S2 , S2 × S2 by a group of fixed point free isometries. As before by running the Ricci flow until it converges and using the uniqueness (Theorem 1.2.4), we see that this group is actually a subgroup of the isometries in the standard metrics. Case 2. G is not reducible (i.e., irreducible).

If the manifold is not Einstein, then by Berger’s classification theorem for holonomy groups [7], G = so(4) or u(2). Since the curvature operator is not strictly ˜ of M is K¨ positive, G = u(2), and the universal cover M ahler and has positive bisec˜ tional curvature. In this case M is biholomorphic to CP2 by the result of AndreottiFrankel [47] (also the resolution of the Frankel conjecture by Mori [96] and Siu-Yau [120]). If the manifold is Einstein, then by the block decomposition of the curvature operator matrix in four-manifolds (see the third section of Chapter 1), Rm(Λ2+ , Λ2− ) = 0. Let ϕ 6= 0, and ϕ = ϕ+ + ϕ− ∈ Λ2+ ⊕ Λ2− , lies in the kernel of the curvature operator, then 0 = Rm(ϕ+ , ϕ+ ) + Rm(ϕ− , ϕ− ). It follows that Rm(ϕ+ , ϕ+ ) = 0, and Rm(ϕ− , ϕ− ) = 0. We may assume ϕ+ 6= 0 (the argument for the other case is similar). We consider the restriction of Rm to Λ2+ , since Λ2+ is an invariant subspace of Rm and the intersection of Λ2+ with the null space of Rm is nontrivial. By considering the null space of Rm and its orthogonal complement in Λ2+ , we obtain a parallel distribution of rank one in Λ2+ . This parallel distribution gives a parallel translation invariant two-form ˜ of M . This two-form is nondegenerate, so it ω ∈ Λ2+ on the universal cover M ˜ . Since the K¨ induces a K¨ ahler structure of M ahler metric is parallel with respect to the original metric and the manifold is irreducible, the K¨ ahler metric is proportional ˜ is K¨ to the original metric. Hence the manifold M ahler-Einstein with nonnegative


335

˜ is curvature operator. Taking into account the irreducibility of G, it follows that M biholomorphic to CP2 . Therefore the proof of Theorem 5.2.3 is completed. To end this section, we mention the generalizations of Hamilton’s differential sphere theorem (Theorem 5.2.1 and Theorem 5.2.2) to higher dimensions. Using minimal surface theory, Micallef and Moore [88] proved that any compact simply connected n-dimensional manifold with positive curvature operator is homeomorphic to the sphere Sn . But it is still an open question whether a compact simply connected n-dimensional manifold with positive curvature operator is diffeomorphic to the sphere Sn . It is well-known that the curvature tensor Rm = {Rijkl } of a Riemannian manifold can be decomposed into three orthogonal components which have the same symmetries as Rm: Rm = W + V + U. Here W = {Wijkl } is the Weyl conformal curvature tensor, whereas V = {Vijkl } and U = {Uijkl } denote the traceless Ricci part and the scalar curvature part respectively. The following pointwisely pinching sphere theorem under the additional assumption that the manifold is compact was first obtained by Huisken [71], Margerin [83], [84] and Nishikawa [101] by using the Ricci flow. The compactness assumption was later removed by Chen and the second author in [30]. Theorem 5.2.9. Let n ≥ 4. Suppose M is a complete n-dimensional manifold with positive and bounded scalar curvature and satisfies the pointwisely pinching condition |W |2 + |V |2 ≤ δn (1 − ε)2 |U |2 , where ε > 0, δ4 = 51 , δ5 =

1 10 ,

and δn =

2 , n ≥ 6. (n − 2)(n + 1)

Then M is diffeomorphic to the sphere Sn or a quotient of it by a finite group of fixed point free isometries in the standard metric. In [30], Chen and the second author also used the Ricci flow to obtain the following flatness theorem for noncompact three-manifolds. Theorem 5.2.10. Let M be a three-dimensional complete noncompact Riemannian manifold with bounded and nonnegative sectional curvature. Suppose M satisfies the following Ricci pinching condition Rij ≥ εRgij ,

on M,

for some ε > 0. Then M is flat. The basic idea of proofs of these two theorems is to analyze the asymptotic behavior of the solution to the Ricci flow. For the details, one can consult the above cited literatures.

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5.3. Nonsingular Solutions on Three-manifolds. We have seen in the previous section that a good understanding of the long time behaviors for solutions to the Ricci flow could lead to remarkable topological or geometric consequences for the underlying manifolds. Since one of the central themes of the Ricci flow is to study the geometry and topology of three-manifolds, we will start to analyze the long time behavior of the Ricci flow on a compact three-manifold by first considering a special class of solutions (i.e., the nonsingular solutions defined below) in this section. Our presentation follows closely the paper of Hamilton [65]. Let M be a compact three-manifold. We will consider the (unnormalized) Ricci flow ∂ gij = −2Rij , ∂t and the normalized Ricci flow ∂ 2 gij = rgij − 2Rij ∂t 3 where r = r(t) is the function of the average of the scalar curvature. Recall that the normalized flow differs from theRunnormalized flow only by rescaling in space and time so that the total volume V = M dµ remains constant. In this section we only consider a special class of solutions that we now define. Definition 5.3.1. A nonsingular solution of the Ricci flow is one where the solution of the normalized flow exists for all time 0 ≤ t < ∞, and the curvature remains bounded |Rm| ≤ C < +∞ for all time with some constant C independent of t. Clearly any solution to the Ricci flow on a compact three-manifold with nonnegative Ricci curvature is nonsingular. Currently there are few conditions which guarantee a solution will remain nonsingular. Nevertheless, the ideas and arguments of this section is extremely important. One will see in Chapter 7 that these arguments will be modified to analyze the long-time behavior of arbitrary solutions, or even the solutions with surgery, to the Ricci flow on three-manifolds. We begin with an improvement of Hamilton-Ivey pinching result (Theorem 2.4.1). Theorem 5.3.2 (Hamilton [65]). Suppose we have a solution to the (unnormalized ) Ricci flow on a three–manifold which is complete with bounded curvature for each t ≥ 0. Assume at t = 0 the eigenvalues λ ≥ µ ≥ ν of the curvature operator at each point are bounded below by ν ≥ −1. Then at all points and all times t ≥ 0 we have the pinching estimate R ≥ (−ν)[log(−ν) + log(1 + t) − 3] whenever ν < 0. Proof. As before, we study the ODE system  dλ   = λ2 + µν,   dt    dµ = µ2 + λν,  dt       dν = ν 2 + λµ. dt


337

Consider again the function y = f (x) = x(log x − 3) for e2 ≤ x < +∞, which is increasing and convex with range −e2 ≤ y < +∞. Its inverse function x = f −1 (y) is increasing and concave on −e2 ≤ y < +∞. For each t ≥ 0, we consider the set K(t) of 3×3 symmetric matrices defined by the inequalities: λ+µ+ν ≥−

(5.3.1)

3 , 1+t

and (5.3.2)

ν(1 + t) + f −1 ((λ + µ + ν)(1 + t)) ≥ 0,

which is closed and convex (as we saw in the proof of Theorem 2.4.1). By the assumptions at t = 0 and the advanced maximum principle Theorem 2.3.5, we only need to check that the set K(t) is preserved by the ODE system. Since R = λ + µ + ν, we get from the ODE that dR 2 1 ≥ R2 ≥ R2 dt 3 3 which implies that R≥−

3 , 1+t

for all t ≥ 0.

Thus the first inequality (5.3.1) is preserved. Note that the second inequality (5.3.2) is automatically satisfied when (−ν) ≤ 3/(1 + t). Now we compute from the ODE system, d R 1 dR d(−ν) ( − log(−ν)) = (−ν) · − (R + (−ν)) dt (−ν) (−ν)2 dt dt =

1 [(−ν)3 + (−ν)µ2 + λ2 ((−ν) + µ) − λµ(ν − µ)] (−ν)2

≥ (−ν) ≥

3 (1 + t)

≥

d [log(1 + t) − 3] dt

whenever R = (−ν)[log(−ν) + log(1 + t) − 3] and (−ν) ≥ 3/(1 + t). Thus the second inequality (5.3.2) is also preserved under the system of ODE. Therefore we have proved the theorem. Denote by ρˆ(t) = max{inj (x, gij (t)) | x ∈ M } where inj (x, gij (t)) is the injectivity radius of the manifold M at x with respect to the metric gij (t).

338


Definition 5.3.3. We say a solution to the normalized Ricci flow is collapsed if there is a sequence of times tk → +∞ such that ρˆ(tk ) → 0 as k → +∞. When a nonsingular solution of the Ricci flow on M is collapsed, it follows from the work of Cheeger-Gromov [24] [25] or Cheeger-Gromov-Fukaya [26] that the manifold M has an F -structure and then its topology is completely understood. In the following we always assume nonsingular solutions are not collapsed. Now suppose that we have a nonsingular solution which does not collapse. Then for arbitrary sequence of times tj → ∞, we can find a sequence of points xj and some δ > 0 so that the injectivity radius of M at xj in the metric at time tj is at least δ. Clearly the Hamilton’s compactness theorem (Theorem 4.1.5) also holds for the normalized Ricci flow. Then by taking the xj as origins and the tj as initial times, we can extract a convergent subsequence. We call such a limit a noncollapsing limit. Of course the limit has also finite volume. However the volume of the limit may be smaller than the original one if the diameter goes to infinity. The main result of this section is the following theorem of Hamilton [65]. Theorem 5.3.4 (Hamilton [65]). Let gij (t), 0 ≤ t < +∞, be a noncollapsing nonsingular solution of the normalized Ricci flow on a compact three-manifold M . Then either (i) there exist a sequence of times tk → +∞ and a sequence of diffeomorphisms ϕk : M → M so that the pull-back of the metric gij (tk ) by ϕk converges in the C ∞ topology to a metric on M with constant sectional curvature; or (ii) we can find a finite collection of complete noncompact hyperbolic threemanifolds H1 , . . . , Hm with finite volume, and for all t beyond some time T < +∞ we can find compact subsets K1 , . . . , Km of H1 , . . . , Hm respectively obtained by truncating each cusp of the hyperbolic manifolds along constant mean curvature torus of small area, and diffeomorphisms ϕl (t), 1 ≤ l ≤ m, of Kl into M so that as long as t sufficiently large, the pull-back of the solution metric gij (t) by ϕl (t) is as close as to the hyperbolic metric as we like on the compact sets K1 , . . . , Km ; and moreover if we call the exceptional part of M those points where they are not in the image of any ϕl , we can take the injectivity radii of the exceptional part at everywhere as small as we like and the boundary tori of each Kl are incompressible in the sense that each ϕl injects π1 (∂Kl ) into π1 (M ). Remark 5.3.5. The exceptional part has bounded curvature and arbitrarily small injectivity radii everywhere as t large enough. Moreover the boundary of the exceptional part consists of a finite disjoint union of tori with sufficiently small area and is convex. Then by the work of Cheeger-Gromov [24], [25] or Cheeger-GromovFukaya [26], there exists an F -structure on the exceptional part. In particular, the exceptional part is a graph manifold, which have been topologically classified. Hence any nonsingular solution to the normalized Ricci flow is geometrizable in the sense of Thurston (see the last section of Chapter 7 for details). The rest of this section is devoted to the proof of Theorem 5.3.4. We will divide the proof into three parts. Part I: Subsequence Convergence According to Lemma 5.1.1, the scalar curvature of the normalized flow evolves by


339

the equation ∂ 2 R = ∆R + 2|Ric|2 − rR ∂t 3 ◦ 2 = ∆R + 2| Ric | + R(R − r) 3

(5.3.3)

◦

where Ric is the traceless part of the Ricci tensor. As before, we denote by Rmin (t) = minx∈M R(x, t). It then follows from the maximum principle that (5.3.4)

d 2 Rmin ≥ Rmin (Rmin − r), dt 3

which implies that if Rmin ≤ 0 it must be nondecreasing, and if Rmin ≥ 0 it cannot go negative again. We can then divide the noncollapsing solutions of the normalized Ricci flow into three cases. Case (1): Rmin (t) > 0 for some t > 0; Case (2): Rmin (t) ≤ 0 for all t ∈ [0, +∞) and lim Rmin (t) = 0; t→+∞

Case (3): Rmin (t) ≤ 0 for all t ∈ [0, +∞) and lim Rmin (t) < 0. t→+∞

Let us first consider Case (1). In this case the maximal time interval [0, T ) of the corresponding solution of the unnormalized flow is finite, since the unnormalized ˜ satisfies scalar curvature R ∂ ˜ ˜ + 2|Ric| ˜ 2 R = ∆R ∂t ˜ + 2R ˜2 ≥ ∆R 3 which implies that the curvature of the unnormalized solution blows up in finite time. Without loss of generality, we may assume that for the initial metric at t = 0, the ˜ ≥µ eigenvalues λ ˜ ≥ ν˜ of the curvature operator are bounded below by ν˜ ≥ −1. It follows from Theorem 5.3.2 that the pinching estimate ˜ ≥ (−˜ R ν )[log(−˜ ν ) + log(1 + t) − 3] holds whenever ν˜ < 0. This shows that when the unnormalized curvature big, the negative ones are not nearly as large as the positive ones. Note that the unnormalized curvature becomes unbounded in finite time. Thus when we rescale the unnormalized flow to the normalized flow, the scaling factor must go to infinity. In the nonsingular case the rescaled positive curvature stay finite, so the rescaled negative curvature (if any) go to zero. Hence we can take a noncollapsing limit for the nonsingular solution of the normalized flow so that it has nonnegative sectional curvature. Since the volume of the limit is finite, it follows from a result of Calabi and Yau [112] that the limit must be compact and the limiting manifold is the original one. Then by the strong maximum principle as in the proof of Theorem 5.2.3 (i), either the limit is flat, or it is a compact metric quotient of the product of a positively curved surface Σ2 with R, or it has strictly positive curvature. By the work of Schoen-Yau [110], a flat three-manifold cannot have a metric of positive scalar curvature, but our manifold does in Case (1). This rules out the possibility of a flat limit. Clearly the limit is also a nonsingular solution to the normalized Ricci flow. Note that the

340


curvature of the surface Σ2 has a positive lower bound and is compact since it comes from the lifting of the compact limiting manifold. From Theorem 5.1.11, we see the metric of the two-dimensional factor Σ2 converges to the round two-sphere S2 in the normalized Ricci flow. Note also that the normalized factors in two-dimension and three-dimension are different. This implies that the compact quotient of the product Σ2 × R cannot be nonsingular, which is also ruled out for the limit. Thus the limit must have strictly positive sectional curvature. Since the convergence takes place everywhere for the compact limit, it follows that as t large enough the original nonsingular solution has strictly positive sectional curvature. This in turn shows that the corresponding unnormalized flow has strictly positive sectional curvature after some finite time. Then in views of the proof of Theorem 5.2.1, in particular the pinching estimate in Proposition 5.2.5, the limit has constant Ricci curvature and then constant sectional curvature for three-manifolds. This finishes the proof in Case (1). We next consider Case (2). In this case we only need to show that we can take a noncollapsing limit which has nonnegative sectional curvature. Indeed, if this is true, then as in the previous case, the limit is compact and either it is flat, or it splits as a product (or a quotient of a product) of a positively curved S 2 with a circle S 1 , or it has strictly positive curvature. But the assumption Rmin (t) ≤ 0 for all times t ≥ 0 in this case implies the limit must be flat. Let us consider the corresponding unnormalized flow g˜ij (t) associated to the noncollapsing nonsingular solution. The pinching estimate in Theorem 5.3.2 tells us that we may assume the unnormalized flow g˜ij (t) exists for all times 0 ≤ t < +∞, for otherwise, the scaling factor approaches infinity as in the previous case which implies the limit has nonnegative sectional curvature. The volume V˜ (t) of the unnormalized solution g˜ij (t) now changes. We divide the discussion into three subcases. Subcase (2.1): there is a sequence of times t˜k → +∞ such that V˜ (t˜k ) → +∞; Subcase (2.2): there is a sequence of times t˜k → +∞ such that V˜ (t˜k ) → 0; Subcase (2.3): there exist two positive constants C1 , C2 such that C1 ≤ V˜ (t) ≤ C2 for all 0 ≤ t < +∞. For Subcase (2.1), because dV˜ = −rV˜ dt we have Z t˜k V˜ (t˜k ) log =− r(t)dt → +∞, as k → +∞, V˜ (0) 0 which implies that there exists another sequence of times, still denoted by t˜k , such that t˜k → +∞ and r(t˜k ) ≤ 0. Let tk be the corresponding times for the normalized flow. Thus there holds for the normalized flow r(tk ) → 0,

as k → ∞,

since 0 ≥ r(tk ) ≥ Rmin (tk ) → 0 as k → +∞. Then Z (R − Rmin )dµ(tk ) = (r(tk ) − Rmin (tk ))V → 0, M

as k → ∞.


341

As we take a noncollapsing limit along the time sequence tk , we get Z Rdµ∞ = 0 M∞

for the limit of the normalized solutions at the new time t = 0. But R ≥ 0 for the limit because lim Rmin (t) = 0 for the nonsingular solution. So R = 0 at t = 0 for t→+∞

the limit. Since the limit flow exists for −∞ < t < +∞ and the scalar curvature of the limit flow evolves by ∂ 2 R = ∆R + 2|Ric |2 − r∞ R, ∂t 3

t ∈ (−∞, +∞)

where r∞ is the limit of the function r(t) by translating the times tk as the new time t = 0. It follows from the strong maximum principle that R ≡ 0,

on M ∞ × (−∞, +∞).

This in turn implies, in view of the above evolution equation, that Ric ≡ 0,

on M ∞ × (−∞, +∞).

Hence this limit must be flat. Since the limit M ∞ is complete and has finite volume, the flat manifold M ∞ must be compact. Thus the underlying manifold M ∞ must agree with the original M (as a topological manifold). This says that the limit was taken on M . For Subcase (2.2), we may assume as before that for the initial metric at t = 0 ˜≥µ of the unnormalized flow g˜ij (t), the eigenvalues λ ˜ ≥ ν˜ of the curvature operator satisfy ν˜ ≥ −1. It then follows from Theorem 5.3.2 that ˜ ≥ (−˜ R ν )[log(−˜ ν ) + log(1 + t) − 3],

for all t ≥ 0

whenever ν˜ < 0. Let tk be the sequence of times in the normalized flow which corresponds to the sequence of times t˜k . Take a noncollapsing limit for the normalized flow along the times tk . Since V˜ (t˜k ) → 0, the normalized curvatures at the times tk are reduced by 2 multiplying the factor (V˜ (t˜k )) 3 . We claim the noncollapsing limit has nonnegative sectional curvature. Indeed if the maximum value of (−˜ ν ) at the time t˜k does not go to infinity, the normalized eigenvalue −ν at the corresponding time tk must get rescaled to tend to zero; while if the maximum value of (−˜ ν ) at the time t˜k does go to ˜ ˜ infinity, the maximum value of R at tk will go to infinity even faster from the pinching estimate, and when we normalize to keep the normalized scalar curvature R bounded at the time tk so the normalized (−ν) at the time tk will go to zero. Thus in either case the noncollapsing limit has nonnegative sectional curvature at the initial time t = 0 and then has nonnegative sectional curvature for all times t ≥ 0. For Subcase (2.3), normalizing the flow only changes quantities in a bounded way. As before we have the pinching estimate R ≥ (−ν)[log(−ν) + log(1 + t) − C] for the normalized Ricci flow, where C is a positive constant depending only on the constants C1 , C2 in the assumption of Subcase (2.3). If (−ν) ≤

A 1+t

342


for any fixed positive constant A, then (−ν) → 0 as t → +∞ and we can take a noncollapsing limit which has nonnegative sectional curvature. On the other hand if we can pick a sequence of times tk → ∞ and points xk where (−ν)(xk , tk ) = max(−ν)(x, tk ) satisfies x∈M

(−ν)(xk , tk )(1 + tk ) → +∞,

as k → +∞,

then from the pinching estimate, we have R(xk , tk ) → +∞, (−ν)(xk , tk )

as

k → +∞.

But R(xk , tk ) are uniformly bounded since normalizing the flow only changes quantities in bounded way. This shows sup(−ν)(·, tk ) → 0 as k → +∞. Thus we can take a noncollapsing limit along tk which has nonnegative sectional curvature. Hence we have completed the proof of Case (2). We now come to the most interesting Case (3) where Rmin increases monotonically to a limit strictly less than zero. By scaling we can assume Rmin (t) → −6 as t → +∞. Lemma 5.3.6. In Case (3) where Rmin → −6 as t → +∞, all noncollapsing limit are hyperbolic with constant sectional curvature −1. Proof. By (5.3.4) and the fact Rmin (t) ≤ −6, we have d Rmin (t) ≥ 4(r(t) − Rmin (t)) dt and Z

∞ 0

(r(t) − Rmin (t))dt < +∞.

Since r(t) − Rmin (t) ≥ 0 and Rmin (t) → −6 as t → +∞, it follows that the function r(t) has the limit r = −6, for any convergent subsequence. And since Z (R − Rmin (t))dµ = (r(t) − Rmin (t)) · V, M

it then follows that R ≡ −6 for the limit. The limit still has the following evolution equation for the limiting scalar curvature ◦ ∂ 2 R = ∆R + 2|Ric|2 + R(R − r). ∂t 3 ◦

Since R ≡ r ≡ −6 in space and time for the limit, it follows directly that |Ric| ≡ 0 for the limit. Thus the limit metric has λ = µ = ν = −2, so it has constant sectional curvature −1 as desired.


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If in the discussion above there exists a compact noncollapsing limit, then we know that the underlying manifold M is compact and we fall into the conclusion of Theorem 5.3.4(i) for the constant negative sectional curvature limit. Thus it remains to show when every noncollapsing limit is a complete noncompact hyperbolic manifold with finite volume, we have conclusion (ii) in Theorem 5.3.4. Now we first want to find a finite collection of persistent complete noncompact hyperbolic manifolds as stated in Theorem 5.3.4 (ii). Part II: Persistence of Hyperbolic Pieces We begin with the definition of the topology of C ∞ convergence on compact sets for maps F : M → N of one Riemannian manifold to another. For any compact set K ⊂⊂ M and any two maps F, G : M → N , we define dK (F, G) = sup d(F (x), G(x)) x∈K 0 where d(y, z) is the geodesic distance from y to z on N . This gives the Cloc topology k for maps between M and N . To define Cloc topology for any positive integer k ≥ 1, we consider the k-jet space J k M of a manifold M which is the collection of all

(x, J 1 , J 2 , . . . , J k ) where x is a point on M and J i is a tangent vector for 1 ≤ i ≤ k defined by the ith covariant derivative J i = ∇i∂ γ(0) for a path γ passing through the point x with ∂t

γ(0) = x. A smooth map F : M → N induces a map J kF :

J kM → J kN

defined by J k F (x, J 1 , . . . , J k ) = (F (x), ∇ ∂ (F (γ))(0), . . . , ∇k∂ (F (γ))(0)) ∂t

∂t

where γ is a path passing through the point x with J i = ∇i∂ γ(0), 1 ≤ i ≤ k. ∂t Define the k-jet distance between F and G on a compact set K ⊂⊂ M by dC k (K) (F, G) = dBJ k K (J k F, J k G) where BJ k K consists of all k-jets (x, J 1 , . . . , J k ) with x ∈ K and |J 1 |2 + |J 2 |2 + · · · + |J k |2 ≤ 1. Then the convergence in the metric dC k (K) for all positive integers k and all compact sets K defines the topology of C ∞ convergence on compact sets for the space of maps. We will need the following Mostow type rigidity result. Lemma 5.3.7. For any complete noncompact hyperbolic three-manifold H with finite volume with metric h, we can find a compact set K of H such that for every integer k and every ε > 0, there exist an integer q and a δ > 0 with the following property: if F is a diffeomorphism of K into another complete noncompact hyperbolic ˜ such that ˜ with no fewer cusps (than H), finite volume with metric h three-manifold H ˜ − hkC q (K) < δ kF ∗ h

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˜ such that then there exists an isometry I of H to H dC k (K) (F, I) < ε. ˜ for an appropriate choice of Proof. First we claim that H is isometric to H compact set K, positive integer q and positive number δ. Let l : H → R be a function defined at each point by the length of the shortest non-contractible loop starting and ending at this point. Denote the Margulis constant by µ. Then by Margulis lemma (see for example [55] or [76]), for any 0 < ε0 < 21 µ, the set l−1 ([0, ε0 ]) ⊂ H consists of finitely many components and each of these components is isometric to a cusp or to a tube. Topologically, a tube is just a solid torus. Let ε0 be even smaller than one half of the minimum of the lengths of the all closed geodesics on the tubes. Then l−1 ([0, ε0 ]) consists of finite number of cusps. Set K0 = l−1 ([ε0 , ∞)). The boundary of K0 consists of flat tori with constant mean curvatures. Note that each embedded torus in a complete hyperbolic three-manifold with finite volume either bounds a solid torus or is isotopic to a standard torus in a cusp. The diffeomorphism F implies the boundary F (∂K0 ) are embedded tori. If one of components bounds a solid torus, then ˜ would have fewer cusps than H, which as δ sufficiently small and q sufficiently large, H o ˜ is diffeomorphic to F (K0 ). Here contradicts with our assumption. Consequently, H o

o

K0 is the interior of the set K0 . Since H is diffeomorphic to K0 , H is diffeomorphic ˜ Hence by Mostow’s rigidity theorem (see [97] and [107]), H is isometric to H. ˜ to H. ˜ So we can assume H = H. For K = K0 , we argue by contradiction. Suppose there is some k > 0 and ε > 0 so that there exist sequences of integers qj → ∞, δj → 0+ and diffeomorphisms Fj mapping K into H with kFj∗ h − hkC qj (K) < δj and dC k (K) (Fj , I) ≥ ε

for all isometries I of H to itself. We can extract a subsequence of Fj convergent to ∗ a map F∞ with F∞ h = h on K. We need to check that F∞ is still a diffeomorphism on K. Since F∞ is a local diffeomorphism and is the limit of diffeomorphisms, we can find an inverse of F∞ on o

o

F∞ (K). So F∞ is a diffeomorphism on K. We claim the image of the boundary can not touch the image of the interior. Indeed, if F∞ (x1 ) = F∞ (x2 ) with x1 ∈ ∂K and o

o

o

x2 ∈ K, then we can find x3 ∈ K near x1 and x4 ∈ K near x2 with F∞ (x3 ) = F∞ (x4 ), since F∞ is a local diffeomorphism. This contradicts with the fact that F∞ is a o

diffeomorphism on K. This proves our claim. Hence, the only possible overlap is at the boundary. But the image F∞ (∂K) is strictly concave, this prevents the boundary from touching itself. We conclude that the mapping F∞ is a diffeomorphism on K, hence an isometry. To extend F∞ to a global isometry, we argue as follows. For each truncated cusp end of K, the area of constant mean curvature flat torus is strictly decreasing. Since F∞ takes each such torus to another of the same area, we see that F∞ takes the foliation of an end by constant mean curvature flat tori to another such foliation. So F∞ takes cusps to cusps and preserves their foliations. Note that the isometric type


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of a cusp is just the isometric type of the torus, more precisely, let (N, dr2 + e−2r gV ) be a cusp (where gV is the flat metric on the torus V), 0 < a < b are two constants, any isometry of N ∩ l−1 [a, b] to itself is just an isometry of V. Hence the isometry F∞ can be extended to the whole cusps. This gives a global isometry I contradicting our assumption when j large enough. The proof of the Lemma 5.3.7 is completed. In order to obtain the persistent hyperbolic pieces stated in Theorem 5.3.4 (ii), we will need to use a special parametrization given by harmonic maps. Lemma 5.3.8. Let (X, g) be a compact Riemannian manifold with strictly negative Ricci curvature and with strictly concave boundary. Then there are positive integer l0 and small number ε0 > 0 such that for each positive integer l ≥ l0 and positive number ε ≤ ε0 we can find positive integer q and positive number δ > 0 such that for every metric g˜ on X with ||˜ g − g||C q (X) ≤ δ we can find a unique diffeomorphism F of X to itself so that (a) F : (X, g) → (X, g˜) is harmonic, (b) F takes the boundary ∂X to itself and satisfies the free boundary condition that the normal derivative ∇N F of F at the boundary is normal to the boundary, (c) dC l (X) (F, Id) < ε, where Id is the identity map. Proof. Let Φ(X, ∂X) be the space of maps of X to itself which take ∂X to itself. Then Φ(X, ∂X) is a Banach manifold and the tangent space to Φ(X, ∂X) at the ∂ identity is the space of vector fields V = V i ∂x i tangent to the boundary. Consider the map sending F ∈ Φ(X, ∂X) to the pair {∆F, (∇N F )// } consisting of the harmonic map Laplacian and the tangential component (in the target) of the normal derivative of F at the boundary. By using the inverse function theorem, we only need to check that the derivative of this map is an isomorphism at the identity with g˜ = g. Let {xi }i=1,...,n be a local coordinates of (X, g) and {y α }α=1,...,n be a local coordinates of (X, g˜). The harmonic map Laplacian of F : (X, g) → (X, g˜) is given in local coordinates by ˜α ◦ F ) (∆F )α = ∆(F α ) + g ij (Γ βγ

∂F β ∂F γ ∂xi ∂xj

˜ α is the connection where ∆(F α ) is the Laplacian of the function F α on X and Γ βγ of g˜. Let F be a one-parameter family with F |s=0 = Id and dF ds |s=0 = V , a smooth vector field on X tangent to the boundary (with respect to g). At an arbitrary given point x ∈ X, we choose the coordinates {xi }i=1,...,n so that Γijk (x) = 0. We compute at the point x with g˜ = g, ∂ α d Γ V k. (∆F )α = ∆(V α ) + g ij ds s=0 ∂xk ij Since

β (∇i V )α = ∇i V α + (Γα iβ ◦ F )V ,

we have, at s = 0 and the point x, (∆V )α = ∆(V α ) + g ij

∂ α k Γ V . ∂xi jk

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Thus we obtain (5.3.5)

d |s=0 (∆F )α = (∆V )α + g ij ds

∂ α ∂ α Γ − Γ Vk ∂xk ij ∂xi jk

= (∆V )α + g αi Rik V k . Since (∇N F )(F −1 (x)) = N i (F −1 (x))

∂F j −1 ∂ (F (x)) j (x) ∂xi ∂x

on ∂X,

we have (5.3.6)

d d |s=0 {(∇N F )// } = |s=0 (∇N F − h∇N F, N iN ) ds ds d d = |s=0 (∇N F ) − |s=0 ∇N F, N N ds ds

− h∇N F, ∇V N iN |s=0 − h∇N F, N i∇V N |s=0 d = |s=0 ∇N F − ∇V N ds // ∂ j i ∂ − II(V ) = −V (N ) i + N (V ) j ∂x ∂x //

= [N, V ]// − II(V )

= (∇N V )// − 2II(V )

where II is the second fundamental form of the boundary (as an automorphism of T (∂X)). Thus by (5.3.5) and (5.3.6), the kernel of the map sending F ∈ Φ(X, ∂X) to the pair {∆F, (∇N F )// } is the space of solutions of elliptic boundary value problem   on X ∆V + Ric (V ) = 0 (5.3.7) V⊥ = 0, at ∂X,   (∇N V )// − 2II(V ) = 0, at ∂X, where V⊥ is the normal component of V . Now using these equations and integrating by parts gives Z Z Z Z Z 2 |∇V | = Ric (V, V ) + 2 II(V, V ). X

X

∂X

Since Rc < 0 and II < 0 we conclude that the kernel is trivial. Clearly this elliptic boundary value is self-adjoint because of the free boundary condition. Thus the cokernel is trivial also. This proves the lemma. Now we can prove the persistence of hyperbolic pieces. Let gij (t), 0 ≤ t < +∞, be a noncollapsing nonsingular solution of the normalized Ricci flow on a compact three-manifold M . Assume that any noncollapsing limit of the nonsingular solution is a complete noncompact hyperbolic three-manifold with finite volume. Consider all the possible hyperbolic limits of the given nonsingular solution, and among them choose one such complete noncompact hyperbolic three-manifold H with the least possible number of cusps. In particular, we can find a sequence of times tk → +∞ and a


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sequence of points Pk on M such that the marked three-manifolds (M, gij (tk ), Pk ) ∞ converge in the Cloc topology to H with hyperbolic metric hij and marked point P ∈ H. For any small enough a > 0 we can truncate each cusp of H along a constant mean curvature torus of area a which is uniquely determined; the remainder we denote by Ha . Clearly as a → 0 the Ha exhaust H. Pick a sufficiently small number a > 0 to truncate cusps so that Lemma 5.3.7 is applicable for the compact set K = Ha . Choose an integer l0 large enough and an ε0 sufficiently small to guarantee from Lemma 5.3.8 the uniqueness of the identity map Id among maps close to Id as a harmonic map F from Ha to itself with taking ∂Ha to itself, with the normal derivative of F at the boundary of the domain normal to the boundary of the target, and with dC l0 (Ha ) (F, Id) < ε0 . Then choose positive integer q0 and small number δ0 > 0 from Lemma 5.3.7 such that if F˜ is a diffeomorphism of Ha into another complete ˜ with no fewer cusps (than H), finite volume noncompact hyperbolic three-manifold H ˜ with metric hij satisfying ˜ ij − hij ||C q0 (H ) ≤ δ0 , ||F˜ ∗ h a ˜ such that then there exists an isometry I of H to H (5.3.8)

dC l0 (Ha ) (F˜ , I) < ε0 .

And we further require q0 and δ0 from Lemma 5.3.8 to guarantee the existence of harmonic diffeomorphism from (Ha , g˜ij ) to (Ha , hij ) for any metric g˜ij on Ha with ||˜ gij − hij ||C q0 (Ha ) ≤ δ0 . By definition, there exist a sequence of exhausting compact sets Uk of H (each Uk ⊃ Ha ) and a sequence of diffeomorphisms Fk from Uk into M such that Fk (P ) = Pk and ||Fk∗ gij (tk ) − hij ||C m (Uk ) → 0 as k → +∞ for all positive integers m. Note that ∂Ha is strictly concave and we can foliate a neighborhood of ∂Ha with constant mean curvature hypersurfaces where the area a has a nonzero gradient. As the approximating maps Fk : (Uk , hij ) → (M, gij (tk )) are close enough to isometries on this collar of ∂Ha , the metrics gij (tk ) on M will also admit a unique constant mean curvature hypersurface with the same area a near Fk (∂Ha )(⊂ M ) by the inverse function theorem. Thus we can change the map Fk by an amount which goes to zero as k → ∞ so that now Fk (∂Ha ) has constant mean curvature with the area a. Furthermore, by applying Lemma 5.3.8 we can again change Fk by an amount which goes to zero as k → ∞ so as to make Fk a harmonic diffeomorphism and take ∂Ha to the constant mean curvature hypersurface Fk (∂Ha ) and also satisfy the free boundary condition that the normal derivative of Fk at the boundary of the domain is normal to the boundary of the target. Hence for arbitrarily given positive integer q ≥ q0 and positive number δ < δ0 , there exists a positive integer k0 such that for the modified harmonic diffeomorphism Fk , when k ≥ k0 , ||Fk∗ gij (tk ) − hij ||C q (Ha ) < δ. For each fixed k ≥ k0 , by the implicit function theorem we can first find a constant mean curvature hypersurface near Fk (∂Ha ) in M with the metric gij (t) for t close to tk and with the same area for each component since ∂Ha is strictly concave and a neighborhood of ∂Ha is foliated by constant mean curvature hypersurfaces where the area a has a nonzero gradient and Fk : (Ha , hij ) → (M, gij (tk )) is close enough to an isometry and gij (t) varies smoothly. Then by applying Lemma 5.3.8 we can smoothly

348


continue the harmonic diffeomorphism Fk forward in time a little to a family of harmonic diffeomorphisms Fk (t) from Ha into M with the metric gij (t), with Fk (tk ) = Fk , where each Fk (t) takes ∂Ha into the constant mean curvature hypersurface we just found in (M, gij (t)) and satisfies the free boundary condition, and also satisfies ||Fk∗ (t)gij (t) − hij ||C q (Ha ) < δ. We claim that for all sufficiently large k, we can smoothly extend the harmonic diffeomorphism Fk to the family harmonic diffeomorphisms Fk (t) with ||Fk∗ (t)gij (t) − hij ||C q (Ha ) ≤ δ on a maximal time interval tk ≤ t ≤ ωk (or tk ≤ t < ωk when ωk = +∞); and if ωk < +∞, then (5.3.9)

||Fk∗ (ωk )gij (ωk ) − hij ||C q (Ha ) = δ.

Clearly the above argument shows that the set of t where we can extend the harmonic diffeomorphisms as desired is open. To verify claim (5.3.9), we thus only need to show that if we have a family of harmonic diffeomorphisms Fk (t) such as we desire for tk ≤ t < ω(< +∞), we can take the limit of Fk (t) as t → ω to get a harmonic diffeomorphism Fk (ω) satisfying ||Fk∗ (ω)gij (ω) − hij ||C q (Ha ) ≤ δ, and if ||Fk∗ (ω)gij (ω) − hij ||C q (Ha ) < δ, then we can extend Fk (ω) forward in time a little (i.e., we can find a constant mean curvature hypersurface near Fk (ω)(∂Ha ) in M with the metric gij (t) for each t close to ω and with the same area a for each component). Note that (5.3.10)

||Fk∗ (t)gij (t) − hij ||C q (Ha ) < δ.

for tk ≤ t < ω and the metrics gij (t) for tk ≤ t ≤ ω are uniformly equivalent. We can find a subsequence tn → ω for which Fk (tn ) converge to Fk (ω) in C q−1 (Ha ) and the limit map has ||Fk∗ (ω)gij (ω) − hij ||C q−1 (Ha ) ≤ δ. We need to check that Fk (ω) is still a diffeomorphism. We at least know Fk (ω) is a local diffeomorphism, and Fk (ω) is the limit of diffeomorphisms, so the only possibility of overlap is at the boundary. Hence we use the fact that Fk (ω)(∂Ha ) is still strictly concave since q is large and δ is small to prevent the boundary from touching itself. Thus Fk (ω) is a diffeomorphism. A limit of harmonic maps is harmonic, so Fk (ω) is a harmonic diffeomorphism from Ha into M with the metric gij (ω). Moreover Fk (ω) takes ∂Ha to the constant mean curvature hypersurface ∂(Fk (ω)(Ha )) of the area a in (M, gij (ω)) and continue to satisfy the free boundary condition. As a consequence of the standard regularity result of elliptic partial differential equations (see for example [48]), the map Fk (ω) ∈ C ∞ (Ha ) and then from (5.3.10) we have ||Fk∗ (ω)gij (ω) − hij ||C q (Ha ) ≤ δ. If ||Fk∗ (ω)gij (ω) − hij ||C q (Ha ) = δ, we then finish the proof of the claim. So we may assume that ||Fk∗ (ω)gij (ω) − hij ||C q (Ha ) < δ. We want to show that Fk (ω) can be extended forward in time a little.


349

We argue by contradiction. Suppose not, then we consider the new sequence of the manifolds M with metric gij (ω) and the origins Fk (ω)(P ). Since Fk (ω) are close to isometries, the injectivity radii of the metrics gij (ω) at Fk (ω)(P ) do not go to zero, e with the and we can extract a subsequence which converges to a hyperbolic limit H e e e metric hij and the origin P and with finite volume. The new limit H has at least as many cusps as the old limit H, since we choose H with cusps as few as possible. By ek exhausting the definition of convergence, we can find a sequence of compact sets B e e e ek H and containing P , and a sequence of diffeomorphisms Fk of neighborhoods of B e e e e into M with Fk (P ) = Fk (ω)(P ) such that for each compact set B in H and each integer m ||Fek∗ (gij (ω)) − e hij ||C m (B) e →0

ek ) will contain all points out to any as k → +∞. For large enough k the set Fek (B fixed distance we need from the point Fk (ω)(P ), and then ek ) ⊃ Fk (ω)(Ha ) Fek (B

since the points of Ha have a bounded distance from P and Fk (ω) are reasonably close to preserving the metrics. Hence we can form the composition e Gk = Fek−1 ◦ Fk (ω) : Ha → H.

Arbitrarily fix δ ′ ∈ (δ, δ0 ). Since the Fek are as close to preserving the metric as we like, we have ||G∗k e hij − hij ||C q (Ha ) < δ ′

for all sufficiently large k. By Lemma 5.3.7, we deduce that there exists an isometry e and then (M, gij (ω), Fk (ω)(P )) (on compact subsets) is very close to I of H to H, (H, hij , P ) as long as δ small enough and k large enough. Since Fk (ω)(∂Ha ) is strictly concave and the foliation of a neighborhood of Fk (ω)(∂Ha ) by constant mean curvature hypersurfaces has the area as a function with nonzero gradient, by the implicit function theorem, there exists a unique constant mean curvature hypersurface with the same area a near Fk (ω)(∂Ha ) in M with the metric gij (t) for t close to ω. Hence, when k sufficiently large, Fk (ω) can be extended forward in time a little. This is a contradiction and we have proved claim (5.3.9). We further claim that there must be some k such that ωk = +∞ (i.e., we can smoothly continue the family of harmonic diffeomorphisms Fk (t) for all tk ≤ t < +∞, in other words, there must be at least one hyperbolic piece persisting). We argue by contradiction. Suppose for each k large enough, we can continue the family Fk (t) for tk ≤ t ≤ ωk < +∞ with ||Fk∗ (ωk )gij (ωk ) − hij ||C q (Ha ) = δ. Then as before, we consider the new sequence of the manifolds M with metrics gij (ωk ) and origins Fk (ωk )(P ). For sufficiently large k, we can obtain diffeomorphisms Fek of ek into M with Fek (Pe ) = Fk (ωk )(P ) which are as close to preserving neighborhoods of B ek is a sequence of compact sets, exhausting some the metric as we like, where B ˜ of finite volume and with no fewer cusps (than H), and hyperbolic three-manifold H,

350


ek ) will contain all the points out to any fixed containing P˜ ; moreover, the set Fek (B distance we need from the point Fk (ωk )(P ); and hence ek ) ⊇ Fk (ωk )(Ha ) Fek (B

since Ha is at bounded distance from P and Fk (ωk ) is reasonably close to preserving the metrics. Then we can form the composition Gk = Fek−1 ◦ Fk (ωk ) :

˜ Ha → H.

Since the Fek are as close to preserving the metric as we like, for any δe > δ we have hij − hij ||C q (Ha ) < δe ||G∗k e

for large enough k. Then a subsequence of Gk converges at least in C q−1 (Ha ) topology e By the same reason as in the argument of previous two to a map G∞ of Ha into H. paragraphs, the limit map G∞ is a smooth harmonic diffeomorphism from Ha into ˜ ij , and takes ∂Ha to a constant mean curvature hypersurface e with the metric h H e ˜ G∞ (∂Ha ) of (H, hij ) with the area a, and also satisfies the free boundary condition. Moreover we still have (5.3.11)

||G∗∞ e hij − hij ||C q (Ha ) = δ.

e with Now by Lemma 5.3.7 we deduce that there exists an isometry I of H to H dC l0 (Ha ) (G∞ , I) < ε0 .

ea and Ha , we see that the map I −1 ◦ G∞ is a harmonic By using I to identify H diffeomorphism of Ha to itself which satisfies the free boundary condition and dC l0 (Ha ) (I −1 ◦ G∞ , Id) < ε0 . From the uniqueness in Lemma 5.3.8 we conclude that I −1 ◦ G∞ = Id which contradicts with (5.3.11). This shows at least one hyperbolic piece persists. Moreover the pull-back of the solution metric gij (t) by Fk (t), for tk ≤ t < +∞, is as close to the hyperbolic metric hij as we like. We can continue to form other persistent hyperbolic pieces in the same way as long as there are any points Pk outside of the chosen pieces where the injectivity radius at times tk → ∞ are all at least some fixed positive number ρ > 0. The only modification in the proof is to take the new limit H to have the least possible number of cusps out of all remaining possible limits. Note that the volume of the normalized Ricci flow is constant in time. Therefore by combining with Margulis lemma (see for example [55] [76]), we have proved that there exists a finite collection of complete noncompact hyperbolic three-manifolds H1 , . . . , Hm with finite volume, a small number a > 0 and a time T < +∞ such that for all t beyond T we can find diffeomorphisms ϕl (t) of (Hl )a into M , 1 ≤ l ≤ m, so that the pull-back of the solution metric gij (t) by ϕl (t) is as close to the hyperbolic metrics as we like and the exceptional part of M where the points are not in the image of any ϕl has the injectivity radii everywhere as small as we like.


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Part III: Incompressibility We remain to show that the boundary tori of any persistent hyperbolic piece are incompressible, in the sense that the fundamental group of the torus injects into that of the whole manifold. The argument of this part is a parabolic version of Schoen and Yau’s minimal surface argument in [109, 110, 111]. Let B be a small positive number and assume the above positive number a is much smaller than B. Denote by Ma a persistent hyperbolic piece of the manifold M truncated by boundary tori of area a with constant mean curvature and denote by ◦

Mac = M \ Ma the part of M exterior to Ma . Thus there is a persistent hyperbolic piece MB ⊂ Ma of the manifold M truncated by boundary tori of area B with constant ◦

mean curvature. We also denote by MBc = M \ MB . By Van Kampen’s Theorem, if π1 (∂MB ) injects into π1 (MBc ) then it injects into π1 (M ) also. Thus we only need to show π1 (∂MB ) injects into π1 (MBc ). We will argue by contradiction. Let T be a torus in ∂MB . Suppose π1 (T ) does not inject into π1 (MBc ), then by Dehn’s Lemma the kernel is a cyclic subgroup of π1 (T ) generated by a primitive element. The work of Meeks-Yau [86] or Meeks-Simon-Yau [87] shows that among all disks in MBc whose boundary curve lies in T and generates the kernel, there is a smooth embedded disk normal to the boundary which has the least possible area. Let A = A(t) be the area of this disk. This is defined for all t sufficiently large. We will show that A(t) decreases at a rate bounded away from zero which will be a contradiction. Let us compute the rate at which A(t) changes under the Ricci flow. We need to show A(t) decrease at least at a certain rate, and since A(t) is the minimum area to bound any disk in the given homotopy class, it suffices to find some such disk whose area decreases at least that fast. We choose this disk as follows. Pick the minimal disk at time t0 , and extend it smoothly a little past the boundary torus since the minimal disk is normal to the boundary. For times t a little bigger than t0 , the boundary torus may need to move a little to stay constant mean curvature with area B as the metrics change, but we leave the surface alone and take the bounding disk to be the one cut ˜ of such disk comes from the off from it by the new torus. The change of the area A(t) change in the metric and the change in the boundary. For the change in the metric, we choose an orthonormal frame X, Y, Z at a point x in the disk so that X and Y are tangent to the disk while Z is normal and compute the rate of change of the area element dσ on the disk as T ∂ 1 2 − 2(Rij )T )dσ dσ = (g ij )T r(gij ∂t 2 3 2 = r − Ric (X, X) − Ric (Y, Y ) dσ, 3

since the metric evolves by the normalized Ricci flow. Here (·)T denotes the tangential projections on the disk. Notice the torus T may move in time to preserve constant mean curvature and constant area B. Suppose the boundary of the disk evolves with a normal velocity N . The change of the area at boundary along a piece of length ds ˜ is given by is given by N ds. Thus the total change of the area A(t) dA˜ = dt

Z Z

Z 2 r − Ric (X, X) − Ric (Y, Y ) dσ + N ds. 3 ∂

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Note that Ric (X, X) + Ric (Y, Y ) = R(X, Y, X, Y ) + R(X, Z, X, Z) + R(Y, X, Y, X) + R(Y, Z, Y, Z) 1 = R + R(X, Y, X, Y ). 2 By the Gauss equation, the Gauss curvature K of the disk is given by K = R(X, Y, X, Y ) + det II where II is the second fundamental form of the disk in MBc . This gives at t = t0 , Z Z Z Z Z dA 2 1 ≤ r − R dσ − (K − det II)dσ + N ds dt 3 2 ∂ Since the bounding disk is a minimal surface, we have det II ≤ 0. The Gauss-Bonnet Theorem tells us that for a disk Z Z Z Kdσ + kds = 2π ∂

where k is the geodesic curvature of the boundary. Thus we obtain Z Z Z Z 2 1 dA ≤ r − R dσ + kds + N ds − 2π. (5.3.12) dt 3 2 ∂ ∂ Recall that we are assuming Rmin (t) increases monotonically to −6 as t → +∞. By the evolution equation of the scalar curvature, d Rmin (t) ≥ 4(r(t) − Rmin (t)) dt and then Z

∞ 0

(r(t) − Rmin (t))dt < +∞.

This implies that r(t) → −6 as t → +∞ by using the derivatives estimate for the curvatures. Thus for every ε > 0 we have 1 2 r − R ≤ −(1 − ε) 3 2 for t sufficiently large. And then the first term on RHS of (5.3.12) is bounded above by Z Z 1 2 r − R dσ ≤ −(1 − ε)A. 3 2 The geodesic curvature k of the boundary of the minimal disk is the acceleration of a curve moving with unit speed along the intersection of the disk with the torus;


353

since the disk and torus are normal, this is the same as the second fundamental form of the torus in the direction of the curve of intersection. Now if the metric were actually hyperbolic, the second fundamental form of the torus would be exactly 1 in all directions. Note that the persistent hyperbolic pieces are as close to the standard hyperbolic as we like. This makes that the second term of RHS of (5.3.12) is bounded above by Z kds ≤ (1 + ε0 )L ∂

for some sufficiently small positive number ε0 > 0, where L is the length of the boundary curve. Also since the metric on the persistent hyperbolic pieces are close to the standard hyperbolic as we like, its change under the normalized Ricci flow is as small as we like; So the motion of the constant mean curvature torus of fixed area B will have a normal velocity N as small as we like. This again makes the third term of RHS of (5.3.12) bounded above by Z N ds ≤ ε0 L. ∂

Combining these estimates, we obtain (5.3.13)

dA ≤ (1 + 2ε0 )L − (1 − ε0 )A − 2π dt

on the persistent hyperbolic piece, where ε0 is some sufficiently small positive number. We next need to bound the length L in terms of the area A. Since a is much smaller than B, for large t the metric is as close as we like to the standard hyperbolic one; not just on the persistent hyperbolic piece MB but as far beyond as we like. Thus for a long distance into MBc the metric will look nearly like a standard hyperbolic cusplike collar. Let us first recall a special coordinate system on the standard hyperbolic cusp projecting beyond torus T1 in ∂H1 as follows. The universal cover of the flat torus T1 can be mapped conformally to the x-y plane so that the deck transformation of T1 become translations in x and y, and so that the Euclidean area of the quotient is 1; then these coordinates are unique up to a translation. The hyperbolic cusp projecting beyond the torus T1 in ∂H1 can be parametrized by {(x, y, z) ∈ R3 | z > 0} with the hyperbolic metric (5.3.14)

ds2 =

dx2 + dy 2 + dz 2 . z2

Note that we can make the solution metric, in an arbitrarily large neighborhood of the torus T (of ∂MB ), as close to hyperbolic as we wish (in the sense that there exists a diffeomorphism from a large neighborhood of the torus TB (of ∂HB ) on the standard hyperbolic cusp to the above neighborhood of the torus T (of ∂MB ) such that the pull-back of the solution metric by the diffeomorphism is as close to the hyperbolic metric as we wish). Then by using this diffeomorphism (up to a slight modification) we can parametrize the cusplike tube of MBc projecting beyond the torus T in ∂MB by {(x, y, z) | z ≥ ζ} where the height ζ is chosen so that the torus in the hyperbolic cusp at height ζ has the area B. Now consider our minimal disk, and let L(z) be the length of the curve of the intersection of the disk with the torus at height z in the above coordinate system,

354


and also let A(z) be the area of the part of the disk between ζ and z. We now want to derive a monotonicity formula on the area A(z) for the minimal surface. For almost every z the intersection of the disk with the torus at height z is a smooth embedded curve or a finite union of them by the standard transversality theorem. If there is more than one curve, at least one of them is not homotopic to a point in T and represents the primitive generator in the kernel of π1 (T ) such that a part of the original disk beyond height z continues to a disk that bounds it. We extend this disk back to the initial height ζ by dropping the curve straight down. Let ˜ ˜ L(z) be the length of the curve we picked at height z; of course L(z) ≤ L(z) with ˜ equality if it is the only piece. Let L(w) denote the length of the same curve in the x-y plane dropped down to height w for ζ ≤ w ≤ z. In the hyperbolic space we would have z ˜ ˜ L(w) = L(z) w ˜ exactly. In our case there is a small error proportional to L(z) and we can also take it ˜ ˜ proportional to the distance z − w by which it drops since L(w)| w=z = L(z) and the ∞ solution metric is close to the hyperbolic in the Cloc topology. Thus, for arbitrarily given δ > 0 and ζ ∗ > ζ, as the solution metric is sufficiently close to hyperbolic, we have z ˜ ˜ ˜ ≤ δ(z − w)L(z) |L(w) − L(z)| w for all z and w in ζ ≤ w ≤ z ≤ ζ ∗ . Now given ε and ζ ∗ pick δ = 2ε/ζ ∗ . Then z˜ 2ε(z − w) ˜ (5.3.15) L(w) ≤ L(z) 1+ . w w When we drop the curve vertically for the construction of the new disk we get an area ˜ A(z) between ζ and z given by Z z ˜ L(w) ˜ dw. A(z) = (1 + o(1)) w ζ Here and in the following o(1) denotes various small error quantities as the solution metric close to hyperbolic. On the other hand if we do not drop vertically we pick up even more area, so the area A(z) of the original disk between ζ and z has Z z L(w) (5.3.16) A(z) ≥ (1 − o(1)) dw. w ζ Since the original disk minimized among all disks bounded a curve in the primitive generator of the kernel of π1 (T ), and the new disk beyond the height z is part of the original disk, we have ˜ A(z) ≤ A(z) and then by combining with (5.3.15), Z z Z z L(w) 1 − 2ε 2εz ˜ dw ≤ (1 + o(1))z L(z) + 3 dw w w2 w ζ ζ (z − ζ) z−ζ ≤ (1 + o(1))L(z) 1+ε . ζ ζ


355

˜ Here we used the fact that L(z) ≤ L(z). Since the solution metric is sufficiently close to hyperbolic, we have Z z d L(w) L(z) dw = dz w z ζ Z z ζ z−ζ L(w) 1−ε dw ≥ (1 − o(1)) z(z − ζ) ζ w ζ Z z L(w) 1 + 2ε 1 − dw, ≥ z−ζ z w ζ or equivalently (5.3.17)

d log dz

z 1+2ε (z − ζ)

Z

ζ

z

L(w) dw w

≥ 0.

This is the desired monotonicity formula for the area A(z). It follows directly from (5.3.16) and (5.3.17) that

or equivalently

z 1+2ε A(z) ≥ (1 − o(1))ζ 2ε L(ζ), (z − ζ) 2ε z z A(z) L(ζ) ≤ (1 + o(1)) ζ z−ζ

for all z ∈ [ζ, ζ ∗ ]. Since the solution metric, in an arbitrarily large neighborhood of the torus T (of ∂MB ), as √ close to hyperbolic as we wish, we may assume that√ζ ∗ is so √ ∗ ∗ ∗ large that ζ > ζ and √ζ ∗ζ−ζ is close to 1, and also ε > 0 is so small that ( ζζ )2ε is close to 1. Thus for arbitrarily small δ0 > 0, we have p ζ∗ . (5.3.18) L(ζ) ≤ (1 + δ0 )A

Now recall that (5.3.13) states

dA ≤ (1 + 2ε0 )L − (1 − ε0 )A − 2π. dt We now claim that if (1 + 2ε0 )L − (1 − ε0 )A ≥ 0 then L = L(ζ) is uniformly bounded from above. Indeed by the assumption we have A(ζ ∗ ) ≤

(1 + 2ε0 ) L(ζ) (1 − ε0 )

√ since A(ζ ∗ ) ≤ A. By combining with (5.3.16) we have some z0 ∈ ( ζ ∗ , ζ ∗ ) satisfying Z ζ∗ L(z0 ) ∗ p ∗ L(w) ζ − ζ ≤ √ dw z0 w ∗ ζ ≤ (1 + o(1))A(ζ ∗ ) 1 + 2ε0 ≤ (1 + o(1)) L(ζ). 1 − ε0

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Thus for ζ ∗ suitably large, by noting that the solution metric on a large neighborhood of T (of ∂MB ) is sufficiently close to hyperbolic, we have (5.3.19)

L(z0 ) ≤ (1 + 4ε0 )

z0 L(ζ) ζ∗

√ for some z0 ∈ ( ζ ∗ , ζ ∗ ). It is clear that we may assume the intersection curve between the minimal disk with the torus at this height z0 is smooth and embedded. If the intersection curve at the height z0 has more than one piece, as before one of them will represent the primitive generator in the kernel of π1 (T ), and we can ignore the others. Let us move (the piece of) the intersection curve on the torus at height z0 through as small as possible area in the same homotopy class of π1 (T ) to a curve which is a geodesic circle in the flat torus coming from our special coordinates, and then drop this geodesic circle vertically in the special coordinates to obtain another new disk. We will compare the area of this new disk with the original minimal disk as follows. Denote by G the length of the geodesic circle in the standard hyperbolic cusp at height 1. Then the length of the geodesic circle at height z0 will be G/z0 . Observe that given an embedded curve of length l circling the cylinder S 1 × R of circumference w once, it is possible to deform the curve through an area not bigger than lw into a meridian circle. Note that (the piece of) the intersection curve represents the primitive generator in the kernel of π1 (T ). Note also that the solution metric is sufficiently close to the hyperbolic metric. Then the area of the deformation from (the piece of) the intersection curve on the torus at height z0 to the geodesic circle at height z0 is bounded by G · L(z0 ). (1 + o(1)) z0 The area to drop the geodesic circle from height z0 to height ζ is bounded by Z z0 G (1 + o(1)) dw. w2 ζ Hence comparing the area of the original minimal disk to that of this new disk gives 1 1 L(z0 ) + − . A(z0 ) ≤ (1 + o(1))G z0 ζ z0 √ By (5.3.18), (5.3.19) and the fact that z0 ∈ ( ζ ∗ , ζ ∗ ), this in turn gives L(ζ) ≤ (1 + δ0 )A(z0 ) L(ζ) 1 ≤ (1 + δ0 )G (1 + 4ε0 ) ∗ + . ζ ζ Since ζ ∗ is suitably large, we obtain L(ζ) ≤ 2G/ζ This gives the desired assertion since G is fixed from the geometry of the limit hyperbolic manifold H and ζ is very large as long as the area B of ∂MB small enough. Thus the combination of (5.3.13), (5.3.18) and the assertion implies that either d A ≤ −2π, dt


357

or d A ≤ (1 + 2ε0 )L − (1 − ε0 )A − 2π dt 2G ≤ (1 + 2ε0 ) − 2π ζ ≤ −π, since the solution metric on a very large neighborhood of the torus T (of ∂MB ) is sufficiently close to hyperbolic and ζ is very large as the area B of ∂MB small enough. This is impossible because A ≥ 0 and the persistent hyperbolic pieces go on forever. The contradiction shows that π1 (T ) in fact injects into π1 (MBc ). This proves that π1 (∂MB ) injects into π1 (M ). Therefore we have completed the proof of Theorem 5.3.4. 6. Ancient κ-solutions. Let us consider a solution of the Ricci flow on a compact manifold. If the solution blows up in finite time (i.e., the maximal solution exists only on a finite time interval), then as we saw in Chapter 4 a sequence of rescalings of the solution around the singularities converge to a solution which exists at least on the time interval (−∞, T ) for some finite number T . Furthermore, by Perelman’s no local collapsing theorem I (Theorem 3.3.2), we see that the limit is κ-noncollapsed on all scales for some positive constant κ. In addition, if the dimension n = 3 then the Hamilton-Ivey pinching estimate implies that the limiting solution must have nonnegative curvature operator. We call a solution to the Ricci flow an ancient κ-solution if it is complete (either compact or noncompact) and defined on an ancient time interval (−∞, T ) with T > 0, has nonnegative curvature operator and bounded curvature, and is κ-noncollapsed on all scales for some positive constant κ. In this chapter we study ancient κ-solutions of the Ricci flow. We will obtain crucial curvature estimates of such solutions and determine their structures in lower dimensional cases. 6.1. Preliminaries. We first present a useful geometric property, given by Chen and the second author in [34], for complete noncompact Riemannian manifolds with nonnegative sectional curvature. Let (M, gij ) be an n-dimensional complete Riemannian manifold and let ε be a positive constant. We call an open subset N ⊂ M an ε-neck of radius r if −1 (N, r−2 gij ) is ε-close, in the C [ε ] topology, to a standard neck Sn−1 × I, where Sn−1 is the round (n − 1)-sphere with scalar curvature 1 and I is an interval of length 2ε−1 . The following result is, to some extent, in similar spirit of Yau’s volume lower bound estimate [128]. Proposition 6.1.1 (Chen-Zhu [34]). There exists a positive constant ε0 = ε0 (n) such that every complete noncompact n-dimensional Riemannian manifold (M, gij ) of nonnegative sectional curvature has a positive constant r0 such that any ε-neck of radius r on (M, gij ) with ε ≤ ε0 must have r ≥ r0 . Proof. We argue by contradiction. Suppose there exist a sequence of positive constants εα → 0 and a sequence of n-dimensional complete noncompact Riemannian α manifolds (M α , gij ) such that for each fixed α, there exists a sequence of εα -necks Nk of radius at most 1/k in M α with centers Pk divergent to infinity.

358


Fix a point P on the manifold M α and connect each Pk to P by a minimizing geodesic γk . By passing to a subsequence we may assume the angle θkl between geodesic γk and γl at P is very small and tends to zero as k, l → +∞, and the length of γk+1 is much bigger than the length of γk . Let us connect Pk to Pl by a minimizing geodesic ηkl . For each fixed l > k, let P˜k be a point on the geodesic γl such that the geodesic segment from P to P˜k has the same length as γk and consider the triangle ∆P Pk P˜k in M α with vertices P , Pk and P˜k . By comparing with the corresponding triangle in the Euclidean plane R2 whose sides have the same corresponding lengths, Toponogov’s comparison theorem implies 1 ˜ d(Pk , Pk ) ≤ 2 sin θkl · d(Pk , P ). 2 Since θkl is very small, the distance from Pk to the geodesic γl can be realized by a geodesic ζkl which connects Pk to a point Pk′ on the interior of the geodesic γl and has length at most 2 sin( 12 θkl ) · d(Pk , P ). Clearly the angle between ζkl and γl at the intersection point Pk′ is π2 . Consider α to be fixed and sufficiently large. We claim that for large enough k, each minimizing geodesic γl with l > k, connecting P to Pl , goes through the neck Nk . Suppose not; then the angle between γk and ζkl at Pk is close to either zero or π since Pk is in the center of an εα -neck and α is sufficiently large. If the angle between γk and ζkl at Pk is close to zero, we consider the triangle ∆P Pk Pk′ in M α with vertices P , Pk , and Pk′ . Note that the length between Pk and Pk′ is much smaller than the lengths from Pk or Pk′ to P . By comparing the angles of this triangle with those of the corresponding triangle in the Euclidean plane with the same corresponding lengths and using Toponogov’s comparison theorem, we find that it is impossible. Thus the angle between γk and ζkl at Pk is close to π. We now consider the triangle ∆Pk Pk′ Pl in M α with the three sides ζkl , ηkl and the geodesic segment from Pk′ to Pl on γl . We have seen that the angle of ∆Pk Pk′ Pl at Pk is close to zero and the angle at Pk′ is π2 . ¯ P¯k P¯′ P¯l in the Euclidean plane R2 whose By comparing with corresponding triangle ∆ k sides have the same corresponding lengths, Toponogov’s comparison theorem implies 3 ∠P¯l P¯k P¯k′ + ∠P¯l P¯k′ P¯k ≤ ∠Pl Pk Pk′ + ∠Pl Pk′ Pk < π. 4 This is impossible since the length between P¯k and P¯k′ is much smaller than the length from P¯l to either P¯k or P¯k′ . So we have proved each γl with l > k passes through the neck Nk . Hence by taking a limit, we get a geodesic ray γ emanating from P which passes through all the necks Nk , k = 1, 2, . . . , except a finite number of them. Throwing these finite number of necks away, we may assume γ passes through all necks Nk , k = 1, 2, . . . . Denote the center sphere of Nk by Sk , and their intersection points with γ by pk ∈ Sk ∩ γ, k = 1, 2, . . . . Take a sequence of points γ(m) with m = 1, 2, . . . . For each fixed neck Nk , arbitrarily choose a point qk ∈ Nk near the center sphere Sk and draw a geodesic segment γ km from qk to γ(m). Now we claim that for any neck Nl with l > k, γ km will pass through Nl for all sufficiently large m. We argue by contradiction. Let us place all the necks Ni horizontally so that the geodesic γ passes through each Ni from the left to the right. We observe that the geodesic segment γ km must pass through the right half of Nk ; otherwise γ km cannot be minimal. Then for large enough m, the distance from pl to the geodesic segment


359

γ km must be achieved by the distance from pl to some interior point pk ′ of γ km . Let us draw a minimal geodesic η from pl to the interior point pk ′ with the angle at the intersection point pk ′ ∈ η ∩ γ km to be π2 . Suppose the claim is false. Then the angle between η and γ at pl is close to 0 or π since εα is small. If the angle between η and γ at pl is close to 0, we consider the triangle ∆pl pk ′ γ(m) ¯ pl p¯k ′ γ¯ (m) in the plane with the same correand construct a comparison triangle ∆¯ sponding length. Then by Toponogov’s comparison theorem, we see the sum of the ¯ pl p¯k ′ γ¯ (m) is less than 3π/4, which is imposinner angles of the comparison triangle ∆¯ sible. If the angle between η and γ at pl is close to π, by drawing a minimal geodesic ξ from qk to pl , we see that ξ must pass through the right half of Nk and the left half of Nl ; otherwise ξ cannot be minimal. Thus the three inner angles of the triangle ∆pl pk ′ qk are almost 0, π/2, and 0 respectively. This is also impossible by the Toponogov comparison theorem. Hence we have proved that the geodesic segment γ km passes through Nl for m large enough. Consider the triangle ∆pk qk γ(m) with two long sides pk γ(m)(⊂ γ) and qk γ(m)(= γ km ). For any s > 0, choose points p˜k on pk γ(m) and q˜k on qk γ(m) with d(pk , p˜k ) = d(qk , q˜k ) = s. By Toponogov’s comparison theorem, we have

d(˜p , q˜ ) 2 k

k

d(pk , qk ) =

¯ pk γ(m)˜ d(˜ pk , γ(m))2 + d(˜ qk , γ(m))2 − 2d(˜ pk , γ(m))d(˜ qk , γ(m)) cos ∡(˜ qk ) 2 2 ¯ k γ(m)qk ) d(pk , γ(m)) + d(qk , γ(m)) − 2d(pk , γ(m))d(qk , γ(m)) cos ∡(p

≥

¯ pk γ(m)˜ d(˜ pk , γ(m))2 + d(˜ qk , γ(m))2 − 2d(˜ pk , γ(m))d(˜ qk , γ(m)) cos ∡(˜ qk ) ¯ pk γ(m)˜ d(pk , γ(m))2 + d(qk , γ(m))2 − 2d(pk , γ(m))d(qk , γ(m)) cos ∡(˜ qk )

=

¯ pk γ(m)˜ (d(˜ pk , γ(m)) − d(˜ qk , γ(m)))2 + 2d(˜ pk , γ(m))d(˜ qk , γ(m))(1 − cos ∡(˜ qk )) 2 ¯ (d(˜ pk , γ(m)) − d(˜ qk , γ(m))) + 2d(pk , γ(m))d(qk , γ(m))(1 − cos ∡(˜ pk γ(m)˜ qk ))

≥

d(˜ pk , γ(m))d(˜ qk , γ(m)) d(pk , γ(m))d(qk , γ(m))

→1

¯ k γ(m)qk ) and ∡(˜ ¯ pk γ(m)˜ as m → ∞, where ∡(p qk ) are the corresponding angles of the comparison triangles. Letting m → ∞, we see that γ km has a convergent subsequence whose limit γ k is a geodesic ray passing through all Nl with l > k. Let us denote by pj = γ(tj ), j = 1, 2, . . .. From the above computation, we deduce that d(pk , qk ) ≤ d(γ(tk + s), γ k (s)) for all s > 0. Let ϕ(x) = limt→+∞ (t − d(x, γ(t))) be the Busemann function constructed from the ray γ. Note that the level set ϕ−1 (ϕ(pj )) ∩ Nj is close to the center sphere Sj for any j = 1, 2, . . .. Now let qk be any fixed point in ϕ−1 (ϕ(pk )) ∩ Nk . By the definition of Busemann function ϕ associated to the ray γ, we see that ϕ(γ k (s1 )) − ϕ(γ k (s2 )) = s1 − s2 for any s1 , s2 ≥ 0. Consequently, for each l > k, by choosing s = tl − tk , we see γ k (tl − tk ) ∈ ϕ−1 (ϕ(pl )) ∩ Nl . Since γ(tk + tl − tk ) = pl , it follows that d(pk , qk ) ≤ d(pl , γ k (s)).

360


with s = tl − tk > 0. This implies that the diameter of ϕ−1 (ϕ(pk )) ∩ Nk is not greater than the diameter of ϕ−1 (ϕ(pl )) ∩ Nl for any l > k, which is a contradiction for l much larger than k. Therefore we have proved the proposition. In [63], Hamilton discovered an important repulsion principle (cf. Theorem 21.4 of [63]) about the influence of a bump of strictly positive curvature in a complete noncompact manifold of nonnegative sectional curvature. Namely minimal geodesic paths that go past the bump have to avoid it. As a consequence he obtained a finite bump theorem (cf. Theorem 21.5 of [63]) that gives a bound on the number of bumps of curvature. Let M be a complete noncompact Riemannian manifold with nonnegative sectional curvature K ≥ 0. A geodesic ball B(p, r) of radius r centered at a point p ∈ M is called a curvature β-bump if sectional curvature K ≥ β/r2 at all points in the ball. The ball B(p, r) is called λ-remote from an origin O if d(p, O) ≥ λr. Finite Bump Theorem (Hamilton [63]). For every β > 0 there exists λ < ∞ such that in any complete manifold of nonnegative sectional curvature there are at most a finite number of disjoint balls which are λ-remote curvature β-bumps. This finite bump theorem played an important role in Hamilton’s study of the behavior of singularity models at infinity and in the dimension reduction argument he developed for the Ricci flow (cf. Section 22 of [63], see also [29] for application to the Kähler-Ricci flow and uniformization problem in complex dimension two). A special consequence of the finite bump theorem is that if we have a complete noncompact solution to the Ricci flow on an ancient time interval −∞ < t < T with T > 0 satisfying certain local injectivity radius bound, with curvature bounded at each time and with asymptotic scalar curvature ratio A = lim sup Rs2 = ∞, then we can find a sequence of points pj going to ∞ (as in the following Lemma 6.1.3) such that a cover of the limit of dilations around these points at time t = 0 splits as a product with a flat factor. The following result, obtained by Chen and the second author in [34], is in similar spirit as Hamilton’s finite bumps theorem and its consequence. The advantage is that we will get in the limit of dilations a product of the line with a lower dimensional manifold, instead of a quotient of such a product. Proposition 6.1.2 (Chen-Zhu [34]). Suppose (M, gij ) is a complete ndimensional Riemannian manifold with nonnegative sectional curvature. Let P ∈ M be fixed, and Pk ∈ M a sequence of points and λk a sequence of positive numbers with d(P, Pk ) → +∞ and λk d(P, Pk ) → +∞. Suppose also that the marked manifolds ∞ f. Then the (M, λ2k gij , Pk ) converge in the Cloc topology to a Riemannian manifold M f splits isometrically as the metric product of the form R × N , where N is a limit M Riemannian manifold with nonnegative sectional curvature. Proof. Let us denote by |OQ| = d(O, Q) the distance between two points O, Q ∈ M . Without loss of generality, we may assume that for each k, (6.1.1)

1 + 2|P Pk | ≤ |P Pk+1 |.

Draw a minimal geodesic γk from P to Pk and a minimal geodesic σk from Pk to Pk+1 , both parametrized by arclength. We may further assume (6.1.2)

θk = |∡(γ˙ k (0), γ˙ k+1 (0))|
0,    k    λk d(Pk , Bk ) → B > 0, (6.1.3)  λk d(Ak , Bk ) → C > 0,      but A + B > C.

Pk σk (((( ( ( B ( k ( δk ((((P ((( k+1 ( ( γk Ak ( (( ( ( ( ((( (((( ( ( (((( (((( ( ( P Now draw a minimal geodesic δk from Ak to Bk . Consider comparison triangles ¯ P¯k P¯ P¯k+1 and △ ¯ P¯k A¯k B ¯k in R2 with △ |P¯k P¯ | = |Pk P |, |P¯k P¯k+1 | = |Pk Pk+1 |, |P¯ P¯k+1 | = |P Pk+1 |, ¯k | = |Pk Bk |, |A¯k B ¯k | = |Ak Bk |. and |P¯k A¯k | = |Pk Ak |, |P¯k B By Toponogov’s comparison theorem [8], we have (6.1.4)

¯k ≥ ∡P¯ P¯k P¯k+1 . ∡A¯k P¯k B

On the other hand, by (6.1.2) and using Toponogov’s comparison theorem again, we have (6.1.5)

1 ∡P¯k P¯ P¯k+1 ≤ ∡Pk P Pk+1 < , k

and since |P¯k P¯k+1 | > |P¯ P¯k | by (6.1.1), we further have (6.1.6)

1 ∡P¯k P¯k+1 P¯ ≤ ∡P¯k P¯ P¯k+1 < . k

Thus the above inequalities (6.1.4)-(6.1.6) imply that ¯k > π − 2 . ∡A¯k P¯k B k

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Hence (6.1.7)

¯k |2 ≥ |A¯k P¯k |2 + |P¯k B ¯k |2 − 2|A¯k P¯k | · |P¯k B ¯k | cos π − 2 . |A¯k B k

Multiplying the above inequality by λ2k and letting k → +∞, we get C ≥A+B which contradicts (6.1.3). Therefore we have proved the proposition. Let M be an n-dimensional complete noncompact Riemannian manifold. Pick an origin O ∈ M . Let s be the geodesic distance to the origin O of M , and R the scalar curvature. Recall that in Chapter 4 we have defined the asymptotic scalar curvature ratio A = lim sup Rs2 . s→+∞

We now state a useful lemma of Hamilton (Lemma 22.2 in [63]) about picking local (almost) maximum curvature points at infinity. Lemma 6.1.3. Given a complete noncompact Riemannian manifold with bounded curvature and with asymptotic scalar curvature ratio A = lim sup Rs2 = +∞, s→+∞

we can find a sequence of points xj divergent to infinity, a sequence of radii rj and a sequence of positive numbers δj → 0 such that (i) R(x) ≤ (1 + δj )R(xj ) for all x in the ball B(xj , rj ) of radius rj around xj , (ii) rj2 R(xj ) → +∞, (iii) λj = d(xj , O)/rj → +∞, (iv) the balls B(xj , rj ) are disjoint, where d(xj , O) is the distance of xj from the origin O. Proof. Pick a sequence of positive numbers ǫj → 0, then choose Aj → +∞ so that Aj ǫ2j → +∞. Let σj be the largest number such that sup{R(x)d(x, O)2 | d(x, O) ≤ σj } ≤ Aj . Then there exists some yj ∈ M such that R(yj )d(yj , O)2 = Aj

and d(yj , O) = σj .

Now pick xj ∈ M so that d(xj , O) ≥ σj and R(xj ) ≥

1 sup{R(x) | d(x, O) ≥ σj }. 1 + ǫj

Finally pick rj = ǫj σj . We check the properties (i)-(iv) as follows. (i) If x ∈ B(xj , rj ) ∩ {d(·, O) ≥ σj }, we have R(x) ≤ (1 + ǫj )R(xj )


363

by the choice of the point xj ; while if x ∈ B(xj , rj ) ∩ {d(·, O) ≤ σj }, we have R(x) ≤ Aj /d(x, O)2 ≤

1 (Aj /σj2 ) (1 − ǫj )2

=

1 R(yj ) (1 − ǫj )2

≤

(1 + ǫj ) R(xj ), (1 − ǫj )2

since d(x, O) ≥ d(xj , O) − d(x, xj ) ≥ σj − rj = (1 − ǫj )σj . Thus we have obtained R(x) ≤ (1 + δj )R(xj ), where δj = (ii)

(1+ǫj ) (1−ǫj )2

∀ x ∈ B(xj , rj ),

− 1 → 0 as j → +∞.

By the choices of rj , xj and yj , we have rj2 R(xj ) = ǫ2j σj2 R(xj ) 1 ≥ ǫ2j σj2 R(yj ) 1 + ǫj =

ǫ2j Aj → +∞, 1 + ǫj

as j → +∞.

(iii) Since d(xj , O) ≥ σj = rj /ǫj , it follows that λj = d(xj , O)/rj → +∞ as j → +∞. (iv) For any x ∈ B(xj , rj ), the distance from the origin d(x, O) ≥ d(xj , O) − d(x, xj ) ≥ σj − rj = (1 − ǫj )σj → +∞, as j → +∞. Thus any fixed compact set does not meet the balls B(xj , rj ) for large enough j. If we pass to a subsequence, the balls will all avoid each other. The above point picking lemma of Hamilton, as written down in Lemma 22.2 of [63], requires the curvature of the manifold to be bounded. When the manifold has unbounded curvature, we will appeal to the following simple lemma. Lemma 6.1.4. Given a complete noncompact Riemannian manifold with unbounded curvature, we can find a sequence of points xj divergent to infinity such that for each positive integer j, we have |Rm(xj )| ≥ j, and |Rm(x)| ≤ 4|Rm(xj )| for x ∈ B(xj , √

j ). |Rm(xj )|

Proof. Each xj can be constructed as a limit of a finite sequence {yi }, defined as follows. Let y0 be any fixed point with |Rm(y0 )| ≥ j. Inductively, if yi cannot be

364


taken as xj , then there is a yi+1 such that    |Rm(yi+1 )| > 4|Rm(yi )|, j  .  d(yi , yi+1 ) 6 p |Rm(yi )| Thus we have

|Rm(yi )| > 4i |Rm(y0 )| ≥ 4i j, d(yi , y0 ) ≤ j

i X

k=1

p 1 p < 2 j. k−1 4 j

Since the manifold is smooth, the sequence {yi } must be finite. The last element fits. 6.2. Asymptotic Shrinking Solitons. We begin with the study of the asymptotic behavior of an ancient κ-solution gij (x, t), on M × (−∞, T ) with T > 0, to the Ricci flow as t → −∞. Pick an arbitrary point (p, t0 ) ∈ M × (−∞, 0] and recall from Chapter 3 that τ = t0 − t, 1 l(q, τ ) = √ inf 2 τ

Z

τ 0

for t < t0 ,

√ s R(γ(s), t0 − s) +

and V˜ (τ ) =

Z

2 |γ(s)| ˙ gij (t0 −s)

γ : [0, τ ] → M with ds γ(0) = p, γ(τ ) = q

n

(4πτ )− 2 exp(−l(q, τ ))dVt0 −τ (q). M

We first observe that Corollary 3.2.6 also holds for the general complete manifold ¯ τ ) = 4τ l(·, τ ) M . Indeed, since the scalar curvature is nonnegative, the function L(·, achieves its minimum on M for each fixed τ > 0. Thus the same argument in the proof of Corollary 3.2.6 shows there exists q = q(τ ) such that (6.2.1)

l(q(τ ), τ ) ≤

n 2

for each τ > 0. Recall from (3.2.11)-(3.2.13), the Li-Yau-Perelman distance l satisfies the following (6.2.2) (6.2.3) (6.2.4)

l 1 ∂ l = − + R + 3/2 K, ∂τ τ 2τ 1 l |∇l|2 = −R + − 3/2 K, τ τ n 1 ∆l ≤ −R + − 3/2 K, 2τ 2τ


365

and the equality in (6.2.4) holds R τ everywhere if and only if we are on a gradient shrinking soliton. Here K = 0 s3/2 Q(X)ds, Q(X) is the trace Li-Yau-Hamilton quadratic given by Q(X) = −Rτ −

R − 2h∇R, Xi + 2Ric (X, X) τ

and X is the tangential (velocity) vector field of an L-shortest curve γ : [0, τ ] → M connecting p to q. By applying the trace Li-Yau-Hamilton inequality (Corollary 2.5.5) to the ancient κ-solution, we have Q(X) = −Rτ − ≥−

R − 2h∇R, Xi + 2Ric (X, X) τ

R τ

and hence K=

Z

τ

s3/2 Q(X)ds 0

≥−

Z

τ

√ sRds

0

≥ −L(q, τ ). Thus by (6.2.3) we get |∇l|2 + R ≤

(6.2.5)

3l . τ

We now state and prove a result of Perelman [103] about the asymptotic shapes of ancient κ-solutions as the time t → −∞. Theorem 6.2.1 (Perelman [103]). Let gij (·, t), −∞ < t < T with some T > 0, be a nonflat ancient κ-solution for some κ > 0. Then there exist a sequence of points qk and a sequence of times tk → −∞ such that the scalings of gij (·, t) around qk with factor |tk |−1 and with the times tk shifting to the new time zero converge to a nonflat ∞ gradient shrinking soliton in Cloc topology. Proof. Clearly, we may assume that the nonflat ancient κ-solution is not a gradient shrinking soliton. For the arbitrarily fixed (p, t0 ), let q(τ )(τ = t0 − t) be chosen as in (6.2.1) with l(q(τ ), τ ) ≤ n2 . We only need to show that the scalings of gij (·, t) around q(τ ) with factor τ −1 converge along a subsequence of τ → +∞ to a nonflat gradient ∞ shrinking soliton in the Cloc topology. We first claim that for any A ≥ 1, one can find B = B(A) < +∞ such that for every τ¯ > 1 there holds (6.2.6)

l(q, τ ) ≤ B and τ R(q, t0 − τ ) ≤ B,

τ and d2t0 − τ¯ (q, q( τ2¯ )) ≤ A¯ τ. whenever 12 τ¯ ≤ τ ≤ A¯ 2

366


Indeed, by using (6.2.5) at τ = τ2¯ , we have r r τ¯ √ τ¯ n l(q, ) ≤ (6.2.7) + sup{|∇ l|} · dt0 − τ2¯ q, q 2 2 2 r r n 3 √ τ ≤ + · A¯ 2 2¯ τ r r n 3A = + , 2 2 and

(6.2.8)

τ¯ 3l(q, τ2¯ ) ≤ R q, t0 − 2 ( τ2¯ ) r !2 r 6 n 3A ≤ + , τ¯ 2 2

√ for q ∈ Bt0 − τ2¯ (q( τ2¯ ), A¯ τ ). Recall that the Li-Yau-Hamilton inequality implies that the scalar curvature of the ancient solution is pointwise nondecreasing in time. Thus we know from (6.2.8) that r !2 r n 3A + (6.2.9) τ R(q, t0 − τ ) ≤ 6A 2 2 τ and d2t0 − τ¯ (q, q( τ2¯ )) ≤ A¯ τ. whenever 21 τ¯ ≤ τ ≤ A¯ 2 By (6.2.2) and (6.2.3) we have ∂l 1 l R + |∇l|2 = − + . ∂τ 2 2τ 2 This together with (6.2.9) implies that r !2 r n 3A l 3A ∂l ≤− + + ∂τ 2τ τ 2 2 i.e., r !2 n 3A + 2 2 q, q τ2¯ ≤ A¯ τ . Hence by integrating this differ-

∂ √ 3A ( τ l) ≤ √ ∂τ τ

r

whenever 12 τ¯ ≤ τ ≤ A¯ τ and d2t0 − τ¯ 2 ential inequality, we obtain r !2 r r √ √ τ¯ τ¯ n 3A l q, ≤ 6A + τ l(q, τ ) − τ 2 2 2 2 and then by (6.2.7), (6.2.10)

r !2 r τ¯ n 3A l(q, τ ) ≤ l q, + 6A + 2 2 2 ! r 2 r n 3A ≤ 7A + 2 2


367

τ and d2t0 − τ¯ (q, q( τ2¯ )) ≤ A¯ τ . So we have proved claim (6.2.6). whenever 12 τ¯ ≤ τ ≤ A¯ 2 Recall that gij (τ ) = gij (·, t0 − τ ) satisfies (gij )τ = 2Rij . Let us take the scaling of the ancient κ-solution around q( τ2¯ ) with factor ( τ2¯ )−1 , i.e., g˜ij (s) =

2 τ¯ gij ·, t0 − s τ¯ 2

where s ∈ [0, +∞). Claim (6.2.6) says that for all s ∈ [1, 2A] and all q such that ˜ s) = τ¯ R(q, t0 − s ¯τ ) ≤ B. Now taking into dist2g˜ij (1) (q, q( τ2¯ )) ≤ A, we have R(q, 2 2 account the κ-noncollapsing assumption and Theorem 4.2.2, we can use Hamilton’s compactness theorem (Theorem 4.1.5) to obtain a sequence τ¯k → +∞ such that the (k) (k) marked evolving manifolds (M, g˜ij (s), q( τ¯2k )), with g˜ij (s) = τ¯2k gij (·, t0 − s τ¯2k ) and ¯ , g¯ij (s), q¯) with s ∈ [1, +∞), where g¯ij (s) is s ∈ [1, +∞), converge to a manifold (M ¯ also a solution to the Ricci flow on M . (k) Denote by ˜lk the corresponding Li-Yau-Perelman distance of g˜ij (s). It is easy to see that ˜lk (q, s) = l(q, τ¯2k s), for s ∈ [1, +∞). From (6.2.5), we also have (6.2.11)

˜ (k) ≤ 6˜lk , |∇˜lk |2g˜(k) + R ij

˜ (k) is the scalar curvature of the metric g˜(k) . Claim (6.2.6) says that ˜lk are where R ij uniformly bounded on compact subsets of M ×[1, +∞) (with the corresponding origins q( τ¯2k )). Thus the above gradient estimate (6.2.11) implies that the functions ˜lk tend ¯. (up to a subsequence) to a function ¯l which is a locally Lipschitz function on M ˜ We know from (6.2.2)-(6.2.4) that the Li-Yau-Perelman distance lk satisfies the following inequalities: (6.2.12)

˜ (k) + n ≥ 0, (˜lk )s − ∆˜lk + |∇˜lk |2 − R 2s

(6.2.13)

˜ ˜ (k) + lk − n ≤ 0. 2∆˜lk − |∇˜lk |2 + R s

We next show that the limit ¯l also satisfies the above two inequalities in the sense of distributions. Indeed the above two inequalities can be rewritten as n ∂ e (k) (4πs)− 2 exp(−e −△+R lk ) ≤ 0, (6.2.14) ∂s (6.2.15)

e lk

e(k) )e− 2 + −(4△ − R

˜lk − n lek e− 2 ≤ 0, s

in the sense of distributions. Note that the estimate (6.2.11) implies that ˜lk → ¯l in 0,α the Cloc norm for any 0 < α < 1. Thus the inequalities (6.2.14) and (6.2.15) imply that the limit l satisfies n ∂ − △ + R (4πs)− 2 exp(−l) ≤ 0, (6.2.16) ∂s (6.2.17)

l

−(4△ − R)e− 2 +

¯l − n l e− 2 ≤ 0, s

368


in the sense of distributions. (k) Denote by V˜ (k) (s) Perelman’s reduced volume of the scaled metric g˜ij (s). Since ˜lk (q, s) = l(q, τ¯2k s), we see that V˜ (k) (s) = V˜ ( τ¯2k s) where V˜ is Perelman’s reduced volume of the ancient κ-solution. The monotonicity of Perelman’s reduced volume (Theorem 3.2.8) then implies that (6.2.18)

lim V˜ (k) (s) = V¯ , for s ∈ [1, 2],

k→∞

for some nonnegative constant V¯ . (We remark that by the Jacobian comparison theorem (Theorem 3.2.7), (3.2.18) and (3.2.19), the integrand of V˜ (k) (s) is bounded by n n (4πs)− 2 exp(−˜lk (X, s))J˜(k) (s) ≤ (4π)− 2 exp(−|X|2 )

on Tp M , where J˜(k) (s) is the L-Jacobian of the L-exponential map of the metric (k) g˜ij (s) at Tp M . Thus we can apply the dominant convergence theorem to get the convergence in (6.2.18). But we are not sure whether the limiting V¯ is exactly Perel¯ , g¯ij (s)), because the points q( τ¯k ) man’s reduced volume of the limiting manifold (M 2 may diverge to infinity. Nevertheless, we can ensure that V¯ is not less than Perelman’s reduced volume of the limit.) Note by (6.2.5) that (6.2.19)

V˜ (k) (2) − V˜ (k) (1) Z 2 d ˜ (k) (V (s))ds = ds 1 Z 2 Z ∂ n (k) ˜ = ds −∆+R (4πs)− 2 exp(−˜lk ) dVg˜(k) (s) . ij ∂s 1 M

Thus we deduce that in the sense of distributions, ∂ ¯ (4πs)− n2 exp(−¯l) = 0, −∆+R (6.2.20) ∂s and

l¯

¯ −2 = (4∆ − R)e

¯l − n ¯l e− 2 s

or equivalently, (6.2.21)

¯ ¯ + l − n = 0, 2∆¯l − |∇¯l|2 + R s

¯ × [1, 2]. Thus by applying standard parabolic equation theory to (6.2.20) we on M find that ¯l is actually smooth. Here we used (6.2.2)-(6.2.4) to show that the equality in (6.2.16) implies the equality in (6.2.17). Set ¯ + ¯l − n] · (4πs)− n2 e−¯l . v = [s(2∆¯l − |∇¯l|2 + R) Then by applying Lemma 2.6.1, we have ∂ ¯ v = −2s|R ¯ ij + ∇i ∇j ¯l − 1 g¯ij |2 · (4πs)− n2 e−¯l . (6.2.22) −∆+R ∂s 2s


369

We see from (6.2.21) that the LHS of the equation (6.2.22) is identically zero. Thus the limit metric g¯ij satisfies (6.2.23)

¯ ij + ∇i ∇j ¯l − 1 g¯ij = 0, R 2s

so we have shown the limit is a gradient shrinking soliton. To show that the limiting gradient shrinking soliton is nonflat, we first show that the constant function V¯ (s) is strictly less than 1. Consider Perelman’s reduced volume V˜ (τ ) of the ancient κ-solution. By using Perelman’s Jacobian comparison theorem (Theorem 3.2.7), (3.2.18) and (3.2.19) as before, we have Z n V˜ (τ ) = (4πτ )− 2 e−l(X,τ ) J (τ )dX Z 2 n ≤ (4π)− 2 e−|X| dX Tp M

= 1.

Recall that we have assumed the nonflat ancient κ-solution is not a gradient shrinking soliton. Thus for τ > 0, we must have V˜ (τ ) < 1. Since the limiting function V¯ (s) is the limit of V˜ ( τ¯2k s) with τ¯k → +∞, we deduce that the constant V¯ (s) is strictly less than 1, for s ∈ [1, 2]. We now argue by contradiction. Suppose the limiting gradient shrinking soliton g¯ij (s) is flat. Then by (6.2.23), 1 n ∇i ∇j ¯l = g¯ij and ∆¯l = . 2s 2s Putting these into the identity (6.2.21), we get ¯l |∇¯l|2 = s √ Since the function ¯l is strictly convex, it follows that 4s¯l is a distance function (from ¯ . From the smoothness of the function some point) on the complete flat manifold M ¯l, we conclude that the flat manifold M ¯ must be Rn . In this case we would have its reduced distance to be ¯l and its reduced volume to be 1. Since V¯ is not less than the reduced volume of the limit, this is a contradiction. Therefore the limiting gradient shrinking soliton g¯ij is not flat. To conclude this section, we use the above theorem to derive the classification of all two-dimensional ancient κ-solutions which was obtained earlier by Hamilton in Section 26 of [63]. Theorem 6.2.2. The only nonflat ancient κ-solutions to Ricci flow on twodimensional manifolds are the round sphere S2 and the round real projective plane RP2 . Proof. Let gij (x, t) be a nonflat ancient κ-solution defined on M × (−∞, T ) (for some T > 0), where M is a two-dimensional manifold. Note that the ancient κ-solution satisfies the Li-Yau-Hamilton inequality (Corollary 2.5.5). In particular by Corollary 2.5.8, the scalar curvature of the ancient κ-solution is pointwise nondecreasing in

370


time. Moreover by the strong maximum principle, the ancient κ-solution has strictly positive curvature everywhere. By the above Theorem 6.2.1, we know that the scalings of the ancient κ-solution along a sequence of points qk in M and a sequence of times tk → −∞ converge to a ¯ , g¯ij (x, t)) with −∞ < t ≤ 0. nonflat gradient shrinking soliton (M ¯ , g¯ij (x, t)) has uniWe first show that the limiting gradient shrinking soliton (M formly bounded curvature. Clearly, the limiting soliton has nonnegative curvature and is κ-noncollapsed on all scales, and its scalar curvature is still pointwise nondecreasing in time. Thus we only need to show that the limiting soliton has bounded curvature at t = 0. We argue by contradiction. Suppose the curvature of the limiting soliton ¯ is noncompact. is unbounded at t = 0. Of course in this case the limiting soliton M Then by applying Lemma 6.1.4, we can choose a sequence of points xj , j = 1, 2, . . . , ¯ of the limit satisfies divergent to infinity such that the scalar curvature R ¯ j , 0) ≥ j and R(x, ¯ 0) ≤ 4R(x ¯ j , 0) R(x p ¯ j , 0)). And then by the nondecreasing (in for all j = 1, 2, . . . , and x ∈ B0 (xj , j/ R(x time) of the scalar curvature, we have ¯ t) ≤ 4R(x ¯ j , 0), R(x,

p ¯ j , 0)) and t ≤ 0. By combining with Hamilfor all j = 1, 2, . . ., x ∈ B0 (xj , j/ R(x ton’s compactness theorem (Theorem 4.1.5) and the κ-noncollapsing, we know that a subsequence of the rescaling solutions ¯ , R(x ¯ j , 0)¯ ¯ j , 0)), xj ), (M gij (x, t/R(x

j = 1, 2, . . . ,

∞ converges in the Cloc topology to a nonflat smooth solution of the Ricci flow. Then Proposition 6.1.2 implies that the new (two-dimensional) limit must be flat. This arrives at a contradiction. So we have proved that the limiting gradient shrinking soliton has uniformly bounded curvature. We next show that the limiting soliton is compact. Suppose the limiting soliton is (complete and) noncompact. By the strong maximum principle we know that the limiting soliton also has strictly positive curvature everywhere. After a shift of the time, we may assume that the limiting soliton satisfies the following equation

(6.2.24)

¯ ij + 1 g¯ij = 0, ∇i ∇j f + R 2t

on − ∞ < t < 0,

everywhere for some function f . Differentiating the equation (6.2.24) and switching the order of differentiations, as in the derivation of (1.1.14), we get (6.2.25)

¯ = 2R ¯ ij ∇j f. ∇i R

Fix some t < 0, say t = −1, and consider a long shortest geodesic γ(s), 0 ≤ s ≤ s. Let x0 = γ(0) and X(s) = γ(s). ˙ Let V (0) be any unit vector orthogonal to γ(0) ˙ and translate V (0) along γ(s) to get a parallel vector field V (s), 0 ≤ s ≤ s on γ. Set   for 0 ≤ s ≤ 1, sV (s), b V (s) = V (s), for 1 ≤ s ≤ s − 1,   (s − s)V (s), for s − 1 ≤ s ≤ s.


371

It follows from the second variation formula of arclength that Z

0

Thus we clearly have

s

˙ ¯ (|Vb (s)|2 − R(X, Vb , X, Vb ))ds ≥ 0. Z

s

¯ R(X, Vb , X, Vb )ds ≤ const.,

0

and then (6.2.26)

Z

s

0

¯ Ric(X, X)ds ≤ const..

By integrating the equation (6.2.24) we get X(f (γ(s))) − X(f (γ(0))) +

Z

s 0

1 ¯ Ric(X, X)ds − s = 0 2

and then by (6.2.26), we deduce d s (f ◦ γ(s)) ≥ − const., ds 2

and f ◦ γ(s) ≥

s2 − const. · s − const. 4

for s > 0 large enough. Thus at large distances from the fixed point x0 the function f has no critical points and is proper. It then follows from the Morse theory that any two high level sets of f are diffeomorphic via the gradient curves of f . Since by (6.2.25), d ¯ ¯ η(s)i R(η(s), −1) = h∇R, ˙ ds ¯ ij ∇i f ∇j f = 2R ≥0

¯ −1) has for any integral curve η(s) of ∇f , we conclude that the scalar curvature R(x, ¯ , which contradicts the Bonnet-Myers Theorem. So we a positive lower bound on M have proved that the limiting gradient shrinking soliton is compact. By Proposition 5.1.10, the compact limiting gradient shrinking soliton has constant curvature. This says that the scalings of the ancient κ-solution (M, gij (x, t)) along a sequence of points qk ∈ M and a sequence of times tk → −∞ converge in the C ∞ topology to the round S2 or the round RP2 . In particular, by looking at the time derivative of the volume and the Gauss-Bonnet theorem, we know that the ancient κ-solution (M, gij (x, t)) exists on a maximal time interval (−∞, T ) with T < +∞. Consider the scaled entropy of Hamilton [60] Z E(t) = R log[R(T − t)]dVt . M

372


We compute (6.2.27)

d E(t) = dt

Z

Z

log[R(T − t)]∆RdVt + M |∇R|2 + R2 − rR dVt = − R M Z |∇R|2 2 + (R − r) dVt = − R M Z

M

R ∆R + R − dVt (T − t) 2

R R where r = M RdVt /V olt (M ) and we have used Vol t (M ) = ( M RdVt ) · (T − t) (by the Gauss-Bonnet theorem). For a smooth function f on the surface M , one can readily check 2 Z Z Z 1 2 R|∇f |2 , (∆f ) = 2 ∇i ∇j f − 2 (∆f )gij + M M M Z Z Z Z |∇R|2 |∇R + R∇f |2 = −2 R(∆f ) + R|∇f |2 , R R M M M M and then Z

Z |∇R|2 + (∆f )(∆f − 2R) R M M 2 Z Z 1 =2 ∇i ∇j f − 2 (∆f )gij + M

By choosing ∆f = R − r, we get Z Z |∇R|2 − (R − r)2 R M M 2 Z Z ∇i ∇j f − 1 (∆f )gij + =2 2 M

M

M

|∇R + R∇f |2 . R

|∇R + R∇f |2 ≥ 0. R

If the equality holds, then we have

1 ∇i ∇j f − (∆f )gij = 0 2 i.e., ∇f is conformal. By the Kazdan-Warner identity [77], it follows that Z ∇R · ∇f = 0, M

so

0=− =−

Z

R∆f

M

Z

M

(R − r)2 .

Hence we have proved the following inequality due to Chow [37] Z Z |∇R|2 (6.2.28) ≥ (R − r)2 , R M M


373

and the equality holds if and only if R ≡ r. The combination (6.2.27) and (6.2.28) shows that the scaled entropy E(t) is strictly decreasing along the Ricci flow unless we are on the round sphere S2 or its quotient RP2 . Moreover the convergence result in Theorem 5.1.11 shows that the scaled entropy E has its minimum value at the constant curvature metric (round S2 or round RP2 ). We had shown that the scalings of the nonflat ancient κ-solution along a sequence of times tk → −∞ converge to the constant curvature metric. Then E(t) has its minimal value at t = −∞, so it was constant all along, hence the ancient κ-solution must have constant curvature for each t ∈ (−∞, T ). This proves the theorem. 6.3. Curvature Estimates via Volume Growth. For solutions to the Ricci flow, Perelman’s no local collapsing theorems tell us that the local curvature upper bounds imply the local volume lower bounds. Conversely, one would expect to get local curvature upper bounds from local volume lower bounds. If this is the case, one will be able to establish an elliptic type estimate for the curvatures of solutions to the Ricci flow. This will provide the key estimate for the canonical neighborhood structure and thick-thin decomposition of the Ricci flow on three-manifolds. In this section we derive such curvature estimates for nonnegatively curved solutions. In the next chapter we will derive similar estimates for all smooth solutions, as well as surgically modified solutions, of the Ricci flow on three-manifolds. Let M be an n-dimensional complete noncompact Riemannian manifold with nonnegative Ricci curvature. Pick an origin O ∈ M . The well-known Bishop-Gromov volume comparison theorem tells us the ratio V ol(B(O, r))/rn is monotone nonincreasing in r ∈ [0, +∞). Thus there exists a limit νM = lim

r→+∞

Vol (B(O, r)) . rn

Clearly the number νM is invariant under dilation and is independent of the choice of the origin. νM is called the asymptotic volume ratio of the Riemannian manifold M. The following result obtained by Perelman in [103] shows that any ancient κsolution must have zero asymptotic volume ratio. This result for the Ricci flow on Kähler manifolds was obtained by Chen and the second author in [32] independently. Moreover in the Kähler case, as shown by Chen, Tang and the second author in [29] (for complex two dimension) and by Ni in [98] (for all dimensions), the condition of nonnegative curvature operator can be replaced by the weaker condition of nonnegative bisectional curvature. Lemma 6.3.1. Let M be an n-dimensional complete noncompact Riemannian manifold. Suppose gij (x, t), x ∈ M and t ∈ (−∞, T ) with T > 0, is a nonflat ancient solution of the Ricci flow with nonnegative curvature operator and bounded curvature. Then the asymptotic volume ratio of the solution metric satisfies νM (t) = lim

r→+∞

Vol t (Bt (O, r)) =0 rn

for each t ∈ (−∞, T ). Proof. The proof is by induction on the dimension. When the dimension is two, the lemma is valid by Theorem 6.2.2. For dimension ≥ 3, we argue by contradiction.

374


Suppose the lemma is valid for dimensions ≤ n − 1 and suppose νM (t0 ) > 0 for some n-dimensional nonflat ancient solution with nonnegative curvature operator and bounded curvature at some time t0 ≤ 0. Fixing a point x0 ∈ M , we consider the asymptotic scalar curvature ratio A=

lim sup dt0 (x,x0 )→+∞

R(x, t0 )d2t0 (x, x0 ).

We divide the proof into three cases. Case 1: A = +∞. By Lemma 6.1.3, there exist sequences of points xk ∈ M divergent to infinity, of radii rk → +∞, and of positive constants δk → 0 such that (i) R(x, t0 ) ≤ (1+δk )R(xk , t0 ) for all x in the ball Bt0 (xk , rk ) of radius rk around xk , (ii) rk2 R(xk , t0 ) → +∞ as k → +∞, (iii) dt0 (xk , x0 )/rk → +∞. By scaling the solution around the points xk with factor R(xk , t0 ), and shifting the time t0 to the new time zero, we get a sequence of rescaled solutions s gk (s) = R(xk , t0 )g ·, t0 + R(xk , t0 ) to the Ricci flow. Since the ancient solution has nonnegative curvature operator and bounded curvature, there holds the Li-Yau-Hamilton inequality (Corollary 2.5.5). Thus the rescaled solutions satisfy Rk (x, s) ≤ (1 + δk ) p for all s ≤ 0 and x ∈ Bgk (0) (xk , rk R(xk , t0 )). Since νM (t0 ) > 0, it follows from the standard volume comparison and Theorem 4.2.2 that the injectivity radii of the rescaled solutions gk at the points xk and the new time zero is uniformly bounded below by a positive number. Then by Hamilton’s compactness theorem (Theorem 4.1.5), ˜ , g˜(s), O) after passing to a subsequence, (M, gk (s), xk ) will converge to a solution (M to the Ricci flow with ˜ s) ≤ 1, for all s ≤ 0 and y ∈ M ˜, R(y, and ˜ R(O, 0) = 1. Since the metric is shrinking, by (ii) and (iii), we get R(xk , t0 )d2g(·,t0 +

s R(xk ,t0 ) )

(x0 , xk ) ≥ R(xk , t0 )d2g(·,t0 ) (x0 , xk )

which tends to +∞, as k → +∞, for all s ≤ 0. Thus by Proposition 6.1.2, for ˜ , g˜(s)) splits off a line. We now consider the lifting of the solution each s ≤ 0, (M ˜˜ g˜˜(s)), s ≤ 0. Clearly we ˜ , g˜(s)), s ≤ 0, to its universal cover and denote it by (M, (M still have νM˜ (0) > 0 and νM˜˜ (0) > 0.


375

By applying Hamilton’s strong maximum principle and the de Rham decomposi˜˜ splits isometrically as X × R for some (n − 1)tion theorem, the universal cover M dimensional nonflat (complete) ancient solution X with nonnegative curvature operator and bounded curvature. These imply that νX (0) > 0, which contradicts the induction hypothesis. Case 2: 0 < A < +∞. Take a sequence of points xk divergent to infinity such that R(xk , t0 )d2t0 (xk , x0 ) → A, as k → +∞. Consider the rescaled solutions (M, gk (s)) (around the fixed point x0 ), where s gk (s) = R(xk , t0 )g ·, t0 + , s ∈ (−∞, 0]. R(xk , t0 ) Then there is a constant C > 0 such that  Rk (x, 0) ≤ C/d2k (x, x0 , 0),    Rk (xk , 0) = 1, (6.3.1)   √  dk (xk , x0 , 0) → A > 0,

where dk (·, x0 , 0) is the distance function from the point x0 in the metric gk (0). Since νM (t0 ) > 0, it is a basic result in Alexandrov space theory (see for example Theorem 7.6 of [20]) that a subsequence of (M, gk (s), x0 ) converges in the Gromov˜ , g˜(0), x0 ) with vertex x0 . Hausdorff sense to an n-dimensional metric cone (M By (6.3.1), the standard volume comparison and Theorem 4.2.2, we know that the injectivity radius of (M, gk (0)) at xk is uniformly bounded from below by a positive number ρ0 . After taking a subsequence, we may assume xk converges to a point x∞ ˜ \ {x0 }. Then by Hamilton’s compactness theorem (Theorem 4.1.5), we can take in M a subsequence such that the metrics gk (s) on the metric balls B0 (xk , 21 ρ0 )(⊂ M with ∞ respect to the metric gk (0)) converge in the Cloc topology to a solution of the Ricci 1 ∞ flow on a ball B0 (x∞ , 2 ρ0 ). Clearly the Cloc limit has nonnegative curvature operator and it is a piece of the metric cone at the time s = 0. By (6.3.1), we have (6.3.2)

˜ ∞ , 0) = 1. R(x

Let x be any point in the limiting ball B0 (x∞ , 21 ρ0 ) and e1 be any radial direction ˜ 1 , e1 ) = 0. Recall that the evolution equation of the Ricci tensor at x. Clearly Ric(e in frame coordinates is ∂ ˜ ˜R ˜ ab + 2R ˜ acbd R ˜ cd . Rab = △ ∂t Since the curvature operator is nonnegative, by applying Hamilton’s strong maximum ˜ principle (Theorem 2.2.1) to the above equation, we deduce that the null space of Ric is invariant under parallel translation. In particular, all radial directions split off locally and isometrically. While by (6.3.2), the piece of the metric cone is nonflat. This gives a contradiction. Case 3: A = 0.

376


The gap theorem as was initiated by Mok-Siu-Yau [93] and established by GreeneWu [49, 50], Eschenberg-Shrader-Strake [45], and Drees [44] shows that a complete noncompact n-dimensional (except n = 4 or 8) Riemannian manifold with nonnegative sectional curvature and the asymptotic scalar curvature ratio A = 0 must be flat. So the present case is ruled out except in dimension n = 4 or 8. Since in our situation the asymptotic volume ratio is positive and the manifold is the solution of the Ricci flow, we can give an alternative proof for all dimensions as follows. We claim the sectional curvature of (M, gij (x, t0 )) is positive everywhere. Indeed, by Theorem 2.2.2, the image of the curvature operator is just the restricted holonomy algebra G of the manifold. If the sectional curvature vanishes for some two-plane, then the holonomy algebra G cannot be so(n). We observe the manifold is not Einstein since it is noncompact, nonflat and has nonnegative curvature operator. If G is irreducible, then by Berger’s Theorem [7], G = u( n2 ). Thus the manifold is Kähler with bounded and nonnegative bisectional curvature and with curvature decay faster than quadratic. Then by the gap theorem obtained by Chen and the second author in [31], this Kähler manifold must be flat. This contradicts the assumption. Hence the holonomy algebra ˜ 1 ×M ˜ 2 nontrivially. G is reducible and the universal cover of M splits isometrically as M ˜ 1 and Clearly the universal cover of M has positive asymptotic volume ratio. So M ˜ M2 still have positive asymptotic volume ratio and at least one of them is nonflat. By the induction hypothesis, this is also impossible. Thus our claim is proved. Now we know that the sectional curvature of (M, gij (x, t0 )) is positive everywhere. Choose a sequence of points xk divergent to infinity such that  R(xk , t0 )d2t0 (xk , x0 ) = sup{R(x, t0 )d2t0 (x, x0 ) | dt0 (x, x0 ) ≥ dt0 (xk , x0 )},    dt0 (xk , x0 ) ≥ k,    R(xk , t0 )d2t0 (xk , x0 ) = εk → 0. Consider the rescaled metric

gk (0) = R(xk , t0 )g(·, t0 ) as before. Then ( (6.3.3)

Rk (x, 0) ≤ εk /d2k (x, x0 , 0), √ dk (xk , x0 , 0) = εk → 0.

for dk (x, x0 , 0) ≥

√ εk ,

As in Case 2, the rescaled marked solutions (M, gk (0), x0 ) will converge in the Gromov˜ , g˜(0), x0 ). And by the virtue of Hamilton’s Hausdorff sense to a metric cone (M compactness theorem (Theorem 4.1.5), up to a subsequence, the convergence is in the ∞ ˜ \ {x0 }. We next claim the metric cone (M ˜ , g˜(0), x0 ) is isometric Cloc topology in M to Rn . ˜ as a warped product R+ ×r X n−1 for Indeed, let us write the metric cone M n−1 some (n − 1)-dimensional manifold X . By (6.3.3), the metric cone must be flat and X n−1 is isometric to a quotient of the round sphere Sn−1 by fixed point free ˜ is isometric to Rn , we only need to isometries in the standard metric. To show M n−1 verify that X is simply connected. Let ϕ be the Busemann function of (M, gij (·, t0 )) with respect to the point x0 . Since (M, gij (·, t0 )) has nonnegative sectional curvature, it is easy to see that for any small ε > 0, there is a r0 > 0 such that (1 − ε)dt0 (x, x0 ) ≤ ϕ(x) ≤ dt0 (x, x0 )

377


for all x ∈ M \ Bt0 (x0 , r0 ). The strict positivity of the sectional curvature of the manifold (M, gij (·, t0 )) implies that the square of the Busemann function is strictly convex (and exhausting). Thus every level set ϕ−1 (a), with a > inf{ϕ(x) | x ∈ M }, of the Busemann function ϕ is diffeomorphic to the (n − 1)-sphere Sn−1 . In particular, ϕ−1 ([a, 32 a]) is simply connected for a > inf{ϕ(x) | x ∈ M } since n ≥ 3. ˜ = R+ ×r X n−1 . Consider an annulus portion [1, 2] × X n−1 of the metric cone M It is the limit of (Mk , gk (0)), where ) ( 2 1 ≤ dt0 (x, x0 ) ≤ p . Mk = x ∈ M p R(xk , t0 ) R(xk , t0 ) It is clear that " ϕ

−1

2(1 − ε) p ,p R(xk , t0 ) R(xk , t0 ) 1

#!

⊂ Mk ⊂ ϕ

−1

"

p

1−ε

R(xk , t0 )

,p

2 R(xk , t0 )

#!

for k large enough. Thus any closed loop in { 23 } × X n−1 can be shrunk to a point by a homotopy in [1, 2] × X n−1 . This shows that X n−1 is simply connected. Hence we ˜ , g˜(0), x0 ) is isometric to Rn . Consequently, have proven that the metric cone (M \ Bg(t0 ) x0 , √ σr Vol g(t0 ) Bg(t0 ) x0 , √ r R(xk ,t0 ) R(xk ,t0 ) n = αn (1−σ n ) lim k→+∞ r √ R(xk ,t0 )

for any r > 0 and 0 < σ < 1, where αn is the volume of the unit ball in the Euclidean space Rn . Finally, by combining with the monotonicity of the Bishop-Gromov volume comparison, we conclude that the manifold (M, gij (·, t0 )) is flat and isometric to Rn . This contradicts the assumption. Therefore, we have proved the lemma. Finally we would like to include an alternative simpler argument, inspired by Ni [98], for the above Case 2 and Case 3 to avoid the use of the gap theorem, holonomy groups, and asymptotic cone structure. Alternative Proof for Case 2 and Case 3. Let us consider the situation of 0 ≤ A < +∞ in the above proof. Observe that νM (t) is nonincreasing in time t by using Lemma 3.4.1(ii) and the fact that the metric is shrinking in t. Suppose νM (t0 ) > 0, then the solution gij (·, t) is κ-noncollapsed for t ≤ t0 for some uniform κ > 0. By combining with Theorem 6.2.1, there exist a sequence of points qk and a sequence of times tk → −∞ such that the scalings of gij (·, t) around qk with factor |tk |−1 and with the times tk shifting to the new time zero converge to a nonflat gradient shrinking ¯ in the C ∞ topology. This gradient soliton also has maximal volume growth soliton M loc (i.e. νM¯ (t) > 0) and satisfies the Li-Yau-Hamilton estimate (Corollary 2.5.5). If the ¯ at the time −1 is bounded, then we see from curvature of the shrinking soliton M the proof of Theorem 6.2.2 that by using the equations (6.2.24)-(6.2.26), the scalar ¯ at the time −1. In particular, curvature has a positive lower bound everywhere on M this implies the asymptotic scalar curvature ratio A = ∞ for the soliton at the time −1, which reduces to Case 1 and arrives at a contradiction by the dimension reduction argument. On the other hand, if the scalar curvature is unbounded, then by Lemma

378


6.1.4, the Li-Yau-Hamilton estimate (Corollary 2.5.5) and Lemma 6.1.2, we can do the same dimension reduction as in Case 1 to arrive at a contradiction also. The following lemma is a local and space-time version of Lemma 6.1.4 on picking local (almost) maximum curvature points. We formulate it from Perelman’s arguments in the section 10 of [103]. Lemma 6.3.2. For any positive constants B, C with B > 4 and C > 1000, there 1 exists 1 ≤ A < min{ 41 B, 1000 C} which tends to infinity as B and C tend to infinity and satisfies the following property. Suppose we have a (not necessarily complete) solution gij (t) to the Ricci flow, defined on M × [−t0 , 0], so that at each time t ∈ [−t0 , 0] the metric ball Bt (x0 , 1) is compactly contained in M . Suppose there exists a point (x′ , t′ ) ∈ M × (−t0 , 0] such that dt′ (x′ , x0 ) ≤

1 and |Rm(x′ , t′ )| > C + B(t′ + t0 )−1 . 4

Then we can find a point (¯ x, t¯) ∈ M × (−t0 , 0] such that dt¯(¯ x, x0 )
C + B(t¯ + t0 )−1 , 3

and |Rm(x, t)| ≤ 4Q for all (−t0 C + B(t¯ + t0 )−1 , and |Rm(x, t)| ≤ 4Q

(6.3.4)

1

wherever t¯ − AQ−1 ≤ t ≤ t¯, dt (x, x0 ) ≤ dt¯(¯ x, x0 ) + (AQ−1 ) 2 . We will construct such (¯ x, t¯) as a limit of a finite sequence of points. Take an arbitrary (x1 , t1 ) such that dt1 (x1 , x0 ) ≤

1 , 4

−t0 < t1 ≤ 0 and |Rm(x1 , t1 )| > C + B(t1 + t0 )−1 .

Such a point clearly exists by our assumption. Assume we have already constructed (xk , tk ). If we cannot take the point (xk , tk ) to be the desired point (¯ x, t¯), then there exists a point (xk+1 , tk+1 ) such that tk − A|Rm(xk , tk )|−1 ≤ tk+1 ≤ tk , and 1

dtk+1 (xk+1 , x0 ) ≤ dtk (xk , x0 ) + (A|Rm(xk , tk )|−1 ) 2 , but |Rm(xk+1 , tk+1 )| > 4|Rm(xk , tk )|.


379

It then follows that dtk+1 (xk+1 , x0 ) ≤ dt1 (x1 , x0 ) + A 1 1 ≤ + A2 4

k X

2

1 2

k X i=1

−(i−1)

i=1

− 12

|Rm(xi , ti )|

− 12

|Rm(x1 , t1 )|

!

!

1 1 + 2(AC −1 ) 2 4 1 < , 3

≤

tk+1 − (−t0 ) =

k X i=1

≥−

(ti+1 − ti ) + (t1 − (−t0 ))

k X i=1

≥ −A ≥− ≥

A|Rm(xi , ti )|−1 + (t1 − (−t0 ))

k X i=1

4−(i−1) |Rm(x1 , t1 )|−1 + (t1 − (−t0 ))

2A (t1 + t0 ) + (t1 + t0 ) B

1 (t1 + t0 ), 2

and |Rm(xk+1 , tk+1 )| > 4k |Rm(x1 , t1 )|

≥ 4k C → +∞ as k → +∞.

Since the solution is smooth, the sequence {(xk , tk )} is finite and its last element fits. Thus we have proved assertion (6.3.4). From the above construction we also see that the chosen point (¯ x, t¯) satisfies dt¯(¯ x, x0 )
C + B(t¯ + t0 )−1 . Clearly, up to some adjustment of the constant A, we only need to show that |Rm(x, t)| ≤ 4Q

(6.3.4)′ 1

1

1

1 1 wherever t¯ − 200n A 2 Q−1 ≤ t ≤ t¯ and dt (x, x¯) ≤ 10 A 2 Q− 2 . 1 1 1 − ¯ For any point (x, t) with dt¯(x, x¯) ≤ 10 A 2 Q 2 , we have

dt¯(x, x0 ) ≤ dt¯(¯ x, x0 ) + dt¯(x, x¯)

1

≤ dt¯(¯ x, x0 ) + (AQ−1 ) 2

380


and then by (6.3.4) |Rm(x, t¯)| ≤ 4Q. 1 1 A 2 Q−1 , t¯] such that Thus by continuity, there is a minimal t¯′ ∈ [t¯ − 200n 1 1 −1 ′ ¯ ¯ 2 2 ≤ 5Q. (6.3.5) sup |Rm(x, t)| | t ≤ t ≤ t, dt (x, x¯) ≤ A Q 10

For any point (x, t) with t¯′ ≤ t ≤ t¯ and dt (x, x¯) ≤ discussion into two cases.

1 −1 12 ) , 10 (AQ

we divide the

1

3 Case (1): dt (¯ x, x0 ) ≤ 10 (AQ−1 ) 2 . From assertion (6.3.4) we see that

(6.3.5)′

1 sup{|Rm(x, t)| | t¯′ ≤ t ≤ t¯, dt (x, x0 ) ≤ (AQ−1 ) 2 } ≤ 4Q.

Since dt (¯ x, x0 ) ≤

3 −1 21 ) , 10 (AQ

we have

dt (x, x0 ) ≤ dt (x, x¯) + dt (¯ x, x0 ) 1 1 1 3 ≤ (AQ−1 ) 2 + (AQ−1 ) 2 10 10 1 ≤ (AQ−1 ) 2 which implies the estimate |Rm(x, t)| ≤ 4Q from (6.3.5)′ . 1

3 Case (2): dt (¯ x, x0 ) > 10 (AQ−1 ) 2 . From the curvature bounds in (6.3.5) and (6.3.5)′ , we can apply Lemma 3.4.1 (ii) 1 1 Q− 2 to get with r0 = 10

d 1 (dt (¯ x, x0 )) ≥ −40(n − 1)Q 2 dt and then 1

dt (¯ x, x0 ) ≤ dtˆ(¯ x, x0 ) + 40n(Q 2 )

1 1 A 2 Q−1 200n

1 1 ≤ dtˆ(¯ x, x0 ) + (AQ−1 ) 2 5

where tˆ ∈ (t, t¯] satisfies the property that ds (¯ x, x0 ) ≥ So we have either

3 −1 12 ) 10 (AQ

whenever s ∈ [t, tˆ].

dt (x, x0 ) ≤ dt (x, x¯) + dt (¯ x, x0 ) 1 1 1 3 1 1 ≤ (AQ−1 ) 2 + (AQ−1 ) 2 + (AQ−1 ) 2 10 10 5 1 ≤ (AQ−1 ) 2 , or dt (x, x0 ) ≤ dt (x, x¯) + dt (¯ x, x0 ) 1 1 1 1 x, x0 ) + (AQ−1 ) 2 ≤ (AQ−1 ) 2 + dt¯(¯ 10 5 1 ≤ dt¯(¯ x, x0 ) + (AQ−1 ) 2 .


381

It then follows from (6.3.4) that |Rm(x, t)| ≤ 4Q. Hence we have proved |Rm(x, t)| ≤ 4Q 1 1 for any point (x, t) with t¯′ ≤ t ≤ t¯ and dt (x, x ¯) ≤ 10 (AQ−1 ) 2 . By combining with 1 1 the choice of t¯′ in (6.3.5), we must have t¯′ = t¯ − 200n A 2 Q−1 . This proves assertion ′ (6.3.4) . Therefore we have completed the proof of the lemma.

We now use the volume lower bound assumption to establish the crucial curvature upper bound estimate of Perelman [103] for the Ricci flow. For the Ricci flow on Kähler manifolds, a global version of this estimate (i.e., curvature decaying linear in time and quadratic in space) was independently obtained in [29] and [32]. Note that the volume estimate conclusion in the following Theorem 6.3.3 (ii) was not stated in Corollary 11.6 (b) of Perelman [103]. The estimate will be used later in the proof of Theorem 7.2.2 and Theorem 7.5.2. Theorem 6.3.3 (Perelman [103]). For every w > 0 there exist B = B(w) < +∞, C = C(w) < +∞, τ0 = τ0 (w) > 0, and ξ = ξ(w) > 0 (depending also on the dimension) with the following properties. Suppose we have a (not necessarily complete) solution gij (t) to the Ricci flow, defined on M × [−t0 r02 , 0], so that at each time t ∈ [−t0 r02 , 0] the metric ball Bt (x0 , r0 ) is compactly contained in M. (i) If at each time t ∈ [−t0 r02 , 0], Rm(., t) ≥ −r0−2 on Bt (x0 , r0 ) and Vol t (Bt (x0 , r0 )) ≥ wr0n , then we have the estimate |Rm(x, t)| ≤ Cr0−2 + B(t + t0 r02 )−1 whenever −t0 r02 < t ≤ 0 and dt (x, x0 ) ≤ 41 r0 . (ii) If for some 0 < τ¯ ≤ t0 , Rm(x, t) ≥ −r0−2 for t ∈ [−¯ τ r02 , 0], x ∈ Bt (x0 , r0 ), and Vol 0 (B0 (x0 , r0 )) ≥ wr0n , then we have the estimates Vol t (Bt (x0 , r0 )) ≥ ξr0n for all max{−¯ τ r02 , −τ0 r02 } ≤ t ≤ 0, and |Rm(x, t)| ≤ Cr0−2 + B(t − max{−¯ τ r02 , −τ0 r02 })−1 whenever max{−¯ τ r02 , −τ0 r02 } < t ≤ 0 and dt (x, x0 ) ≤ 14 r0 .

382


Proof. By scaling we may assume r0 = 1. (i) By the standard (relative) volume comparison, we know that there exists some w′ > 0, with w′ ≤ w, depending only on w, such that for each point (x, t) with −t0 ≤ t ≤ 0 and dt (x, x0 ) ≤ 13 , and for each r ≤ 13 , there holds Vol t (Bt (x, r)) ≥ w′ rn .

(6.3.6)

We argue by contradiction. Suppose there are sequences B, C → +∞, of solutions gij (t) and points (x′ , t′ ) such that dt′ (x′ , x0 ) ≤

1 , 4

−t0 < t′ ≤ 0 and |Rm(x′ , t′ )| > C + B(t′ + t0 )−1 .

Then by Lemma 6.3.2, we can find a sequence of points (¯ x, t¯) such that dt¯(¯ x, x0 )
C + B(t¯ + t0 )−1 , and |Rm(x, t)| ≤ 4Q 1 1 1 A 2 Q− 2 , where A tends to infinity wherever (−t0 0 for each t ∈ (−∞, 0] (by (6.3.6)). This contradicts Lemma 6.3.1.

(ii) Let B(w), C(w) be good for the first part of the theorem. By the volume assumption at t = 0 and the standard (relative) volume comparison, we still have the estimate (6.3.6)′

Vol 0 (B0 (x, r)) ≥ w′ rn

for each x ∈ M with d0 (x, x0 ) ≤ 31 and r ≤ 13 . We will show that ξ = 5−n w′ , B = B(5−n w′ ) and C = C(5−n w′ ) are good for the second part of the theorem. By continuity and the volume assumption at t = 0, there is a maximal subinterval [−τ, 0] of the time interval [−¯ τ , 0] such that Vol t (Bt (x0 , 1)) ≥ w ≥ 5−n w′

for all t ∈ [−τ, 0].

This says that the assumptions of (i) hold with 5−n w′ in place of w and with τ in place of t0 . Thus the conclusion of the part (i) gives us the estimate (6.3.7)

|Rm(x, t)| ≤ C + B(t + τ )−1

whenever t ∈ (−τ, 0] and dt (x, x0 ) ≤ 14 . We need to show that one can choose a positive τ0 depending only on w and the dimension such that the maximal τ ≥ min{¯ τ , τ0 }.


For t ∈ (−τ, 0] and

1 8

383

≤ dt (x, x0 ) ≤ 14 , we use (6.3.7) and Lemma 3.4.1(ii) to get

√ √ √ d dt (x, x0 ) ≥ −10(n − 1)( C + ( B/ t + τ )) dt which further gives √ √ d0 (x, x0 ) ≥ d−τ (x, x0 ) − 10(n − 1)(τ C + 2 Bτ ). This means (6.3.8)

√ √ 1 1 B(−τ ) (x0 , ) ⊃ B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) . 4 4

Note that the scalar curvature R ≥ −C(n) for some constant C(n) depending only on the dimension since Rm ≥ −1. We have √ √ d 1 Vol t B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) dt 4 Z (−R)dVt = √ √ B0 (x0 , 41 −10(n−1)(τ

C+2 Bτ ))

√ √ 1 ≤ C(n)Vol t B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) 4

and then (6.3.9)

√ √ 1 Vol t B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) 4 √ √ 1 . ≤ eC(n)τ Vol (−τ ) B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) 4

Thus by (6.3.6)′ , (6.3.8) and (6.3.9), Vol (−τ ) (B(−τ ) )(x0 , 1) 1 ≥ Vol (−τ ) (B(−τ ) ) x0 , 4 √ √ 1 ≥ Vol (−τ ) B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) 4 √ √ 1 −C(n)τ ≥e Vol 0 B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) 4 n √ √ 1 −C(n)τ ′ ≥e w − 10(n − 1)(τ C + 2 Bτ ) . 4 So it suffices to choose τ0 = τ0 (w) small enough so that e−C(n)τ0

n n p √ 1 1 ≥ − 10(n − 1)(τ0 C + 2 Bτ0 ) . 4 5

Therefore we have proved the theorem.

384


6.4. Ancient κ-solutions on Three-manifolds. In this section we will determine the structures of ancient κ-solutions on three-manifolds. First of all, we consider a special class of ancient solutions — gradient shrinking Ricci solitons. Recall that a solution gij (t) to the Ricci flow is said to be a gradient shrinking Ricci soliton if there exists a smooth function f such that 1 gij = 0 for − ∞ < t < 0. 2t A gradient shrinking Ricci soliton moves by the one parameter group of diffeomorphisms generated by ∇f and shrinks by a factor at the same time. The following result of Perelman [103] gives a complete classification for all threedimensional complete κ-noncollapsed gradient shrinking solitons with bounded and nonnegative sectional curvature. (6.4.1)

∇i ∇j f + Rij +

Lemma 6.4.1 (Classification of three-dimensional shrinking solitons). Let (M, gij (t)) be a nonflat gradient shrinking soliton on a three-manifold. Suppose (M, gij (t)) has bounded and nonnegative sectional curvature and is κ-noncollapsed on all scales for some κ > 0. Then (M, gij (t)) is one of the following: (i) the round three-sphere S3 , or a metric quotient of S3 ; (ii) the round infinite cylinder S2 × R, or one of its Z2 quotients. Proof. We first consider the case that the sectional curvature of the nonflat gradient shrinking soliton is not strictly positive. Let us pull back the soliton to its universal cover. Then the pull-back metric is again a nonflat ancient κ-solution. By Hamilton’s strong maximum principle (Theorem 2.2.1), we know that the pull-back solution splits as the metric product of a two-dimensional nonflat ancient κ-solution and R. Since the two-dimensional nonflat ancient κ-solution is simply connected, it follows from Theorem 6.2.2 that it must be the round sphere S2 . Thus, the gradient shrinking soliton must be S2 × R/Γ, a metric quotient of the round cylinder. For each σ ∈ Γ and (x, s) ∈ S2 × R, we write σ(x, s) = (σ1 (x, s), σ2 (x, s)) ∈ S2 × R. Since σ sends lines to lines, and sends cross spheres to cross spheres, we have σ2 (x, s) = σ2 (y, s), for all x, y ∈ S2 . This says that σ2 reduces to a function of s alone on R. Moreover, for any (x, s), (x′ , s′ ) ∈ S2 × R, since σ preserves the distances between cross spheres S2 × {s} and S2 × {s′ }, we have |σ2 (x, s) − σ2 (x′ , s′ )| = |s − s′ |. So the projection Γ2 of Γ to the second factor R is an isometry subgroup of R. If the metric quotient S2 × R/Γ were compact, it would not be κ-noncollapsed on sufficiently large scales as t → −∞. Thus the metric quotient S2 × R/Γ is noncompact. It follows that Γ2 = {1} or Z2 . In particular, there is a Γ-invariant cross sphere S2 in the round cylinder S2 × R. Denote it by S2 × {0}. Then Γ acts on the round two-sphere S2 × {0} isometrically without fixed points. This implies Γ is either {1} or Z2 . Hence we conclude that the gradient shrinking soliton is either the round cylinder S2 × R, or ˜ where Z2 flips both S2 and R. RP2 × R, or the twisted product S2 ×R We next consider the case that the gradient shrinking soliton is compact and has strictly positive sectional curvature everywhere. By the proof of Theorem 5.2.1 (see also Remark 5.2.8) we see that the compact gradient shrinking soliton is getting round and tends to a space form (with positive constant curvature) as the time approaches the maximal time t = 0. Since the shape of a gradient shrinking Ricci soliton does not change up to reparametrizations and homothetical scalings, the gradient shrinking soliton has to be the round three-sphere S3 or a metric quotient of S3 . Finally we want to exclude the case that the gradient shrinking soliton is noncompact and has strictly positive sectional curvature everywhere.


385

Suppose there is a (complete three-dimensional) noncompact κ-noncollapsed gradient shrinking soliton gij (t), −∞ < t < 0, with bounded and positive sectional curvature at each t ∈ (−∞, 0) and satisfying the equation (6.4.1). Then as in (6.2.25), we have ∇i R = 2Rij ∇j f.

(6.4.2)

Fix some t < 0, say t = −1, and consider a long shortest geodesic γ(s), 0 ≤ s ≤ s¯. Let x0 = γ(0) and X(s) = γ(s). ˙ Let U (0) be any unit vector orthogonal to γ(0) ˙ and translate U (0) along γ(s) to get a parallel vector field U (s), 0 ≤ s ≤ s¯, on γ. Set  sU (s), for 0 ≤ s ≤ 1,    e U (s), for 1 ≤ s ≤ s¯ − 1 U (s) =    (¯ s − s)U (s), for s¯ − 1 ≤ s ≤ s¯. It follows from the second variation formula of arclength that Z s¯ e˙ (s)|2 − R(X, U e , X, U e ))ds ≥ 0. (|U 0

Since the curvature of the metric gij (−1) is bounded, we clearly have Z s¯ R(X, U, X, U )ds ≤ const. 0

and then Z

(6.4.3)

0

s¯

Ric (X, X)ds ≤ const..

Moreover, since the curvature of the metric gij (−1) is positive, it follows from the Cauchy-Schwarz inequality that for any unit vector field Y along γ and orthogonal to X(= γ(s)), ˙ we have Z s¯ Z s¯ Ric (X, X)Ric (Y, Y )ds |Ric (X, Y )|2 ds ≤ 0 0 Z s¯ ≤ const. · Ric (X, X)ds 0

≤ const.

and then Z

(6.4.4)

s¯ 0

√ |Ric (X, Y )|ds ≤ const. · ( s¯ + 1).

From (6.4.1) we know ∇X ∇X f + Ric (X, X) −

1 =0 2

and by integrating this equation we get X(f (γ(¯ s))) − X(f (γ(0))) +

Z

s¯ 0

1 Ric (X, X)ds − s¯ = 0. 2

386


Thus by (6.4.3) we deduce (6.4.5)

s¯ s¯ − const. ≤ hX, ∇f (γ(¯ s))i ≤ + const.. 2 2

Similarly by integrating (6.4.1) and using (6.4.4) we can deduce √ (6.4.6) |hY, ∇f (γ(¯ s))i| ≤ const. · ( s¯ + 1). These two inequalities tell us that at large distances from the fixed point x0 the function f has no critical point, and its gradient makes a small angle with the gradient of the distance function from x0 . Now from (6.4.2) we see that at large distances from x0 , R is strictly increasing along the gradient curves of f , in particular ¯= R

lim sup d(−1) (x,x0 )→+∞

R(x, −1) > 0.

¯ By the noncolLet us choose a sequence of points (xk , −1) where R(xk , −1) → R. lapsing assumption we can take a limit along this sequence of points of the gradient soliton and get an ancient κ-solution defined on −∞ < t < 0. By Proposition 6.1.2, we deduce that the limiting ancient κ-solution splits off a line. Since the soliton has positive sectional curvature, we know from Gromoll-Meyer [52] that it is orientable. Then it follows from Theorem 6.2.2 that the limiting solution is the shrinking round ¯ at time t = −1. Since the limiting solution infinite cylinder with scalar curvature R ¯ exists on (−∞, 0), we conclude that R ≤ 1. Hence R(x, −1) < 1 when the distance from x to the fixed x0 is large enough on the gradient shrinking soliton. Let us consider the level surface {f = a} of f . The second fundamental form of the level surface is given by ∇f hij = ∇i , ej |∇f | = ∇i ∇j f /|∇f |,

i, j = 1, 2,

where {e1 , e2 } is an orthonormal basis of the level surface. By (6.4.1), we have ∇ei ∇ei f =

1 1 R − Ric (ei , ei ) ≥ − > 0, 2 2 2

i = 1, 2,

since for a three-manifold the positivity of sectional curvature is equivalent to R ≥ 2Ric . It then follows from the first variation formula that Z ∇f d (6.4.7) Area {f = a} = div da |∇f | {f =a} Z 1 ≥ (1 − R) |∇f | {f =a} Z 1 ¯ (1 − R) > |∇f | {f =a} ≥0


387

for a large enough. We conclude that Area {f = a} strictly increases as a increases. From (6.4.5) we see that for s large enough df − s ≤ const., ds 2 and then

Thus we get from (6.4.7)

2 f − s ≤ const. · (s + 1). 4

¯ 1−R d Area {f = a} > √ Area {f = a} da 2 a for a large enough. This implies that √ ¯ a − const. log Area {f = a} > (1 − R) for a large enough. But it is clear from (6.4.7) that Area {f = a} is uniformly bounded ¯ for all large a. Thus from above by the area of the round sphere of scalar curvature R ¯ we deduce that R = 1. So Area {f = a} < 8π

(6.4.8)

for a large enough. Denote by X the unit normal vector to the level surface {f = a}. By using the Gauss equation and (6.4.1), the intrinsic curvature of the level surface {f = a} can be computed as (6.4.9)

intrinsic curvature = R1212 + det(hij ) = ≤ = =
0 and |∇f | is large when a is large. Thus the combination of (6.4.8) and (6.4.9) gives a contradiction to the Gauss-Bonnet formula. Therefore we have proved the lemma. As a direct consequence, there is a universal positive constant κ0 such that any nonflat three-dimensional gradient shrinking soliton, which is also an ancient

388


κ-solution, to the Ricci flow must be κ0 -noncollapsed on all scales unless it is a metric quotient of round three-sphere. The following result, claimed by Perelman in the section 1.5 of [104], shows that this property actually holds for all nonflat threedimensional ancient κ-solutions. Proposition 6.4.2 (Universal noncollapsing). There exists a positive constant κ0 with the following property. Suppose we have a nonflat three-dimensional ancient κ-solution for some κ > 0. Then either the solution is κ0 -noncollapsed on all scales, or it is a metric quotient of the round three-sphere. Proof. Let gij (x, t), x ∈ M and t ∈ (−∞, 0], be a nonflat ancient κ-solution for some κ > 0. For an arbitrary point (p, t0 ) ∈ M × (−∞, 0], we define as in Chapter 3 that τ = t0 − t,

for t < t0 , Z τ √ 1 2 l(q, τ ) = √ inf s(R(γ(s), t0 − s) + |γ(s)| ˙ gij (t0 −s) )ds| 2 τ 0 γ : [0, τ ] → M with γ(0) = p, γ(τ ) = q , Z 3 and Ve (τ ) = (4πτ )− 2 exp(−l(q, τ ))dVt0 −τ (q). M

Recall from (6.2.1) that for each τ > 0 we can find q = q(τ ) such that l(q, τ ) ≤ 32 . In view of Lemma 6.4.1, we may assume that the ancient κ-solution is not a gradient shrinking Ricci soliton. Thus by (the proof of) Theorem 6.2.1, the scalings of gij (t0 −τ ) at q(τ ) with factor τ −1 converge along a subsequence of τ → +∞ to a nonflat gradient shrinking soliton with nonnegative curvature operator which is κ-noncollapsed on all scales. We now show that the limit has bounded curvature. ¯ , g¯ij (x, t)) with −∞ < Denote the limiting nonflat gradient shrinking soliton by (M t ≤ 0. Note that there holds the Li-Yau-Hamilton inequality (Theorem 2.5.4) on any ancient κ-solution and in particular, the scalar curvature of the ancient κ-solution is pointwise nondecreasing in time. This implies that the scalar curvature of the limiting ¯ , g¯ij (x, t)) is still pointwise nondecreasing in time. Thus we only need to soliton (M show that the limiting soliton has bounded curvature at t = 0. We argue by contradiction. By lifting to its orientable cover, we may assume ¯ is orientable. Suppose the curvature of the limiting soliton is unbounded at that M ¯ is noncompact. Then by applying t = 0. Of course in this case the limiting soliton M Lemma 6.1.4, we can choose a sequence of points xj , j = 1, 2, . . . , divergent to infinity ¯ of the limit satisfies such that the scalar curvature R ¯ j , 0) ≥ j and R(x, ¯ 0) ≤ 4R(x ¯ j , 0) R(x p ¯ j , 0)) and j = 1, 2, . . .. Since the scalar curvature is nondefor all x ∈ B0 (xj , j/ R(x creasing in time, we have (6.4.10) p

¯ t) ≤ 4R(x ¯ j , 0), R(x,

¯ j , 0)), all t ≤ 0 and j = 1, 2, . . .. By combining with for all x ∈ B0 (xj , j/ R(x Hamilton’s compactness theorem (Theorem 4.1.5) and the κ-noncollapsing, we know that a subsequence of the rescaled solutions ¯ , R(x ¯ j , 0)¯ ¯ j , 0)), xj ), j = 1, 2, . . . , (M gij (x, t/R(x


389

∞ converges in the Cloc topology to a nonflat smooth solution of the Ricci flow. Then Proposition 6.1.2 implies that the new limit at the new time {t = 0} must split off a line. By pulling back the new limit to its universal cover and applying Hamilton’s strong maximum principle, we deduce that the pull-back of the new limit on the universal cover splits off a line for all time t ≤ 0. Thus by combining with Theorem 6.2.2 and the argument in the proof of Lemma 6.4.1, we further deduce that the new ¯ is orientable, limit is either the round cylinder S2 × R or the round RP2 × R. Since M ¯ , g¯ij (x, 0)) has nonnegative curvature operator the new limit must be S2 × R. Since (M ¯ j , 0) → +∞, this gives a contradiction and the points {xj } going to infinity and R(x to Proposition 6.1.1. So we have proved that the limiting gradient shrinking soliton has bounded curvature at each time. Hence by Lemma 6.4.1, the limiting gradient shrinking soliton is either the round three-sphere S3 or its metric quotients, or the infinite cylinder S2 × R or one of its Z2 quotients. If the asymptotic gradient shrinking soliton is the round three-sphere S3 or its metric quotients, it follows from Lemma 5.2.4 and Proposition 5.2.5 that the ancient κ-solution must be round. Thus in the following we may assume the asymptotic gradient shrinking soliton is the infinite cylinder S2 × R or a Z2 quotient of S2 × R. We now come back to consider the original ancient κ-solution (M, gij (x, t)). By rescaling, we can assume that R(x, t) ≤ 1 for all (x, t) satisfying dt0 (x, p) ≤ 2 and t ∈ [t0 − 1, t0 ]. We will argue as in the proof of Theorem 3.3.2 (Perelman’s no local collapsing theorem I) to obtain a positive lower bound for Vol t0 (Bt0 (p, 1)). 1

3 Denote by ξ = Vol t0 (Bt0 (p, 1)) √ . For any v ∈ Tp M we can find an L-geodesic γ(τ ), starting at p, with limτ →0+ τ γ(τ ˙ ) = v. It follows from the L-geodesic equation (3.2.1) that

√ 1√ d √ ˙ − τ ∇R + 2Ric ( τ γ, ˙ ·) = 0. ( τ γ) dτ 2 By integrating as before we see that for τ ≤ ξ with the property γ(σ) ∈ Bt0 (p, 1) as long as σ < τ , there holds √ | τ γ(τ ˙ ) − v| ≤ Cξ(|v| + 1) where C is some positive constant depending only on the dimension. Without loss of 1 1 . Then for v ∈ Tp M with |v| ≤ 14 ξ − 2 generality, we may assume Cξ ≤ 14 and ξ ≤ 100 and for τ ≤ ξ with the property γ(σ) ∈ Bt0 (p, 1) as long as σ < τ , we have dt0 (p, γ(τ )) ≤

Z

τ

|γ(σ)|dσ ˙ Z 1 − 1 τ dσ 2 √ < ξ 2 σ 0 = 1. 0

This shows (6.4.11)

1 1 L exp |v| ≤ ξ − 2 (ξ) ⊂ Bt0 (p, 1). 4

390


We decompose Perelman’s reduced volume Ve (ξ) as

(6.4.12)

Ve (ξ) =

Z

n

1

L exp |v|≤ 41 ξ − 2

+

Z

o

(ξ)

n

M\L exp

1 |v|≤ 41 ξ − 2

3

o

(ξ)

(4πξ)− 2 exp(−l(q, ξ))dVt0 −ξ (q).

By using (6.4.11) and the metric evolution equation of the Ricci flow, the first term on the RHS of (6.4.12) can be estimated by Z

3

−1 L exp{|v|≤ 41 ξ 2

Z

≤

(4πξ)− 2 exp(−l(q, ξ))dVt0 −ξ (q) }(ξ) 3

(4πξ)− 2 e3ξ dVt0 (q) Bt0 (p,1) 3

3

= (4π)− 2 e3ξ ξ 2 3

< ξ2, while by using Theorem 3.2.7 (Perelman’s Jacobian comparison theorem), the second term on the RHS of (6.4.12) can be estimated by (6.4.13)

Z

n

M\L exp

≤

Z

1 |v|≤ 14 ξ − 2

3

(4πξ)− 2 exp(−l(q, ξ))dVt0 −ξ (q)

o

(ξ) 3

1 {|v|> 41 ξ − 2

− 32

= (4π)

Z

(4πτ )− 2 exp(−l(τ ))J (τ )|τ =0 dv }

1

exp(−|v|2 )dv

{|v|> 41 ξ − 2 }

3

< ξ2 3

since limτ →0+ τ − 2 J (τ ) = 1 and limτ →0+ l(τ ) = |v|2 by (3.2.18) and (3.2.19) respectively. Thus we obtain (6.4.14)

3 Ve (ξ) < 2ξ 2 .

On the other hand, we recall that there exist a sequence τk → +∞ and a sequence of points q(τk ) ∈ M with l(q(τk ), τk ) ≤ 32 so that the scalings of the ancient κ-solution at q(τk ) with factor τk−1 converge to either round S2 × R or one of its Z2 quotients. For sufficiently large k, we construct a path γ : [0, 2τk ] → M , connecting p to any given point q ∈ M , as follows: the first half path γ|[0,τk ] connects p to q(τk ) such that 1 l(q(τk ), τk ) = √ 2 τk

Z

0

τk

√

τ (R + |γ(τ ˙ )|2 )dτ ≤ 2,

and the second half path γ|[τk ,2τk ] is a shortest geodesic connecting q(τk ) to q with respect to the metric gij (t0 − τk ). Note that the rescaled metric τk−1 gij (t0 − τ ) over √ the domain Bt0 −τk (q(τk ), τk ) × [t0 − 2τk , t0 − τk ] is sufficiently close to the round


391

S2 × R or its Z2 quotients. Then there is a universal positive constant β such that Z τk Z 2τk √ 1 + τ (R + |γ(τ ˙ )|2 )dτ l(q, 2τk ) ≤ √ 2 2τk 0 τk Z 2τk √ √ 1 ≤ 2+ √ τ (R + |γ(τ ˙ )|2 )dτ 2 2τk τk ≤β √ for all q ∈ Bt0 −τk (q(τk ), τk ). Thus Z 3 Ve (2τk ) = (4π(2τk ))− 2 exp(−l(q, 2τk ))dVt0 −2τk (q) M Z 3 −β (4π(2τk ))− 2 dVt0 −2τk (q) ≥e √ Bt0 −τk (q(τk ), τk )

≥ β˜

˜ Here we have used the curvature estimate for some universal positive constant β. (6.2.6). By combining with the monotonicity of Perelman’s reduced volume (Theorem 3.2.8) and (6.4.14), we deduce that

This proves

3 β˜ ≤ Ve (2τk ) ≤ Ve (ξ) < 2ξ 2 .

Vol t0 (Bt0 (p, 1)) ≥ κ0 > 0 for some universal positive constant κ0 . So we have proved that the ancient κ-solution is also an ancient κ0 -solution. The important Li-Yau-Hamilton inequality gives rise to a parabolic Harnack estimate (Corollary 2.5.7) for solutions of the Ricci flow with bounded and nonnegative curvature operator. As explained in the previous section, the no local collapsing theorem of Perelman implies a volume lower bound from a curvature upper bound, while the estimate in the previous section implies a curvature upper bound from a volume lower bound. The combination of these two estimates as well as the Li-Yau-Hamilton inequality will give an important elliptic type property for three-dimensional ancient κ-solutions. This elliptic type property was first implicitly given by Perelman in [103] and it will play a crucial role in the analysis of singularities. Theorem 6.4.3 (Elliptic type estimate). There exist a positive constant η and a positive increasing function ω : [0, +∞) → (0, +∞) with the following properties. Suppose we have a three-dimensional ancient κ-solution (M, gij (t)), −∞ < t ≤ 0, for some κ > 0. Then (i) for every x, y ∈ M and t ∈ (−∞, 0], there holds R(x, t) ≤ R(y, t) · ω(R(y, t)d2t (x, y)); (ii) for all x ∈ M and t ∈ (−∞, 0], there hold ∂R 3 (x, t) ≤ ηR2 (x, t). 2 |∇R|(x, t) ≤ ηR (x, t) and ∂t

392


Proof. (i) Consider a three-dimensional nonflat ancient κ-solution gij (x, t) on M × (−∞, 0]. In view of Proposition 6.4.2, we may assume that the ancient solution is universal κ0 -noncollapsed. Obviously we only need to establish the estimate at t = 0. Let y be an arbitrarily fixed point in M . By rescaling, we can assume R(y, 0) = 1. Let us first consider the case that sup{R(x, 0)d20 (x, y) | x ∈ M } > 1. Define z to be the closest point to y (at time t = 0) satisfying R(z, 0)d20 (z, y) = 1. We want to 1 bound R(x, 0)/R(z, 0) from above for x ∈ B0 (z, 2R(z, 0)− 2 ). Connect y and z by a shortest geodesic and choose a point z˜ lying on the geo1 desic satisfying d0 (˜ z , z) = 14 R(z, 0)− 2 . Denote by B the ball centered at z˜ and with 1 radius 14 R(z, 0)− 2 (with respect to the metric at t = 0). Clearly the ball B lies in 1 1 B0 (y, R(z, 0)− 2 ) and lies outside B0 (y, 21 R(z, 0)− 2 ). Thus for x ∈ B, we have R(x, 0)d20 (x, y) ≤ 1 and d0 (x, y) ≥

1 1 R(z, 0)− 2 2

and hence R(x, 0) ≤

1 1 ( 12 R(z, 0)− 2 )2

for all x ∈ B.

Then by the Li-Yau-Hamilton inequality and the κ0 -noncollapsing, we have 3 1 − 21 , Vol 0 (B) ≥ κ0 R(z, 0) 4 and then 1

Vol 0 (B0 (z, 8R(z, 0)− 2 )) ≥

1 κ0 (8R(z, 0)− 2 )3 . 215

So by Theorem 6.3.3(ii), there exist positive constants B(κ0 ), C(κ0 ), and τ0 (κ0 ) such that (6.4.15)

R(x, 0) ≤ (C(κ0 ) +

B(κ0 ) )R(z, 0) τ0 (κ0 )

1

for all x ∈ B0 (z, 2R(z, 0)− 2 ). We now consider the remaining case. If R(x, 0)d20 (x, y) ≤ 1 everywhere, we choose a point z satisfying sup{R(x, 0) | x ∈ M } ≤ 2R(z, 0). Obviously we also have the estimate (6.4.15) in this case. We next want to bound R(z, 0) for the chosen z ∈ M . By (6.4.15) and the Li-Yau-Hamilton inequality, we have R(x, t) ≤ (C(κ0 ) +

B(κ0 ) )R(z, 0) τ0 (κ0 )

1

for all x ∈ B0 (z, 2R(z, 0)− 2 ) and all t ≤ 0. It then follows from the local derivative estimates of Shi that

which implies (6.4.16)

∂R e 0 )R(z, 0)2 , (z, t) ≤ C(κ ∂t

for all − R−1 (z, 0) ≤ t ≤ 0

R(z, −cR−1(z, 0)) ≥ cR(z, 0)


393

for some small positive constant c depending only on κ0 . On the other hand, by using the Harnack estimate in Corollary 2.5.7, we have (6.4.17)

1 = R(y, 0) ≥ e cR(z, −cR−1 (z, 0))

1

for some small positive constant e c depending only on κ0 , since d0 (y, z) ≤ R(z, 0)− 2 and the metric gij (t) is equivalent on 1

B0 (z, 2R(z, 0)− 2 ) × [−cR−1 (z, 0), 0]

with c > 0 small enough. Thus we get from (6.4.16) and (6.4.17) that e R(z, 0) ≤ A

(6.4.18)

e depending only on κ0 . for some positive constant A 1 1 1 e − 12 , the combiSince B0 (z, 2R(z, 0)− 2 ) ⊃ B0 (y, R(z, 0)− 2 ) and R(z, 0)− 2 ≥ (A) nation of (6.4.15) and (6.4.18) gives R(x, 0) ≤ (C(κ0 ) +

(6.4.19)

B(κ0 ) e )A τ0 (κ0 )

e − 2 ). Then by the κ0 -noncollapsing there exists a positive whenever x ∈ B0 (y, (A) constant r0 depending only on κ0 such that 1

Vol 0 (B0 (y, r0 )) ≥ κ0 r03 .

For any fixed R0 ≥ r0 , we then have Vol 0 (B0 (y, R0 )) ≥ κ0 r03 = κ0 (

r0 3 ) · R03 . R0

By applying Theorem 6.3.3 (ii) again and noting that the constant κ0 is universal, there exists a positive constant ω(R0 ) depending only on R0 such that R(x, 0) ≤ ω(R02 )

1 for all x ∈ B0 (y, R0 ). 4

This gives the desired estimate. (ii) This follows immediately from conclusion (i), the Li-Yau-Hamilton inequality and the local derivative estimate of Shi. As a consequence, we have the following compactness result due to Perelman [103]. Corollary 6.4.4 (Compactness of ancient κ0 -solutions). The set of nonflat three-dimensional ancient κ0 -solutions is compact modulo scaling in the sense that for any sequence of such solutions and marking points (xk , 0) with R(xk , 0) = 1, we ∞ can extract a Cloc converging subsequence whose limit is also an ancient κ0 -solution. Proof. Consider any sequence of three-dimensional ancient κ0 -solutions and marking points (xk , 0) with R(xk , 0) = 1. By Theorem 6.4.3(i), the Li-Yau-Hamilton inequality and Hamilton’s compactness theorem (Theorem 4.1.5), we can extract a ∞ ¯ , g¯ij (x, t)), with −∞ < t ≤ 0, Cloc converging subsequence such that the limit (M is an ancient solution to the Ricci flow with nonnegative curvature operator and κ0 noncollapsed on all scales. Since any ancient κ0 -solution satisfies the Li-Yau-Hamilton

394


¯ t) of the limit (M ¯ , g¯ij (x, t)) is inequality, it implies that the scalar curvature R(x, pointwise nondecreasing in time. Thus it remains to show that the limit solution has bounded curvature at t = 0. ¯ is noncompact. By pulling Obviously we may assume the limiting manifold M back the limiting solution to its orientable cover, we can assume that the limiting man¯ is orientable. We now argue by contradiction. Suppose the scalar curvature ifold M ¯ of the limit at t = 0 is unbounded. R ¯,j = By applying Lemma 6.1.4, we can choose a sequence of points xj ∈ M ¯ of the limit satisfies 1, 2, . . . , divergent to infinity such that the scalar curvature R ¯ j , 0) ≥ j and R(x, ¯ 0) ≤ 4R(x ¯ j , 0) R(x p ¯ j , 0)). Then from the fact that the limiting for all j = 1, 2, . . . , and x ∈ B0 (xj , j/ R(x ¯ scalar curvature R(x, t) is pointwise nondecreasing in time, we have ¯ t) ≤ 4R(x ¯ j , 0) R(x, p ¯ j , 0)) and t ≤ 0. By combining with Hamilton’s for all j = 1, 2, . . ., x ∈ B0 (xj , j/ R(x compactness theorem (Theorem 4.1.5) and the κ0 -noncollapsing, we know that a subsequence of the rescaled solutions

(6.4.20)

¯ , R(x ¯ j , 0)¯ ¯ j , 0)), xj ), j = 1, 2, . . . , (M gij (x, t/R(x ∞ converges in the Cloc topology to a nonflat smooth solution of the Ricci flow. Then Proposition 6.1.2 implies that the new limit at the new time {t = 0} must split off a line. By pulling back the new limit to its universal cover and applying Hamilton’s strong maximum principle, we deduce that the pull-back of the new limit on the universal cover splits off a line for all time t ≤ 0. Thus by combining with Theorem 6.2.2 and the argument in the proof of Lemma 6.4.1, we further deduce that the new ¯ is orientable, limit is either the round cylinder S2 × R or the round RP2 × R. Since M 2 ¯ the new limit must be S ×R. Moreover, since (M , g¯ij (x, 0)) has nonnegative curvature ¯ j , 0) → +∞, this gives a operator and the points {xj } are going to infinity and R(x ¯ , g¯ij (x, t)) has contradiction to Proposition 6.1.1. So we have proved that the limit (M uniformly bounded curvature.

Arbitrarily fix ε > 0. Let gij (x, t) be a nonflat ancient κ-solution on a threemanifold M for some κ > 0. We say that a point x0 ∈ M is the center of an evolving ε-neck at t = 0, if the solution gij (x, t) in the set {(x, t) | − ε−2 Q−1 < t ≤ 0, d2t (x, x0 ) < ε−2 Q−1 }, where Q = R(x0 , 0), is, after scaling with factor Q, ε-close (in −1 the C [ε ] topology) to the corresponding set of the evolving round cylinder having scalar curvature one at t = 0. As another consequence of the elliptic type estimate, we have the following global structure result obtained by Perelman in [103] for noncompact ancient κ-solutions. Corollary 6.4.5. For any ε > 0 there exists C = C(ε) > 0, such that if gij (t) is a nonflat ancient κ-solution on a noncompact three-manifold M for some κ > 0, and Mε denotes the set of points in M which are not centers of evolving ε-necks at t = 0, then at t = 0, either the whole manifold M is the round cylinder S2 × R or its Z2 metric quotients, or Mε satisfies the following (i) Mε is compact, 1 (ii) diam Mε ≤ CQ− 2 and C −1 Q ≤ R(x, 0) ≤ CQ, whenever x ∈ Mε , where Q = R(x0 , 0) for some x0 ∈ ∂Mε .


395

Proof. We first consider the easy case that the curvature operator of the ancient κ-solution has a nontrivial null vector somewhere at some time. Let us pull back the solution to its universal cover. By applying Hamilton’s strong maximum principle and Theorem 6.2.2, we see that the universal cover is the evolving round cylinder S2 × R. Thus in this case, by the argument in the proof of Lemma 6.4.1, we conclude that the ancient κ-solution is either isometric to the round cylinder S2 × R or one of its Z2 ˜ where Z2 flips both S2 , metric quotients (i.e., RP2 × R, or the twisted product S2 ×R or R). We then assume that the curvature operator of the nonflat ancient κ-solution is positive everywhere. Firstly we want to show Mε is compact. We argue by contradiction. Suppose there exists a sequence of points zk , k = 1, 2, . . ., going to infinity (with respect to the metric gij (0)) such that each zk is not the center of any evolving ε-neck. For an arbitrarily fixed point z0 ∈ M , it follows from Theorem 6.4.3(i) that 0 < R(z0 , 0) ≤ R(zk , 0) · ω(R(zk , 0)d20 (zk , z0 ))

which implies that lim R(zk , 0)d20 (zk , z0 ) = +∞.

k→∞

Since the sectional curvature of the ancient κ-solution is positive everywhere, the underlying manifold is diffeomorphic to R3 , and in particular, orientable. Then as before, by Proposition 6.1.2, Theorem 6.2.2 and Corollary 6.4.4, we conclude that zk is the center of an evolving ε-neck for k sufficiently large. This is a contradiction, so we have proved that Mε is compact. Again, we notice that M is diffeomorphic to R3 since the curvature operator is positive. According to the resolution of the Schoenflies conjecture in three-dimensions, every approximately round two-sphere cross-section through the center of an evolving ε-neck divides M into two parts such that one of them is diffeomorphic to the three-ball B3 . Let ϕ be the Busemann function on M , it is a standard fact that ϕ is convex and proper. Since Mε is compact, Mε is contained in a compact set K = ϕ−1 ((−∞, A]) for some large A. We note that each point x ∈ M \Mε is the center of an ε-neck. It is clear that there is an ε-neck N lying entirely outside K. Consider a point x on one of the boundary components of the ε-neck N . Since x ∈ M \ Mε , there is an ε-neck adjacent to the initial ε-neck, producing a longer neck. We then take a point on the boundary of the second ε-neck and continue. This procedure can either terminate when we get into Mε or go on infinitely to produce a semi-infinite (topological) cylinder. The same procedure can be repeated for the other boundary component of the initial ˜ . If N ˜ never touches ε-neck. This procedure will give a maximal extended neck N Mε , the manifold will be diffeomorphic to the standard infinite cylinder, which is ˜ touch Mε , then there is a geodesic connecting a contradiction. If both ends of N two points of Mε and passing through N . This is impossible since the function ϕ ˜ will touch Mε and the other end will is convex. So we conclude that one end of N tend to infinity to produce a semi-infinite (topological) cylinder. Thus we can find an approximately round two-sphere cross-section which encloses the whole set Mε and 1 touches some point x0 ∈ ∂Mε . We next want to show that R(x0 , 0) 2 · diam(Mε ) is bounded from above by some positive constant C = C(ε) depending only on ε. Suppose not; then there exists a sequence of nonflat noncompact threedimensional ancient κ-solutions with positive curvature operator such that for the above chosen points x0 ∈ ∂Mε there would hold (6.4.21)

1

R(x0 , 0) 2 · diam (Mε ) → +∞.

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By Proposition 6.4.2, we know that the ancient solutions are κ0 -noncollapsed on all scales for some universal positive constant κ0 . Let us dilate the ancient solutions around the points x0 with the factors R(x0 , 0). By Corollary 6.4.4, we can extract a convergent subsequence. From the choice of the points x0 and (6.4.21), the limit has at least two ends. Then by Toponogov’s splitting theorem the limit is isometric to X × R for some nonflat two-dimensional ancient κ0 -solution X. Since M is orientable, we conclude from Theorem 6.2.2 that limit must be the evolving round cylinder S2 ×R. This contradicts the fact that each chosen point x0 is not the center of any evolving ε-neck. Therefore we have proved 1

diam (Mε ) ≤ CQ− 2

for some positive constant C = C(ε) depending only on ε, where Q = R(x0 , 0). Finally by combining this diameter estimate with Theorem 6.4.3(i), we immediately deduce e −1 Q ≤ R(x, 0) ≤ CQ, e C whenever x ∈ Mε ,

e depending only on ε. for some positive constant C

We now can describe the canonical structures for three-dimensional nonflat (compact or noncompact) ancient κ-solutions. The following theorem was given by Perelman in the section 1.5 of [104]. Recently in [34], this canonical neighborhood result has been extended to four-dimensional ancient κ-solutions with isotropic curvature pinching. Theorem 6.4.6 (Canonical neighborhood theorem). For any ε > 0 one can find positive constants C1 = C1 (ε) and C2 = C2 (ε) with the following property. Suppose we have a three-dimensional nonflat (compact or noncompact) ancient κsolution (M, gij (x, t)). Then either the ancient solution is the round RP2 × R, or every point (x, t) has an open neighborhood B, with Bt (x, r) ⊂ B ⊂ Bt (x, 2r) for 1 some 0 < r < C1 R(x, t)− 2 , which falls into one of the following three categories: (a) B is an evolving ε-neck (in the sense that it is the slice at the time t of the parabolic region {(x′ , t′ ) | x′ ∈ B, t′ ∈ [t − ε−2 R(x, t)−1 , t]} which is, after scaling with factor R(x, t) and shifting the time t to zero, ε-close (in −1 the C [ε ] topology) to the subset (S2 × I) × [−ε−2 , 0] of the evolving standard round cylinder with scalar curvature 1 and length 2ε−1 to I at the time zero), or (b) B is an evolving ε-cap (in the sense that it is the time slice at the time t of an evolving metric on B3 or RP3 \ B¯3 such that the region outside some suitable compact subset of B3 or RP3 \ B¯3 is an evolving ε-neck), or (c) B is a compact manifold (without boundary) with positive sectional curvature (thus it is diffeomorphic to the round three-sphere S3 or a metric quotient of S3 ); furthermore, the scalar curvature of the ancient κ-solution on B at time t is between C2−1 R(x, t) and C2 R(x, t), and the volume of B in case (a) and case (b) satisfies 3

(C2 R(x, t))− 2 ≤ Vol t (B) ≤ εr3 . Proof. As before, we first consider the easy case that the curvature operator has a nontrivial null vector somewhere at some time. By pulling back the solution to


397

its universal cover and applying Hamilton’s strong maximum principle and Theorem 6.2.2, we deduce that the universal cover is the evolving round cylinder S2 × R. Then exactly as before, by the argument in the proof of Lemma 6.4.1, we conclude that the ancient κ-solution is isometric to the round S2 × R, RP2 × R, or the twisted product ˜ where Z2 flips both S2 and R. Clearly each point of the round cylinder S2 × R S2 ×R ˜ has a neighborhood falling into the category (a) or (b) or the twisted product S2 ×R (over RP3 \ B¯3 ). We now assume that the curvature operator of the nonflat ancient κ-solution is positive everywhere. Then the manifold is orientable by the Cheeger-Gromoll theorem [23] for the noncompact case or the Synge theorem [22] for the compact case. 1 Without loss of generality, we may assume ε is suitably small, say 0 < ε < 100 . If the nonflat ancient κ-solution is noncompact, the conclusions follow immediately from the combination of Corollary 6.4.5 and Theorem 6.4.3(i). Thus we may assume the nonflat ancient κ-solution is compact. By Proposition 6.4.2, either the compact ancient κ-solution is isometric to a metric quotient of the round S3 , or it is κ0 noncollapsed on all scales for the universal positive constant κ0 . Clearly each point of a metric quotient of the round S3 has a neighborhood falling into category (c). Thus we may further assume the ancient κ-solution is also κ0 -noncollapsing. 1 ), there exist a We argue by contradiction. Suppose that for some ε ∈ (0, 100 sequence of compact orientable ancient κ0 -solutions (Mk , gk ) with positive curvature operator, a sequence of points (xk , 0) with xk ∈ Mk and sequences of positive constants 2 C1k → ∞ and C2k = ω(4C1k ), with the function ω given in Theorem 6.4.3, such 1 that for every radius r, 0 < r < C1k R(xk , 0)− 2 , any open neighborhood B, with B0 (xk , r) ⊂ B ⊂ B0 (xk , 2r), does not fall into one of the three categories (a), (b) and (c), where in the case (a) and case (b), we require the neighborhood B to satisfy the volume estimate 3

(C2k R(xk , 0))− 2 ≤ Vol 0 (B) ≤ εr3 . By Theorem 6.4.3(i) and the choice of the constants C2k we see that the diameter 1 of each Mk at t = 0 is at least C1k R(xk , 0)− 2 ; otherwise we can choose suitable 1 r ∈ (0, C1k R(xk , 0)− 2 ) and B = Mk , which falls into the category (c) with the scalar −1 curvature between C2k R(x, 0) and C2k R(x, 0) on B. Now by scaling the ancient κ0 -solutions along the points (xk , 0) with factors R(xk , 0), it follows from Corollary ∞ 6.4.4 that a sequence of the ancient κ0 -solutions converge in the Cloc topology to a noncompact orientable ancient κ0 -solution. If the curvature operator of the noncompact limit has a nontrivial null vector somewhere at some time, it follows exactly as before by using the argument in the proof of Lemma 6.4.1 that the orientable limit is isometric to the round S2 × R, or ˜ where Z2 flips both S2 and R. Then for k large enough, the twisted product S2 ×R a suitable neighborhood B (for suitable r) of the point (xk , 0) would fall into the category (a) or (b) (over RP3 \ B¯3 ) with the desired volume estimate. This is a contradiction. If the noncompact limit has positive sectional curvature everywhere, then by using Corollary 6.4.5 and Theorem 6.4.3(i) for the noncompact limit we see that for k large enough, a suitable neighborhood B (for suitable r) of the point (xk , 0) would fall into category (a) or (b) (over B3 ) with the desired volume estimate. This is also a contradiction. Finally, the statement on the curvature estimate in the neighborhood B follows directly from Theorem 6.4.3(i).

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7. Ricci Flow on Three-manifolds. We will use the Ricci flow to study the topology of compact orientable three-manifolds. Let M be a compact threedimensional orientable manifold. Arbitrarily given a Riemannian metric on the manifold, we evolve it by the Ricci flow. The basic idea is to understand the topology of the underlying manifold by studying long-time behavior of the solution of the Ricci flow. We have seen in Chapter 5 that for a compact three-manifold with positive Ricci curvature as initial data, the solution to the Ricci flow tends, up to scalings, to a metric of positive constant curvature. Consequently, a compact three-manifold with positive Ricci curvature is diffeomorphic to the round three-sphere or a metric quotient of it. However, for general initial metrics, the Ricci flow may develop singularities in some parts while it keeps smooth in other parts. Naturally one would like to cut off the singularities and continue to run the Ricci flow. If the Ricci flow still develops singularities after a while, one can do the surgeries and run the Ricci flow again. By repeating this procedure, one will get a kind of “weak” solution to the Ricci flow. Furthermore, if the “weak” solution has only a finite number of surgeries at any finite time interval and one can remember what had been cut during the surgeries, and if the “weak” solution has a well-understood long-time behavior, then one will also get the topology structure of the initial manifold. This theory of surgically modified Ricci flow was first developed by Hamilton [64] for compact four-manifolds and further developed more recently by Perelman [104] for compact orientable three-manifolds. The main purpose of this chapter is to give a complete and detailed discussion of the Ricci flow with surgery on three-manifolds. 7.1. Canonical Neighborhood Structures. Let us call a Riemannian metric on a compact orientable three-dimensional manifold normalized if the eigenvalues 1 1 ≥ λ ≥ µ ≥ ν ≥ − 10 , and of its curvature operator at every point are bounded by 10 every geodesic ball of radius one has volume at least one. By the evolution equation of the curvature and the maximum principle, it is easy to see that any solution to the Ricci flow with (compact and three-dimensional) normalized initial metric exists on a maximal time interval [0, tmax ) with tmax > 1. Consider a smooth solution gij (x, t) to the Ricci flow on M × [0, T ), where M is a compact orientable three-manifold and T < +∞. After rescaling, we may always assume the initial metric gij (·, 0) is normalized. By Theorem 5.3.2, the solution gij (·, t) then satisfies the pinching estimate R ≥ (−ν)[log(−ν) + log(1 + t) − 3]

(7.1.1)

whenever ν < 0 on M × [0, T ). Recall the function y = f (x) = x(log x − 3), for e2 ≤ x < +∞, is increasing and convex with range −e2 ≤ y < +∞, and its inverse function is also increasing and satisfies lim f −1 (y)/y = 0.

y→+∞

We can rewrite the pinching estimate (7.1.1) as (7.1.2)

Rm(x, t) ≥ −[f −1 (R(x, t)(1 + t))/(R(x, t)(1 + t))]R(x, t)

on M × [0, T ).


399

Suppose that the solution gij (·, t) becomes singular as t → T . Let us take a sequence of times tk → T , and a sequence of points pk ∈ M such that for some positive constant C, |Rm|(x, t) ≤ CQk with Qk = |Rm(pk , tk )| for all x ∈ M and t ∈ [0, tk ]. Thus, (pk , tk ) is a sequence of (almost) maximum points. By applying Hamilton’s compactness theorem and Perelman’s no local collapsing theorem I as well as the pinching estimate (7.1.2), a sequence of the scalings of the solution gij (x, t) around the points pk with factors Qk converges to a nonflat complete three-dimensional orientable ancient κ-solution (for some κ > 0). For an arbitrarily given ε > 0, the canonical neighborhood theorem (Theorem 6.4.6) in the previous chapter implies that each point in the ancient κ-solution has a neighborhood which is either an evolving ε-neck, or an evolving ε-cap, or a compact (without boundary) positively curved manifold. This gives the structure of singularities coming from a sequence of (almost) maximum points. However the above argument does not work for singularities coming from a sequence of points (yk , τk ) with τk → T and |Rm(yk , τk )| → +∞ when |Rm(yk , τk )| is not comparable with the maximum of the curvature at the time τk , since we cannot take a limit directly. In [103], Perelman developed a refined rescaling argument to obtain the following singularity structure theorem. We remark that our statement of the singularity structure theorem below is slightly different from Perelman’s original statement (cf. Theorem 12.1 of [103]). While Perelman assumed the condition of κ-noncollapsing on scales less than r0 , we assume that the initial metric is normalized so that from the rescaling argument one can get the κ-noncollapsing on all scales for the limit solutions. Theorem 7.1.1 (Singularity structure theorem). Given ε > 0 and T0 > 1, one can find r0 > 0 with the following property. If gij (x, t), x ∈ M and t ∈ [0, T ) with 1 < T ≤ T0 , is a solution to the Ricci flow on a compact orientable three-manifold M with normalized initial metric, then for any point (x0 , t0 ) with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , the solution in {(x, t) | d2t0 (x, x0 ) < ε−2 Q−1 , t0 − ε−2 Q−1 ≤ t ≤ t0 } −1 is, after scaling by the factor Q, ε-close (in the C [ε ] -topology) to the corresponding subset of some orientable ancient κ-solution (for some κ > 0). Proof. Since the initial metric is normalized, it follows from the no local collapsing theorem I or I’ (and their proofs) that there is a positive constant κ, depending only on T0√ , such that the solution in Theorem 7.1.1 is κ-noncollapsed on all scales less than T0 . Let C(ε) be a positive constant larger than or equal to ε−2 . It suffices to prove that there exists r0 > 0 such that for any point (x0 , t0 ) with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , the solution in the parabolic region {(x, t) ∈ M × [0, T ) | d2t0 (x, x0 ) < C(ε)Q−1 , t0 − C(ε)Q−1 ≤ t ≤ t0 } is, after scaling by the factor Q, ε-close to the corresponding subset of some orientable ancient κ-solution. The constant C(ε) will be determined later. We argue by contradiction. Suppose for some ε > 0, there exist a sequence of solutions (Mk , gk (·, t)) to the Ricci flow on compact orientable three-manifolds with normalized initial metrics, defined on the time intervals [0,Tk ) with 1 < Tk ≤ T0 , a sequence of positive numbers rk → 0, and a sequence of points xk ∈ Mk and times tk ≥ 1 with Qk = Rk (xk , tk ) ≥ rk−2 such that each solution (Mk , gk (·, t)) in the −1 parabolic region {(x, t) ∈ Mk × [0, Tk ) | d2tk (x, xk ) < C(ε)Q−1 k , tk − C(ε)Qk ≤ t ≤ tk } is not, after scaling by the factor Qk , ε-close to the corresponding subset of any orientable ancient κ-solution, where Rk denotes the scalar curvature of (Mk , gk ). For each solution (Mk , gk (·, t)), we may adjust the point (xk , tk ) with tk ≥ 21

400


and with Qk = Rk (xk , tk ) to be as large as possible so that the conclusion of the theorem fails at (xk , tk ), but holds for any (x, t) ∈ Mk × [tk − Hk Q−1 k , tk ] satisfying Rk (x, t) ≥ 2Qk , where Hk = 41 rk−2 → +∞ as k → +∞. Indeed, suppose not, by setting (xk1 , tk1 ) = (xk , tk ), we can choose a sequence of points (xkl , tkl ) ∈ Mk × [tk(l−1) − Hk Rk (xk(l−1) , tk(l−1) )−1 , tk(l−1) ] such that Rk (xkl , tkl ) ≥ 2Rk (xk(l−1) , tk(l−1) ) and the conclusion of the theorem fails at (xkl , tkl ) for each l = 2, 3, . . . . Since the solution is smooth, but Rk (xkl , tkl ) ≥ 2Rk (xk(l−1) , tk(l−1) ) ≥ · · · ≥ 2l−1 Rk (xk , tk ), and tkl ≥ tk(l−1) − Hk Rk (xk(l−1) , tk(l−1) )−1 ≥ t k − Hk ≥

1 , 2

l−1 X 1 Rk (xk , tk )−1 i−1 2 i=1

this process must terminate after a finite number of steps and the last element fits. Let (Mk , g˜k (·, t), xk ) be the rescaled solutions obtained by rescaling (Mk , gk (·, t)) around xk with the factors Qk = Rk (xk , tk ) and shifting the time tk ˜ k the rescaled scalar curvature. We will show that to the new time zero. Denote by R a subsequence of the orientable rescaled solutions (Mk , g˜k (·, t), xk ) converges in the ∞ Cloc topology to an orientable ancient κ-solution, which is a contradiction. In the following we divide the argument into four steps. Step 1. First of all, we need a local bound on curvatures. ¯ ¯ Lemma 7.1.2. For each (¯ x, t¯) with tk − 21 Hk Q−1 k ≤ t ≤ tk , we have Rk (x, t) ≤ 4Qk −1 −1 ¯ ≤ t ≤ t¯ and d2¯ (x, x ¯ , where Q ¯ k = Qk + Rk (¯ whenever t¯− cQ ¯ ) ≤ cQ x, t¯) and c > 0 k k t is a small universal constant. ¯ −1 , t¯] with c > 0 to be ¯ −1 ) 12 ) × [t¯− cQ Proof. Consider any point (x, t) ∈ Bt¯(¯ x, (cQ k k determined. If Rk (x, t) ≤ 2Qk , there is nothing to show. If Rk (x, t) > 2Qk , consider a space-time curve γ from (x, t) to (¯ x, t¯) that goes straight from (x, t) to (x, t¯) and goes from (x, t¯) to (¯ x, t¯) along a minimizing geodesic (with respect to the metric gk (·, t¯)). If there is a point on γ with the scalar curvature 2Qk , let y0 be the nearest such point to (x, t). If not, put y0 = (¯ x, t¯). On the segment of γ from (x, t) to y0 , the scalar curvature is at least 2Qk . According to the choice of the point (xk , tk ), the solution along the segment is ε-close to that of some ancient κ-solution. It follows from Theorem 6.4.3 (ii) that ∂ − 12 −1 |∇(Rk )| ≤ 2η and (Rk ) ≤ 2η ∂t

on the segment. (Here, without loss of generality, we may assume ε is suitably small). Then by choosing c > 0 (depending only on η) small enough we get the desired curvature bound by integrating the above derivative estimates along the segment. This proves the lemma. Step 2. Next we want to show that for each A < +∞, there exist a positive constant C(A) (independent of k) such that the curvatures of the rescaled solutions


401

g˜k (·, t) at the new time t = 0 (corresponding to the original times tk ) satisfy the estimate g k |(y, 0) ≤ C(A) |Rm

whenever dg˜k (·,0) (y, xk ) ≤ A and k ≥ 1. For each ρ ≥ 0, set

˜ k (x, 0) | k ≥ 1, x ∈ Mk with d0 (x, xk ) ≤ ρ} M (ρ) = sup{R and ρ0 = sup{ρ ≥ 0 | M (ρ) < +∞}. By the pinching estimate (7.1.1), it suffices to show ρ0 = +∞. Note that ρ0 > 0 by applying Lemma 7.1.2 with (¯ x, t¯) = (xk , tk ). We now argue by contradiction to show ρ0 = +∞. Suppose not, we may find (after passing to a subsequence if necessary) a sequence of points yk ∈ Mk with d0 (xk , yk ) → ρ0 < +∞ ˜ k (yk , 0) → +∞. Let γk (⊂ Mk ) be a minimizing geodesic segment from xk to and R ˜ k (zk , 0) = 2, and let βk be the yk . Let zk ∈ γk be the point on γk closest to yk with R subsegment of γk running from zk to yk . By Lemma 7.1.2 the length of βk is bounded away from zero independent of k. By the pinching estimate (7.1.1), for each ρ < ρ0 , we have a uniform bound on the curvatures on the open balls B0 (xk , ρ) ⊂ (Mk , g˜k ). The injectivity radii of the rescaled solutions g˜k at the points xk and the time t = 0 are also uniformly bounded from below by the κ-noncollapsing property. Therefore by Lemma 7.1.2 and Hamilton’s compactness theorem (Theorem 4.1.5), after passing to a subsequence, we can assume that the marked sequence (B0 (xk , ρ0 ), g˜k (·, 0), xk ) ∞ converges in the Cloc topology to a marked (noncomplete) manifold (B∞ , g˜∞ , x∞ ), the segments γk converge to a geodesic segment (missing an endpoint) γ∞ ⊂ B∞ ¯∞ denote emanating from x∞ , and βk converges to a subsegment β∞ of γ∞ . Let B ¯∞ the limit point of γ∞ . the completion of (B∞ , g˜∞ ), and y∞ ∈ B ˜ ∞ the scalar curvature of (B∞ , g˜∞ ). Since the rescaled scalar curvaDenote by R ˜ k along βk are at least 2, it follows from the choice of the points (xk , 0) that for tures R ˜∞ (q0 ))−1 } any q0 ∈ β∞ , the manifold (B∞ , g˜∞ ) in {q ∈ B∞ | dist2g˜∞ (q, q0 ) < C(ε)(R is 2ε-close to the corresponding subset of (a time slice of) some orientable ancient κ-solution. Then by Theorem 6.4.6, we know that the orientable ancient κ-solution 1 at each point (x, t) has a radius r, 0 < r < C1 (2ε)R(x, t)− 2 , such that its canonical neighborhood B, with Bt (x, r) ⊂ B ⊂ Bt (x, 2r), is either an evolving 2ε-neck, or an evolving 2ε-cap, or a compact manifold (without boundary) diffeomorphic to a metric quotient of the round three-sphere S3 , and moreover the scalar curvature is between (C2 (2ε))−1 R(x, t) and C2 (2ε)R(x, t), where C1 (2ε) and C2 (2ε) are the positive constants in Theorem 6.4.6. We now choose C(ε) = max{2C12 (2ε), ε−2 }. By the local curvature estimate ˜ ∞ becomes unbounded when in Lemma 7.1.2, we see that the scalar curvature R approaching y∞ along γ∞ . This implies that the canonical neighborhood around q0 cannot be a compact manifold (without boundary) diffeomorphic to a metric quotient of the round three-sphere S3 . Note that γ∞ is shortest since it is the limit of a sequence of shortest geodesics. Without loss of generality, we may assume ε is suitably small 1 (say, ε ≤ 100 ). These imply that as q0 gets sufficiently close to y∞ , the canonical neighborhood around q0 cannot be an evolving 2ε-cap. Thus we conclude that each q0 ∈ γ∞ sufficiently close to y∞ is the center of an evolving 2ε-neck.

402


Let U=

[

˜ ∞ (q0 ))− 21 ) (⊂ (B∞ , g˜∞ )), B(q0 , 4π(R

q0 ∈γ∞

˜ ∞ (q0 ))− 21 ) is the ball centered at q0 ∈ B∞ with the radius where B(q0 , 4π(R ˜ ∞ (q0 ))− 12 . Clearly U has nonnegative sectional curvature by the pinching es4π(R timate (7.1.1). Since the metric g˜∞ is cylindrical at any point q0 ∈ γ∞ which is sufficiently close to y∞ , we see that the metric space U = U ∪ {y∞ } by adding in the point y∞ , is locally complete and intrinsic near y∞ . Furthermore y∞ cannot be an interior point of any geodesic segment in U . This implies the curvature of U at y∞ is nonnegative in the Alexandrov sense. It is a basic result in Alexandrov space theory (see for example Theorem 10.9.3 and Corollary 10.9.5 of [9]) that there exists a three-dimensional tangent cone Cy∞ U at y∞ which is a metric cone. It is clear that its aperture is ≤ 10ε, thus the tangent cone is nonflat. Pick a point p ∈ Cy∞ U such that the distance from the vertex y∞ to p is one and it is nonflat around p. Then the ball B(p, 21 ) ⊂ Cy∞ U is the Gromov-Hausdorff limit of the scalings of a sequence of balls B0 (pk , sk ) ⊂ (Mk , g˜k (·, 0)) by some factors ak , where sk → 0+ . Since the tangent cone is three-dimensional and nonflat around ˜ k (pk , 0). By using the local curvature p, the factors ak must be comparable with R ∞ estimate in Lemma 7.1.2, we actually have the convergence in the Cloc topology for the solutions g˜k (·, t) on the balls B0 (pk , sk ) and over some time interval t ∈ [−δ, 0] for some sufficiently small δ > 0. The limiting ball B(p, 12 ) ⊂ Cy∞ U is a piece of the nonnegative curved and nonflat metric cone whose radial directions are all Ricci flat. On the other hand, by applying Hamilton’s strong maximum principle to the evolution equation of the Ricci curvature tensor as in the proof of Lemma 6.3.1, the limiting ball B(p, 21 ) would split off all radial directions isometrically (and locally). Since the limit is nonflat around p, this is impossible. Therefore we have proved that the curvatures of the rescaled solutions g˜k (·, t) at the new times t = 0 (corresponding to the original times tk ) stay uniformly bounded at bounded distances from xk for all k. We have proved that for each A < +∞, the curvature of the marked manifold (Mk , g˜k (·, 0), xk ) at each point y ∈ Mk with distance from xk at most A is bounded by C(A). Lemma 7.1.2 extends this curvature control to a backward parabolic neighborhood centered at y whose radius depends only on the distance from y to xk . Thus by Shi’s local derivative estimates (Theorem 1.4.2) we can control all derivatives of the curvature in such backward parabolic neighborhoods. Then by using the κ-noncollapsing and Hamilton’s compactness theorem (Theorem 4.1.5), we can take ∞ a Cloc subsequent limit to obtain (M∞ , g˜∞ (·, t), x∞ ), which is κ-noncollapsed on all scales and is defined on a space-time open subset of M∞ × (−∞, 0] containing the time slice M∞ × {0}. Clearly it follows from the pinching estimate (7.1.1) that the limit (M∞ , g˜∞ (·, 0), x∞ ) has nonnegative curvature operator (and hence nonnegative sectional curvature). Step 3. We further claim that the limit (M∞ , g˜∞ (·, 0), x∞ ) at the time slice {t = 0} has bounded curvature. We know that the sectional curvature of the limit (M∞ , g˜∞ (·, 0), x∞ ) is nonnegative everywhere. Argue by contradiction. Suppose the curvature of (M∞ , g˜∞ (·, 0), x∞ ) is not bounded, then by Lemma 6.1.4, there exists a sequence of points qj ∈ M∞ di˜ ∞ (qj , 0) → +∞ as j → +∞ verging to infinity such that their scalar curvatures R


403

and ˜ ∞ (x, 0) ≤ 4R ˜ ∞ (qj , 0) R q ˜ ∞ (qj , 0)) ⊂ (M∞ , g˜∞ (·, 0)). By combining with Lemma 7.1.2 for x ∈ B(qj , j/ R and the κ-noncollapsing, a subsequence of the rescaled and marked manifolds ∞ ˜ ∞ (qj , 0)˜ (M∞ , R g∞ (·, 0), qj ) converges in the Cloc topology to a smooth nonflat limit Y . By Proposition 6.1.2, the new limit Y is isometric to a metric product N × R for some two-dimensional manifold N . On the other hand, in view of the choice of the points (xk , tk ), the original limit (M∞ , g˜∞ (·, 0), x∞ ) at the point qj has a canonical neighborhood which is either a 2ε-neck, a 2ε-cap, or a compact manifold (without boundary) diffeomorphic to a metric quotient of the round S3 . It follows that for j ˜ ∞ (qj , 0))− 12 . Without loss of large enough, qj is the center of a 2ε-neck of radius (R generality, we may further assume that 2ε < ε0 , where ε0 is the positive constant ˜∞ (qj , 0))− 21 → 0 as j → +∞, this contradicts given in Proposition 6.1.1. Since (R Proposition 6.1.1. So the curvature of (M∞ , g˜∞ (·, 0)) is bounded. Step 4. Finally we want to extend the limit backwards in time to −∞. By Lemma 7.1.2 again, we now know that the limiting solution (M∞ , g˜∞ (·, t)) is defined on a backward time interval [−a, 0] for some a > 0. Denote by t′ = inf{t˜| we can take a smooth limit on (t˜, 0] (with bounded curvature at each time slice) from a subsequence of the convergent rescaled solutions g˜k }. We first claim that there is a subsequence of the rescaled solutions g˜k which converges ∞ in the Cloc topology to a smooth limit (M∞ , g˜∞ (·, t)) on the maximal time interval ′ (t , 0]. Indeed, let t′k be a sequence of negative numbers such that t′k → t′ and there exist k smooth limits (M∞ , g˜∞ (·, t)) defined on (t′k , 0]. For each k, the limit has nonnegative sectional curvature and has bounded curvature at each time slice. Moreover by Lemma 7.1.2, the limit has bounded curvature on each subinterval [−b, 0] ⊂ (t′k , 0]. Denote ˜ the scalar curvature upper bound of the limit at time zero (where Q ˜ is the same by Q for all k). Then we can apply the Li-Yau-Hamilton estimate (Corollary 2.5.7) to get −t′k k ˜ ˜ , R∞ (x, t) ≤ Q t − t′k k k ˜∞ where R (x, t) are the scalar curvatures of the limits (M∞ , g˜∞ (·, t)). Hence by the definition of convergence and the above curvature estimates, we can find a subsequence ∞ of the above convergent rescaled solutions g˜k which converges in the Cloc topology to ′ a smooth limit (M∞ , g˜∞ (·, t)) on the maximal time interval (t , 0]. We next claim that t′ = −∞. Suppose not, then by Lemma 7.1.2, the curvature of the limit (M∞ , g˜∞ (·, t)) becomes unbounded as t → t′ > −∞. By applying the maximum principle to the evolution equation of the scalar curvature, we see that the infimum of the scalar ˜ ∞ (x∞ , 0) = 1. Thus there exists curvature is nondecreasing in time. Note that R some point y∞ ∈ M∞ such that ˜ ∞ y ∞ , t′ + c < 3 R 10 2

404


where c > 0 is the universal constant in Lemma 7.1.2. By using Lemma 7.1.2 again c ) we see that the limit (M∞ , g˜∞ (·, t)) in a small neighborhood of the point (y∞ , t′ + 10 c c ′ ′ extends backwards to the time interval [t − 10 , t + 10 ]. We remark that the distances at time t and time 0 are roughly equivalent in the following sense (7.1.3)

dt (x, y) ≥ d0 (x, y) ≥ dt (x, y) − const.

for any x, y ∈ M∞ and t ∈ (t′ , 0]. Indeed from the Li-Yau-Hamilton inequality (Corollary 2.5.7) we have the estimate −t′ ˜ ˜ R∞ (x, t) ≤ Q , on M∞ × (t′ , 0]. t − t′ By applying Lemma 3.4.1 (ii), we have q ˜ dt (x, y) ≤ d0 (x, y) + 30(−t′ ) Q

for any x, y ∈ M∞ and t ∈ (t′ , 0]. On the other hand, since the curvature of the limit metric g˜∞ (·, t) is nonnegative, we have dt (x, y) ≥ d0 (x, y) for any x, y ∈ M∞ and t ∈ (t′ , 0]. Thus we obtain the estimate (7.1.3). Let us still denote by (Mk , g˜k (·, t)) the subsequence which converges on the maximal time interval (t′, 0]. Consider the rescaled sequence (Mk , g˜k (·, t)) with the marked points xk replaced by the associated sequence of points yk → y∞ and the (original −1 c c ′ unshifted) times tk replaced by any sk ∈ [tk + (t′ − 20 )Q−1 k , tk + (t + 20 )Qk ]. It follows from Lemma 7.1.2 that for k large enough, the rescaled solutions (Mk , g˜k (·, t)) at yk satisfy ˜ k (yk , t) ≤ 10 R c c , t′ + 10 ]. By applying the same arguments as in the above Step 2, for all t ∈ [t′ − 10 we conclude that for any A > 0, there is a positive constant C(A) < +∞ such that

˜ k (x, t) ≤ C(A) R c c for all (x, t) with dt (x, yk ) ≤ A and t ∈ [t′ − 20 , t′ + 20 ]. The estimate (7.1.3) implies c that there is a positive constant A0 such that for arbitrarily given small ǫ′ ∈ (0, 100 ), for k large enough, there hold

dt (xk , yk ) ≤ A0 for all t ∈ [t′ + ǫ′ , 0]. By combining with Lemma 7.1.2, we then conclude that for any ˜ A > 0, there is a positive constant C(A) such that for k large enough, the rescaled solutions (Mk , g˜k (·, t)) satisfy ˜ k (x, t) ≤ C(A) ˜ R ˜0 (xk , A) and t ∈ [t′ − c (C(A))−1 , 0]. for all x ∈ B 100 Now, by taking convergent subsequences from the (original) rescaled solutions (Mk , g˜k (·, t), xk ), we see that the limiting solution (M∞ , g˜∞ (·, t)) is defined on a spacetime open subset of M∞ ×(−∞, 0] containing M∞ ×[t′ , 0]. By repeating the argument


405

of Step 3 and using Lemma 7.1.2, we further conclude the limit (M∞ , g˜∞ (·, t)) has uniformly bounded curvature on M∞ × [t′ , 0]. This is a contradiction. Therefore we have proved a subsequence of the rescaled solutions (Mk , g˜k (·, t), xk ) converges to an orientable ancient κ-solution, which gives the desired contradiction. This completes the proof of the theorem. We remark that this singularity structure theorem has been extended by Chen and the second author in [34] to the Ricci flow on compact four-manifolds with positive isotropic curvature. 7.2. Curvature Estimates for Smooth Solutions. Let us consider solutions to the Ricci flow on compact orientable three-manifolds with normalized initial metrics. The above singularity structure theorem (Theorem 7.1.1) tells us that the solutions around high curvature points are sufficiently close to ancient κ-solutions. It is thus reasonable to expect that the elliptic type estimate (Theorem 6.4.3) and the curvature estimate via volume growth (Theorem 6.3.3) for ancient κ-solutions are heritable to general solutions of the Ricci flow on three-manifolds. The main purpose of this section is to establish such curvature estimates. In the fifth section of this chapter, we will further extend these estimates to surgically modified solutions. The first result of this section is an extension of the elliptic type estimate (Theorem 6.4.3). This result is reminiscent of the second step in the proof of Theorem 7.1.1. Theorem 7.2.1 (Perelman [103]). For any A < +∞, there exist K = K(A) < +∞ and α = α(A) > 0 with the following property. Suppose we have a solution to the Ricci flow on a three-dimensional, compact and orientable manifold M with normalized initial metric. Suppose that for some x0 ∈ M and some r0 > 0 with r0 < α, the solution is defined for 0 ≤ t ≤ r02 and satisfies |Rm|(x, t) ≤ r0−2 ,

for 0 ≤ t ≤ r02 , d0 (x, x0 ) ≤ r0 ,

and Vol 0 (B0 (x0 , r0 )) ≥ A−1 r03 . Then R(x, r02 ) ≤ Kr0−2 whenever dr02 (x, x0 ) < Ar0 . Proof. Given any large A > 0 and letting α > 0 be chosen later, by Perelman’s no local collapsing theorem II (Theorem 3.4.2), there exists a positive constant κ = κ(A) (independent of α) such that any complete solution satisfying the assumptions of the theorem is κ-noncollapsed on scales ≤ r0 over the region {(x, t) | 15 r02 ≤ t ≤ r02 , dt (x, x0 ) ≤ 5Ar0 }. Set 1 1 ε0 , , ε = min 4 100 where ε0 is the positive constant in Proposition 6.1.1. We first prove the following assertion. Claim. For the above fixed ε > 0, one can find K = K(A, ε) < +∞ such that if we have a three-dimensional complete orientable solution with normalized initial metric and satisfying |Rm|(x, t) ≤ r0−2

for 0 ≤ t ≤ r02 , d0 (x, x0 ) ≤ r0 ,

406


and Vol 0 (B0 (x0 , r0 )) ≥ A−1 r03 for some x0 ∈ M and some r0 > 0, then for any point x ∈ M with dr02 (x, x0 ) < 3Ar0 , either R(x, r02 ) < Kr0−2 or the subset {(y, t) | d2r2 (y, x) ≤ ε−2 R(x, r02 )−1 , r02 − ε−2 R(x, r02 )−1 ≤ t ≤ r02 } around 0 the point (x, r02 ) is ε-close to the corresponding subset of an orientable ancient κsolution. Notice that in this assertion we don’t impose the restriction of r0 < α, so we can consider for the moment r0 > 0 to be arbitrary in proving the above claim. Note that the assumption on the normalization of the initial metric is just to ensure the pinching estimate. By scaling, we may assume r0 = 1. The proof of the claim is essentially adapted from that of Theorem 7.1.1. But we will meet the difficulties of adjusting points and verifying a local curvature estimate. Suppose that the claim is not true. Then there exist a sequence of solutions (Mk , gk (·, t)) to the Ricci flow satisfying the assumptions of the claim with the origins x0k , and a sequence of positive numbers Kk → ∞, times tk = 1 and points xk ∈ Mk with dtk (xk , x0k ) < 3A such that Qk = Rk (xk , tk ) ≥ Kk and the solution in −1 2 {(x, t) | tk − C(ε)Q−1 k ≤ t ≤ tk , dtk (x, xk ) ≤ C(ε)Qk } is not, after scaling by the factor Qk , ε-close to the corresponding subset of any orientable ancient κ-solution, where Rk denotes the scalar curvature of (Mk , gk (·, t)) and C(ε)(≥ ε−2 ) is the constant defined in the proof of Theorem 7.1.1. As before we need to first adjust the point (xk , tk ) with tk ≥ 12 and dtk (xk , x0k ) < 4A so that Qk = Rk (xk , tk ) ≥ Kk and the conclusion of the claim fails at (xk , tk ), but holds for any (x, t) satisfying Rk (x, t) ≥ 1

−1

2Qk , tk − Hk Q−1 ≤ t ≤ tk and dt (x, x0k ) < dtk (xk , x0k ) + Hk2 Qk 2 , where Hk = k 1 4 Kk → ∞, as k → +∞. Indeed, by starting with (xk1 , tk1 ) = (xk , 1) we can choose (xk2 , tk2 ) ∈ Mk × (0, 1] with tk1 − Hk Rk (xk1 , tk1 )−1 ≤ tk2 ≤ tk1 , and dtk2 (xk2 , x0k ) < dtk1 (xk1 , x0k ) + 1

1

Hk2 Rk (xk1 , tk1 )− 2 such that Rk (xk2 , tk2 ) ≥ 2Rk (xk1 , tk1 ) and the conclusion of the claim fails at (xk2 , tk2 ); otherwise we have the desired point. Repeating this process, we can choose points (xki , tki ), i = 2, . . . , j, such that Rk (xki , tki ) ≥ 2Rk (xki−1 , tki−1 ), tki−1 − Hk Rk (xki−1 , tki−1 )−1 ≤ tki ≤ tki−1 , 1

1

dtki (xki , x0k ) < dtki−1 (xki−1 , x0k ) + Hk2 Rk (xki−1 , tki−1 )− 2 , and the conclusion of the claim fails at the points (xki , tki ), i = 2, . . . , j. These inequalities imply Rk (xkj , tkj ) ≥ 2j−1 Rk (xk1 , tk1 ) ≥ 2j−1 Kk , 1 ≥ tkj ≥ tk1 − Hk

j−2 X 1 1 Rk (xk1 , tk1 )−1 ≥ , 2i 2 i=0


407

and 1

dtkj (xkj , x0k ) < dtk1 (xk1 , x0k ) + Hk2

j−2 X i=0

1 1 √ Rk (xk1 , tk1 )− 2 < 4A. ( 2)i

Since the solutions are smooth, this process must terminate after a finite number of steps to give the desired point, still denoted by (xk , tk ). For each adjusted (xk , tk ), let [t′ , tk ] be the maximal subinterval of [tk − 1 −2 −1 ε Qk , tk ] so that the conclusion of the claim with K = 2Qk holds on 2 1 1 −1 P xk , tk , Hk2 Qk 2 , t′ − tk 10 1 1 −1 = (x, t) | x ∈ Bt xk , Hk2 Qk 2 , t ∈ [t′ , tk ] 10 for all sufficiently large k. We now want to show t′ = tk − 12 ε−2 Q−1 k . Consider the scalar curvature Rk at the point xk over the time interval [t′ , tk ]. If there is a time t˜ ∈ [t′ , tk ] satisfying Rk (xk , t˜) ≥ 2Qk , we let t˜ be the first of such time from tk . Then the solution (Mk , gk (·, t)) around the point xk over the time ˜ interval [t˜ − 12 ε−2 Q−1 k , t] is ε-close to some orientable ancient κ-solution. Note from the Li-Yau-Hamilton inequality that the scalar curvature of any ancient κ-solution is pointwise nondecreasing in time. Consequently, we have the following curvature estimate Rk (xk , t) ≤ 2(1 + ε)Qk ′ ˜ for t ∈ [t˜ − 21 ε−2 Q−1 k , tk ] (or t ∈ [t , tk ] if there is no such time t). By combining with the elliptic type estimate for ancient κ-solutions (Theorem 6.4.3) and the HamiltonIvey pinching estimate, we further have

|Rm(x, t)| ≤ 5ω(1)Qk

(7.2.1)

1 ′ for all x ∈ Bt (xk , (3Qk )− 2 ) and t ∈ [t˜− 21 ε−2 Q−1 k , tk ] (or t ∈ [t , tk ]) and all sufficiently large k, where ω is the positive function in Theorem 6.4.3. ′ For any point (x, t) with t˜ − 12 ε−2 Q−1 k ≤ t ≤ tk (or t ∈ [t , tk ]) and dt (x, xk ) ≤ 1 − 21 1 2 10 Hk Qk ,

we divide the discussion into two cases.

Case (1): dt (xk , x0k ) ≤

1 − 12 3 2 10 Hk Qk .

dt (x, x0k ) ≤ dt (x, xk ) + dt (xk , x0k ) 3 1 −1 1 12 − 12 Hk Qk + Hk2 Qk 2 ≤ 10 10 1 12 − 12 ≤ Hk Q k . 2

(7.2.2)

1

−1

3 Hk2 Qk 2 . Case (2): dt (xk , x0k ) > 10 From the curvature bound (7.2.1) and the assumption, we apply Lemma 3.4.1(ii) − 21

with r0 = Qk

to get 1 d (dt (xk , x0k )) ≥ −20(ω(1) + 1)Qk2 , dt

408


and then for k large enough, − 12

dt (xk , x0k ) ≤ dtˆ(xk , x0k ) + 20(ω(1) + 1)ε−2 Qk 1 1 −1 ≤ dtˆ(xk , x0k ) + Hk2 Qk 2 , 10 where tˆ ∈ (t, tk ] satisfies the property that ds (xk , x0k ) ≥ So we have (7.2.3)

1 − 12 3 2 10 Hk Qk

whenever s ∈ [t, tˆ].

dt (x, x0k ) ≤ dt (x, xk ) + dt (xk , x0k ) 1 12 − 12 1 1 −1 ≤ Hk Qk + dtˆ(xk , x0k ) + Hk2 Qk 2 10 10 1 12 − 12 ≤ dtk (xk , x0k ) + Hk Qk , 2

for all sufficiently large k. Then the combination of (7.2.2), (7.2.3) and the choice of the points (xk , tk ) implies t′ = tk − 21 ε−2 Q−1 k for all sufficiently large k. (Here we also used the maximality of the subinterval [t′ , tk ] in the case that there is no time in [t′ , tk ] with Rk (xk , ·) ≥ 2Qk .) Now we rescale the solutions (Mk , gk (·, t)) into (Mk , g˜k (·, t)) around the points xk by the factors Qk = Rk (xk , tk ) and shift the times tk to the new times zero. Then the same arguments from Step 1 to Step 3 in the proof of Theorem 7.1.1 prove that ∞ a subsequence of the rescaled solutions (Mk , g˜k (·, t)) converges in the Cloc topology to a limiting (complete) solution (M∞ , g˜∞ (·, t)), which is defined on a backward time interval [−a, 0] for some a > 0. (The only modification is in Lemma 7.1.2 of Step 1 ¯ by further requiring tk − 41 ε−2 Q−1 k ≤ t ≤ tk ). We next study how to adapt the argument of Step 4 in the proof of Theorem 7.1.1. As before, we have a maximal time interval (t∞ , 0] for which we can take a smooth limit (M∞ , g˜∞ (·, t), x∞ ) from a subsequence of the rescaled solutions (Mk , g˜k (·, t), xk ). We want to show t∞ = −∞. Suppose not; then t∞ > −∞. Let c > 0 be a positive constant much smaller than 1 −2 ε . Note that the infimum of the scalar curvature is nondecreasing in time. Then 10 we can find some point y∞ ∈ M∞ and some time t = t∞ + θ with 0 < θ < 3c such ˜ ∞ (y∞ , t∞ + θ) ≤ 3 . that R 2 Consider the (unrescaled) scalar curvature Rk of (Mk , gk (·, t)) at the point xk ˜ over the time interval [tk + (t∞ + 2θ )Q−1 k , tk ]. Since the scalar curvature R∞ of the θ limit on M∞ × [t∞ + 3 , 0] is uniformly bounded by some positive constant C, we have the curvature estimate Rk (xk , t) ≤ 2CQk for all t ∈ [tk + (t∞ + θ2 )Q−1 k , tk ] and all sufficiently large k. Then by repeating the same arguments as in deriving (7.2.1), (7.2.2) and (7.2.3), we deduce that the conclusion of the claim with K = 2Qk holds on the parabolic neighborhood 1 −1 1 Hk2 Qk 2 , (t∞ + θ2 )Q−1 P (xk , tk , 10 k ) for all sufficiently large k. Let (yk , tk + (t∞ + θk )Q−1 ) be a sequence of associated points and times in the k (unrescaled) solutions (Mk , gk (·, t)) so that after rescaling, the sequence converges to the (y∞ , t∞ + θ) in the limit. Clearly θ2 ≤ θk ≤ 2θ for all sufficiently large k. Then, by considering the scalar curvature Rk at the point yk over the time interval −1 [tk + (t∞ − 3c )Q−1 k , tk + (t∞ + θk )Qk ], the above argument (as in deriving the similar


409

estimates (7.2.1)-(7.2.3)) implies that the conclusion of the claim with K = 2Qk holds 1

−1

1 on the parabolic neighborhood P (yk , tk , 10 Hk2 Qk 2 , (t∞ − 3c )Q−1 k ) for all sufficiently large k. In particular, we have the curvature estimate

Rk (yk , t) ≤ 4(1 + ε)Qk −1 for t ∈ [tk + (t∞ − 3c )Q−1 k , tk + (t∞ + θk )Qk ] for all sufficiently large k. We now consider the rescaled sequence (Mk , g˜k (·, t)) with the marked points re−1 c placed by yk and the times replaced by sk ∈ [tk + (t∞ − 4c )Q−1 k , tk + (t∞ + 4 )Qk ]. By applying the same arguments from Step 1 to Step 3 in the proof of Theorem 7.1.1 and the Li-Yau-Hamilton inequality as in Step 4 of Theorem 7.1.1, we conclude that there is some small constant a′ > 0 such that the original limit (M∞ , g˜∞ (·, t)) is actually well defined on M∞ × [t∞ − a′ , 0] with uniformly bounded curvature. This is a contradiction. Therefore we have checked the claim. To finish the proof, we next argue by contradiction. Suppose there exist sequences of positive numbers Kk → +∞, αk → 0, as k → +∞, and a sequence of solutions (Mk , gk (·, t)) to the Ricci flow satisfying the assumptions of the theorem with origins x0k and with radii r0k satisfying r0k < αk such that for some points xk ∈ Mk with dr02 (xk , x0k ) < Ar0k we have k

R(xk , r02k ) > Kk r0−2 k

(7.2.4)

for all k. Let (Mk , gˆk (·, t), x0k ) be the rescaled solutions of (Mk , gk (·, t)) around the origins x0k by the factors r0−2 and shifting the times r02k to the new times zero. k The above claim tells us that for k large, any point (y, 0) ∈ (Mk , gˆk (·, 0), x0k ) with ˆ k (y, 0) > Kk has a dgˆk (·,0) (y, x0k ) < 3A and with the rescaled scalar curvature R canonical neighborhood which is either a 2ε-neck, or a 2ε-cap, or a compact manifold (without boundary) diffeomorphic to a metric quotient of the round three-sphere. Note that the pinching estimate (7.1.1) and the condition αk → 0 imply any subsequential limit of the rescaled solutions (Mk , gˆk (·, t), x0k ) must have nonnegative sectional curvature. Thus the same argument as in Step 2 of the proof of Theorem 7.1.1 shows that for all sufficiently large k, the curvatures of the rescaled solutions at the time zero stay uniformly bounded at those points whose distances from the origins x0k do not exceed 2A. This contradicts (7.2.4) for k large enough. Therefore we have completed the proof of the theorem. The next result is a generalization of the curvature estimate via volume growth in Theorem 6.3.3 (ii) where the condition on the curvature lower bound over a time interval is replaced by that at a time slice only. Theorem 7.2.2 (Perelman [103]). For any w > 0 there exist τ = τ (w) > 0, K = K(w) < +∞, α = α(w) > 0 with the following property. Suppose we have a three-dimensional, compact and orientable solution to the Ricci flow defined on M × [0, T ) with normalized initial metric. Suppose that for some radius r0 > 0 with r0 < α and a point (x0 , t0 ) ∈ M × [0, T ) with T > t0 ≥ 4τ r02 , the solution on the ball Bt0 (x0 , r0 ) satisfies Rm(x, t0 ) ≥ −r0−2 and

on Bt0 (x0 , r0 ),

Vol t0 (Bt0 (x0 , r0 )) ≥ wr03 .

410


Then R(x, t) ≤ Kr0−2 whenever t ∈ [t0 − τ r02 , t0 ] and dt (x, x0 ) ≤ 41 r0 . Proof. If we knew that Rm(x, t) ≥ −r0−2 for all t ∈ [0, t0 ] and dt (x, x0 ) ≤ r0 , then we could just apply Theorem 6.3.3 (ii) and take τ (w) = τ0 (w)/2, K(w) = C(w) + 2B(w)/τ0 (w). Now fix these values of τ and K. We argue by contradiction. Consider a three-dimensional, compact and orientable solution gij (t) to the Ricci flow with normalized initial metric, a point (x0 , t0 ) and some radius r0 > 0 with r0 < α, for α > 0 a sufficiently small constant to be determined later, such that the assumptions of the theorem do hold whereas the conclusion does not. We first claim that we may assume that any other point (x′ , t′ ) and radius r′ > 0 with the same property has either t′ > t0 or t′ < t0 − 2τ r02 , or 2r′ > r0 . Indeed, suppose otherwise. Then there exist (x′0 , t′0 ) and r0′ with t′0 ∈ [t0 −2τ r02 , t0 ] and r0′ ≤ 21 r0 , for which the assumptions of the theorem hold but the conclusion does not. Thus, there is a point (x, t) such that i h τ t ∈ [t′0 − τ (r0′ )2 , t′0 ] ⊂ t0 − 2τ r02 − r02 , t0 4 and R(x, t) > K(r0′ )−2 ≥ 4Kr0−2 . If the point (x′0 , t′0 ) and the radius r0′ satisfy the claim then we stop, and otherwise we iterate the procedure. Since t0 ≥ 4τ r02 and the solution is smooth, the iteration must terminate in a finite number of steps, which provides the desired point and the desired radius. Let τ ′ ≥ 0 be the largest number such that (7.2.5)

Rm(x, t) ≥ −r0−2

whenever t ∈ [t0 − τ ′ r02 , t0 ] and dt (x, x0 ) ≤ r0 . If τ ′ ≥ 2τ , we are done by Theorem 6.3.3 (ii). Thus we may assume τ ′ < 2τ . By applying Theorem 6.3.3(ii), we know that at time t′ = t0 − τ ′ r02 , the ball Bt′ (x0 , r0 ) has (7.2.6)

Vol t′ (Bt′ (x0 , r0 )) ≥ ξ(w)r03

for some positive constant ξ(w) depending only on w. We next claim that there exists a ball (at time t′ = t0 − τ ′ r02 ) Bt′ (x′ , r′ ) ⊂ Bt′ (x0 , r0 ) with (7.2.7)

Vol t′ (Bt′ (x′ , r′ )) ≥

1 α3 (r′ )3 2

and with (7.2.8)

r0 > r′ ≥ c(w)r0 2

for some small positive constant c(w) depending only on w, where α3 is the volume of the unit ball B3 in the Euclidean space R3 . Indeed, suppose that it is not true. Then after rescaling, there is a sequence of Riemannian manifolds Mi , i = 1, 2, . . . , with balls B(xi , 1) ⊂ Mi so that (7.2.5)′

Rm ≥ −1 on B(xi , 1)


411

and (7.2.6)′

Vol (B(xi , 1)) ≥ ξ(w)

for all i, but all balls B(x′i , ri′ ) ⊂ B(xi , 1) with (7.2.9)

Vol (B(x′i , ri′ ))
ri′ ≥

1 i

satisfy

1 α3 (ri′ )3 . 2

It follows from basic results in Alexandrov space theory (see for example Theorem 10.7.2 and Theorem 10.10.10 of [9]) that, after taking a subsequence, the marked balls (B(xi , 1), xi ) converge in the Gromov-Hausdorff topology to a marked length space (B∞ , x∞ ) with curvature bounded from below by −1 in the Alexandrov space sense, and the associated Riemannian volume forms dVol Mi over (B(xi , 1), xi ) converge weakly to the Hausdorff measure µ of B∞ . It is well-known that the Hausdorff dimension of any Alexandrov space is either an integer or infinity (see for example Theorem 10.8.2 of [9]). Then by (7.2.6)′ , we know the limit (B∞ , x∞ ) is a threedimensional Alexandrov space of curvature ≥ −1. In the Alexandrov space theory, a point p ∈ B∞ is said to be regular if the tangent cone of B∞ at p is isometric to R3 . It is also a basic result in Alexandrov space theory (see for example Corollary 10.9.13 of [9]) that the set of regular points in B∞ is dense and for each regular point there is a small neighborhood which is almost isometric to an open set of the Euclidean ′ ′ space R3 . Thus for any ε > 0, there are balls B(x′∞ , r∞ ) ⊂ B∞ with 0 < r∞ < 13 and satisfying ′ ′ 3 µ(B(x′∞ , r∞ )) ≥ (1 − ε)α3 (r∞ ) .

This is a contradiction with (7.2.9). Without loss of generality, we may assume w ≤ 41 α3 . Since τ ′ < 2τ , it follows from the choice of the point (x0 , t0 ) and the radius r0 and (7.2.5), (7.2.7), (7.2.8) that the conclusion of the theorem holds for (x′ , t′ ) and r′ . Thus we have the estimate R(x, t) ≤ K(r′ )−2 whenever t ∈ [t′ − τ (r′ )2 , t′ ] and dt (x, x′ ) ≤ 14 r′ . For α > 0 small, by combining with the pinching estimate (7.1.1), we have |Rm(x, t)| ≤ K ′ (r′ )−2 whenever t ∈ [t′ − τ (r′ )2 , t′ ] and dt (x, x′ ) ≤ 14 r′ , where K ′ is some positive constant depending only on K. Note that this curvature estimate implies the evolving metrics are equivalent over a suitable subregion of {(x, t) | t ∈ [t′ − τ (r′ )2 , t′ ] and dt (x, x′ ) ≤ 1 ′ 4 r }. Now we can apply Theorem 7.2.1 to choose α = α(w) > 0 so small that (7.2.10)

′ −2 −2 −2 ˜ ˜ R(x, t) ≤ K(w)(r ) ≤ K(w)c(w) r0

whenever t ∈ [t′ − τ2 (r′ )2 , t′ ] and dt (x, x′ ) ≤ 10r0 . Then the combination of (7.2.10) with the pinching estimate (7.1.2) would imply Rm(x, t) ≥ −[f −1 (R(x, t)(1 + t))/(R(x, t)(1 + t))]R(x, t) 1 ≥ − r0−2 2

412


on the region {(x, t) | t ∈ [t′ − τ2 (r′ )2 , t′ ] and dt (x, x0 ) ≤ r0 } when α = α(w) > r0 small enough. This contradicts the choice of τ ′ . Therefore we have proved the theorem. The combination of the above two theorems immediately gives the following consequence. Corollary 7.2.3. For any w > 0 and A < +∞, there exist τ = τ (w, A) > 0, K = K(w, A) < +∞, and α = α(w, A) > 0 with the following property. Suppose we have a three-dimensional, compact and orientable solution to the Ricci flow defined on M × [0, T ) with normalized initial metric. Suppose that for some radius r0 > 0 with r0 < α and a point (x0 , t0 ) ∈ M × [0, T ) with T > t0 ≥ 4τ r02 , the solution on the ball Bt0 (x0 , r0 ) satisfies Rm(x, t0 ) ≥ −r0−2 and

on Bt0 (x0 , r0 ),

Vol t0 (Bt0 (x0 , r0 )) ≥ wr03 .

Then R(x, t) ≤ Kr0−2 whenever t ∈ [t0 − τ r02 , t0 ] and dt (x, x0 ) ≤ Ar0 . We can also state the previous corollary in the following version.

Corollary 7.2.4 (Perelman [103]). For any w > 0 one can find ρ = ρ(w) > 0 such that if gij (t) is a complete solution to the Ricci flow defined on M × [0, T ) with T > 1 and with normalized initial metric, where M is a three-dimensional, compact and orientable manifold, and if Bt0 (x0 , r0 ) is a metric ball at time t0 ≥ 1, with r0 < ρ, such that min{Rm(x, t0 ) | x ∈ Bt0 (x0 , r0 )} = −r0−2 , then Vol t0 (Bt0 (x0 , r0 )) ≤ wr03 . Proof. We argue by contradiction. Suppose for any ρ > 0, there is a solution and a ball Bt0 (x0 , r0 ) satisfying the assumption of the corollary with r0 < ρ, t0 ≥ 1, and with min{Rm(x, t0 ) | x ∈ Bt0 (x0 , r0 )} = −r0−2 , but Vol t0 (Bt0 (x0 , r0 )) > wr03 . We can apply Corollary 7.2.3 to get R(x, t) ≤ Kr0−2 whenever t ∈ [t0 −τ r02 , t0 ] and dt (x, x0 ) ≤ 2r0 , provided ρ > 0 is so small that 4τ ρ2 ≤ 1 and ρ < α, where τ, α and K are the positive constants obtained in Corollary 7.2.3. Then for r0 < ρ and ρ > 0 sufficiently small, it follows from the pinching estimate (7.1.2) that Rm(x, t) ≥ −[f −1 (R(x, t)(1 + t))/(R(x, t)(1 + t))]R(x, t) 1 ≥ − r0−2 2


413

in the region {(x, t) | t ∈ [t0 − τ (r0 )2 , t0 ] and dt (x, x0 ) ≤ 2r0 }. In particular, this would imply min{Rm(x, t0 )|x ∈ Bt0 (x0 , r0 )} > −r0−2 . This contradicts the assumption. 7.3. Ricci Flow with Surgery. One of the central themes of the Ricci flow theory is to give a classification of all compact orientable three-manifolds. As we mentioned before, the basic idea is to obtain long-time behavior of solutions to the Ricci flow. However the solutions will in general become singular in finite time. Fortunately, we now understand the precise structures of the solutions around the singularities, thanks to Theorem 7.1.1. When a solution develops singularities, one can perform geometric surgeries by cutting off the canonical neighborhoods around the singularities and gluing back some known pieces, and then continue running the Ricci flow. By repeating this procedure, one hopes to get a kind of “weak” solution. In this section we will give a detailed description of this surgery procedure and define a global “weak” solution to the Ricci flow. Given any ε > 0, based on the singularity structure theorem (Theorem 7.1.1), we can get a clear picture of the solution near the singular time as follows. Let (M, gij (·, t)) be a maximal solution to the Ricci flow on the maximal time interval [0, T ) with T < +∞, where M is a connected compact orientable threemanifold and the initial metric is normalized. For the given ε > 0 and the solution (M, gij (·, t)), we can find r0 > 0 such that each point (x, t) with R(x, t) ≥ r0−2 satisfies the derivative estimates ∂ 3 2 (7.3.1) |∇R(x, t)| < ηR (x, t) and R(x, t) < ηR2 (x, t), ∂t

where η > 0 is a universal constant, and has a canonical neighborhood which is either an evolving ε-neck, or an evolving ε-cap, or a compact positively curved manifold (without boundary). In the last case the solution becomes extinct at the maximal time T and the manifold M is diffeomorphic to the round three-sphere S3 or a metric quotient of S3 by Theorem 5.2.1. Let Ω denote the set of all points in M where the curvature stays bounded as t → T . The gradient estimates in (7.3.1) imply that Ω is open and that R(x, t) → ∞ as t → T for each x ∈ M \ Ω. If Ω is empty, then the solution becomes extinct at time T . In this case, either the manifold M is compact and positively curved, or it is entirely covered by evolving ε-necks and evolving ε-caps shortly before the maximal time T . So the manifold M is diffeomorphic to either S3 , or a metric quotient of the round S3 , or S2 × S1 , or RP3 #RP3 . The reason is as follows. Clearly, we only need to consider the situation that the manifold M is entirely covered by evolving ε-necks and evolving ε-caps shortly before the maximal time T . If M contains a cap C, then there is a cap or a neck adjacent to the neck-like end of C. The former case implies that M is diffeomorphic to S3 , RP3 , or RP3 #RP3 . In the latter case, we get a new longer cap and continue. Finally, we must end up with a cap, producing a S3 , RP3 , or RP3 #RP3 . If M contains no caps, we start with a neck N . By connecting with the necks that are adjacent to the boundary of N , we get a longer neck and continue. After a finite number of steps, the resulting neck must repeat itself. Since M is orientable, we conclude that M is diffeomorphic to S2 × S1 .

414


We can now assume that Ω is nonempty. By using the local derivative estimates of Shi (Theorem 1.4.2), we see that as t → T the solution gij (·, t) has a smooth limit ¯ g¯ij (·) on Ω. Let R(x) denote the scalar curvature of g¯ij . For any ρ < r0 , let us consider the set ¯ Ωρ = {x ∈ Ω | R(x) ≤ ρ−2 }. By the evolution equation of the Ricci flow, we see that the initial metric gij (·, 0) and the limit metric gij (·) are equivalent over any fixed region where the curvature remains uniformly bounded. Note that for any fixed x ∈ ∂Ω, and any sequence of points xj ∈ Ω with xj → x with respect to the initial metric gij (·, 0), we have R(xj ) → +∞. In fact, if there were a subsequence xjk so that limk→∞ R(xjk ) exists and is finite, then it would follow from the gradient estimates (7.3.1) that R is uniformly bounded in some small neighborhood of x ∈ ∂Ω (with respect to the induced topology of the initial metric gij (·, 0)); this is a contradiction. From this observation and the compactness of the initial manifold, we see that Ωρ is compact (with respect to the metric gij (·)). For further discussions, let us introduce the following terminologies. Denote by I an interval. Recall that an ε-neck (of radius r) is an open set with a Riemannian metric, which is, after scaling the metric with factor r−2 , ε-close to the standard neck S2 × I with the product metric, where S2 has constant scalar curvature one and I has length −1 2ε−1 and the ε-closeness refers to the C [ε ] topology. 2 A metric on S × I, such that each point is contained in some ε-neck, is called an ε-tube, or an ε-horn, or a double ε-horn, if the scalar curvature stays bounded on both ends, or stays bounded on one end and tends to infinity on the other, or tends to infinity on both ends, respectively. ¯ 3 is called a ε-cap if the region outside some suitable A metric on B3 or RP3 \ B ¯ 3 is called an capped ε-horn compact subset is an ε-neck. A metric on B3 or RP3 \ B if each point outside some compact subset is contained in an ε-neck and the scalar curvature tends to infinity on the end. Now take any ε-neck in (Ω, g¯ij ) and consider a point x on one of its boundary components. If x ∈ Ω \ Ωρ , then there is either an ε-cap or an ε-neck, adjacent to the initial ε-neck. In the latter case we can take a point on the boundary of the second ε-neck and continue. This procedure can either terminate when we get into Ωρ or an ε-cap, or go on indefinitely, producing an ε-horn. The same procedure can be repeated for the other boundary component of the initial ε-neck. Therefore, taking into account that Ω has no compact components, we conclude that each ε-neck of (Ω, g¯ij ) is contained in a subset of Ω of one of the following types: (a) an ε-tube with boundary components in Ωρ , or

(7.3.2)

(b)

an ε-cap with boundary in Ωρ , or

(c)

an ε-horn with boundary in Ωρ , or

(d)

a capped ε-horn, or

(e)

a double ε-horn.

Similarly, each ε-cap of (Ω, g¯ij ) is contained in a subset of Ω of either type (b) or type (d). It is clear that there is a definite lower bound (depending on ρ) for the volume of subsets of types (a), (b) and (c), so there can be only a finite number of them. Thus


415

we conclude that there is only a finite number of components of Ω containing points of Ωρ , and every such component has a finite number of ends, each being an ε-horn. On the other hand, every component of Ω, containing no points of Ωρ , is either a capped ε-horn, or a double ε-horn. If we look at the solution at a slightly earlier time, the above argument shows each ε-neck or ε-cap of (M, gij (·, t)) is contained in a subset of types (a) or (b), while the ε-horns, capped ε-horns and double ε-horns (at the maximal time T) are connected together to form ε-tubes and ε-caps at the times t shortly before T .

Ωρ R @

Ωρ R @

@ I ε-horn

6 double ε-horn

@ I ε-tube

6 capped ε-horn

Hence, by looking at the solution at times shortly before T , we see that the topology of M can be reconstructed as follows: take the components Ωj , 1 ≤ j ≤ k, of Ω which contain points of Ωρ , truncate their ε-horns, and glue to the boundary ¯ 3. components of truncated Ωj a finite collection of tubes S2 × I and caps B3 or RP3 \ B ¯ j , 1 ≤ j ≤ k, with a finite number Thus, M is diffeomorphic to a connected sum of Ω of copies of S2 × S1 (which correspond to gluing a tube to two boundary components ¯ j denotes Ωj with each of the same Ωj ), and a finite number of copies of RP3 . Here Ω ¯ j in the following ε-horn one point compactified. More geometrically, one can get Ω way: in every ε-horn of Ωj one can find an ε-neck, cut it along the middle two-sphere, remove the horn-shaped end, and glue back a cap (or more precisely, a differentiable three-ball). Thus to understand the topology of M , one only needs to understand the ¯ j , 1 ≤ j ≤ k. topologies of the compact orientable three-manifolds Ω ¯ j by the Ricci flow again and, when singularities Naturally one can evolve each Ω develop again, perform the above surgery for each ε-horn to get new compact orientable three-manifolds. By repeating this procedure indefinitely, it will likely give a long-time “weak” solution to the Ricci flow. The following abstract definition for this kind of “weak” solution was introduced by Perelman in [104] . Definition 7.3.1. Suppose we are given a (finite or countably infinite) collection + k of three-dimensional smooth solutions gij (t) to the Ricci flow defined on Mk × [t− k , tk )

416


and go singular as t → t+ k , where each manifold Mk is compact and orientable, possibly k disconnected with only a finite number of connected components. Let (Ωk , g¯ij ) be the + k limits of the corresponding solutions gij (t) as t → tk , as above. Suppose also that for + k−1 k − each k we have t− ¯ij ) and (Mk , gij (tk )) contain compact k = tk−1 , and that (Ωk−1 , g (possibly disconnected) three-dimensional submanifolds with smooth boundary which are isometric. Then by identifying these isometric submanifolds, we say the collection k of solutions gij (t) is a solution to the Ricci flow with surgery (or a surgically modified solution to the Ricci flow) on the time interval which is the union of all + + [t− k , tk ), and say the times tk are surgery times. To get the topology of the initial manifold from the solution to the Ricci flow with surgery, one has to overcome the following two difficulties: (i) how to prevent the surgery times from accumulating? (ii) how to obtain the long time behavior of the solution to the Ricci flow with surgery? Thus it is natural to consider those solutions having “good” properties. For any arbitrarily fixed positive number ε, we will only consider those solutions to the Ricci flow with surgery which satisfy the following a priori assumptions (with accuracy ε). Pinching assumption. The eigenvalues λ ≥ µ ≥ ν of the curvature operator of the solution to the Ricci flow with surgery at each point and each time satisfy (7.3.3)

R ≥ (−ν)[log(−ν) + log(1 + t) − 3]

whenever ν < 0. Canonical neighborhood assumption (with accuracy ε). For any given ε > 0, there exist positive constants C1 and C2 depending only on ε, and a nonincreasing positive function r : [0, +∞) → (0, +∞) such that at each time t > 0, every point x where scalar curvature R(x, t) is at least r−2 (t) has a neighborhood B, with 1 Bt (x, σ) ⊂ B ⊂ Bt (x, 2σ) for some 0 < σ < C1 R− 2 (x, t), which falls into one of the following three categories: (a) B is a strong ε-neck (in the sense B is the slice at time t of the parabolic neighborhood {(x′ , t′ ) | x′ ∈ B, t′ ∈ [t − R(x, t)−1 , t]}, where the solution is well defined on the whole parabolic neighborhood and is, after scaling with −1 factor R(x, t) and shifting the time to zero, ε-close (in the C [ε ] topology) 2 to the subset (S × I) × [−1, 0] of the evolving standard round cylinder with scalar curvature 1 to S2 and length 2ε−1 to I at time zero), or (b) B is an ε-cap, or (c) B is a compact manifold (without boundary) of positive sectional curvature. Furthermore, the scalar curvature in B at time t is between C2−1 R(x, t) and C2 R(x, t), satisfies the gradient estimates ∂R 3 < ηR2 , (7.3.4) |∇R| < ηR 2 and ∂t and the volume of B in case (a) and case (b) satisfies 3

(C2 R(x, t))− 2 ≤ Vol t (B) ≤ εσ 3 . Here η is a universal positive constant.


417

Without loss of generality, we always assume the above constants C1 and C2 are twice bigger than the corresponding constants C1 ( 2ε ) and C2 ( 2ε ) in Theorem 6.4.6 with the accuracy 2ε . We remark that the above definition of the canonical neighborhood assumption is slightly different from that of Perelman in [104] in two aspects: (1) it allows the parameter r to depend on time; (2) it also includes an volume upper bound for the canonical neighborhoods of types (a) and (b). Arbitrarily given a compact orientable three-manifold with a Riemannian metric, by scaling, we may assume the Riemannian metric is normalized. In the rest of this section and the next section, we will show the Ricci flow with surgery, with the normalized metric as initial data, has a long-time solution which satisfies the above a priori assumptions and has only a finite number of surgery times at each finite time interval. The construction of the long-time solution will be given by an induction argument. First, for the arbitrarily given compact orientable normalized three-dimensional Riemannian manifold (M, gij (x)), the Ricci flow with it as initial data has a maximal solution gij (x, t) on a maximal time interval [0, T ) with T > 1. It follows from Theorem 5.3.2 and Theorem 7.1.1 that the a priori assumptions (with accuracy ε) hold for the smooth solution on [0, T ). If T = +∞, we have the desired long time solution. Thus, without loss of generality, we may assume the maximal time T < +∞ so that the solution goes singular at time T . Suppose that we have a solution to the Ricci flow with surgery, with the normalized metric as initial data, satisfying the a priori assumptions (with accuracy ε), defined on [0, T ) with T < +∞, going singular at time T and having only a finite number of surgery times on [0, T ). Let Ω denote the set of all points where the curvature stays bounded as t → T . As we have seen before, the canonical neighborhood assumption implies that Ω is open and that R(x, t) → ∞ as t → T for all x lying outside Ω. Moreover, as t → T , the solution gij (x, t) has a smooth limit g¯ij (x) on Ω. For some δ > 0 to be chosen much smaller than ε, we let ρ = δr(T ) where r(t) is the positive nonincreasing function in the definition of the canonical neighborhood assumption. We consider the corresponding compact set ¯ Ωρ = {x ∈ Ω | R(x) ≤ ρ−2 }, ¯ where R(x) is the scalar curvature of g¯ij . If Ωρ is empty, the manifold (near the maximal time T ) is entirely covered by ε-tubes, ε-caps and compact components with positive curvature. Clearly, the number of the compact components is finite. Then in this case the manifold (near the maximal time T ) is diffeomorphic to the union of a finite number of copies of S3 , or metric quotients of the round S3 , or S2 × S1 , or a connected sum of them. Thus when Ωρ is empty, the procedure stops here, and we say the solution becomes extinct. We now assume Ωρ is not empty. Then we know that every point x ∈ Ω \ Ωρ lies in one of the subsets of Ω listed in (7.3.2), or in a compact component with positive curvature, or in a compact component which is contained in Ω \ Ωρ and is diffeomorphic to either S3 , or S2 × S1 or RP3 #RP3 . Note again that the number of the compact components is finite. Let us throw away all the compact components lying in Ω \ Ωρ and all the compact components with positive curvature, and then consider those components Ωj , 1 ≤ j ≤ k, of Ω which contain points of Ωρ . (We will consider those components of Ω \ Ωρ consisting of capped ε-horns and double ε-horns later). We will perform surgical procedures, as we roughly described before, by finding an ε-neck in every horn of Ωj , 1 ≤ j ≤ k,

418


and then cutting it along the middle two-sphere, removing the horn-shaped end, and gluing back a cap. In order to maintain the a priori assumptions with the same accuracy after the surgery, we will need to find sufficient “fine” necks in the ε-horns and to glue sufficient “fine” caps. Note that δ > 0 will be chosen much smaller than ε > 0. The following lemma due to Perelman [104] gives us the “fine” necks in the ε-horns. (At the first sight, we should also cut off all those ε-tubes and ε-caps in the surgery procedure. However, in general we are not able to find a “fine” neck in an ε-tube or in an ε-cap, and surgeries at “rough” ε-necks will certainly lose some accuracy. If we perform surgeries at the necks with some fixed accuracy ε in the high curvature region at each surgery time, then it is possible that the errors of surgeries may accumulate to a certain amount so that at some later time we cannot recognize the structure of very high curvature regions. This prevents us from carrying out the whole process in finite time with a finite number of steps. This is the reason why we will only perform the surgeries at the ε-horns.) 1 Lemma 7.3.2 (Perelman [104]). Given 0 < ε ≤ 100 , 0 < δ < ε and 0 < T < +∞, there exists a radius 0 < h < δρ, depending only on δ and r(T ), such that if we have a solution to the Ricci flow with surgery, with a normalized metric as initial data, satisfying the a priori assumptions (with accuracy ε), defined on [0, T ), going singular at time T and having only a finite number of surgery times on [0, T ), then for each ¯ − 12 (x) ≤ h in an ε-horn of (Ω, g¯ij ) with boundary in Ωρ , the point x with h(x) = R neighborhood BT (x, δ −1 h(x)) , {y ∈ Ω | dT (y, x) ≤ δ −1 h(x)} is a strong δ-neck (i.e., BT (x, δ −1 h(x)) × [T − h2 (x), T ] is, after scaling with factor h−2 (x), δ-close (in the −1 C [δ ] topology) to the corresponding subset of the evolving standard round cylinder 2 S × R over the time interval [−1, 0] with scalar curvature 1 at time zero).

strong δ-neck

Ωρ

Proof. We argue by contradiction. Suppose that there exists a sequence of soluk tions gij (·, t), k = 1, 2, . . ., to the Ricci flow with surgery, satisfying the a priori ask sumptions (with accuracy ε), defined on [0, T ) with limit metrics (Ωk , g¯ij ), k = 1, 2, . . ., k k k and points x , lying inside an ε-horn of Ω with boundary in Ωρ , and having h(xk ) → 0 such that the neighborhoods BT (xk , δ −1 h(xk )) = {y ∈ Ωk | dT (y, xk ) ≤ δ −1 h(xk )} are not strong δ-necks. k ¯ k ) = h−2 (xk ) Let geij (·, t) be the solutions obtained by rescaling by the factor R(x k around x and shifting the time T to the new time zero. We now want to show that a k subsequence of geij (·, t), k = 1, 2, . . ., converges to the evolving round cylinder, which will give a contradiction. k Note that e gij (·, t), k = 1, 2, . . . , are solutions modified by surgery. So, we cannot apply Hamilton’s compactness theorem directly since it is stated only for smooth k solutions. For each (unrescaled) surgical solution g¯ij (·, t), we pick a point z k , with k 2 −2 k k ¯ ) = 2C (ε)ρ , in the ε-horn of (Ω , g¯ ) with boundary in Ωk , where C2 (ε) is R(z 2 ρ ij the positive constant in the canonical neighborhood assumption. From the definition


419

of ε-horn and the canonical neighborhood assumption, we know that each point x k k k k lying inside the ε-horn of (Ωk , g¯ij ) with dg¯ij k (x, Ωρ ) ≥ dg k (z , Ωρ ) has a strong ε-neck ¯ij

as its canonical neighborhood. Since h(xk ) → 0, each xk lies deeply inside an ε-horn. k Thus for each positive A < +∞, the rescaled (surgical) solutions e gij (·, t) with the k k k marked origins x over the geodesic balls Begij k (·,0) (x , A), centered at x of radii A

k (with respect to the metrics geij (·, 0)), will be smooth on some uniform (size) small k time intervals for all sufficiently large k, if the curvatures of the rescaled solutions e gij k at t = 0 in Begij k (·,0) (x , A) are uniformly bounded. In such a situation, Hamilton’s compactness theorem is applicable. Then we can apply the same argument as in the second step of the proof of Theorem 7.1.1 to conclude that for each A < +∞, there exists a positive constant C(A) such that the curvatures of the rescaled solutions k geij (·, t) at the new time 0 satisfy the estimate

e k |(y, 0) ≤ C(A) |Rm

k whenever degij k (·,0) (y, x ) ≤ A and k ≥ 1; otherwise we would get a piece of a non-flat nonnegatively curved metric cone as a blow-up limit, which contradicts Hamilton’s strong maximum principle. Moreover, by Hamilton’s compactness theorem (Theorem k ∞ ∞ 4.1.5), a subsequence of the rescaled solutions geij (·, t) converges to a Cloc limit e gij (·, t), defined on a spacetime set which is relatively open in the half spacetime {t ≤ 0} and contains the time slice t = 0. By the pinching assumption, the limit is a complete manifold with nonnegative sectional curvature. Since xk was contained in an ε-horn with boundary in Ωkρ and h(xk )/ρ → 0, the limiting manifold has two ends. Thus, by Toponogov’s splitting theorem, the limiting manifold admits a metric splitting Σ2 × R, where Σ2 is diffeomorphic to the two-sphere S2 because xk was the center of a strong ε-neck. By combining with the canonical neighborhood assumption (with accuracy ε), we see that the limit is defined on the time interval [−1, 0] and is ε-close to the evolving standard round cylinder. In particular, the scalar curvature of the limit at time t = −1 is ε-close to 1/2. Since h(xk )/ρ → 0, each point in the limiting manifold at time t = −1 also has a strong ε-neck as its canonical neighborhood. Thus the limit is defined at least on the time interval [−2, 0] and the limiting manifold at time t = −2 is, after rescaling, ε-close to the standard round cylinder. By using the canonical neighborhood assumption again, every point in the limiting manifold at time t = −2 still has a strong ε-neck as its canonical neighborhood. Also note that the scalar curvature of the limit at t = −2 is not bigger than 1/2 + ε. Thus the limit is defined at least on the time interval [−3, 0] and the limiting manifold at time t = −3 is, after rescaling, ε-close to the standard round cylinder. By repeating this argument we prove that the limit exists on the ancient time interval (−∞, 0]. The above argument also shows that at every time, each point of the limit has a strong ε-neck as its canonical neighborhood. This implies that the limit is κnoncollaped on all scales for some κ > 0. Therefore, by Theorem 6.2.2, the limit is the evolving round cylinder S2 × R, which gives the desired contradiction.

In the above lemma, the property that the radius h depends only on δ and the time T but is independent of the surgical solution is crucial; otherwise we will not be able to cut off enough volume at each surgery to guarantee the number of surgeries being finite in each finite time interval. We also remark that the above proof actually

420


proves a stronger result: the parabolic region {(y, t) | y ∈ BT (x, δ −1 h(x)), t ∈ [T − −1 δ −2 h2 (x), T ]} is, after scaling with factor h−2 (x), δ-close (in the C [δ ] topology) to the corresponding subset of the evolving standard round cylinder S2 × R over the time interval [−δ −2 , 0] with scalar curvature 1 at the time zero. This fact will be used later in the proof of Proposition 7.4.1. We next want to construct “fine” caps. Take a rotationally symmetric metric on R3 with nonnegative sectional curvature and positive scalar curvature such that outside some compact set it is a semi-infinite standard round cylinder (i.e. the metric product of a ray with the round two-sphere of scalar curvature 1). We call such a metric on R3 a standard capped infinite cylinder. By the short-time existence theorem of Shi (Theorem 1.2.3), the Ricci flow with a standard capped infinite cylinder as initial data has a complete solution on a maximal time interval [0, T ) such that the curvature of the solution is bounded on R3 × [0, T ′] for each 0 < T ′ < T . Such a solution is called a standard solution by Perelman [104]. The following result proved by Chen and the second author in [34] gives the curvature estimate for standard solutions. Proposition 7.3.3. Let gij be a complete Riemannian metric on Rn (n ≥ 3) with nonnegative curvature operator and positive scalar curvature which is asymptotic to a round cylinder of scalar curvature 1 at infinity. Then there is a complete solution gij (·, t) to the Ricci flow, with gij as initial metric, which exists on the time interval n−1 [0, n−1 2 ), has bounded curvature at each time t ∈ [0, 2 ), and satisfies the estimate R(x, t) ≥

C −1 n−1 2 −t

for some C depending only on the initial metric gij . Proof. Since the initial metric has bounded curvature operator and a positive lower bound on its scalar curvature, the Ricci flow has a solution gij (·, t) defined on a maximal time interval [0, T ) with T < ∞ which has bounded curvature on Rn × [0, T ′] for each 0 < T ′ < T . By Proposition 2.1.4, the solution gij (·, t) has nonnegative curvature operator for all t ∈ [0, T ). Note that the injectivity radius of the initial metric has a positive lower bound. As we remarked at the beginning of Section 3.4, the same proof of Perelman’s no local√collapsing theorem I concludes that gij (·, t) is κ-noncollapsed on all scales less than T for some κ > 0 depending only on the initial metric. We will first prove the following assertion. Claim 1. There is a positive function ω : [0, ∞) → [0, ∞) depending only on the initial metric and κ such that R(x, t) ≤ R(y, t)ω(R(y, t)d2t (x, y)) for all x, y ∈ Rn , t ∈ [0, T ). The proof is similar to that of Theorem 6.4.3. Notice that the initial metric has nonnegative curvature operator and its scalar curvature satisfies the bounds (7.3.5)

C −1 6 R(x) 6 C

for some positive constant C. By the maximum principle, we know T ≥ 1 1 R(x, t) ≤ 2C for t ∈ [0, 4nC ]. The assertion is clearly true for t ∈ [0, 4nC ].

1 2nC

and

421


1 . Let z be the closest point to y with Now fix (y, t0 ) ∈ Rn × [0, T ) with t0 ≥ 4nC 2 the property R(z, t0 )dt0 (z, y) = 1 (at time t0 ). Draw a shortest geodesic from y to z 1 and choose a point z˜ on the geodesic satisfying dt0 (z, z˜) = 14 R(z, t0 )− 2 , then we have

R(x, t0 ) ≤

1 1

( 21 R(z, t0 )− 2 )2

on Bt0

− 12 1 z˜, R(z, t0 ). 4

Note that R(x, t) > C −1 everywhere by the evolution equation of the scalar curvature. Then by the Li-Yau-Hamilton inequality (Corollary 2.5.5), for all (x, t) ∈ 1 1 1 1 Bt0 (˜ z , 8nC R(z, t0 )− 2 ) × [t0 − ( 8nC R(z, t0 )− 2 )2 , t0 ], we have    R(x, t) ≤ 

t0 −

t0

1 √ 8n C

 2 

1 1 R(z, t0 )− 2 ≤ 8nC

−2

1

1 − 12 2 R(z, t0 )

2

.

Combining this with the κ-noncollapsing property, we have n 1 1 1 1 ≥κ Vol Bt0 z˜, R(z, t0 )− 2 R(z, t0 )− 2 8nC 8nC and then

− 21 Vol Bt0 z, 8R(z, t0) ≥κ

1 64nC

n

1

8R(z, t0 )− 2

n

.

So by Theorem 6.3.3 (ii), we have 1 R(x, t0 ) ≤ C(κ)R(z, t0 ) for all x ∈ Bt0 z, 4R(z, t0)− 2 .

Here and in the following we denote by C(κ) various positive constants depending only on κ, n and the initial metric. Now by the Li-Yau-Hamilton inequality (Corollary 2.5.5) and local gradient estimate of Shi (Theorem 1.4.2), we obtain ∂ R(x, t) ≤ C(κ)R(z, t0 ) and R (x, t) ≤ C(κ)(R(z, t0 ))2 ∂t 1

1

1 for all (x, t) ∈ Bt0 (z, 2R(z, t0)− 2 )) × [t0 − ( 8nC R(z, t0 )− 2 )2 , t0 ]. Therefore by combining with the Harnack estimate (Corollary 2.5.7), we obtain

R(y, t0 ) ≥ C(κ)−1 R(z, t0 − C(κ)−1 R(z, t0 )−1 ) ≥ C(κ)−2 R(z, t0 )

and

Consequently, we have showed that there is a constant C(κ) such that n 1 1 Vol Bt0 y, R(y, t0 )− 2 ≥ C(κ)−1 R(y, t0 )− 2 1 R(x, t0 ) ≤ C(κ)R(y, t0 ) for all x ∈ Bt0 y, R(y, t0 )− 2 .

422

H.-D. CAO AND X.-P. ZHU 1

In general, for any r ≥ R(y, t0 )− 2 , we have Vol (Bt0 (y, r)) ≥ C(κ)−1 (r2 R(y, t0 ))− 2 rn . n

By applying Theorem 6.3.3(ii) again, there exists a positive ω(r2 R(y, t0 )) depending only on the constant r2 R(y, t0 ) and κ such that 1 2 R(x, t0 ) ≤ R(y, t0 )ω(r R(y, t0 )) for all x ∈ Bt0 y, r . 4

constant

This proves the desired Claim 1. Now we study the asymptotic behavior of the solution at infinity. For any 0 < t0 < T , we know that the metrics gij (x, t) with t ∈ [0, t0 ] has uniformly bounded curvature. Let xk be a sequence of points with d0 (x0 , xk ) → ∞. After passing to a subsequence, gij (x, t) around xk will converge to a solution to the Ricci flow on R × Sn−1 with round cylinder metric of scalar curvature 1 as initial data. Denote the limit by g˜ij . Then by the uniqueness theorem (Theorem 1.2.4), we have ˜ t) = R(x, It follows that T ≤ assertion.

n−1 2 .

n−1 2 n−1 2 −

t

for all t ∈ [0, t0 ].

In order to show T =

n−1 2 ,

it suffices to prove the following

n Claim 2. Suppose T < n−1 2 . Fix a point x0 ∈ R , then there is a δ > 0, such −1 that for any x ∈ M with d0 (x, x0 ) ≥ δ , we have

R(x, t) ≤ 2C +

n−1 −t

n−1 2

for all t ∈ [0, T ),

where C is the constant in (7.3.5). In view of Claim 1, if Claim 2 holds, then n−1 −2 2C + sup R(y, t) ≤ ω δ 2C + n−1 M n ×[0,T ) 2 −T 0, there is a point (xδ , tδ ) with 0 < tδ < T such that R(xδ , tδ ) > 2C +

n−1 and d0 (xδ , x0 ) ≥ δ −1 . − tδ

n−1 2

Let (

Since

lim

t¯δ = sup t

d0 (y,x0 )→∞

R(y, t) =

n−1 sup R(y, t) < 2C + n−1 n −1 M \B0 (x0 ,δ ) 2 −t

n−1 2 n−1 2 −t

)

.

1 and supM×[0, 4nC ] R(y, t) ≤ 2C, we know

tδ and there is a x ¯δ such that d0 (x0 , x ¯δ ) ≥ δ −1 and R(¯ xδ , t¯δ ) = 2C +

n−1 n−1 ¯ 2 −tδ

1 4nC

≤ t¯δ ≤

. By Claim


423

1 and Hamilton’s compactness theorem (Theorem 4.1.5), for δ → 0 and after taking −1 a subsequence, the metrics gij (x, t) on B0 (¯ xδ , δ 2 ) over the time interval [0, t¯δ ] will converge to a solution g˜ij on R × Sn−1 with the standard metric of scalar curvature 1 as initial data over the time interval [0, t¯∞ ], and its scalar curvature satisfies n−1 , − t¯∞ ˜ t) 6 2C + n − 1 , R(x, n−1 ¯ 2 − t∞

˜ x∞ , t¯∞ ) = 2C + R(¯

n−1 2

for all t ∈ [0, t¯∞ ],

where (¯ x∞ , t¯∞ ) is the limit of (¯ xδ , t¯δ ). On the other hand, by the uniqueness theorem (Theorem 1.2.4) again, we know ˜ x∞ , t¯∞ ) = R(¯

n−1 2 n−1 ¯ 2 − t∞

which is a contradiction. Hence we have proved Claim 2 and then have verified T = n−1 2 . Now we are ready to show (7.3.6)

R(x, t) ≥

C˜ −1 , −t

n−1 2

h n − 1 for all (x, t) ∈ Rn × 0, , 2

for some positive constant C˜ depending only on the initial metric. For any (x, t) ∈ Rn × [0, n−1 2 ), by Claim 1 and κ-noncollapsing, there is a constant C(κ) > 0 such that 1

1

Vol t (Bt (x, R(x, t)− 2 )) ≥ C(κ)−1 (R(x, t)− 2 )n . Then by the well-known volume estimate of Calabi-Yau (see for example [128] or [112]) for complete manifolds with Ric ≥ 0, for any a ≥ 1, we have Vol t (Bt (x, aR(x, t)

− 21

)) ≥ C(κ)−1

a −1 (R(x, t) 2 )n . 8n

On the other hand, since (Rn , gij (·, t)) is asymptotic to a cylinder of scalar curvature n−1 ( n−1 2 )/( 2 − t), for sufficiently large a > 0, we have !! r n2 n−1 n−1 −t ≤ C(n)a −t Vol t Bt x, a . 2 2 Combining the two inequalities, for all sufficiently large a, we have:     q n2 n−1 1 n−1 2 −t −  R(x, t) 2  C(n)a −t ≥ Vol t Bt x, a  1 2 R(x, t)− 2 q  n−1 n − t a 2 − 12   ≥ C(κ)−1 R(x, t) , 8n R(x, t)− 12

which gives the desired estimate (7.3.6). Therefore the proof of the proposition is complete.

424


We now fix a standard capped infinite cylinder for dimension n = 3 as follows. Consider the semi-infinite standard round cylinder N0 = S2 × (−∞, 4) with the metric g0 of scalar curvature 1. Denote by z the coordinate of the second factor (−∞, 4). Let f be a smooth nondecreasing convex function on (−∞, 4) defined by  f (z) = 0, z ≤ 0,        f (z) = ce− Pz , z ∈ (0, 3], (7.3.7)   f (z) is strictly convex on z ∈ [3, 3.9],      f (z) = − 21 log(16 − z 2 ), z ∈ [3.9, 4),

where the (small) constant c > 0 and (big) constant P > 0 will be determined later. Let us replace the standard metric g0 on the portion S2 × [0, 4) of the semi-infinite cylinder by gˆ = e−2f g0 . Then the resulting metric gˆ will be smooth on R3 obtained by adding a point to S2 × (−∞, 4) at z = 4. We denote by C(c, P ) = (R3 , gˆ). Clearly, C(c, P ) is a standard capped infinite cylinder. We next use a compact portion of the standard capped infinite cylinder C(c, P ) and the δ-neck obtained in Lemma 7.3.2 to perform the following surgery procedure due to Hamilton [64]. Consider the metric g¯ at the maximal time T < +∞. Take an ε-horn with boundary in Ωρ . By Lemma 7.3.2, there exists a δ-neck N of radius 0 < h < δρ in the −1 ε-horn. By definition, (N, h−2 g¯) is δ-close (in the C [δ ] topology) to the standard round neck S2 × I of scalar curvature 1 with I = (−δ −1 , δ −1 ). Using the parameter z ∈ I, we see the above function f is defined on the δ-neck N . T Let us cut the δ-neck N along the middle (topological) two-sphere N {z = 0}.T Without loss of generality, we may assume that the right hand half portion N {z ≥ 0} is contained in the horn-shaped end. Let ϕ be a smooth bump function with ϕ = 1 for z ≤ 2, and ϕ = 0 for z ≥ 3. Construct a new metric g˜ on a (topological) three-ball B3 as follows   g¯, z = 0,     e−2f g¯, z ∈ [0, 2], (7.3.8) g˜ = −2f −2f 2  ϕe g¯ + (1 − ϕ)e h g0 , z ∈ [2, 3],     h2 e−2f g , z ∈ [3, 4]. 0 The surgery is to replace the horn-shaped end by the cap (B3 , g˜). We call such surgery procedure a δ-cutoff surgery. The following lemma determines the constants c and P in the δ-cutoff surgery so that the pinching assumption is preserved under the surgery.

Lemma 7.3.4 (Justification of the pinching assumption). There are universal positive constants δ0 , c0 and P0 such that if we take a δ-cutoff surgery at a δ-neck of radius h at time T with δ ≤ δ0 and h−2 ≥ 2e2 log(1 + T ), then we can choose c = c0 and P = P0 in the definition of f (z) such that after the surgery, there still holds the pinching condition (7.3.9)

˜ ≥ (−˜ R ν )[log(−˜ ν ) + log(1 + T ) − 3]

˜ is the scalar curvature of the metric g˜ and ν˜ is the least whenever ν˜ < 0, where R eigenvalue of the curvature operator of g˜. Moreover, after the surgery, any metric ball


425

1

of radius δ − 2 h with center near the tip (i.e., the origin of the attached cap) is, after −1

1

scaling with factor h−2 , δ 2 -close (in the C [δ 2 ] topology) to the corresponding ball of the standard capped infinite cylinder C(c0 , P0 ). Proof. First, we consider the metric g˜ on the portion {0 ≤ z ≤ 2}. Under the ˜ ijkl is given by conformal change g˜ = e−2f g¯, the curvature tensor R h ˜ ijkl = e−2f R ¯ ijkl + |∇f ¯ |2 (¯ R gil g¯jk − g¯ik g¯jl ) + (fik + fi fk )¯ gjl i + (fjl + fj fl )¯ gik − (fil + fi fl )¯ gjk − (fjk + fj fk )¯ gil .

∂ ∂ If {F¯a = F¯ai ∂x ¯ij , then {Fã = ef F¯a = Fãi ∂x i } is an orthonormal frame for g i } is an orthonormal frame for g˜ij . Let

¯ abcd = R ¯ ijkl F¯ai F¯ j F¯ck F¯dl , R b ˜ abcd = R ˜ ijkl F˜ i F˜ j F˜ k F˜ l , R a b c d then (7.3.10)

h ˜ abcd = e2f R ¯ abcd + |∇f ¯ |2 (δad δbc − δac δbd ) + (fac + fa fc )δbd R

i + (fbd + fb fd )δac − (fad + fa fd )δbc − (fbc + fb fc )δad ,

and

¯ − 2|∇f ˜ = e2f (R ¯ + 4△f ¯ |2 ). R

(7.3.11) Since

df P P d2 f P = ce− z 2 , = ce− z dz z dz 2

P2 2P − 3 z4 z

,

then for any small θ > 0, we may choose c > 0 small and P > 0 large such that for z ∈ [0, 3], we have 2 df df d2 f d2 f 2f (7.3.12) |e − 1| + + < θ. < θ 2, dz dz dz dz 2 On the other hand, by the definition of δ-neck of radius h, we have |¯ g − h2 g0 |g0 < δh2 , o

|∇j g¯|g0 < δh2 , for 1 ≤ j ≤ [δ −1 ], where g0 is the standard metric of the round cylinder S2 × R. Note that in three dimensions, we can choose the orthonormal frame {F¯1 , F¯2 , F¯3 } for √ g¯ so √ the metric that√its curvature operator is diagonal in the orthonormal frame { 2F¯2 ∧ F¯3 , 2F¯3 ∧ ¯ and F¯1 , 2F¯1 ∧ F¯2 } with eigenvalues ν¯ ≤ µ ¯≤λ ¯ = 2R ¯ 2323 , µ ¯ 3131 , λ ¯ 1212 . ν¯ = 2R ¯ = 2R

426


Since h−2 g¯ is δ-close to the standard round cylinder metric g0 on the δ-neck, we have  ¯3131 | + |R ¯ 2323 | < δ 78 h−2 ,  |R   ¯1212 − 1 h−2 | < δ 78 h−2 , |R (7.3.13) 2    |F¯3 − h−1 ∂ |g0 < δ 78 h−1 , ∂z

¯ a z = ∇z( ¯ F¯a ) and ∇ ¯ a∇ ¯ bz = ∇ ¯ 2 z(F¯a , F¯b ), it follows for suitably small δ > 0. Since ∇ that 7

¯ 3 z − h−1 | < δ 8 h−1 , |∇ 7

¯ 1 z| + |∇ ¯ 2 z| < δ 8 h−1 , |∇ and 7

¯ a∇ ¯ b z| < δ 8 h−2 , for 1 ≤ a, b ≤ 3. |∇ By combining with 2 ¯ a z, ∇ ¯ a∇ ¯ b f = df ∇ ¯ a∇ ¯ bz + d f ∇ ¯ az∇ ¯ bz ¯ a f = df ∇ ∇ dz dz dz 2

and (7.3.12), we get  ¯ f | < 2θh−1 d2 f2 ,  for 1 ≤ a ≤ 3, |∇  dz  a 2 3 d f −2 ¯ ¯ (7.3.14) |∇a ∇b f | < δ 4 h dz2 , unless a = b = 3,   2 2 3  ¯ ¯ d f d f |∇3 ∇3 f − h−2 dz2 | < δ 4 h−2 dz2 .

By combining (7.3.10) and (7.3.14), we have  ˜ 1212 ≥ R ¯ 1212 − (θ 21 + δ 58 )h−2 d2 f2 ,  R  dz     R ˜ 3131 ≥ R ¯ 3131 + (1 − θ 12 − δ 58 )h−2 d2 f2 , dz (7.3.15) 1 5 d2 f −2  ˜ ¯ 2 8  R2323 ≥ R2323 + (1 − θ − δ )h dz2 ,    2 1 5  ˜ otherwise, |Rabcd | ≤ (θ 2 + δ 8 )h−2 d f2 , dz

where θ and δ are suitably small. Then it follows that

2

˜≥R ¯ + [4 − 6(θ 31 + δ 12 )]h−2 d f , R dz 2 1

1

−˜ ν ≤ −¯ ν − [2 − 2(θ 3 + δ 2 )]h−2

d2 f , dz 2

for suitably small θ and δ. If 0 < −˜ ν ≤ e2 , then by the assumption that h−2 ≥ 2e2 log(1 + T ), we have ˜≥R ¯ R 1 ≥ h−2 2 ≥ e2 log(1 + T ) ≥ (−˜ ν )[log(−˜ ν ) + log(1 + T ) − 3].


427

While if −˜ ν > e2 , then by the pinching estimate of g¯, we have ˜≥R ¯ R ≥ (−¯ ν )[log(−¯ ν ) + log(1 + T ) − 3]

≥ (−˜ ν )[log(−˜ ν ) + log(1 + T ) − 3].

So we have verified the pinching condition on the portion {0 ≤ z ≤ 2}. Next, we consider the metric g˜ on the portion {2 ≤ z ≤ 4}. Let θ be a fixed suitably small positive number. Then the constant c = c0 and P = P0 are fixed. 2 So ζ = minz∈[1,4] ddzf2 > 0 is also fixed. By the same argument as in the derivation of (7.3.15) from (7.3.10), we see that the curvature of the metric gˆ = e−2f g0 of the standard capped infinite cylinder C(c0 , P0 ) on the portion {1 ≤ z ≤ 4} is bounded from below by 32 ζ > 0. Since h−2 g¯ is δ-close to the standard round metric g0 , the 3 metric h−2 g˜ defined by (7.3.8) is clearly δ 4 -close to the metric gˆ = e−2f g0 of the standard capped infinite cylinder on the portion {1 ≤ z ≤ 4}. Thus as δ is sufficiently small, the curvature operator of g˜ on the portion {2 ≤ z ≤ 4} is positive. Hence the pinching condition (7.3.9) holds trivially on the portion {2 ≤ z ≤ 4}. The last statement in Lemma 7.3.4 is obvious from the definition (7.3.8). Recall from Lemma 7.3.2 that the δ-necks at a time t > 0, where we performed Hamilton’s surgeries, have their radii 0 < h < δρ = δ 2 r(t). Without loss of generality, we may assume the positive nonincreasing function r(t) in the definition of the canonical neighborhood assumption is less than 1 and the universal constant δ0 in Lemma ¯ by 7.3.4 is also less than 1. We define a positive function δ(t) (7.3.16)

¯ = min δ(t)

1 , δ0 2 2e log(1 + t)

for t ∈ [0, +∞).

¯ for any δ-cutoff surgery at time From now on, we always assume 0 < δ < δ(t) t > 0 and assume c = c0 and P = P0 . As a result, the standard capped infinite cylinder and the standard solution are also fixed. The following lemma, which will be used in the next section, gives the canonical neighborhood structure for the fixed standard solution. Lemma 7.3.5. Let gij (x, t) be the above fixed standard solution to the Ricci flow on R3 × [0, 1). Then for any ε > 0, there is a positive constant C(ε) such that each point (x, t) ∈ R3 × [0, 1) has an open neighborhood B, with Bt (x, r) ⊂ B ⊂ Bt (x, 2r) 1 for some 0 < r < C(ε)R(x, t)− 2 , which falls into one of the following two categories: either (a) B is an ε-cap, or (b) B is an ε-neck and it is the slice at the time t of the parabolic neighborhood 1 Bt (x, ε−1 R(x, t)− 2 ) × [t − min{R(x, t)−1 , t}, t], on which the standard solution is, after scaling with the factor R(x, t) and shifting the time t to zero, −1 ε-close (in the C [ε ] topology) to the corresponding subset of the evolving standard cylinder S2 × R over the time interval [− min{tR(x, t), 1}, 0] with scalar curvature 1 at the time zero. Proof. The proof of the lemma is reduced to two assertions. We now state and prove the first assertion which takes care of those points with times close to 1.

428


Assertion 1. For any ε > 0, there is a positive number θ = θ(ε) with 0 < θ < 1 such that for any (x0 , t0 ) ∈ R3 × [θ, 1), the standard solution on the parabolic neigh1 borhood Bt0 (x, ε−1 R(x0 , t0 )− 2 ) × [t0 − ε−2 R(x0 , t0 )−1 , t0 ] is well-defined and is, after −1 scaling with the factor R(x0 , t0 ), ε-close (in the C [ε ] topology) to the corresponding subset of some orientable ancient κ-solution. We argue by contradiction. Suppose Assertion 1 is not true, then there exist ε¯ > 0 and a sequence of points (xk , tk ) with tk → 1, such that for each k, the standard solution on the parabolic neighborhood 1

Btk (xk , ε¯−1 R(xk , tk )− 2 ) × [tk − ε¯−2 R(xk , tk )−1 , tk ] is not, after scaling by the factor R(xk , tk ), ε¯-close to the corresponding subset of any ancient κ-solution. Note that by Proposition 7.3.3, there is a constant C > 0 (depending only on the initial metric, hence it is universal ) such that R(x, t) ≥ C −1 /(1 − t). This implies ε¯−2 R(xk , tk )−1 ≤ C ε¯−2 (1 − tk ) < tk , and then the standard solution on the parabolic neighborhood Btk (xk , 1 ε¯−1 R(xk , tk )− 2 ) × [tk − ε¯−2 R(xk , tk )−1 , tk ] is well-defined for k large. By Claim 1 in the proof of Proposition 7.3.3, there is a positive function ω : [0, ∞) → [0, ∞) such that R(x, tk ) ≤ R(xk , tk )ω(R(xk , tk )d2tk (x, xk )) for all x ∈ R3 . Now by scaling the standard solution gij (·, t) around xk with the factor R(xk , tk ) and shifting the time tk to zero, we get a sequence of the rescaled k solutions g˜ij (x, t˜) = R(xk , tk )gij (x, tk + t˜/R(xk , tk )) to the Ricci flow defined on R3 with t˜ ∈ [−R(xk , tk )tk , 0]. We denote the scalar curvature and the distance of the k ˜ By combining with Claim 1 in the proof of Proposition ˜ k and d. rescaled metric g˜ij by R 7.3.3 and the Li-Yau-Hamilton inequality, we get ˜ k (x, 0) ≤ ω(d˜2 (x, xk )) R 0 ˜ k (x, t˜) ≤ R

R(xk , tk )tk ω(d˜20 (x, xk )) t˜ + R(xk , tk )tk

for any x ∈ R3 and t˜ ∈ (−R(xk , tk )tk , 0]. Note that R(xk , tk )tk → ∞ by Proposition 7.3.3. We have shown in the proof of Proposition 7.3.3 that the standard solution is κnoncollapsed on all scales less than 1 for some κ > 0. Then from the κ-noncollapsing property, the above curvature estimates and Hamilton’s compactness theorem, we k know g˜ij (x, t˜) has a convergent subsequence (as k → ∞) whose limit is an ancient, κnoncollapsed, complete and orientable solution with nonnegative curvature operator. This limit must have bounded curvature by the same proof of Step 3 in the proof of Theorem 7.1.1. This gives a contradiction. Hence Assertion 1 is proved. We now fix the constant θ(ε) obtained in Assertion 1. Let O be the tip of the standard capped infinite cylinder R3 (it is rotationally symmetric about O at time 0, and it remains so as t > 0 by the uniqueness Theorem 1.2.4). Assertion 2. There are constants B1 (ε) and B2 (ε) depending only on ε, such that if (x0 , t0 ) ∈ R3 × [0, θ) with dt0 (x0 , O) ≤ B1 (ε), then there is a 0 < r < B2 (ε)


429

such that Bt0 (x0 , r) is an ε-cap; if (x0 , t0 ) ∈ R3 × [0, θ) with dt0 (x0 , O) ≥ B1 (ε), then 1 the parabolic neighborhood Bt0 (x0 , ε−1 R(x0 , t0 )− 2 ) ×[t0 − min{R(x0 , t0 )−1 , t0 }, t0 ] is after scaling with the factor R(x0 , t0 ) and shifting the time t0 to zero, ε-close (in the −1 C [ε ] topology) to the corresponding subset of the evolving standard cylinder S2 × R over the time interval [− min{t0 R(x0 , t0 ), 1}, 0] with scalar curvature 1 at time zero. Since the standard solution exists on the time interval [0, 1), there is a constant B0 (ε) such that the curvatures on [0, θ(ε)] are uniformly bounded by B0 (ε). This implies that the metrics in [0, θ(ε)] are equivalent. Note that the initial metric is asymptotic to the standard capped infinite cylinder. For any sequence of points xk with d0 (O, xk ) → ∞, after passing to a subsequence, gij (x, t) around xk will converge to a solution to the Ricci flow on R × S2 with round cylinder metric of scalar curvature 1 as initial data. By the uniqueness theorem (Theorem 1.2.4), the limit solution must be the standard evolving round cylinder. This implies that there is a constant B1 (ε) > 0 depending on ε such that for any (x0 , t0 ) with t0 ≤ θ(ε) and dt0 (x0 , O) ≥ B1 (ε), the standard solution on the parabolic neighbor1 hood Bt0 (x0 , ε−1 R(x0 , t0 )− 2 ) × [t0 − min{R(x0 , t0 )−1 , t0 }, t0 ] is, after scaling with the factor R(x0 , t0 ), ε-close to the corresponding subset of the evolving round cylinder. Since the solution is rotationally symmetric around O, the cap neighborhood structures of those points x0 with dt0 (x0 , O) ≤ B1 (ε) follow directly. Hence Assertion 2 is proved. The combination of these two assertions proves the lemma. Since there are only a finite number of horns with the other end connected to Ωρ , we perform only a finite number of such δ-cutoff surgeries at time T . Besides those horns, there could be capped horns and double horns which lie in Ω\Ωρ . As explained before, they are connected to form tubes or capped tubes at any time slightly before T . So we can regard the capped horns and double horns (of Ω \ Ωρ ) to be extinct and throw them away at time T . We only need to remember that the connected sums were broken there. Remember that we have thrown away all compact components, either lying in Ω \ Ωρ or with positive sectional curvature, each of which is diffeomorphic to either S3 , or a metric quotient of S3 , or S2 × S1 or RP3 #RP3 . So we have also removed a finite number of copies of S3 , or metric quotients of S3 , or S2 × S1 or RP3 #RP3 at the time T . Let us agree to declare extinct every compact component either with positive sectional curvature or lying in Ω \ Ωρ ; in particular, this allows us to exclude the components with positive sectional curvature from the list of canonical neighborhoods. In summary, our surgery at time T consists of the following four procedures: (1) perform δ-cutoff surgeries for all ε-horns, whose other ends are connected to Ωρ ; (2) declare extinct every compact component which has positive sectional curvature; (3) throw away all capped horns and double horns lying in Ω \ Ωρ ; (4) declare extinct all compact components lying in Ω \ Ωρ . (In Sections 7.6 and 7.7, we will add one more procedure by declaring extinct every compact component which has nonnegative scalar curvature.) By Lemma 7.3.4, after performing surgeries at time T , the pinching assumption (7.3.3) still holds for the surgically modified manifold. With this surgically modified manifold (possibly disconnected) as initial data, we now continue our solution under

430


the Ricci flow until it becomes singular again at some time T ′ (> T ). Therefore, we have extended the solution to the Ricci flow with surgery, originally defined on [0, T ) with T < +∞, to the new time interval [0, T ′ ) with T ′ > T . By the proof of Theorem 5.3.2, we see that the solution to the Ricci flow with surgery also satisfies the pinching assumption on [0, T ′ ). It remains to verify the canonical neighborhood assumption (with accuracy ε) for the solution on the time interval [T, T ′) and to prove that this extension procedure works indefinitely (unless it becomes extinct at some finite time) and that there exists at most a finite number of surgeries at every finite time interval. We leave these arguments to the next section. Before we end this section, we check the following two assertions of Perelman in [104] which will be used in the next section to estimate the Li-Yau-Perelman distance of space-time curves which stretch to surgery regions. Lemma 7.3.6 (Perelman [104]). For any 0 < ε ≤ 1/100, 1 < A < +∞ and ¯ 0 < θ < 1, one can find δ¯ = δ(A, θ, ε) with the following property. Suppose we have a solution to the Ricci flow with surgery which satisfies the a priori assumptions (with accuracy ε) on [0, T ] and is obtained from a compact orientable three-manifold by a ¯ Suppose we have a cutoff surgery finite number of δ-cutoff surgeries with each δ < δ. at time T0 ∈ (0, T ), let x0 be any fixed point on the gluing caps (i.e., the regions affected by the cutoff surgeries at time T0 ), and let T1 = min{T, T0 + θh2 }, where h is the cutoff radius around x0 obtained in Lemma 7.3.2. Then either (i) the solution is defined on P (x0 , T0 , Ah, T1 − T0 ) , {(x, t) | x ∈ Bt (x0 , Ah), t ∈ [T0 , T1 ]} and is, after scaling with factor h−2 and shifting time T0 to zero, A−1 -close to a corresponding subset of the standard solution, or (ii) the assertion (i) holds with T1 replaced by some time t+ ∈ (T0 , T1 ), where t+ is a surgery time; moreover, for each point in BT0 (x0 , Ah), the solution is defined for t ∈ [T0 , t+ ) but is not defined past t+ (i.e., the whole ball BT0 (x0 , Ah) is cut off at the time t+ ). Proof. Let Q be the maximum of the scalar curvature of the standard solution in −T0 ) the time interval [0, θ] and choose a large positive integer N so that ∆t = (T1N < −1 −1 2 εη Q h , where the positive constant η is given in the canonical neighborhood assumption. Set tk = T0 + k∆t, k = 0, 1, . . . , N . 1 From Lemma 7.3.4, the geodesic ball BT0 (x0 , A0 h) at time T0 , with A0 = δ − 2 1 −2 is, after scaling with factor h , δ 2 -close to the corresponding ball in the standard capped infinite cylinder with the center near the tip. Assume first that for each point in BT0 (x0 , A0 h), the solution is defined on [T0 , t1 ]. By the gradient estimates (7.3.4) in the canonical neighborhood assumption and the choice of ∆t we have a uniform curvature bound on this set for h−2 -scaled metric. Then by the uniqueness theorem 1 1 (Theorem 1.2.4), if δ 2 → 0 (i.e. A0 = δ − 2 → +∞), the solution with h−2 -scaled ∞ metric will converge to the standard solution in the Cloc topology. Therefore we can define A1 , depending only on A0 and tending to infinity with A0 , such that the solution in the parabolic region P (x0 , T0 , A1 h, t1 − T0 ) , {(x, t) | x ∈ Bt (x0 , A1 h), t ∈ [T0 , T0 + (t1 − T0 )]} is, after scaling with factor h−2 and shifting time T0 to zero, A−1 1 -close to the corresponding subset in the standard solution. In particular, the scalar curvature on this subset does not exceed 2Qh−2 . Now if for each point in BT0 (x0 , A1 h) the solution is defined on [T0 , t2 ], then we can repeat the procedure, defining A2 , such that the solution in the parabolic region P (x0 , T0 , A2 h, t2 − T0 ) , {(x, t) | x ∈ Bt (x0 , A2 h), t ∈ [T0 , T0 + (t2 − T0 )]} is, after scaling with factor h−2 and shifting time T0 to zero, A−1 2 -close to the corresponding subset in the standard


431

solution. Again, the scalar curvature on this subset still does not exceed 2Qh−2 . Continuing this way, we eventually define AN . Note that N is depends only on θ ¯ ¯ we have and ε. Thus there exists a positive δ¯ = δ(A, θ, ε) such that for δ < δ, A0 > A1 > · · · > AN > A, and assertion (i) holds when the solution is defined on BT0 (x0 , A(N −1) h) × [T0 , T1 ]. The above argument shows that either assertion (i) holds, or there exists some k (0 ≤ k ≤ N − 1) and a surgery time t+ ∈ (tk , tk+1 ] such that the solution on BT0 (x0 , Ak h) is defined on [T0 , t+ ), but for some point of this set it is not defined past t+ . Now we consider the latter case. Clearly the above argument also shows that the parabolic region P (x0 , T0 , Ak+1 h, t+ − T0 ) , {(x, t) | x ∈ Bt (x, Ak+1 h), t ∈ [T0 , t+ )} is, after scaling with factor h−2 and shifting time T0 to zero, A−1 k+1 -close to the corresponding subset in the standard solution. In particular, as time tends to t+ , the ball BT0 (x0 , Ak+1 h) keeps on looking like a cap. Since the scalar curvature on BT0 (x0 , Ak h) × [T0 , tk ] does not exceed 2Qh−2 , it follows from the pinching assumption, the gradient estimates in the canonical neighborhood assumption and the evolution equation of the metric that the diameter of the set BT0 (x0 , Ak h) at any 1 time t ∈ [T0 , t+ ) is bounded from above by 4δ − 2 h. These imply that no point of the ball BT0 (x0 , Ak h) at any time near t+ can be the center of a δ-neck for any 1 ¯ ¯ 0 < δ < δ(A, θ, ε) with δ(A, θ, ε) > 0 small enough, since 4δ − 2 h is much smaller than δ −1 h. However the solution disappears somewhere in the set BT0 (x0 , Ak h) at time t+ by a cutoff surgery and the surgery is always done along the middle two-sphere of a δ-neck. So the set BT0 (x0 , Ak h) at time t+ is a part of a capped horn. (Recall that we have declared extinct every compact component with positive curvature and every compact component lying in Ω \ Ωρ ). Hence for each point of BT0 (x0 , Ak h) the solution terminates at t+ . This proves assertion (ii). Corollary 7.3.7 (Perelman [104]). For any l < ∞ one can find A = A(l) < ∞ and θ = θ(l), 0 < θ < 1, with the following property. Suppose we are in the ¯ situation of the lemma above, with δ < δ(A, θ, ε). Consider smooth curves γ in the set BT0 (x0 , Ah), parametrized by t ∈ [T0 , Tγ ], such that γ(T0 ) ∈ BT0 (x0 , Ah 2 ) and either Tγ = T1 < T , or Tγ < T1 and γ(Tγ ) ∈ ∂BT0 (x0 , Ah), where x0 is any fixed point on a gluing cap at T0 and T1 = min{T, T0 + θh2 }. Then Z

Tγ

T0

2 (R(γ(t), t) + |γ(t)| ˙ )dt > l.

Proof. We know from Proposition 7.3.3 that on the standard solution, Z

0

θ

Rdt ≥ const.

Z

θ 0

(1 − t)−1 dt

= −const. · log(1 − θ).

By choosing θ = θ(l) sufficiently close to 1 we have the desired estimate for the standard solution. Let us consider the first case: Tγ = T1 < T . For θ = θ(l) fixed above, by Lemma 7.3.6, our solution in the subset BT0 (x0 , Ah) and in the time interval [T0 , Tγ ] is, after scaling with factor h−2 and shifting time T0 to zero, A−1 -close to the corresponding

432


subset in the standard solution for any sufficiently large A. So we have Z Tγ Z θ 2 (R(γ(t), t) + |γ(t)| ˙ )dt ≥ const. (1 − t)−1 dt 0

T0

= −const. · log(1 − θ).

Hence we have obtained the desired estimate in the first case. We now consider the second case: Tγ < T1 and γ(Tγ ) ∈ ∂BT0 (x0 , Ah). Let θ = θ(l) be chosen above and let Q = Q(l) be the maximum of the scalar curvature on the standard solution in the time interval [0, θ]. On the standard solution, we can choose A = A(l) so large that for each t ∈ [0, θ], distt (x0 , ∂B0 (x0 , A)) ≥ dist0 (x0 , ∂B0 (x0 , A)) − 4(Q + 1)t ≥ A − 4(Q + 1)θ 4 ≥ A 5

and A A ≤ , distt x0 , ∂B0 x0 , 2 2 where we have used Lemma 3.4.1(ii) in the first inequality. Now our solution in the subset BT0 (x0 , Ah) and in the time interval [T0 , Tγ ] is, after scaling with factor h−2 and shifting time T0 to zero, A−1 -close to the corresponding subset in the standard solution. This implies that for A = A(l) large enough 1 Ah ≤ 5

Z

Tγ

T0

|γ(t)|dt ˙ ≤

Z

Tγ

T0

2 |γ(t)| ˙ dt

! 12

1

· (Tγ − T0 ) 2 ,

Hence Z

Tγ

T0

2 (R(γ(t), t) + |γ(t)| ˙ )dt ≥

A2 > l. 25θ

This proves the desired estimate. 7.4. Justification of the Canonical Neighborhood Assumptions. We continue the induction argument for the construction of a long-time solution to the Ricci flow with surgery. Let us recall what we have done in the previous section. Let ε be an arbitrarily given positive constant satisfying 0 < ε ≤ 1/100. For an arbitrarily given compact orientable normalized three-manifold, we evolve it by the Ricci flow. We may assume that the solution goes singular at some time 0 < t+ 1 < +∞ and know that the solution satisfies the a priori assumptions (with accuracy ε) on [0, t+ 1 ) for a nonincreasing positive function r = r1 (t) (defined on [0, +∞)). Suppose that we have a solution + + + to the Ricci flow with surgery, defined on [0, t+ k ) with 0 < t1 < t2 < · · · < tk < +∞, satisfying the a priori assumptions (with accuracy ε) for some nonincreasing positive function r = rk (t) (defined on [0, +∞)), going singular at time t+ k and having δi -cutoff + ¯ surgeries at each time t+ , 1 ≤ i ≤ k − 1, where δ < δ(t ) for each 1 ≤ i ≤ k − 1. Then i i i ¯ + ), we can perform δk -cutoff surgeries at the time t+ and extend for any 0 < δk < δ(t k k + + ¯ the solution to the interval [0, t+ k+1 ) with tk+1 > tk . Here δ(t) is the positive function


433

defined in (7.3.16). We have already shown in Lemma 7.3.4 that the extended solution still satisfies the pinching assumption on [0, t+ k+1 ). In view of Theorem 7.1.1, there always is a nonincreasing positive function r = rk+1 (t), defined on [0, +∞), such that the canonical neighborhood assumption (with accuracy ε) holds on the extended time interval [0, t+ k+1 ) with the positive function r = rk+1 (t). Nevertheless, in order to prevent the surgery times from accumulating, the key is to choose the nonincreasing positive functions r = ri (t), i = 1, 2, . . ., uniformly. That is, to justify the canonical neighborhood assumption (with accuracy ε) for the indefinitely extending solution, we need to show that there exists a nonincreasing positive function re(t), defined on [0, +∞), which is independent of k, such that the above chosen nonincreasing positive functions satisfy ri (t) ≥ re(t),

on [0, +∞),

for all i = 1, 2, . . . , k + 1. ¯ By a further restriction on the positive function δ(t), we can verify this after proving the following assertion which was stated by Perelman in [104]. Proposition 7.4.1 (Justification of the canonical neighborhood assumption). Given any small ε > 0, there exist decreasing sequences 0 < rej < ε, κj > 0, and 0 < δej < ε2 , j = 1, 2, · · · , with the following property. Define the positive function e on [0, +∞) by δ(t) e = δej for t ∈ [(j − 1)ε2 , jε2 ). Suppose there is a surgically δ(t) modified solution, defined on [0, T ) with T < +∞, to the Ricci flow which satisfies the following: (1) it starts on a compact orientable three-manifold with normalized initial metric, and (2) it has only a finite number of surgeries such that each surgery at a time t ∈ (0, T ) is a δ(t)-cutoff surgery with e δ(t)}. ¯ 0 < δ(t) ≤ min{δ(t), T Then on each time interval [(j − 1)ε2 , jε2 ] [0, T ), j = 1, 2, · · · , the solution satisfies the κj -noncollapsing condition on all scales less than ε and the canonical neighborhood assumption (with accuracy ε) with r = rej .

Here and in the following, we call a (three-dimensional) surgically modified solution gij (t), 0 ≤ t < T , κ-noncollapsed at (x0 , t0 ) on the scales less than ρ (for some κ > 0, ρ > 0) if it satisfies the following property: whenever r < ρ and |Rm(x, t)| ≤ r−2 for all those (x, t) ∈ P (x0 , t0 , r, −r2 ) = {(x′ , t′ ) | x′ ∈ Bt′ (x0 , r), t′ ∈ [t0 − r2 , t0 ]}, for which the solution is defined, we have Vol t0 (Bt0 (x0 , r)) ≥ κr3 . Before we give the proof of the proposition, we need to verify a κ-noncollapsing estimate which was given by Perelman in [104]. Lemma 7.4.2. Given any 0 < ε ≤ ε¯0 (for some sufficiently small universal constant ε¯0 ), suppose we have constructed the sequences satisfying the proposition for 1 ≤ j ≤ m (for some positive integer m). Then there exists κ > 0, such that for any

434


e ε), 0 < δe < ε2 , which may also depend on the r, 0 < r < ε, one can find δe = δ(r, already constructed sequences, with the following property. Suppose we have a solution with a compact orientable normalized three-manifold as initial data, to the Ricci flow with finite number of surgeries on a time interval [0, T¯] with mε2 ≤ T¯ < (m + 1)ε2 , satisfying the assumptions and the conclusions of Proposition 7.4.1 on [0, mε2 ), and the canonical neighborhood assumption (with accuracy ε) with r on [mε2 , T¯], as well e δ(t)} ¯ as 0 < δ(t) ≤ min{δ, for any δ-cutoff surgery with δ = δ(t) at a time t ∈ 2 ¯ [(m − 1)ε , T ]. Then the solution is κ-noncollapsed on [0, T¯] for all scales less than ε. Proof. Consider a parabolic neighborhood P (x0 , t0 , r0 , −r02 ) , {(x, t) | x ∈ Bt (x0 , r0 ), t ∈ [t0 − r02 , t0 ]} with mε2 ≤ t0 ≤ T¯ and 0 < r0 < ε, where the solution satisfies |Rm| ≤ r0−2 , whenever it is defined. We will use an argument analogous to the proof of Theorem 3.3.2 (no local collapsing theorem I) to prove (7.4.1)

Vol t0 (Bt0 (x0 , r0 )) ≥ κr03 .

Let η be the universal positive constant in the definition of the canonical neighborhood assumption. Without loss of generality, we always assume η ≥ 10. Firstly, 1 we want to show that one may assume r0 ≥ 2η r. Obviously, the curvature satisfies the estimate |Rm(x, t)| ≤ 20r0−2 , 1 1 2 1 1 2 for those (x, t) ∈ P (x0 , t0 , 2η r0 , − 8η r0 ) = {(x, t) | x ∈ Bt (x0 , 2η r0 ), t ∈ [t0 − 8η r0 , t0 ]}, 1 ′ for which the solution is defined. When r0 < 2η r, we can enlarge r0 to some r0 ∈ [r0 , r] so that

|Rm| ≤ 20r0′−2 1 ′2 1 ′ r0 , − 8η r0 ) (whenever it is defined), and either the equality holds someon P (x0 , t0 , 2η

1 ′ 1 ′2 where in P (x0 , t0 , 2η r0 , −( 8η r 0 + ǫ′ )) for any arbitrarily small ǫ′ > 0 or r0′ = r. In the case that the equality holds somewhere, it follows from the pinching assumption that we have

R > 10r0′−2 1 ′2 1 ′ r0 , −( 8η r 0 + ǫ′ )) for any arbitrarily small ǫ′ > 0. Here, somewhere in P (x0 , t0 , 2η without loss of generality, we have assumed r is suitably small. Then by the gradient estimates in the definition of the canonical neighborhood assumption, we know

R(x0 , t0 ) > r0′−2 ≥ r−2 . Hence the desired noncollapsing estimate (7.4.1) in this case follows directly from the canonical neighborhood assumption. (Recall that we have excluded every compact component which has positive sectional curvature in the surgery procedure and then we have excluded them from the list of canonical neighborhoods. Here we also used the standard volume comparison when the canonical neighborhood is an ε-cap.)


435

While in the case that r0′ = r, we have the curvature bound |Rm(x, t)| ≤

1 r 2η

−2

,

1 1 1 r, −( 2η r)2 ) = {(x, t) | x ∈ Bt (x0 , 2η r), t ∈ [t0 − for those (x, t) ∈ P (x0 , t0 , 2η 1 ( 2η r)2 , t0 ]}, for which the solution is defined. It follows from the standard volume 1 comparison that we only need to verify the noncollapsing estimate (7.4.1) for r0 = 2η r. 1 Thus we have reduced the proof to the case r0 ≥ 2η r. Recall from Theorem 3.3.2 that if a solution is smooth everywhere, we can get a lower bound for the volume of the ball Bt0 (x0 , r0 ) as follows: define τ (t) = t0 − t and consider Perelman’s reduced volume function and the Li-Yau-Perelman distance associated to the point x0 ; take a point x ¯ at the time t = ε2 so that the Li-YauPerelman distance l attains its minimum lmin (τ ) = l(¯ x, τ ) ≤ 23 for τ = t0 − ε2 ; use it to obtain an upper bound for the Li-Yau-Perelman distance from x0 to each point of B0 (¯ x, 1), thus getting a lower bound for Perelman’s reduced volume at τ = t0 ; apply the monotonicity of Perelman’s reduced volume to deduce a lower bound for Perelman’s reduced volume at τ near 0, and then get the desired estimate for the volume of the ball Bt0 (x0 , r0 ). Now since our solution has undergone surgeries, we need to localize this argument to the region which is unaffected by surgery. We call a space-time curve in the solution track admissible if it stays in the spacetime region unaffected by surgery, and we call a space-time curve in the solution track a barely admissible curve if it is on the boundary of the set of admissible curves. First of all, we want to estimate the L-length of a barely admissible curve.

¯ r, rem , ε) > 0 with the following Claim. For any L < ∞ one can find δ¯ = δ(L, property. Suppose that we have a curve γ, parametrized by t ∈ [T0 , t0 ], (m − 1)ε2 ≤ T0 < t0 , such that γ(t0 ) = x0 , T0 is a surgery time, and γ(T0 ) lies in the gluing cap. ¯ Then we Suppose also each δ-cutoff surgery at a time in [(m − 1)ε2 , T¯] has δ ≤ δ. have an estimate Z t0 √ 2 (7.4.2) t0 − t(R+ (γ(t), t) + |γ(t)| ˙ )dt ≥ L T0

where R+ = max{R, 0}. 1 r and |Rm| ≤ r0−2 on P (x0 , t0 , r0 , −r02 ) (whenever it is defined), we Since r0 ≥ 2η can require δ¯ > 0, depending on r and rem , to be so small that γ(T0 ) does not lie in the region P (x0 , t0 , r0 , −r02 ). Let ∆t be the maximal number such that γ|[t0 −∆t,t0 ] ⊂ P (x0 , t0 , r0 , −∆t) (i.e., t0 − ∆t is the first time when γ escapes the parabolic region P (x0 , t0 , r0 , −r02 )). Obviously we only need to consider the case:

Z

t0

t0 −∆t

√ 2 t0 − t(R+ (γ(t), t) + |γ(t)| ˙ )dt < L.

We observe that ∆t can be bounded from below in terms of L and r0 . Indeed, if ∆t ≥ r02 , there is nothing to prove. Thus we may assume ∆t < r02 . By the curvature bound |Rm| ≤ r0−2 on P (x0 , t0 , r0 , −r02 ) and the Ricci flow equation we see Z

t0

t0 −∆t

|γ(t)|dt ˙ ≥ cr0

436


for some universal positive constant c. On the other hand, by the Cauchy-Schwarz inequality, we have Z

t0

t0 −∆t

|γ(t)|dt ˙ ≤

Z

t0

t0 −∆t

√ t0 − t(R+ + |γ| ˙ 2 )dt

1

1

12 Z ·

t0

t0 −∆t

√

≤ (2L) 2 (∆t) 4 ,

1 t0 − t

dt

12

which implies c2 r02 . 2L

1

(∆t) 2 ≥

(7.4.3) Thus Z

t0

T0

√ t0 − t(R+ + |γ| ˙ 2 )dt ≥

Z

t0 −∆t √

T0 1

≥ (∆t) 2

Z

t0 − t(R+ + |γ| ˙ 2 )dt

t0 −∆t

T0

(R+ + |γ| ˙ 2 )dt

Z t0 −∆t 2 2 c r0 (R+ + |γ| ˙ 2 )dt, , r0 ≥ min 2L T0 ¯ r, rem , ε) > 0 so small that while by Corollary 7.3.7, we can find δ¯ = δ(L, Z

t0 −∆t

T0

2 2 −1 c r0 (R+ + |γ| ˙ )dt ≥ L min , r0 . 2L 2

Then we have proved the desired assertion (7.4.2). Recall that for a curve γ, parametrized by τ = t0 − t ∈ [0, τ¯], with γ(0) = x0 and R τ¯ √ ˙ 2 )dτ . We can also define L+ (γ) τ¯ ≤ t0 − (m − 1)ε2 , we have L(γ) = 0 τ (R + |γ| by replacing R with R+ in the previous formula. Recall that R ≥ −1 at the initial time t = 0 for the normalized initial manifold. Recall that the surgeries occur at the parts where the scalar curvatures are very large. Thus we can apply the maximum principle to conclude that the solution with surgery still satisfies R ≥ −1 everywhere in space-time. This implies (7.4.4)

3

L+ (γ) ≤ L(γ) + (2ε2 ) 2 .

By applying the assertion (7.4.2), we now choose δ˜ > 0 (depending on r, ε and rem ) ˜ such that as each δ-cutoff surgery at the time interval t ∈ [(m − 1)ε2 , T ] has δ ≤ δ, every barely admissible curve γ from (x0 , t0 ) to a point (x, t) (with t ∈ [(m − 1)ε2 , t0 )) has √ L+ (γ) ≥ 22 2. Thus if the Li-Yau-Perelman distance from (x0 , t0 ) to a point (x, t) (with t ∈ [(m − 1)ε2 , t0 )) is achieved by a space-time curve which is not admissible, then its Li-YauPerelman distance has 3

(7.4.5)

l≥

L+ − (2ε2 ) 2 √ > 10ε−1 . 2 2ε


437

We also observe that the absolute value of l(x0 , τ ) is very small as τ closes to zere. Thus the maximum principle argument in Corollary 3.2.6 still works for our solutions with surgery because barely admissible curves do not attain the minimum. So we conclude that 3 lmin (¯ τ ) = min{l(x, τ¯) | x lies in the solution manifold at t0 − τ¯} ≤ 2 for τ¯ ∈ (0, t0 − (m − 1)ε2 ]. In particular, there exists a minimizing curve γ of lmin(t0 − (m − 1)ε2 ), defined on τ ∈ [0, t0 − (m − 1)ε2 ] with γ(0) = x0 , such that √ 3 √ · 2 2ε + 2 2ε3 2 ≤ 5ε,

L+ (γ) ≤

(7.4.6)

since 0 < ε ≤ ε¯0 with ε¯0 sufficiently small (to be further determined). Consequently, there exists a point (¯ x, t¯) on the minimizing curve γ with t¯ ∈ [(m − 1)ε2 + 41 ε2 , (m − 3 2 2 1)ε + 4 ε ] (i.e., τ ∈ [t0 − (m − 1)ε2 − 43 ε2 , t0 − (m − 1)ε2 − 14 ε2 ]) such that −2 R(¯ x, t¯) ≤ 25e rm .

(7.4.7) Otherwise, we have L+ (γ) ≥

Z

t0 −(m−1)ε2 − 41 ε2

t0 −(m−1)ε2 − 34 ε2

−2 > 25e rm

r

1 2 ε 4

> 5ε,

√ τ R(γ(τ ), t0 − τ )dτ

1 2 ε 2

since 0 < rem < ε. This contradicts (7.4.6). Next we want to get a lower bound for Perelman’s reduced volume of a ball around x ¯ of radius about rem at some time slightly before t¯. 1 −1 1 −1 η and θ2 = 64 η , where η is the universal positive constant Denote by θ1 = 16 in the gradient estimates (7.3.4). Since the solution satisfies the canonical neighborhood assumption on the time interval [(m − 1)ε2 , mε2 ), it follows from the gradient estimates (7.3.4) that (7.4.8)

−2 R(x, t) ≤ 400e rm

2 2 ¯ for those (x, t) ∈ P (¯ x, t¯, θ1 rem , −θ2 rem ) , {(x′ , t′ ) | x′ ∈ Bt′ (¯ x, θ1 rem ), t′ ∈ [t¯− θ2 rem , t]}, for which the solution is defined. And since the scalar curvature at the points where e −2 re−2 , the δ-cutoff surgeries occur in the time interval [(m − 1)ε2 , mε2 ) is at least (δ) m 2 the solution is well-defined on the whole parabolic region P (¯ x, t¯, θ1 rem , −θ2 rem ) (i.e., this parabolic region is unaffected by surgery). Thus by combining (7.4.6) and (7.4.8), we know that the Li-Yau-Perelman distance from (x0 , t0 ) to each point of the ball 2 (¯ Bt¯−θ2 rem x, θ1 rem ) is uniformly bounded by some universal constant. Let us define 2 (¯ Perelman’s reduced volume of the ball Bt¯−θ2 rem x, θ1 rem ), by 2 (Bt 2 (¯ x, θ1 rem )) Vet0 −t¯+θ2 rem ¯−θ2 r em Z 3 2 (4π(t0 − t¯ + θ2 rem ))− 2 =

em ) x,θ1 r Bt¯−θ re2 (¯ 2 m

2 2 (q), · exp(−l(q, t0 − t¯ + θ2 rem ))dVt¯−θ2 rem

438


where l(q, τ ) is the Li-Yau-Perelman distance from (x0 , t0 ). Hence by the κm noncollapsing assumption on the time interval [(m − 1)ε2 , mε2 ), we conclude that 2 (¯ Perelman’s reduced volume of the ball Bt¯−θ2 rem x, θ1 rem ) is bounded from below by a positive constant depending only on κm and rem . Finally we want to apply a local version of the monotonicity of Perelman’s reduced volume to get a lower bound estimate for the volume of the ball Bt0 (x0 , r0 ).

We have seen that the Li-Yau-Perelman distance from (x0 , t0 ) to each point of the 2 (¯ ball Bt¯−θ2 rem x, θ1 rem ) is uniformly bounded by some universal constant. Now we can choose a sufficiently small (universal) positive constant ε¯0 such that when 0 < ε ≤ ε¯0 , 2 (¯ x, θ1 rem ) can be connected to (x0 , t0 ) by (7.4.5), all the points in the ball Bt¯−θ2 rem by shortest L-geodesics, and all of these L-geodesics are admissible (i.e., they stay in the region unaffected by surgery). The union of all shortest L-geodesics from (x0 , t0 ) 2 (¯ 2 (¯ to the ball Bt¯−θ2 rem x, θ1 rem ) defined by CBt¯−θ2 rem x, θ1 rem ) = {(x, t) | (x, t) lies in 2 (¯ a shortest L-geodesic from (x0 , t0 ) to a point in Bt¯−θ2 rem x, θ1 rem )}, forms a conelike subset in space-time with the vertex (x0 , t0 ). Denote B(t) by the intersection of 2 (¯ the cone-like subset CBt¯−θ2 rem x, θ1 rem ) with the time-slice at t. Perelman’s reduced volume of the subset B(t) is given by Vet0 −t (B(t)) =

Z

3

B(t)

(4π(t0 − t))− 2 exp(−l(q, t0 − t))dVt (q).

2 (¯ Since the cone-like subset CBt¯−θ2 rem x, θ1 rem ) lies entirely in the region unaffected by surgery, we can apply Perelman’s Jacobian comparison theorem (Theorem 3.2.7) to conclude that

(7.4.9)

2 (Bt 2 (¯ x, θ1 rem )) Vet0 −t (B(t)) ≥ Vet0 −t¯+θ2 rem ¯−θ2 r em

≥ c(κm , rem ),

2 for all t ∈ [t¯− θ2 rem , t0 ], where c(κm , rem ) is some positive constant depending only on κm and rem . 1

Set ξ = r0−1 Vol t0 (Bt0 (x0 , r0 )) 3 . Our purpose is to give a positive lower bound 2 . for ξ. Without loss of generality, we may assume ξ < 14 , thus 0 < ξr02 < t0 − ¯t + θ2 rem 2 2 e 0 − ξr ) the subset of the time-slice {t = t0 − ξr } of which every Denote by B(t 0 0 point can be connected to (x0 , t0 ) by an admissible shortest L-geodesic. Clearly, e 0 − ξr2 ). We now argue as in the proof of Theorem 3.3.2 to bound B(t0 − ξr02 ) ⊂ B(t 0 e 0 − ξr2 ) from above. Perelman’s reduced volume of B(t 0 1 ˜ ε, rem ) sufficiently small, the whole region P (x0 , t0 , r0 , Since r0 ≥ 2η r and δ˜ = δ(r, 2 −r0 ) is unaffected by surgery. Then by exactly the same argument as in deriving (3.3.5), we see that there exists a universal positive constant ξ0 such that when 0 < ξ ≤ ξ0 , there holds

(7.4.10)

L exp

{|υ|≤ 41 ξ

−1 2

}

(ξr02 ) ⊂ Bt0 (x0 , r0 ).


439

e 0 − ξr2 ) is given by Perelman’s reduced volume of B(t 0 (7.4.11)

e 0 − ξr02 )) Veξr02 (B(t Z 3 (4πξr02 )− 2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q) = e 0 −ξr02 ) B(t

=

Z

3

e 0 −ξr02 )∩L exp B(t

+

Z

{|υ|≤ 1 ξ 4

−1 2}

(ξr02 )

(4πξr02 )− 2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q) 3

e

B(t0 −ξr02 )\L exp (ξr02 ) −1 {|υ|≤ 1 ξ 2 } 4

(4πξr02 )− 2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q).

The first term on the RHS of (7.4.11) can be estimated by Z 3 (7.4.12) (4πξr02 )− 2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q) e 0 −ξr02 )∩L exp B(t

{|υ|≤ 1 ξ 4 3 2

3

−1 2}

(ξr02 )

≤ eCξ (4π)− 2 · ξ

for some universal constant C, as in deriving (3.3.7). While as in deriving (3.3.8), the second term on the RHS of (7.4.11) can be estimated by Z 3 (7.4.13) (4πξr02 )− 2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q) e 0 −ξr02 )\L exp B(t

≤

{|υ|≤ 1 ξ 4

Z

−1 2}

(ξr02 )

3

1 {|υ|> 41 ξ − 2 3

= (4π)− 2

Z

(4πτ )− 2 exp(−l(τ ))J (τ )|τ =0 dυ }

1 {|υ|> 14 ξ − 2

exp(−|υ|2 )dυ, }

where we have used Perelman’s Jacobian comparison theorem (Theorem 3.2.7) in the first inequality. Hence the combination of (7.4.9), (7.4.11), (7.4.12) and (7.4.13) bounds ξ from below by a positive constant depending only on κm and rem . Therefore we have completed the proof of the lemma. Now we can prove the proposition.

Proof of Proposition 7.4.1. The proof of the proposition is by induction: having constructed our sequences for 1 ≤ j ≤ m, we make one more step, defining rem+1 , κm+1 , δem+1 , and redefining δem = δem+1 . In view of the previous lemma, we only need to define rem+1 and δem+1 . In Theorem 7.1.1 we have obtained the canonical neighborhood structure for smooth solutions. When adapting the arguments in the proof of Theorem 7.1.1 to the present surgical solutions, we will encounter the new difficulty of how to take a limit for the surgically modified solutions. The idea to overcome the difficulty consists of two parts. The first part, due to Perelman [104], is to choose δem and δem+1 small enough to push the surgical regions to infinity in space. (This is the reason why we need to redefine δem = δem+1 .) The second part is to show that solutions are smooth on some small, but uniform, time intervals (on compact subsets) so that we can apply Hamilton’s compactness theorem, since we only have curvature bounds;

440


otherwise Shi’s interior derivative estimate may not be applicable. In fact, the second part is more crucial. That is just concerned with the question of whether the surgery times accumulate or not. Our argument will use the canonical neighborhood characterization of the standard solution in Lemma 7.3.5. We now start to prove the proposition by contradiction. Suppose for sequence of 1 positive numbers rα and δeαβ , satisfying rα → 0 as α → ∞ and δeαβ ≤ α·β (→ 0), there

αβ exist sequences of solutions gij to the Ricci flow with surgery, where each of them has only a finite number of cutoff surgeries and has a compact orientable normalized three-manifold as initial data, so that the following two assertions hold: (i) each δ-cutoff at a time t ∈ [(m − 1)ε2 , (m + 1)ε2 ] satisfies δ ≤ δeαβ ; and (ii) the solutions satisfy the statement of the proposition on [0, mε2 ], but violate the canonical neighborhood assumption (with accuracy ε) with r = rα on [mε2 , (m + 1)ε2 ]. αβ For each solution gij , we choose t¯ (depending on α and β) to be the nearly first time for which the canonical neighborhood assumption (with accuracy ε) is violated. More precisely, we choose t¯ ∈ [mε2 , (m + 1)ε2 ] so that the canonical neighborhood assumption with r = rα and with accuracy parameter ε is violated at some (¯ x, t¯), however the canonical neighborhood assumption with accuracy parameter 2ε holds on t ∈ [mε2 , t¯]. After passing to subsequences, we may assume each δeαβ is less than the δe in Lemma 7.4.2 with r = rα when α is fixed. Then by Lemma 7.4.2 we have uniform κ-noncollapsing on all scales less than ε on [0, ¯t] with some κ > 0 independent of α, β. Slightly abusing notation, we will often drop the indices α and β. αβ Let geij be the rescaled solutions around (¯ x, t¯) with factors R(¯ x, t¯)(≥ r−2 → +∞) and shift the times t¯ to zero. We hope to take a limit of the rescaled solutions for subsequences of α, β → ∞ and show the limit is an orientable ancient κ-solution, which will give the desired contradiction. We divide our arguments into the following six steps. αβ e tˆ) ≤ A (for Step 1. Let (y, tˆ) be a point on the rescaled solution e gij with R(y, 2 some A ≥ 1) and tˆ ∈ [−(t¯ − (m − 1)ε )R(¯ x, t¯), 0]. Then we have estimate

(7.4.14)

e t) ≤ 10A R(x,

1 for those (x, t) in the parabolic neighborhood P (y, tˆ, 12 η −1 A− 2 , − 18 η −1 A−1 ) , et′ (y, 1 η −1 A− 21 ), t′ ∈ [tˆ − 1 η −1 A−1 , tˆ]}, for which the rescaled so{(x′ , t′ ) | x′ ∈ B 2 8 lution is defined. Indeed, as in the first step of the proof of Theorem 7.1.1, this follows directly from the gradient estimates (7.3.4) in the canonical neighborhood assumption with parameter 2ε.

Step 2. In this step, we will prove three time extension results. Assertion 1. For arbitrarily fixed α, 0 < A < +∞, 1 ≤ C < +∞ and 0 ≤ B < 12 ε2 (rα )−2 − 81 η −1 C −1 , there is a β0 = β0 (ε, A, B, C) (independent of α) αβ e0 (¯ such that if β ≥ β0 and the rescaled solution e gij on the ball B x, A) is defined on a time interval [−b, 0] with 0 ≤ b ≤ B and the scalar curvature satisfies e t) ≤ C, R(x,

e0 (¯ on B x, A) × [−b, 0],

αβ e0 (¯ then the rescaled solution e gij on the ball B x, A) is also defined on the extended 1 −1 −1 time interval [−b − 8 η C , 0].


441

Before giving the proof, we make a simple observation: once a space point in the Ricci flow with surgery is removed by surgery at some time, then it never appears for later time; if a space point at some time t cannot be defined before the time t , then either the point lies in a gluing cap of the surgery at time t or the time t is the initial time of the Ricci flow. Proof of Assertion 1. Firstly we claim that there exists β0 = β0 (ε, A, B, C) such αβ e0 (¯ that when β ≥ β0 , the rescaled solution geij on the ball B x, A) can be defined before e0 (¯ the time −b (i.e., there are no surgeries interfering in B x, A) × [−b − ǫ′ , −b] for some ′ ǫ > 0). e0 (¯ We argue by contradiction. Suppose not, then there is some point x˜ ∈ B x, A) αβ such that the rescaled solution e gij at x ˜ cannot be defined before the time −b. By the above observation, there is a surgery at the time −b such that the point x ˜ lies in the instant gluing cap. 1 ˜ (= R(¯ Let h x, t¯) 2 h) be the cut-off radius at the time −b for the rescaled ˜ ≤ R(˜ ˜ e x, −b)− 21 ≤ Dh. solution. Clearly, there is a universal constant D such that D−1 h By Lemma 7.3.4 and looking at the rescaled solution at the time −b, the gluing 1 ˜ constitute a (δeαβ ) 2 -cap K. For any fixed cap and the adjacent δ-neck, of radius h, small positive constant δ ′ (much smaller than ε), we see that 1 −1 e e(−b) (˜ B x, (δ ′ ) R(˜ x, −b)− 2 ) ⊂ K

when β large enough. We first verify the following

˜ >0 Claim 1. For any small constants 0 < θ˜ < 1, δ ′ > 0, there exists a β(δ ′ , ε, θ) ˜ we have such that when β ≥ β(δ ′ , ε, θ), −1 ˜ αβ e(−b) (˜ (i) the rescaled solution geij over B x, (δ ′ ) h) is defined on the time interval 2 ˜ ˜ [−b, 0] ∩ [−b, −b + (1 − θ)h ]; 1 −1 ˜ e(−b) (˜ (ii) the ball B x, (δ ′ ) h) in the (δeαβ ) 2 -cap K evolved by the Ricci flow on ˜h ˜ 2 ] is, after scaling with factor h ˜ −2 , the time interval [−b, 0] ∩ [−b, −b + (1 − θ) δ ′ -close (in the C [δ solution.

′−1

]

topology) to the corresponding subset of the standard

This claim essentially follows from Lemma 7.3.6. Indeed, suppose there is a ˜h ˜ 2 ] which removes some point surgery at some time t˜˜ ∈ [−b, 0] ∩ (−b, −b + (1 − θ) −1 ˜ ˜ We assume t˜ ∈ (−b, 0] is the first time with that property. e(−b) (˜ ˜˜ ∈ B x x, (δ ′ ) h). ¯ ′ , ε, θ) ˜ such that if δeαβ < δ, ¯ then the ball Then by Lemma 7.3.6, there is a δ¯ = δ(δ 1

−1 ˜ e(−b) (˜ B x, (δ ′ ) h) in the (δeαβ ) 2 -cap K evolved by the Ricci flow on the time interval ˜ −2 , δ ′ -close to the corresponding subset of the [−b, ˜t˜) is, after scaling with factor h −1 ˜ e(−b) (˜ standard solution. Note that the metrics for times in [−b, t˜˜) on B x, (δ ′ ) h) are −1 ˜ keeps looking like a cap e(−b) (˜ equivalent. By Lemma 7.3.6, the solution on B x, (δ ′ ) h) for t ∈ [−b, ˜˜t). On the other hand, by the definition, the surgery is always done along the middle two-sphere of a δ-neck with δ < δeαβ . Then for β large, all the points in −1 ˜ e(−b) (˜ B x, (δ ′ ) h) are removed (as a part of a capped horn) at the time t˜˜. But x˜ (near the tip of the cap) exists past the time t˜˜. This is a contradiction. Hence we have proved ˜ is defined on the time interval [−b, 0] ∩ [−b, −b + (1 − θ) ˜h ˜ 2 ]. e(−b) (˜ that B x, (δ ′ )−1 h)

442


˜h ˜ 2 ]) e(−b) (˜ The δ ′ -closeness of the solution on B x, (δ ′ ) h)×([−b, 0]∩[−b, −b+(1− θ) with the corresponding subset of the standard solution follows from Lemma 7.3.6. Then we have proved Claim 1. We next verify the following −1

˜ ˜h ˜ 2 when β large. Claim 2. There is θ˜ = θ(CB), 0 < θ˜ < 1, such that b ≤ (1 − θ) Note from Proposition 7.3.3, there is a universal constant D′ > 0 such that the standard solution satisfies the following curvature estimate R(y, s) ≥

2D′ . 1−s

We choose θ˜ = D′ /2(D′ + CB). Then for β large enough, the rescaled solution satisfies (7.4.15)

e t) ≥ R(x,

D′ ˜ −2 1 − (t + b)h

˜ −2 h

−1 ˜ ˜h ˜ 2 ]). e(−b) (˜ on B x, (δ ′ ) h) × ([−b, 0] ∩ [−b, −b + (1 − θ) ˜h ˜ 2 . Then by combining with the assumption R(˜ e x, t) ≤ C for Suppose b ≥ (1 − θ) 2 ˜ ˜ t = (1 − θ)h − b, we have

C≥

D′ ˜ −2 , h ˜ −2 1 − (t + b)h

and then ′ ˜ 1+ D . 1 ≥ (1 − θ) CB This is a contradiction. Hence we have proved Claim 2. The combination of the above two claims shows that there is a positive constant ˜ ˜ such 0 < θ˜ = θ(CB) < 1 such that for any small δ ′ > 0, there is a positive β(δ ′ , ε, θ) ′ 2 ˜ ˜ ˜ that when β ≥ β(δ , ε, θ), we have b ≤ (1 − θ)h and the rescaled solution in the ball ˜ on the time interval [−b, 0] is, after scaling with factor h ˜ −2 , δ ′ -close e(−b) (˜ B x, (δ ′ )−1 h) ′ −1 ( in the C [(δ ) ] topology) to the corresponding subset of the standard solution. e ≤ C on B e0 (¯ By (7.4.15) and the assumption R x, A) × [−b, 0], we know that the ˜ cut-off radius h at the time −b for the rescaled solution satisfies r D′ ˜ h≥ . C Let δ ′ > 0 be much smaller than ε and min{A−1 , A}. Since d˜0 (˜ x, x ¯) ≤ A, it follows ˜ depending only on θ˜ such that d˜(−b) (˜ ˜ ≪ that there is constant C(θ) x, x ¯) ≤ C(θ)A ′ −1 ˜ (δ ) h. We now apply Lemma 7.3.5 with the accuracy parameter ε/2. Let C(ε/2) be the positive constant in Lemma 7.3.5. Without loss of generality, we may assume the positive constant C1 (ε) in the canonical neighborhood assumption is larger than 4C(ε/2). When δ ′ (> 0) is much smaller than ε and min{A−1 , A}, the point x¯ at the time t¯ has a neighborhood which is either a 43 ε-cap or a 34 ε-neck. Since the canonical neighborhood assumption with accuracy parameter ε is violated at (¯ x, t¯), the neighborhood of the point x¯ at the new time zero for the rescaled


443

solution must be a 34 ε-neck. By Lemma 7.3.5 (b), we know the neighborhood is the slice at the time zero of the parabolic neighborhood 4 e x, 0)− 21 , − min{R(¯ e x, 0)−1 , b}) P (¯ x, 0, ε−1 R(¯ 3

e x, 0) = 1) which is 3 ε-close (in the C [ 3 ε ] topology) to the corresponding (with R(¯ 4 subset of the evolving standard cylinder S2 × R over the time interval [− min{b, 1}, 0] with scalar curvature 1 at the time zero. If b ≥ 1, the 34 ε-neck is strong, which is a contradiction. While if b < 1, the 34 ε-neck at time −b is contained in the union of the gluing cap and the adjacent δ-neck where the δ-cutoff surgery took place. Since ε is small (say ε < 1/100), it is clear that the point x¯ at time −b is the center of an ε-neck which is entirely contained in the adjacent δ-neck. By the proof of Lemma 7.3.2, the adjacent δ-neck approximates an ancient κ-solution. This implies the point x ¯ at the time t¯ has a strong ε-neck, which is also a contradiction. Hence we have proved that there exists β0 = β0 (ε, A, B, C) such that when β ≥ β0 , e0 (¯ the rescaled solution on the ball B x, A) can be defined before the time −b. αβ αβ Let [tA , 0] ⊃ [−b, 0] be the largest time interval so that the rescaled solution geij αβ 1 −1 −1 e0 (¯ can be defined on B x, A) × [tαβ C for A , 0]. We finally claim that tA ≤ −b − 8 η β large enough. Indeed, suppose not, by the gradient estimates as in Step 1, we have the curvature estimate 4 −1

e t) ≤ 10C R(x,

e0 (¯ on B x, A) × [tαβ A , −b]. Hence we have the curvature estimate e t) ≤ 10C R(x,

e0 (¯ on B x, A) × [tαβ By the above argument there is a β0 = β0 (ε, A, B + A , 0]. 1 −1 −1 e0 (¯ C , 10C) such that for β ≥ β0 , the solution in the ball B x, A) can be de8η αβ fined before the time tA . This is a contradiction. Therefore we have proved Assertion 1. Assertion 2. For arbitrarily fixed α, 0 < A < +∞, 1 ≤ C < +∞ and 1 −1 0 < B < 21 ε2 (rα )−2 − 50 η , there is a β0 = β0 (ε, A, B, C) (independent of α) such αβ e0 (¯ that if β ≥ β0 and the rescaled solution e gij on the ball B x, A) is defined on a time 1 −1 ′ ′ and the scalar curvature interval [−b + ǫ , 0] with 0 < b ≤ B and 0 < ǫ < 50 η satisfies e t) ≤ C R(x,

e0 (¯ on B x, A) × [−b + ǫ′ , 0],

e0 (¯ e −b + ǫ′ ) ≤ 3 , then the rescaled and there is a point y ∈ B x, A) such that R(y, 2 αβ 1 −1 η , 0] and solution geij at y is also defined on the extended time interval [−b − 50 satisfies the estimate for t ∈ [−b −

1 −1 , −b + ǫ′ ]. 50 η

e t) ≤ 15 R(y,

Proof of Assertion 2. We imitate the proof of Assertion 1. If the rescaled solution αβ 1 −1 η , −b + ǫ′ ), then there is a geij at y cannot be defined for some time in [−b − 50

444


1 −1 surgery at some time t˜˜ ∈ [−b − 50 η , −b + ǫ′ ] such that y lies in the instant gluing 1 ˜ (= R(¯ cap. Let h x, t¯) 2 h) be the cutoff radius at the time t˜˜ for the rescaled solution. ˜ By ˜ ≤ R(y, e t˜˜)− 21 ≤ Dh. Clearly, there is a universal constant D > 1 such that D−1 h the gradient estimates as in Step 1, the cutoff radius satisfies

˜ ≥ D−1 15− 21 . h

As in Claim 1 (i) in the proof of Assertion 1, for any small constants 0 < θ˜ < 12 , ˜ > 0 such that for β ≥ β(δ ′ , ε, θ), ˜ there is no surgery δ ′ > 0, there exists a β(δ ′ , ε, θ) ˜ ˜ ˜ ′ −1 2 ˜ × ([t˜, (1 − θ) ˜h ˜ + t˜] ∩ (t˜, 0]). Without loss of generality, e˜(y, (δ ) h) interfering in B t˜ we may assume that the universal constant η is much larger than D. Then we have ˜h ˜ 2 + t˜ ˜ > −b + 1 η −1 . As in Claim 2 in the proof of Assertion 1, we can use (1 − θ) 50 ˜ ˜h ˜ 2 + t˜˜ ≥ 0; the curvature bound assumption to choose θ˜ = θ(B, C) such that (1 − θ) otherwise C≥

D′ θ˜˜h2

for some universal constant D′ > 1, and |t˜˜ + b| ≤

1 −1 η , 50

which implies D′ ˜ 1+ 1 ≥ (1 − θ) 1 −1 C B + 50 η

!

.

1 −1 This is a contradiction if we choose θ˜ = D′ /2(D′ + C(B + 50 η )). ˜ ˜ So there is a positive constant 0 < θ = θ(B, C) < 1 such that for any δ ′ > 0, ˜ such that when β ≥ β(δ ′ , ε, θ), ˜ we have −t˜˜ ≤ (1 − θ) ˜h ˜2 there is a positive β(δ ′ , ε, θ) ˜ ′ −1 ˜ e and the solution in the ball Bt˜˜(˜ x, (δ ) h) on the time interval [t˜, 0] is, after scaling ′−1 −2 ′ ˜ with factor h , δ -close (in the C [δ ] topology) to the corresponding subset of the standard solution. Then exactly as in the proof of Assertion 1, by using the canonical neighborhood structure of the standard solution in Lemma 7.3.5, this gives the desired contradiction with the hypothesis that the canonical neighborhood assumption with accuracy parameter ε is violated at (¯ x, t¯), for β sufficiently large. The curvature estimate at the point y follows from Step 1. Therefore the proof of Assertion 2 is complete. Note that the standard solution satisfies R(x1 , t) ≤ D′′ R(x2 , t) for any t ∈ [0, 12 ] and any two points x1 , x2 , where D′′ ≥ 1 is a universal constant.

Assertion 3. For arbitrarily fixed α, 0 < A < +∞, 1 ≤ C < +∞, there is a 1 β0 = β0 (ε, AC 2 ) such that if any point (y0 , t0 ) with 0 ≤ −t0 < 21 ε2 (rα )−2 − 18 η −1 C −1 αβ e 0 , t0 ) ≤ C , then either the rescaled of the rescaled solution e gij for β ≥ β0 satisfies R(y 1 −1 −1 η C , t0 ] and the rescaled scalar solution at y0 can be defined at least on [t0 − 16 curvature satisfies i h e 0 , t) ≤ 10C for t ∈ t0 − 1 η −1 C −1 , t0 , R(y 16


445

or we have e 1 , t0 ) ≤ 2D′′ R(x e 2 , t0 ) R(x

et0 (y0 , A), where D′′ is the above universal constant. for any two points x1 , x2 ∈ B

αβ Proof of Assertion 3. Suppose the rescaled solution geij at y0 cannot be defined 1 −1 −1 for some t ∈ [t0 − 16 η C , t0 ); then there is a surgery at some time t˜ ∈ [t0 − 1 1 −1 −1 ˜ (= R(¯ C , t0 ] such that y0 lies in the instant gluing cap. Let h x, t¯) 2 h) be the 16 η αβ cutoff radius at the time t˜ for the rescaled solution geij . By the gradient estimates as in Step 1, the cutoff radius satisfies

˜ ≥ D−1 10− 21 C − 12 , h

where D is the universal constant in the proof of the Assertion 1. Since we assume η ˜ 2 + t˜ > t0 . As in Claim 1 (ii) in the is suitably larger than D as before, we have 12 h ′ proof of Assertion 1, for arbitrarily small δ > 0, we know that for β large enough the ˜ −2 , δ ′ -close e˜(y0 , (δ ′ )−1 ˜h) × [t˜, t0 ] is, after scaling with factor h rescaled solution on B t ′ −1 [(δ ) ] (in the C topology) to the corresponding subset of the standard solution. Since ˜ ≫ A for β large enough, Assertion 3 follows from the curvature estimate of (δ ′ )−1 h standard solution in the time interval [0, 21 ]. Step 3. For any subsequence (αk , βk ) of (α, β) with rαk → 0 and δ αk βk → 0 as k → ∞, we next argue as in the second step of the proof of Theorem 7.1.1 to αk βk show that the curvatures of the rescaled solutions g˜ij at the new times zero (after shifting) stay uniformly bounded at bounded distances from x ¯ for all sufficiently large k. More precisely, we will prove the following assertion: αk βk Assertion 4. Given any subsequence of the rescaled solutions g˜ij with rαk → αk βk 0 and δ → 0 as k → ∞, then for any L > 0, there are constants C(L) > 0 and αk βk k(L) such that the rescaled solutions g˜ij satisfy ˜ 0) ≤ C(L) for all points x with d˜0 (x, x¯) ≤ L and all k ≥ 1; (i) R(x, ˜0 (¯ (ii) the rescaled solutions over the ball B x, L) are defined at least on the time 1 −1 η C(L)−1 , 0] for all k ≥ k(L). interval [− 16

Proof of Assertion 4. For each ρ > 0, set n o ˜ 0) | k ≥ 1 and d˜0 (x, x¯) ≤ ρ in the rescaled solutions g˜αk βk M (ρ) = sup R(x, ij

and

ρ0 = sup{ρ > 0 | M (ρ) < +∞}. Note that the estimate (7.4.14) implies that ρ0 > 0. For (i), it suffices to prove ρ0 = +∞. We argue by contradiction. Suppose ρ0 < +∞. Then there is a sequence of αk βk ˜ 0) → +∞. points y in the rescaled solutions g˜ij with d˜0 (¯ x, y) → ρ0 < +∞ and R(y, ˜0 (¯ Denote by γ a minimizing geodesic segment from x ¯ to y and denote by B x, ρ0 ) the αk βk open geodesic ball centered at x ¯ of radius ρ0 on the rescaled solution g˜ij . First, we claim that for any 0 < ρ < ρ0 with ρ near ρ0 , the rescaled solutions 1 −1 ˜0 (¯ on the balls B x, ρ) are defined on the time interval [− 16 η M (ρ)−1 , 0] for all large

446


k. Indeed, this follows from Assertion 3 or Assertion 1. For the later purpose in Step 6, we now present an argument by using Assertion 3. If the claim is not true, 1 −1 η M (ρ)−1 , 0] such that some point then there is a surgery at some time t˜ ∈ [− 16 ˜0 (¯ y˜ ∈ B x, ρ) lies in the instant gluing cap. We can choose sufficiently small δ ′ > 0 1 ˜ where h ˜ ≥ D−1 20− 12 M (ρ)− 21 is the cutoff radius of the such that 2ρ0 < (δ ′ )− 2 h, rescaled solutions at t˜. By applying Assertion 3 with (˜ y , 0) = (y0 , t0 ), we see that there is a k(ρ0 , M (ρ)) > 0 such that when k ≥ k(ρ0 , M (ρ)), e 0) ≤ 2D′′ R(x,

e0 (¯ for all x ∈ B x, ρ). This is a contradiction as ρ → ρ0 . Since for each fixed 0 < ρ < ρ0 with ρ near ρ0 , the rescaled solutions are defined on 1 −1 ˜0 (¯ η M (ρ)−1 , 0] for all large k, by Step 1 and Shi’s derivative estimate, B x, ρ) × [− 16 we know that the covariant derivatives and higher order derivatives of the curvatures 1 −1 ˜0 (¯ on B x, ρ − (ρ02−ρ) ) × [− 32 η M (ρ)−1 , 0] are also uniformly bounded. By the uniform κ-noncollapsing property and Hamilton’s compactness theorem (Theorem 4.1.5), after passing to a subsequence, we can assume that the marked seαk βk ∞ ˜0 (¯ quence (B x, ρ0 ), geij ,x ¯) converges in the Cloc topology to a marked (noncomplete) ∞ manifold (B∞ , e gij , x ¯) and the geodesic segments γ converge to a geodesic segment (missing an endpoint) γ∞ ⊂ B∞ emanating from x ¯. Clearly, the limit has nonnegative sectional curvature by the pinching assumption. Consider a tubular neighborhood along γ∞ defined by [ e∞ (q0 ))− 21 ), V = B∞ (q0 , 4π(R q0 ∈γ∞

e∞ denotes the scalar curvature of the limit and where R e∞ (q0 ))− 12 ) B∞ (q0 , 4π(R

e∞ (q0 ))− 21 . Let B ¯∞ denote the is the ball centered at q0 ∈ B∞ with the radius 4π(R ∞ ¯ completion of (B∞ , e gij ), and y∞ ∈ B∞ the limit point of γ∞ . Exactly as in the second step of the proof of Theorem 7.1.1, it follows from the canonical neighborhood ∞ assumption with accuracy parameter 2ε that the limiting metric geij is cylindrical at any point q0 ∈ γ∞ which is sufficiently close to y∞ and then the metric space V¯ = V ∪ {y∞ } by adding the point y∞ has nonnegative curvature in the Alexandrov sense. Consequently we have a three-dimensional non-flat tangent cone Cy∞ V¯ at y∞ which is a metric cone with aperture ≤ 20ε. On the other hand, note that by the canonical neighborhood assumption, the canonical 2ε-neck neighborhoods are strong. Thus at each point q ∈ V near y∞ , the ∞ limiting metric e gij actually exists on the whole parabolic neighborhood V

\

P

1 e∞ (q))− 12 , − 1 η −1 (R e∞ (q))−1 , q, 0, η −1 (R 3 10

and is a smooth solution of the Ricci flow there. Pick z ∈ Cy∞ V¯ with distance one from the vertex y∞ and it is nonflat around z. By definition the ball B(z, 12 ) ⊂ Cy∞ V¯ is the Gromov-Hausdorff convergent limit of the scalings of a sequence of balls ∞ ∞ B∞ (zℓ , σℓ )(⊂ (V, e gij )) where σℓ → 0. Since the estimate (7.4.14) survives on (V, e gij ) for all A < +∞, and the tangent cone is three-dimensional and nonflat around z,


447

∞ we see that this convergence is actually in the Cloc topology and over some ancient 1 time interval. Since the limiting B∞ (z, 2 )(⊂ Cy∞ V¯ ) is a piece of nonnegatively curved nonflat metric cone, we get a contradiction with Hamilton’s strong maximum principle (Theorem 2.2.1) as before. So we have proved ρ0 = ∞. This proves (i). By the same proof of Assertion 1 in Step 2, we can further show that for any L, ˜0 (¯ the rescaled solutions on the balls B x, L) are defined at least on the time interval 1 −1 [− 16 η C(L)−1 , 0] for all sufficiently large k. This proves (ii).

Step 4. For any subsequence (αk , βk ) of (α, β) with rαk → 0 and δeαk βk → 0 as k → ∞, by Step 3, the κ-noncollapsing property and Hamilton’s compactness αk βk ∞ theorem, we can extract a Cloc convergent subsequence of g˜ij over some space-time open subsets containing the slice {t = 0}. We now want to show any such limit has bounded curvature at t = 0. We prove by contradiction. Suppose not, then there is a sequence of points zℓ divergent to infinity in the limiting metric at time zero with curvature divergent to infinity. Since the curvature at zℓ is large (comparable to one), zℓ has a canonical neighborhood which is a 2ε-cap or strong 2ε-neck. Note that the boundary of 2ε-cap lies in some 2ε-neck. So we get a sequence of 2ε-necks with radius going to zero. Note also that the limit has nonnegative sectional curvature. Without loss of generality, we may assume 2ε < ε0 , where ε0 is the positive constant in Proposition 6.1.1. Thus this arrives at a contradiction with Proposition 6.1.1. Step 5. In this step, we will choose some subsequence (αk , βk ) of (α, β) so that αk βk we can extract a complete smooth limit of the rescaled solutions geij to the Ricci flow with surgery on a time interval [−a, 0] for some a > 0. Choose αk , βk → ∞ so that rαk → 0, δeαk βk → 0, and Assertion 1, 2, 3 hold with α = αk , β = βk for all A ∈ {p/q | p, q = 1, 2, . . . , k}, and B, C ∈ {1, 2, . . . , k}. By Step αk βk ∞ 3, we may assume the rescaled solutions geij converge in the Cloc topology at the time t = 0. Since the curvature of the limit at t = 0 is bounded by Step 4, it follows from Assertion 1 in Step 2 and the choice of the sequence (αk , βk ) that the limiting ∞ (M∞ , geij (·, t)) is defined at least on a backward time interval [−a, 0] for some positive constant a and is a smooth solution to the Ricci flow there.

Step 6. We further want to extend the limit in Step 5 backwards in time to αk βk infinity to get an ancient κ-solution. Let e gij be the convergent sequence obtained in the above Step 5. Denote by n tmax = sup t′ |we can take a smooth limit on (−t′ , 0] (with bounded curvature at each time slice) from a subsequence of o αk βk the rescaled solutions geij .

αk βk We first claim that there is a subsequence of the rescaled solutions geij which con∞ ∞ verges in the Cloc topology to a smooth limit (M∞ , e gij (·, t)) on the maximal time interval (−tmax , 0]. Indeed, let tℓ be a sequence of positive numbers such that tℓ → tmax and there exist smooth limits (M∞ , geℓ∞ (·, t)) defined on (−tℓ , 0]. For each ℓ, the limit has nonnegative sectional curvature and has bounded curvature at each time slice. Moreover by the gradient estimate in canonical neighborhood assumption with accuracy parameter 2ε, e the limit has bounded curvature on each subinterval [−b, 0] ⊂ (−tℓ , 0]. Denote by Q

448


e is independent of ℓ). the scalar curvature upper bound of the limit at time zero (Q Then we can apply Li-Yau-Hamilton inequality (Corollary 2.5.5) to get eℓ∞ (x, t) ≤ R

tℓ e Q, t + tℓ

e∞ (x, t) are the scalar curvatures of the limits (M∞ , ge∞ (·, t)). Hence by the where R ℓ ℓ definition of convergence and the above curvature estimates, we can find a subsequence αk βk ∞ of the rescaled solutions e gij which converges in the Cloc topology to a smooth limit ∞ (M∞ , geij (·, t)) on the maximal time interval (−tmax , 0]. We need to show −tmax = −∞. Suppose −tmax > −∞, there are only the following two possibilities: either ∞ (1) The curvature of the limiting solution (M∞ , geij (·, t)) becomes unbounded as t ց −tmax ; or (2) For each small constant θ > 0 and each large integer k0 > 0, there is some k ≥ αk βk k0 such that the rescaled solution geij has a surgery time Tk ∈ [−tmax − θ, 0] and a surgery point xk lying in a gluing cap at the times Tk so that d2Tk (xk , x¯) is uniformly bounded from above by a constant independent of θ and k0 . We next claim that the possibility (1) always occurs. Suppose not; then the ∞ curvature of the limiting solution (M∞ , e gij (·, t)) is bounded on M∞ × (−tmax , 0] by ˆ some positive constant C. In particular, for any A > 0, there is a sufficiently large αk βk integer k1 > 0 such that any rescaled solution geij with k ≥ k1 on the geodesic 1 −1 ˆ −1 e η C , 0] and its scalar ball B0 (¯ x, A) is defined on the time interval [−tmax + 50 curvature is bounded by 2Cˆ there. (Here, without loss of generality, we may assume 1 −1 ˆ −1 that the upper bound Cˆ is so large that −tmax + 50 η C < 0.) By Assertion 1 in αk βk e0 (¯ Step 2, for k large enough, the rescaled solution e gij over B x, A) can be defined 1 −1 ˆ −1 on the extended time interval [−tmax − 50 η C , 0] and has the scalar curvature 1 −1 ˆ −1 e ≤ 10C ˆ on B e0 (¯ R x, A) × [−tmax − 50 η C , 0]. So we can extract a smooth limit from the sequence to get the limiting solution which is defined on a larger time interval 1 −1 ˆ −1 [−tmax − 50 η C , 0]. This contradicts the definition of the maximal time −tmax . It remains to exclude the possibility (1). By using Li-Yau-Hamilton inequality (Corollary 2.5.5) again, we have e∞ (x, t) ≤ R

tmax e Q. t + tmax

So we only need to control the curvature near −tmax . Exactly as in Step 4 in the proof of Theorem 7.1.1, it follows from Li-Yau-Hamilton inequality that q ˜ ˜ ˜ e (7.4.16) d0 (x, y) ≤ dt (x, y) ≤ d0 (x, y) + 30tmax Q

for any x, y ∈ M∞ and t ∈ (−tmax , 0]. Since the infimum of the scalar curvature is nondecreasing in time, we have some 1 −1 e∞ (y∞ , t∞ ) < η such that R point y∞ ∈ M∞ and some time −tmax < t∞ < −tmax + 50 e0 > 0 such that d˜t (¯ e0 /2 for all 5/4. By (7.4.16), there is a constant A x , y∞ ) ≤ A t ∈ (−tmax , 0]. αk βk Now we come back to the rescaled solution e gij . Clearly, for arbitrarily given ′ small ǫ > 0, when k large enough, there is a point yk in the underlying manifold of αk βk geij at time 0 satisfying the following properties (7.4.17)

e k , t∞ ) < 3 , R(y 2

e0 det (¯ x , yk ) ≤ A


449

for t ∈ [−tmax + ǫ′ , 0]. By the definition of convergence, we know that for any fixed e0 , for k large enough, the rescaled solution over B e0 (¯ A0 ≥ 2A x, A0 ) is defined on the time interval [t∞ , 0] and satisfies e e t) ≤ 2tmax Q R(x, t + tmax

e0 (¯ on B x, A0 ) × [t∞ , 0]. Then by Assertion 2 of Step 2, we have proved that there is a αk βk sufficiently large integer k¯0 such that when k ≥ k¯0 , the rescaled solutions geij at yk 1 −1 can be defined on [−tmax − 50 η , 0], and satisfy e k , t) ≤ 15 R(y

1 −1 η , t∞ ]. for t ∈ [−tmax − 50 We now prove a statement analogous to Assertion 4 (i) of Step 3. αk βk Assertion 5. For the above rescaled solutions geij and k¯0 , we have that for αk βk any L > 0, there is a positive constant ω(L) such that the rescaled solutions e gij satisfy

e t) ≤ ω(L) R(x,

for all (x, t) with d˜t (x, yk ) ≤ L and t ∈ [−tmax −

1 −1 , t∞ ], 50 η

and for all k ≥ k¯0 .

Proof of Assertion 5. We slightly modify the argument in the proof of Assertion 4 (i). Let n e t)|d˜t (x, yk ) ≤ ρ and t ∈ [−tmax − 1 η −1 , t∞ ] M (ρ) = sup R(x, 50 o αk βk in the rescaled solutions e g , k ≥ k¯0 ij

and

ρ0 = sup{ρ > 0 | M (ρ) < +∞}. Note that the estimate (7.4.14) implies that ρ0 > 0. We only need to show ρ0 = +∞. We argue by contradiction. Suppose ρ0 < +∞. Then, after passing to αk βk a subsequence, there is a sequence (˜ yk , tk ) in the rescaled solutions e gij with 1 −1 ˜ e yk , tk ) → +∞. Detk ∈ [−tmax − 50 η , t∞ ] and dtk (yk , y˜k ) → ρ0 < +∞ such that R(˜ note by γk a minimizing geodesic segment from yk to y˜k at the time tk and denote by et (yk , ρ0 ) the open geodesic ball centered at yk of radius ρ0 on the rescaled solution B k αk βk geij (·, tk ). For any 0 < ρ < ρ0 with ρ near ρ0 , by applying Assertion 3 as before, we get et (yk , ρ) are defined on the time interval that the rescaled solutions on the balls B k 1 −1 −1 [tk − 16 η M (ρ) , tk ] for all large k. By Step 1 and Shi’s derivative estimate, we further know that the covariant derivatives and higher order derivatives of the curvaet (yk , ρ− (ρ0 −ρ) )×[tk − 1 η −1 M (ρ)−1 , tk ] are also uniformly bounded. Then tures on B k 2 32 by the uniform κ-noncollapsing property and Hamilton’s compactness theorem (Theorem 4.1.5), after passing to a subsequence, we can assume that the marked sequence αk βk ∞ ˜t (yk , ρ0 ), e (B gij (·, tk ), yk ) converges in the Cloc topology to a marked (noncomplete) k

450


∞ manifold (B∞ , e gij , y∞ ) and the geodesic segments γk converge to a geodesic segment (missing an endpoint) γ∞ ⊂ B∞ emanating from y∞ . Clearly, the limit also has nonnegative sectional curvature by the pinching assumption. Then by repeating the same argument as in the proof of Assertion 4 (i) in the rest, we derive a contradiction with Hamilton’s strong maximum principle. This proves Assertion 5. We then apply the second estimate of (7.4.17) and Assertion 5 to conclude that for any large constant 0 < A < +∞, there is a positive constant C(A) such that for αk βk any small ǫ′ > 0, the rescaled solutions geij satisfy

(7.4.18)

e t) ≤ C(A), R(x,

e0 (¯ for all x ∈ B x, A) and t ∈ [−tmax + ǫ′ , 0], and for all sufficiently large k. Then by αk βk applying Assertion 1 in Step 2, we conclude that the rescaled solutions e gij on the e geodesic balls B0 (¯ x, A) are also defined on the extended time interval [−tmax + ǫ′ − 1 −1 −1 C(A) , 0] for all sufficiently large k. Furthermore, by the gradient estimates as 8η in Step 1, we have e t) ≤ 10C(A), R(x,

e0 (¯ for x ∈ B x, A) and t ∈ [−tmax + ǫ′ − 18 η −1 C(A)−1 , 0]. Since ǫ′ > 0 is arbitrarily small and the positive constant C(A) is independent of ǫ′ , we conclude that the αk βk e0 (¯ rescaled solutions e gij on B x, A) are defined on the extended time interval [−tmax − 1 −1 −1 C(A) , 0] and satisfy 16 η (7.4.19)

e t) ≤ 10C(A), R(x,

1 −1 e0 (¯ for x ∈ B x, A) and t ∈ [−tmax − 16 η C(A)−1 , 0], and for all sufficiently large k. αk βk Now, by taking convergent subsequences from the rescaled solutions geij , we see that the limit solution is defined smoothly on a space-time open subset of M∞ × (−∞, 0] containing M∞ × [−tmax , 0]. By Step 4, we see that the limiting metric ∞ geij (·, −tmax ) at time −tmax has bounded curvature. Then by combining with the canonical neighborhood assumption of accuracy 2ε, we conclude that the curvature of the limit is uniformly bounded on the time interval [−tmax , 0]. So we have excluded the possibility (1). Hence we have proved a subsequence of the rescaled solutions converges to an orientable ancient κ-solution. Finally by combining with the canonical neighborhood theorem (Theorem 6.4.6), we see that (¯ x, t¯) has a canonical neighborhood with parameter ε, which is a contradiction. Therefore we have completed the proof of the proposition. Summing up, we have proved that for any ε > 0, (without loss of generality, we e may assume ε ≤ ε¯0 ), there exist nonincreasing (continuous) positive functions δ(t) and re(t), defined on [0, +∞) with 1 e ¯ δ(t) ≤ δ(t) = min , δ0 , 2e2 log(1 + t)

e such that for arbitrarily given (continuous) positive function δ(t) with δ(t) < δ(t) on [0, +∞), and arbitrarily given a compact orientable normalized three-manifold as


451

initial data, the Ricci flow with surgery has a solution on [0, T ) obtained by evolving the Ricci flow and by performing δ-cutoff surgeries at a sequence of times 0 < t1 < e i ) at each time ti , so that the pinching t2 < · · · < ti < · · · < T , with δ(ti ) ≤ δ ≤ δ(t assumption and the canonical neighborhood assumption (with accuracy ε) with r = re(t) are satisfied. (At this moment we still do not know whether the surgery times ti are discrete.) Since the δ-cutoff surgeries occur at the points lying deeply in the ε-horns, the minimum of the scalar curvature Rmin (t) of the solution to the Ricci flow with surgery at each time-slice is achieved in the region unaffected by the surgeries. Thus we know from the evolution equation of the scalar curvature that d 2 2 Rmin (t) ≥ Rmin (t). dt 3 In particular, the minimum of the scalar curvature Rmin (t) is nondecreasing in time. Also note that each δ-cutoff surgery decreases volume. Then the upper derivative of the volume in time satisfies ¯ d V (t + △t) − V (t) V (t) , lim sup dt △t △t→0

(7.4.20)

≤ −Rmin (0)V (t)

which implies that V (t) ≤ V (0)e−Rmin (0)t . On the other hand, by Lemma 7.3.2 and the δ-cutoff procedure given in the previous section, we know that at each time ti , each δ-cutoff surgery cuts down the volume at least at an amount of h3 (ti ) with h(ti ) depending only on δ(ti ) and re(ti ). Thus the surgery times ti cannot accumulate in any finite interval. When the solution becomes extinct at some finite time T , the solution at time near T is entirely covered by canonical neighborhoods and then the initial manifold is diffeomorphic to a connected sum of a finite copies of S2 ×S1 and S3 /Γ (the metric quotients of round three-sphere). So we have proved the following long-time existence result which was proposed by Perelman in [104]. Theorem 7.4.3 (Long-time existence theorem). For any fixed constant ε > 0, e there exist nonincreasing (continuous) positive functions δ(t) and re(t), defined on [0, +∞), such that for an arbitrarily given (continuous) positive function δ(t) with e on [0, +∞), and arbitrarily given a compact orientable normalized threeδ(t) ≤ δ(t) manifold as initial data, the Ricci flow with surgery has a solution with the following properties: either (i) it is defined on a finite interval [0, T ) and obtained by evolving the Ricci flow and by performing a finite number of cutoff surgeries, with each δ-cutoff at a time t ∈ (0, T ) having δ = δ(t), so that the solution becomes extinct at the finite time T , and the initial manifold is diffeomorphic to a connected sum of a finite copies of S2 ×S1 and S3 /Γ (the metric quotients of round three-sphere) ; or (ii) it is defined on [0, +∞) and obtained by evolving the Ricci flow and by performing at most countably many cutoff surgeries, with each δ-cutoff at a time t ∈ [0, +∞) having δ = δ(t), so that the pinching assumption and the canonical neighborhood assumption (with accuracy ε) with r = re(t) are satisfied, and there exist at most a finite number of surgeries on every finite time interval.

452


In particular, if the initial manifold has positive scalar curvature, say R ≥ a > 0, then by (7.4.20), the solution becomes extinct at T ≤ 32 a. Hence we have the following topological description of compact three-manifolds with nonnegative scalar curvature which improves the well-known work of Schoen-Yau [109], [110]. Corollary 7.4.4 (Perelman [104]). Let M be a compact orientable threemanifold with nonnegative scalar curvature. Then either M is flat or it is diffeomorphic to a connected sum of a finite copies of S2 × S1 and S3 /Γ (the metric quotients of the round three-sphere). The famous Poincar´ e conjecture states that every compact three-manifold with trivial fundamental group is diffeomorphic to S3 . Developing tools to attack the conjecture formed the basis for much of the works in three-dimensional topology over the last one hundred years. Now we use the Ricci flow to discuss the Poincar´ e conjecture. Let M be a compact three-manifold with trivial fundamental group. In particular, the three-manifold M is orientable. Arbitrarily given a Riemannian metric on M , by scaling we may assume the metric is normalized. With this normalized metric as initial data, we consider the solution to the Ricci flow with surgery. If one can show the solution becomes extinct in finite time, it will follow from Theorem 7.4.3 (i) that the three-manifold M is diffeomorphic to the three-sphere S3 . Such finite extinction time result was first proposed by Perelman in [105], and recently, Colding-Minicozzi has published a proof to it in [42]. So the combination of Theorem 7.4.3 (i) and Colding-Minicozzi’s finite extinction result gives a complete proof of the Poincar´ e conjecture. We also remark that the above long-time existence result has been extended to compact four-manifolds with positive isotropic curvature by Chen and the second author in [34]. As a consequence it gave a complete proof of the following classification theorem of compact four-manifolds, with no essential incompressible space-form and with a metric of positive isotropic curvature. The theorem was first proved by Hamilton in ([64]), though it was later found that the proof contains some gaps (see for example the comment of Perelman in Page 1, the second paragraph, of [104]). Theorem 7.4.5. A compact four-manifold with no essential incompressible spaceform and with a metric of positive isotropic curvature is diffeomorphic to S4 , or RP4 , e 1 (the Z2 quotient of S3 × S1 where Z2 flips S3 antipodally and or S3 × S1 , or S3 ×S 1 0 rotates S by 180 ), or a connected sum of them. 7.5. Curvature Estimates for Surgically Modified Solutions. In this section we will generalize the curvature estimates for smooth solutions in Section 7.2 to that of solutions with cutoff surgeries. We first state and prove a version of Theorem 7.2.1. Theorem 7.5.1 (Perelman [104]). For any ε > 0 and 1 ≤ A < +∞, one can find κ = κ(A, ε) > 0, K1 = K1 (A, ε) < +∞, K2 = K2 (A, ε) < +∞ and r¯ = r¯(A, ε) > 0 such that for any t0 < +∞ there exists δ¯A = δ¯A (t0 ) > 0 (depending also on ε), nonincreasing in t0 , with the following property. Suppose we have a solution, constructed by Theorem 7.4.3 with the nonincreasing (continuous) positive functions e and re(t), to the Ricci flow with δ-cutoff surgeries on time interval [0, T ] and with δ(t) a compact orientable normalized three-manifold as initial data, where each δ-cutoff e on [0, T ] and δ = δ(t) ≤ δ¯A on [ t0 , t0 ]; assume at a time t satisfies δ = δ(t) ≤ δ(t) 2


453

that the solution is defined on the whole parabolic neighborhood P (x0 , t0 , r0 , −r02 ) , {(x, t) | x ∈ Bt (x0 , r0 ), t ∈ [t0 − r02 , t0 ]}, 2r02 < t0 , and satisfies |Rm| ≤ r0−2 on P (x0 , t0 , r0 , −r02 ), and

Vol t0 (Bt0 (x0 , r0 )) ≥ A−1 r03 .

Then (i) the solution is κ-noncollapsed on all scales less than r0 in the ball Bt0 (x0 , Ar0 ); (ii) every point x ∈ Bt0 (x0 , Ar0 ) with R(x, t0 ) ≥ K1 r0−2 has a canonical neighbor1 hood B, with Bt0 (x, σ) ⊂ B ⊂ Bt0 (x, 2σ) for some 0 < σ < C1 (ε)R− 2 (x, t0 ), which is √ either a strong ε-neck or an ε-cap; (iii) if r0 ≤ r¯ t0 then R ≤ K2 r0−2 in Bt0 (x0 , Ar0 ). Here C1 (ε) is the positive constant in the canonical neighborhood assumption. Proof. Without loss of generality, we may assume 0 < ε ≤ ε¯0 , where ε¯0 is the sufficiently small (universal) positive constant in Lemma 7.4.2. (i) This is analog of no local collapsing theorem II (Theorem 3.4.2). In comparison with the no local collapsing theorem II, this statement gives κ-noncollapsing property no matter how big the time is and it also allows the solution to be modified by surgery. Let η(≥ 10) be the universal constant in the definition of the canonical neighborhood assumption. Recall that we had removed every component which has positive sectional curvature in our surgery procedure. By the same argument as in the first part of the proof of Lemma 7.4.2, the canonical neighborhood assumption of the so1 lution implies the κ-noncollapsing on the scales less than 2η re(t0 ) for some positive constant κ depending only on C1 (ε) and C2 (ε) (in the definition q of the canonical neighborhood assumption). So we may assume

scales ρ,

1 e(t0 ) 2η r

1 e(t0 ) 2η r

≤ r0 ≤

t0 2,

and study the

≤ ρ ≤ r0 . Let x ∈ Bt0 (x0 , Ar0 ) and assume that the solution satisfies |Rm| ≤ ρ−2

for those points in P (x, t0 , ρ, −ρ2 ) , {(y, t) | y ∈ Bt (x, ρ), t ∈ [t0 − ρ2 , t0 ]} for which the solution is defined. We want to bound the ratio Vol t0 (Bt0 (x, ρ))/ρ3 from below. Recall that a space-time curve is called admissible if it stays in the region unaffected by surgery, and a space-time curve on the boundary of the set of admissible curves is called a barely admissible curve. Consider any barely admissible curve γ, parametrized by t ∈ [tγ , t0 ], t0 − r02 ≤ tγ ≤ t0 , with γ(t0 ) = x. The same proof for the assertion (7.4.2) (in the proof of Lemma 7.4.2) shows that for arbitrarily large L > 0 ¯ t0 , re(t0 ), re( t0 ), ε) > 0 (to be determined later), one can find a sufficiently small δ(L, 2 t0 ¯ such that when each δ-cutoff in [ 2 , t0 ] satisfies δ ≤ δ(L, t0 , re(t0 ), re( t20 ), ε), there holds (7.5.1)

Z

t0

tγ

√

2 t0 − t(R+ (γ(t), t) + |γ(t)| ˙ )dt ≥ Lr0 .

From now on, we assume that each δ-cutoff of the solution in the time interval ¯ t0 , re(t0 ), re( t0 ), ε). [ t20 , t0 ] satisfies δ ≤ δ(L, 2

454


Let us scale the solution, still denoted by gij (·, t), to make r0 = 1 and the time as t0 = 1. By the maximum principle, it is easy to see that the (rescaled) scalar curvature satisfies R≥−

3 2t

on (0, 1]. Let us consider the time interval [ 12 , 1] and define a function of the form √ ¯ τ) + 2 τ) h(y, t) = φ(dt (x0 , y) − A(2t − 1))(L(y, where τ = 1 − t, φ is the function of one variable chosen in the proof of Theorem 1 1 1 ), rapidly increasing to infinity on ( 20 , 10 ), and 3.4.2 which is equal to one on (−∞, 20 (φ′ )2 ′′ ′ ¯ satisfies 2 φ − φ ≥ (2A + 300)φ − C(A)φ for some constant C(A) < +∞, and L is the function defined by n √ Z τ√ ¯ τ ) = inf 2 τ L(q, s(R + |γ| ˙ 2 )ds | (γ(s), s), s ∈ [0, τ ] 0 o is a space-time curve with γ(0) = x and γ(τ ) = q . Note that

(7.5.2)

√ ¯ τ) ≥ 2 τ L(y,

Z

τ

√

sRds

0

≥ −4τ 2 √ > −2 τ

since R ≥ −3 and 0 < τ ≤ 21 . This says h is positive for t ∈ [ 12 , 1]. Also note that (7.5.3)

∂ ¯ ¯≤6 L + △L ∂τ

¯ is achieved by admissible curves. Then as long as the shortest L-geodesics as long as L from (x0 , 0) to (y, τ ) are admissible, there holds at y and t = 1 − τ , √ ∂ ∂ ¯ + 2 τ) − △ h ≥ φ′ − △ dt − 2A − φ′′ · (L ∂t ∂t 1 ¯ − 6+ √ φ − 2h∇φ, ∇Li. τ Firstly, we may assume the constant L in (7.5.1) is not less than 2 exp(C(A) + 100). We claim that Lemma 3.4.1(i) is applicable for d = dt (·, x0 ) ¯ τ ) is achieved by admissible curves at y and t = 1 − τ (with τ ∈ [0, 12 ]) whenever L(y, and satisfies the estimate √ ¯ τ ) ≤ 3 τ exp(C(A) + 100). L(y, Indeed, since the solution is defined on the whole neighborhood P (x0 , t0 , r0 , −r02 ) with r0 = 1 and t0 = 1, the point x0 at the time t = 1 − τ lies on the region unaffected ¯ τ ) is achieved by admissible by surgery. Note that R ≥ −3 √ for t ∈ [ 21 , 1]. When L(y, ¯ τ ) ≤ 3 τ exp(C(A) + 100), the estimate (7.5.1) implies that curves and satisfies L(y,


455

the point y at the time t = 1 − τ does not lie in the collars of the gluing caps. Thus any minimal geodesic (with respect to the metric gij (·, t) with t = 1 − τ ) connecting x0 and y also lies in the region unaffected by surgery; otherwise the geodesic is not minimal. Then from the proof of Lemma 3.4.1(i), we see that it is applicable. Assuming the minimum of h at a time, say t = 1 − τ , is achieved at a point, ¯ τ ) is achieved by admissible curves and satisfies L(y, ¯ τ) ≤ say y, and assuming L(y, √ 3 τ exp(C(A) + 100), we have √ ¯ + 2 τ )∇φ = −φ∇L, ¯ (L and then by the computations and estimates in the proof of Theorem 3.4.2, ∂ −△ h ∂t √ (φ′ )2 ∂ ¯ + 2 τ ) − 6 + √1 φ − △ dt − 2A − φ′′ + 2 · (L ≥ φ′ ∂t φ τ h 1 √ , ≥ −C(A)h − 6 + √ τ (2 τ − 4τ 2 ) at y and t = 1 − τ . Here we used (7.5.2) and Lemma 3.4.1(i). As before, denoting by hmin (τ ) = minz h(z, 1 − τ ), we obtain √ d hmin (τ ) 6 τ +1 1 √ √ − (7.5.4) log ≤ C(A) + 2 dτ τ 2τ − 4τ τ 2τ 50 ≤ C(A) + √ , τ ¯ ≤ 3√τ exp(C(A)+ as long as the associated shortest L-geodesics are admissible with L 100). On the other hand, by definition, we have (7.5.5)

lim+

τ →0

hmin (τ ) √ ≤ φ(d1 (x0 , x) − A) · 2 = 2. τ

The combination of (7.5.4) and (7.5.5) gives the following assertion: ¯ s) | dt (x0 , y) ≤ A(2t − 1) + Let τ ∈ [0, 21 ]. If for each s ∈ [0, τ ], inf{L(y, with s = 1 − t} is achieved by admissible curves, then we have ¯ τ ) | dt (x0 , y) ≤ A(2t − 1) + 1 with τ = 1 − t inf L(y, (7.5.6) 10 √ ≤ 2 τ exp(C(A) + 100). 1 10

Note again that R ≥ −3 for t ∈ [ 21 , 1]. By combining with (7.5.1), we know that any barely admissible curve γ, parametrized by s ∈ [0, τ ], 0 ≤ τ ≤ 12 , with γ(0) = x, satisfies Z τ √ 7 s(R + |γ| ˙ 2 )ds ≥ exp(C(A) + 100), 4 0 by assuming L ≥ 2 exp(C(A) + 100).

456


1 re(t0 ) (and t0 = 1) Since |Rm| ≤ ρ−2 on P (x, t0 , ρ, −ρ2 ) with ρ ≥ 2η t0 ¯ and δ(L, t0 , re(t0 ), re( 2 ), ε) > 0 is sufficiently small, the parabolic neighborhood P (x, 1, ρ, −ρ2 ) around the point (x, 1) is contained in the region unaffected by the 1 ¯ can be bounded from inf L surgery. Thus as τ = 1 − t is sufficiently close to zero, 2√ τ ¯ τ ) | dt (x0 , y) ≤ above by a small positive constant and then the infimum inf{L(y, 1 A(2t − 1) + 10 with τ = 1 − t} is achieved by admissible curves. Hence we conclude that for each τ ∈ [0, 21 ], any minimizing curve γτ of ¯ τ )| dt (x0 , y) ≤ A(2t − 1) + 1 with τ = 1 − t} is admissible and satisfies inf{L(y, 10 Z τ √ s(R + |γ˙ τ |2 )ds ≤ exp(C(A) + 100). 0

Now we come back to the unrescaled solution. It then follows that the Li-YauPerelman distance l from (x, t0 ) satisfies the following estimate 1 1 ≤ exp(C(A) + 100), (7.5.7) min l y, t0 − r02 y ∈ Bt0 − 12 r02 x0 , r0 2 10

by noting the (parabolic) scaling invariance of the Li-Yau-Perelman distance. By the assumption that |Rm| ≤ r0−2 on P (x0 , t0 , r0 , −r02 ), exactly as before, for any q ∈ Bt0 −r02 (x0 , r0 ), we can choose a path γ parametrized by τ ∈ [0, r02 ] with 1 γ(0) = x, γ(r02 ) = q, and γ( 21 r02 ) = y ∈ Bt0 − 21 r02 (x0 , 10 r0 ), where γ|[0, 21 r02 ] achieves the 1 1 2 minimum min{l(y, t0 − 2 r0 ) | y ∈ Bt0 − 12 r02 (x0 , 10 r0 )} and γ|[ 12 r02 ,r02 ] is a suitable curve satisfying γ|[ 21 r02 ,r02 ] (τ ) ∈ Bt0 −τ (x0 , r0 ), for each τ ∈ [ 12 r02 , r02 ], so that the L-length of γ is uniformly bounded from above by a positive constant (depending only on A) multiplying r0 . This implies that the Li-Yau-Perelman distance from (x, t0 ) to the ball Bt0 −r02 (x0 , r0 ) is uniformly bounded by a positive constant L(A) (depending only on A). Now we can choose the constant L in (7.5.1) by L = max{2L(A), 2 exp(C(A) + 100)}. Thus every shortest L-geodesic from (x, t0 ) to the ball Bt0 −r02 (x0 , r0 ) is necessarily admissible. By combining with the assumption that Vol t0 (Bt0 (x0 , r0 )) ≥ A−1 r03 , we conclude that Perelman’s reduced volume of the ball Bt0 −r02 (x0 , r0 ) satisfies the estimate Z 3 e (7.5.8) Vr02 (Bt0 −r02 (x0 , r0 )) = (4πr02 )− 2 exp(−l(q, r02 ))dVt0 −r02 (q) 2 (x0 ,r0 ) 0 −r0

Bt

≥ c(A)

for some positive constant c(A) depending only on A. We can now argue as in the last part of the proof of Lemma 7.4.2 to get a lower bound estimate for the volume of the ball Bt0 (x, ρ). The union of all shortest Lgeodesics from (x, t0 ) to the ball Bt0 −r02 (x0 , r0 ), defined by CBt0 −r02 (x0 , r0 ) = {(y, t) | (y, t) lies in a shortest L-geodesic from

(x, t0 ) to a point in Bt0 −r02 (x0 , r0 )},

forms a cone-like subset in space-time with vertex (x, t0 ). Denote by B(t) the intersection of the cone-like subset CBt0 −r02 (x0 , r0 ) with the time-slice at t. Perelman’s


457

reduced volume of the subset B(t) is given by Z 3 e (4π(t0 − t))− 2 exp(−l(q, t0 − t))dVt (q). Vt0 −t (B(t)) = B(t)

Since the cone-like subset CBt0 −r02 (x0 , r0 ) lies entirely in the region unaffected by surgery, we can apply Perelman’s Jacobian comparison Theorem 3.2.7 and the estimate (7.5.8) to conclude that Vet0 −t (B(t)) ≥ Ver02 (Bt0 −r02 (x0 , r0 ))

(7.5.9)

≥ c(A)

for all t ∈ [t0 − r02 , t0 ]. 1 As before, denoting by ξ = ρ−1 Vol t0 (Bt0 (x, ρ)) 3 , we only need to get a positive e 0 − ξρ2 ), the subset lower bound for ξ. Of course we may assume ξ < 1. Consider B(t 2 at the time-slice {t = t0 − ξρ } where every point can be connected to (x, t0 ) by an e 0 − ξρ2 ) is given by admissible shortest L-geodesic. Perelman’s reduced volume of B(t (7.5.10)

e 0 − ξρ2 )) Veξρ2 (B(t Z 3 (4πξρ2 )− 2 exp(−l(q, ξρ2 ))dVt0 −ξρ2 (q) = e −ξρ2 ) B(t Z 0 3 (4πξρ2 )− 2 exp(−l(q, ξρ2 ))dVt0 −ξρ2 (q) = e 0 −ξρ2 )∩L exp B(t

+

{|υ|≤ 1 ξ 4

Z

−1 2}

(ξρ2 )

3

e 0 −ξρ2 )\L exp B(t

{|υ|≤ 1 ξ 4

−1 2}

(ξρ2 )

(4πξρ2 )− 2 exp(−l(q, ξρ2 ))dVt0 −ξρ2 (q).

Note that the whole region P (x, t0 , ρ, −ρ2 ) is unaffected by surgery because ρ ≥ 1 ¯ t0 , re(t0 ), re( t0 ), ε) > 0 is sufficiently small. Then exactly as before, e(t0 ) and δ(L, 2η r 2 there is a universal positive constant ξ0 such that when 0 < ξ ≤ ξ0 , there holds L exp

1

{|υ|≤ 14 ξ − 2 }

(ξρ2 ) ⊂ Bt0 (x, ρ)

and the first term on RHS of (7.5.10) can be estimated by Z 3 (7.5.11) (4πξρ2 )− 2 exp(−l(q, ξρ2 ))dVt0 −ξρ2 (q) e 0 −ξρ2 )∩L exp B(t 3

≤ eCξ (4π)− 2 ξ

{|υ|≤ 1 ξ 4 3 2

−1 2}

(ξρ2 )

for some universal constant C; while the second term on RHS of (7.5.10) can be estimated by Z 3 (7.5.12) (4πξρ2 )− 2 exp(−l(q, ξρ2 ))dVt0 −ξρ2 (q) e 0 −ξρ2 )\L exp B(t 3

≤ (4π)− 2

Z

{|υ|≤ 1 ξ 4

1

−1 2}

(ξρ2 )

exp(−|υ|2 )dυ.

{|υ|> 14 ξ − 2 }

e 0 − ξρ2 ), the combination of (7.5.9)-(7.5.12) bounds ξ from Since B(t0 − ξρ2 ) ⊂ B(t below by a positive constant depending only on A. This proves the statement (i).

458


(ii) This is analogous to the claim in the proof of Theorem 7.2.1. We argue by contradiction. Suppose that for some A < +∞ and a sequence K1α → ∞, there exists ¯αβ > 0 with δ¯αβ → 0 for fixed α, we a sequence tα 0 such that for any sequences δ αβ have sequences of solutions gij to the Ricci flow with surgery and sequences of points αβ xαβ 0 , of radii r0 , which satisfy the assumptions but violate the statement (ii) at some αβ α αβ −2 xαβ ∈ Btα0 (x0 , Ar0αβ ) with R(xαβ , tα . Slightly abusing notation, we 0 ) ≥ K1 (r0 ) will often drop the indices α, β in the following argument. Exactly as in the proof of Theorem 7.2.1, we need to adjust the point (x, t0 ). More r2 ¯, precisely, we claim that there exists a point (¯ x, t¯) ∈ Bt¯(x0 , 2Ar0 ) ×[t0 − 20 , t0 ] with Q R(¯ x, t¯) ≥ K1 r0−2 such that the point (¯ x, t¯) does not satisfy the canonical neighborhood ¯ does, where P¯ is the set of all statement, but each point (y, t) ∈ P¯ with R(y, t) ≥ 4Q 1 1 −1 ′ ′ ′ ′ ¯ ≤ t ≤ t¯, dt′ (x0 , x ) ≤ dt¯(x0 , x ¯ − 12 . Indeed as ¯) + K12 Q (x , t ) satisfying t¯ − 4 K1 Q before, the point (¯ x, t¯) is chosen by an induction argument. We first choose (x1 , t1 ) = (x, t0 ) which satisfies dt1 (x0 , x1 ) ≤ Ar0 and R(x1 , t1 ) ≥ K1 r0−2 , but does not satisfy the canonical neighborhood statement. Now if (xk , tk ) is already chosen and is not the desired (¯ x, t¯), then some point (xk+1 , tk+1 ) satisfies tk − 41 K1 R(xk , tk )−1 ≤ tk+1 ≤ tk , 1

1

dtk+1 (x0 , xk+1 ) ≤ dtk (x0 , xk ) + K12 R(xk , tk )− 2 , and R(xk+1 , tk+1 ) ≥ 4R(xk , tk ), but (xk+1 , tk+1 ) does not satisfy the canonical neighborhood statement. Then we have R(xk+1 , tk+1 ) ≥ 4k R(x1 , t1 ) ≥ 4k K1 r0−2 , 1

dtk+1 (x0 , xk+1 ) ≤ dt1 (x0 , x1 ) + K12

k X i=1

1

R(xi , ti )− 2 ≤ Ar0 + 2r0 ,

and k X 1 1 t0 ≥ tk+1 ≥ t0 − K1 R(xi , ti )−1 ≥ t0 − r02 . 4 2 i=1

So the sequence must be finite and its last element is the desired (¯ x, t¯). −2 ¯ ¯ Rescale the solutions along (¯ x, t) with factor R(¯ x, t)(≥ K1 r0 ) and shift the times t¯ to zero. We will adapt both the proof of Proposition 7.4.1 and that of Theorem αβ 7.2.1 to show that a sequence of the rescaled solutions geij converges to an ancient κ-solution, which will give the desired contradiction. Since we only need to consider the scale of the curvature less than re(t¯)−2 , the present situation is much easier than that of Proposition 7.4.1. Firstly as before, we need to get a local curvature estimate. 1 −1 ¯ −1 ¯ η Q , t] For each adjusted (¯ x, t¯), let [t′ , t¯] be the maximal subinterval of [t¯ − 20 so that for each sufficiently large α and then sufficiently large β, the canonical 1 1 ¯ − 12 , t′ − t¯) = {(x, t) | x ∈ neighborhood statement holds for any (y, t) in P (¯ x, t¯, 10 K12 Q 1 1 ¯ − 12 ), t ∈ [t′ , t¯]} with R(y, t) ≥ 4Q, ¯ where η is the universal positive Bt (¯ x, 10 K12 Q constant in the definition of canonical neighborhood assumption. We want to show (7.5.13)

t′ = t¯ −

1 −1 ¯ −1 η Q . 20

Consider the scalar curvature R at the point x ¯ over the time interval [t′ , t¯]. If ′ ¯ we let t˜ be the first of such time from there is a time t˜ ∈ [t , t¯] satisfying R(¯ x, t˜) ≥ 4Q, t¯. Since the chosen point (¯ x, t¯) does not satisfy the canonical neighborhood statement,


459

we know R(¯ x, t¯) ≤ re(t¯)−2 . Recall from our designed surgery procedure that if there is a cutoff surgery at a point x at a time t, the scalar curvature at (x, t) is at least αβ (δ¯αβ )−2 re(t)−2 . Then for each fixed α, for β large enough, the solution gij (·, t) around 1 −1 ¯ −1 ˜ the point x ¯ over the time interval [t˜ − 20 η Q , t] is well defined and satisfies the following curvature estimate ¯ R(¯ x, t) ≤ 8Q, 1 −1 ¯ −1 ¯ for t ∈ [t˜ − 20 η Q , t] (or t ∈ [t′ , t¯] if there is no such time t˜). By the assumption 2 that t0 > 2r0 , we have

t0 x, t¯) t¯R(¯ x, t¯) ≥ R(¯ 2 ≥ r02 (K1 r0−2 )

= K1 → +∞.

Thus by using the pinching assumption and the gradient estimates in the canonical neighborhood assumption, we further have ¯ |Rm(x, t)| ≤ 30Q, 1 −1 ¯ − 21 1 −1 ¯ −1 ¯ for all x ∈ Bt (¯ x, 10 η Q ) and t ∈ [t˜− 20 η Q , t] (or t ∈ [t′ , t¯]) and all sufficiently large α and β. Observe that Lemma 3.4.1 (ii) is applicable for dt (x0 , x ¯) with t ∈ 1 −1 ¯ −1 ¯ [t˜− 20 η Q , t] (or t ∈ [t′ , t¯]) since any minimal geodesic, with respect to the metric gij (·, t), connecting x0 and x¯ lies in the region unaffected by surgery; otherwise the geodesic is not minimal. After having obtained the above curvature estimate, we can argue as deriving (7.2.2) and (7.2.3) in the proof of Theorem 7.2.1 to conclude that 1 1 1 −1 ¯ −1 ¯ − 12 , η Q ≤ t ≤ t¯ (or t ∈ [t′ , t¯]) and dt (x, x¯) ≤ 10 K12 Q any point (x, t), with t˜ − 20 satisfies

1 1 ¯− 1 2, dt (x, x0 ) ≤ dt¯(¯ x, x0 ) + K12 Q 2 for all sufficiently large α and β. Then by combining with the choice of the points 1 −1 ¯ −1 (¯ x, t¯), we prove t′ = t¯ − 20 η Q (i.e., the canonical neighborhood statement holds 1 1 ¯ − 12 ,− 1 η −1 Q ¯ −1 ) with for any point (y, t) in the parabolic neighborhood P (¯ x, t¯, 10 K12 Q 20 ¯ R(y, t) ≥ 4Q) for all sufficiently large α and then sufficiently large β. Now it follows from the gradient estimates in the canonical neighborhood assumpαβ tion that the scalar curvatures of the rescaled solutions e gij satisfy e t) ≤ 40 R(x,

1 −1 1 −1 1 −1 et′ (¯ η , − 20 η ) , {(x′ , t′ ) | x′ ∈ B x, 10 η ), t′ ∈ for those (x, t) ∈ P (¯ x, 0, 10 1 −1 e [− 20 η , 0]}, for which the rescaled solution is defined. (Here Bt′ denotes the geodesic ball in the rescaled solution at time t′ ). Note again that R(¯ x, t¯) ≤ re(t¯)−2 and recall from our designed surgery procedure that if there is a cutoff surgery at a point x at a time t, the scalar curvature at (x, t) is at least (δ¯αβ )−2 re(t)−2 . Then for each fixed αβ sufficiently large α, for β large enough, the rescaled solution geij is defined on the 1 −1 1 −1 whole parabolic neighborhood P (¯ x, 0, 10 η , − 20 η ). More generally, for arbitrarily e < +∞, there is a positive integer α0 so that for each α ≥ α0 we can fixed 0 < K

460


find β0 > 0 (depending on α) such that if β ≥ β0 and (y, 0) is a point on the rescaled αβ e 0) ≤ K e and de0 (y, x¯) ≤ K, e we have estimate solution geij with R(y,

(7.5.14)

e t) ≤ 40K e R(x,

1 −1 e − 12 1 −1 e −1 et′ (y, for (x, t) ∈ P (y, 0, 10 η K , − 20 η K ) , {(x′ , t′ ) | x′ ∈ B 1 −1 e −1 1 −1 e − 12 ′ K ), t ∈ [− 20 η K , 0]}. In particular, the rescaled solution is defined 10 η 1 −1 e − 21 1 −1 e −1 on the whole parabolic neighborhood P (y, 0, 10 η K , − 20 η K ). Next, we want to show the curvature of the rescaled solutions at the new times zero (after shifting) stay uniformly bounded at bounded distances from x¯ for some subsequences of α and β. Let αm , βm → +∞ be chosen so that the estimate (7.5.14) e = m. For all ρ ≥ 0, set holds with K

and

e 0) | m ≥ 1, d0 (x, x¯) ≤ ρ in the rescaled solutions geαm βm } M (ρ) = sup{R(x, ij ρ0 = sup{ρ ≥ 0 | M (ρ) < +∞}.

Clearly the estimate (7.5.14) yields ρ0 > 0. As we consider the unshifted time t¯, by combining with the assumption that t0 > 2r02 , we have (7.5.15)

t0 t¯R(¯ x, t¯) ≥ R(¯ x, t¯) 2 ≥ r02 (K1 r0−2 )

= K1 → +∞.

It then follows from the pinching assumption that we only need to show ρ0 = +∞. As before, we argue by contradiction. Suppose we have a sequence of points ym in αm βm e m , 0) → +∞. Dethe rescaled solutions e gij with de0 (¯ x, ym ) → ρ0 < +∞ and R(y e0 (¯ note by γm a minimizing geodesic segment from x ¯ to ym and denote by B x, ρ0 ) the open geodesic balls centered at x ¯ of radius ρ0 of the rescaled solutions. By applying the assertion in statement (i), we have uniform κ-noncollapsing at the points (¯ x, t¯). By combining with the local curvature estimate (7.5.14) and Hamilton’s compactness theorem, we can assume that, after passing to a subsequence, the marked sequence αm βm ∞ e0 (¯ (B x, ρ0 ), geij ,x ¯) converges in the Cloc topology to a marked (noncomplete) man∞ ifold (B∞ , e gij , x∞ ) and the geodesic segments γm converge to a geodesic segment (missing an endpoint) γ∞ ⊂ B∞ emanating from x∞ . Moreover, by the pinching assumption and the estimate (7.5.15), the limit has nonnegative sectional curvature. Then exactly as before, we consider the tubular neighborhood along γ∞ [ e∞ (q0 ))− 21 ) V = B∞ (q0 , 4π(R q0 ∈γ∞

¯∞ of (B∞ , ge∞ ) with y∞ ∈ B ¯∞ the limit point of γ∞ . As and the completion B ij ∞ before, by the choice of the points (¯ x, t¯), we know that the limiting metric e gij is cylindrical at any point q0 ∈ γ∞ which is sufficiently close to y∞ . Then by the same reason as before the metric space V¯ = V ∪ {y∞ } has nonnegative curvature in Alexandrov sense, and we have a three-dimensional nonflat tangent cone Cy∞ V¯ at y∞ . Pick z ∈ Cy∞ V¯ with distance one from the vertex and it is nonflat around z. By


461

definition B(z, 12 )(⊂ Cy∞ V¯ ) is the Gromov-Hausdoff convergent limit of the scalings ∞ of a sequence of balls B∞ (zk , σk )(⊂ (V, e gij )) with σk → 0. Since the estimate (7.5.14) ∞ e survives on (V, geij ) for all K < +∞, we know that this convergence is actually in ∞ the Cloc topology and over some time interval. Since the limit B(z, 21 )(⊂ Cy∞ V¯ ) is a piece of a nonnegatively curved nonflat metric cone, we get a contradiction with Hamilton’s strong maximum Principle (Theorem 2.2.1) as before. Hence we have αm βm proved that a subsequence of the rescaled solution e gij has uniformly bounded curvatures at bounded distance from x ¯ at the new times zero. Further, by the uniform κ-noncollapsing at the points (¯ x, t¯) and the estimate ∞ ∞ (7.5.14) again, we can take a Cloc limit (M∞ , e gij , x∞ ), defined on a space-time subset which contains the time slice {t = 0} and is relatively open in M∞ × (−∞, 0], for the subsequence of the rescaled solutions. The limit is a smooth solution to the Ricci flow, and is complete at t = 0, as well as has nonnegative sectional curvature by the pinching assumption and the estimate (7.5.15). Thus by repeating the same argument as in the Step 4 of the proof Proposition 7.4.1, we conclude that the curvature of the limit at t = 0 is bounded. Finally we try to extend the limit backwards in time to get an ancient κ-solution. Since the curvature of the limit is bounded at t = 0, it follows from the estimate (7.5.14) that the limit is a smooth solution to the Ricci flow defined at least on a backward time interval [−a, 0] for some positive constant a. Let (t∞ , 0] be the maximal time interval over which we can extract a smooth limiting solution. It suffices to show t∞ = −∞. If t∞ > −∞, there are only two possibilities: either there exist surgeries in finite distance around the time t∞ or the curvature of the limiting solution becomes unbounded as t ց t∞ . 1 −1 η . Note again that the Let c > 0 be a positive constant much smaller than 100 infimum of the scalar curvature is nondecreasing in time. Then we can find some point e∞ (y∞ , t∞ + θ) ≤ 2. y∞ ∈ M∞ and some time t = t∞ + θ with 0 < θ < 3c such that R αm βm Consider the (unrescaled) scalar curvature R of gij (·, t) at the point x¯ over ¯ −1 , t¯]. Since the scalar curvature R∞ of the limit on the time interval [t¯ + (t∞ + 2θ )Q M∞ × [t∞ + 3θ , 0] is uniformly bounded by some positive constant C, we have the curvature estimate ¯ R(¯ x, t) ≤ 2C Q ¯ −1 , t¯] and all sufficiently large m. For each fixed m and αm , for all t ∈ [t¯ + (t∞ + 2θ )Q we may require the chosen βm to satisfy αm −2 t αm βm −2 m −2 ¯ ¯ ≥ me r (tα ≥ mQ. (δ ) re 0 0 ) 2

When m is large enough, we observe again that Lemma 3.4.1 (ii) is applicable for ¯ −1 , t¯]. Then by repeating the argument as in the dt (x0 , x¯) with t ∈ [t¯ + (t∞ + θ2 )Q derivation of (7.2.1), (7.2.2) and (7.2.3), we deduce that for all sufficiently large m, the canonical neighborhood statement holds for any (y, t) in the parabolic neighborhood 1 1 ¯ − 12 , (t∞ + θ )Q ¯ −1 ). K12 Q P (¯ x, t¯, 10 2 ¯ ¯ Let (ym , t + (t∞ + θm )Q−1 ) be a sequence of associated points and times in the αm βm (unrescaled) solutions gij (·, t) so that after rescaling, the sequence converges to (y∞ , t∞ + θ) in the limit. Clearly θ2 ≤ θm ≤ 2θ for all sufficiently large m. Then by the argument as in the derivation of (7.5.13), we know that for all sufficiently large

462


αm βm m, the solution gij (·, t) at ym is defined on the whole time interval [t¯+ (t∞ + θm − 1 −1 ¯ −1 ¯ ¯ −1 ] and satisfies the curvature estimate )Q , t + (t∞ + θm )Q 20 η

¯ R(ym , t) ≤ 8Q there; moreover the canonical neighborhood statement holds for any (y, t) with 1 ¯ −1 ). ¯ in the parabolic neighborhood P (ym , t¯, 1 K 2 Q ¯ − 12 , (t∞ − c )Q R(y, t) ≥ 4Q 10

1

3

αm βm We now consider the rescaled sequence geij (·, t) with the marked points re¯ −1 , t¯ + (t∞ + c )Q ¯ −1 ]. placed by ym and the times replaced by sm ∈ [t¯ + (t∞ − 4c )Q 4 As before the Li-Yau-Hamilton inequality implies the rescaling limit around (ym , sm ) agrees with the original one. Then the arguments in previous paragraphs imply the limit is well-defined and smooth on a space-time open neighborhood of the maximal time slice {t = t∞ }. Particularly this excludes the possibility of existing surgeries in finite distance around the time t∞ . Moreover, the limit at t = t∞ also has bounded curvature. By using the gradient estimates in the canonical neighborhood assump1 1 ¯ − 12 , (t∞ − c )Q ¯ −1 ), we see that tion on the parabolic neighborhood P (ym , t¯, 10 K12 Q 3 the second possibility is also impossible. Hence we have proved a subsequence of the rescaled solutions converges to an ancient κ-solution. Therefore we have proved the canonical neighborhood statement (ii).

(iii) This is analogous to Theorem 7.2.1. We also argue by contradiction. Suppose for some A < +∞ and sequences of positive numbers K2α → +∞, r¯α → 0 there exists ¯αβ > 0 with δ¯αβ → 0 for fixed α, we a sequence of times tα 0 such that for any sequences δ αβ have sequences of solutions gij to the Ricci flow with surgery and sequences of points p αβ αβ xαβ ¯α tα 0 which satisfy the assumptions, but 0 , of positive constants r0 with r0 ≤ r for all α, β there hold (7.5.16)

αβ α αβ −2 R(xαβ , tα , for some xαβ ∈ Btα0 (xαβ 0 ) > K2 (r0 ) 0 , Ar0 ).

α ¯ We may assume that δ¯αβ ≤ δ¯4A (tα 0 ) for all α, β, where δ4A (t0 ) is chosen so that αβ αβ αβ the statements (i) and (ii) hold on Btα0 (x0 , 4Ar0 ). Let gîj be the rescaled solutions αβ αβ −2 of gij around the origins xαβ and shift the times tα 0 to zero. Then 0 with factor (r0 ) by applying the statement (ii), we know that the regions, where the scalar curvature αβ of the rescaled solutions gîj is at least K1 (= K1 (4A)), are canonical neighborhood regions. Note that canonical ε-neck neighborhoods are strong. Also note that the pinching assumption and the assertion αβ −2 ≥ (¯ rα )−2 → +∞, as α → +∞, tα 0 (r0 ) αβ imply that any subsequent limit of the rescaled solutions gîj must have nonnegative sectional curvature. Thus by the above argument in the proof of the statement (ii) (or the argument in Step 2 of the proof of Theorem 7.1.1), we conclude that there exist subsequences α = αm , β = βm such that the curvatures of the rescaled solutions αm βm m βm gîj stay uniformly bounded at distances from the origins xα not exceeding 2A. 0 This contradicts (7.5.16) for m sufficiently large. This proves the statements (iii). Clearly for fixed A, after defining the δ¯A (t0 ) for each t0 , one can adjust the δ¯A (t0 ) so that it is nonincreasing in t0 . Therefore we have completed the proof of the theorem. e so that it is also less than δ¯2(t+1) (2t) From now on we redefine the function δ(t) and then the above theorem always holds for A ∈ [1, 2(t0 + 1)]. Particularly, we still


463

have e ≤ δ(t) ¯ = min{ δ(t)

1 2e2 log(1

+ t)

, δ0 },

which tends to zero as t → +∞. We may also require that re(t) tends to zero as t → +∞. The next result is a version of Theorem 7.2.2 for solutions with surgery. Theorem 7.5.2 (Perelman [104]). For any ε > 0 and w > 0, there exist τ = τ (w, ε) > 0, K = K(w, ε) < +∞, r¯ = r¯(w, ε) > 0, θ = θ(w, ε) > 0 and T = T (w) < +∞ with the following property. Suppose we have a solution, constructed by e and re(t), Theorem 7.4.3 with the nonincreasing (continuous) positive functions δ(t) to the Ricci flow with surgery on the time interval [0, t0 ] with a compact orientable normalized three-manifold as initial data, where each δ-cutoff at a time t ∈ [0, t0 ] has e re(2t)}. Let r0 , t0 satisfy θ−1 h ≤ r0 ≤ r¯√t0 and t0 ≥ T , where h δ = δ(t) ≤ min{δ(t), is the maximal cutoff radius for surgeries in [ t20 , t0 ] (if there is no surgery in the time interval [ t20 , t0 ], we take h = 0), and assume that the solution on the ball Bt0 (x0 , r0 ) satisfies Rm(x, t0 ) ≥ −r0−2 , on Bt0 (x0 , r0 ), and

Vol t0 (Bt0 (x0 , r0 )) ≥ wr03 .

Then the solution is well defined and satisfies R(x, t) < Kr0−2 in the whole parabolic neighborhood n o r0 r0 P x0 , t0 , , −τ r02 = (x, t) | x ∈ Bt x0 , , t ∈ [t0 − τ r02 , t0 ] . 4 4 Proof. We are given that Rm(x, t0 ) ≥ −r0−2 for x ∈ Bt0 (x0 , r0 ), and Vol t0 (Bt0 (x0 , r0 )) ≥ wr03 . The same argument in the derivation of (7.2.7) and (7.2.8) (by using the Alexandrov space theory) implies that there exists a ball Bt0 (x′ , r′ ) ⊂ Bt0 (x0 , r0 ) with (7.5.17)

Vol t0 (Bt0 (x′ , r′ )) ≥

1 α3 (r′ )3 2

and with (7.5.18)

r′ ≥ c(w)r0

for some small positive constant c(w) depending only on w, where α3 is the volume of the unit ball in R3 . As in (7.1.2), we can rewrite the pinching assumption (7.3.3) as Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R, where y = f (x) = x(log x − 3),

for e2 ≤ x < +∞,

464


is increasing and convex with range −e2 ≤ y < +∞, and its inverse function is also increasing and satisfies lim f −1 (y)/y = 0.

y→+∞

Note that t0 r0−2 ≥ r¯−2 by the hypotheses. We may require T (w) ≥ 8c(w)−1 . Then by applying Theorem 7.5.1 (iii) with A = 8c(w)−1 and combining with the pinching assumption, we can reduce the proof of the theorem to the special case w = 12 α3 . In the following we simply assume w = 21 α3 . Let us first consider the case r0 < re(t0 ). We claim that R(x, t0 ) ≤ C02 r0−2 on Bt0 (x0 , r30 ), for some sufficiently large positive constant C0 depending only on ε. If not, then there is a canonical neighborhood around (x, t0 ). Note that the type (c) canonical neighborhood has already been ruled out by our design of cutoff surgeries. Thus (x, t0 ) belongs to an ε-neck or an ε-cap. This tells us that there is a nearby point y, 1 with R(y, t0 ) ≥ C2−1 R(x, t0 ) > C2−1 C02 r0−2 and dt0 (y, x) ≤ C1 R(x, t0 )− 2 ≤ C1 C0−1 r0 , 1 which is the center of the ε-neck Bt0 (y, ε−1 R(y, t0 )− 2 ). (Here C1 , C2 are the positive constants in the definition of canonical neighborhood assumption). Clearly, when we choose C0 to be much larger than C1 , C2 and ε−1 , the whole ε-neck 1 Bt0 (y, ε−1 R(y, t0 )− 2 ) is contained in Bt0 (x0 , r20 ) and we have 1

(7.5.19)

Vol t0 (Bt0 (y, ε−1 R(y, t0 )− 2 )) 1

(ε−1 R(y, t0 )− 2 )3

≤ 8πε2 .

Without loss of generality, we may assume ε > 0 is very small. Since we have assumed that Rm ≥ −r0−2 on Bt0 (x0 , r0 ) and Vol t0 (Bt0 (x0 , r0 )) ≥ 12 α3 r03 , we then get a contradiction by applying the standard Bishop-Gromov volume comparison. Thus we have the desired curvature estimate R(x, t0 ) ≤ C02 r0−2 on Bt0 (x0 , r30 ). Furthermore, by using the gradient estimates in the definition of canonical neigh1 −1 −2 borhood assumption, we can take K = 10C02 , τ = 100 η C0 and θ = 15 C0−1 in this −2 case. And since r0 ≥ θ−1 h, we have R < 10C02 r0 ≤ 21 h−2 and the surgeries do not interfere in P (x0 , t0 , r40 , −τ r02 ). √ We now consider the remaining case re(t0 ) ≤ r0 ≤ r¯ t0 . Let us redefine τ¯0 1 −1 −2 τ = min , η C0 , 2 100 ¯ 2B 2 ¯ , 25C0 , K = max 2 C + τ¯0 and θ=

1 − 12 K 2

¯ = B(w) and C¯ = C(w) are the positive constants in Theorem where τ¯0 = τ0 (w), B 1 6.3.3(ii) with w = 2 α3 , and C0 is the positive constant chosen above. We will show there is a sufficiently small r¯ > 0 such that the conclusion of the theorem for w = 21 α3 holds for the chosen τ, K and θ. Argue by contradiction. Suppose not, then there exist a sequence of r¯α → 0, α α and a sequence of solutions gij with points (xα radii r0α such that the as0 , t0 ) and pα α α α sumptions of the theorem do hold with re(t0 ) ≤ r0 ≤ r¯ t0 whereas the conclusion


465

does not. Similarly as in the proof of Theorem 7.2.2, we claim that we may assume that for all sufficiently large α, any other point (xα , tα ) and radius rα > 0 α α with that property has either tα > tα = tα ≥ r0α ; moreover tα 0 or t 0 with r α tends to +∞ as α → +∞. Indeed, for fixed α and the solution gij , let tα min be α α α the infimum of all possible times t with some point x and some radius r hav√ ing that property. Since each such tα satisfies r¯α tα ≥ rα ≥ re(tα ), it follows that when α is large, tα min must be positive and very large. Clearly for each fixed sufficiently large α, by passing to a limit, there exist some point xα min and some radius α α rmin (≥ re(tα ) > 0) so that all assumptions of the theorem still hold for (xα min min , tmin ) α α and rmin , whereas the conclusion of the theorem does not hold with R ≥ K(rmin )−2 1 α α α 2 somewhere in P (xα min , tmin , 4 rmin , −τ (rmin ) ) for all sufficiently large α. Here we 1 α α −2 α α 2 used the fact that if R < K(rmin ) on P (xα min , tmin , 4 rmin , −τ (rmin ) ), then there is no δ-cutoff surgery there; otherwise there must be a point there with scalar curtα tα tα 1 −2 α −2 −2 α min min )(e r ( )) ≥ (e r (t )) (e r ( )) ≫ K(rmin )−2 since vature at least 21 δ −2 ( min min 2 2 2 2 p α r¯α tα e(tα ¯α → 0, which is a contradiction. min ) and r min ≥ rmin ≥ r After choosing the first time tα min , by passing to a limit again, we can then choose α α α rmin to be the smallest radius for all possible (xα min , tmin )’s and rmin ’s with that property. Thus we have verified the claim. For simplicity, we will drop the index α in the following arguments. By the assumption and the standard volume comparison, we have 1 Vol t0 Bt0 x0 , r0 ≥ ξ0 r03 2 for some universal positive ξ0 . As in deriving (7.2.7) and (7.2.8), we can find a ball Bt0 (x′0 , r0′ ) ⊂ Bt0 (x0 , r20 ) with Vol t0 (Bt0 (x′0 , r0′ )) ≥

1 1 α3 (r0′ )3 and r0 ≥ r0′ ≥ ξ0′ r0 2 2

for some universal positive constant ξ0′ . Then by what we had proved in the previous case and by the choice of the first time t0 and the smallest radius r0 , we know that r′ the solution is defined in P (x′0 , t0 , 40 , −τ (r0′ )2 ) with the curvature bound R < K(r0′ )−2 ≤ K(ξ0′ )−2 r0−2 .

√ Since r¯ t0 ≥ r0 ≥ re(t0 ) and r¯ → 0 as α → ∞, we see that t0 → +∞ and t0 r0−2 → +∞ ¯ for some suitable large universal positive as α → +∞. Define T (w) = 8c(w)−1 + ξ, ¯ Then for α sufficiently large, we can apply Theorem 7.5.1(iii) and the constant ξ. pinching assumption to conclude that τ (7.5.20) R ≤ K ′ r0−2 , on P x0 , t0 , 4r0 , − (ξ0′ )2 r02 , 2 for some positive constant K ′ depending only on K and ξ0′ . Furthermore, by combining with the pinching assumption, we deduce that when α sufficiently large, (7.5.21)

Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R ≥ −r0−2

466


on P (x0 , t0 , r0 , − τ2 (ξ0′ )2 r02 ). So by applying Theorem 6.3.3(ii) with w = 12 α3 , we have that when α sufficiently large, (7.5.22)

Vol t (Bt (x0 , r0 )) ≥ ξ1 r03 ,

for all t ∈ [t0 − τ2 (ξ0′ )2 r02 , t0 ], and (7.5.23)

R≤

¯ 2B 1 C¯ + r0−2 ≤ Kr0−2 τ¯0 2

on P (x0 , t0 , r40 , − τ2 (ξ0′ )2 r02 ), where ξ1 is some universal positive constant. Next we want to extend the estimate (7.5.23) backwards in time. Denote by t1 = t0 − τ2 (ξ0′ )2 r02 . The estimate (7.5.22) gives Vol t1 (Bt1 (x0 , r0 )) ≥ ξ1 r03 . By the same argument in the derivation of (7.2.7) and (7.2.8) again, we can find a ball Bt1 (x1 , r1 ) ⊂ Bt1 (x0 , r0 ) with Vol t1 (Bt1 (x1 , r1 )) ≥

1 α3 r13 2

and with r1 ≥ ξ1′ r0 for some universal positive constant ξ1′ . Then by what we had proved in the previous case and by the lower bound (7.5.21) at t1 and the choice of the first time t0 , we know that the solution is defined on P (x1 , t1 , r41 , −τ r12 ) with the curvature bound R < Kr1−2 . By applying Theorem 7.5.1(iii) and the pinching assumption again we get that for α sufficiently large, R ≤ K ′′ r0−2

(7.5.20)′

on P (x0 , t1 , 4r0 , − τ2 (ξ1′ )2 r02 ), for some positive constant K ′′ depending only on K and ξ1′ . Moreover, by combining with the pinching assumption, we have (7.5.21)′

Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R ≥ −r0−2

on P (x0 , t1 , 4r0 , − τ2 (ξ1′ )2 r02 ), for α sufficiently large. So by applying Theorem 6.3.3 (ii) with w = 12 α3 again, we have that for α sufficiently large, (7.5.22)′

Vol t (Bt (x0 , r0 )) ≥ ξ1 r03 ,

for all t ∈ [t0 − τ2 (ξ0′ )2 r02 − τ2 (ξ1′ )2 r02 , t0 ], and (7.5.23)

′

¯ 2B 1 ¯ R≤ C+ r0−2 ≤ Kr0−2 τ¯0 2

on P (x0 , t0 , r40 , − τ2 (ξ0′ )2 r02 − τ2 (ξ1′ )2 r02 ).


467

Note that the constants ξ0 , ξ0′ , ξ1 and ξ1′ are universal, independent of the time t1 and the choice of the ball Bt1 (x1 , r1 ). Then we can repeat the above procedure as many times as we like, until we reach the time t0 − τ r02 . Hence we obtain the estimate (7.5.23)′′

R ≤ (C¯ +

¯ −2 1 2B )r0 ≤ Kr0−2 τ¯0 2

on P (x0 , t0 , r40 , −τ r02 ), for sufficiently large α. This contradicts the choice of the point (x0 , t0 ) and the radius r0 which make R ≥ Kr0−2 somewhere in P (x0 , t0 , r40 , −τ r02 ). Therefore we have completed the proof of the theorem. Consequently, we have the following result which is analog of Corollary 7.2.4. Corollary 7.5.3. For any ε > 0 and w > 0, there exist r¯ = r¯(w, ε) > 0, θ = θ(w, ε) > 0 and T = T (w) with the following property. Suppose we have a e and re(t), to solution, constructed by Theorem 7.4.3 with the positive functions δ(t) the Ricci flow with surgery with a compact orientable normalized three-manifold as e re(2t)}. If initial data, where each δ-cutoff at a time t has δ = δ(t)√ ≤ min{δ(t), Bt0 (x0 , r0 ) is a geodesic ball at time t0 , with θ−1 h ≤ r0 ≤ r¯ t0 and t0 ≥ T , where h is the maximal cutoff radii for surgeries in [ t20 , t0 ] (if there is no surgery in the time interval [ t20 , t0 ], we take h = 0), and satisfies min{Rm(x, t0 ) | x ∈ Bt0 (x0 , r0 )} = −r0−2 , then Vol t0 (Bt0 (x0 , r0 )) < wr03 .

Proof. We argue by contradiction. Let θ = θ(w, ε) and T = 2T (w), where θ(w, ε) and T (w) are the positive constant in Theorem 7.5.2. Suppose for any r¯ > 0 there is a solution and a geodesic ball Bt0 (x0 , r0 ) satisfying the assumptions of the corollary √ with θ−1 h ≤ r0 ≤ r¯ t0 and t0 ≥ T , and with min{Rm(x, t0 ) | x ∈ Bt0 (x0 , r0 )} = −r0−2 , but Vol t0 (Bt0 (x0 , r0 )) ≥ wr03 . Without loss of generality, we may assume that r¯ is less than the corresponding constant in Theorem 7.5.2. We can then apply Theorem 7.5.2 to get R(x, t) ≤ Kr0−2 whenever t ∈ [t0 −τ r02 , t0 ] and dt (x, x0 ) ≤ r40 , where τ and K are the positive constants in Theorem 7.5.2. Note that t0 r0−2 ≥ r¯−2 → +∞ as r¯ → 0. By combining with the pinching assumption we have Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R 1 ≥ − r0−2 2

468


in the region P (x0 , t0 , r40 , −τ r02 ) = {(x, t) | x ∈ Bt (x0 , r40 ), t ∈ [t0 − τ r02 , t0 ]}, provided r¯ > 0 is sufficiently small. Thus we get the estimate |Rm| ≤ K ′ r0−2 in P (x0 , t0 , r40 , −τ r02 ), where K ′ is a positive constant depending only on w and ε. We can now apply Theorem 7.5.1 (iii) to conclude that e −2 R(x, t) ≤ Kr 0

e is a positive constant depending whenever t ∈ [t0 − τ2 r02 , t0 ] and dt (x, x0 ) ≤ r0 , where K only on w and ε. By using the pinching assumption again we further have 1 Rm(x, t) ≥ − r0−2 2

in the region P (x0 , t0 , r0 , − τ2 r02 ) = {(x, t) | x ∈ Bt (x0 , r0 ), t ∈ [t0 − τ2 r02 , t0 ]}, as long as r¯ is sufficiently small. In particular, this would imply min{Rm(x, t0 ) | x ∈ Bt0 (x0 , r0 )} > −r0−2 , which is a contradiction. Remark 7.5.4. In section 7.3 of [104], Perelman claimed a stronger statement than the above Corollary 7.5.3 that allows r0 < θ−1 h in the assumptions. Nevertheless, the above weaker statement is sufficient to deduce the geometrization result. 7.6. Long Time Behavior. In Section 5.3, we obtained the long time behavior for smooth (compact) solutions to the three-dimensional Ricci flow with bounded normalized curvature. The purpose of this section is to adapt the arguments there to solutions of the Ricci flow with surgery and to drop the bounded normalized curvature assumption. Recall from Corollary 7.4.4 that we have completely understood the topological structure of a compact, orientable three-manifold with nonnegative scalar curvature. From now on we assume that our initial manifold does not admit any metric with nonnegative scalar curvature, and that once we get a compact component with nonnegative scalar curvature, it is immediately removed. Furthermore, if a solution to the Ricci flow with surgery becomes extinct in a finite time, we have also obtained the topological structure of the initial manifold. So in the following we only consider those solutions to the Ricci flow with surgery which exist for all times t ≥ 0. Let gij (t), 0 ≤ t < +∞, be a solution to the Ricci flow with δ-cutoff surgeries, constructed by Theorem 7.4.3 with normalized initial data. Let 0 < t1 < t2 < · · · < tk < · · · be the surgery times, where each δ-cutoff at a time tk has δ = δ(tk ) ≤ e k ), re(2tk )}. On each time interval (tk−1 , tk ) (denote by t0 = 0), the scalar min{δ(t curvature satisfies the evolution equation ◦ ∂ 2 R = ∆R + 2| Ric |2 + R2 ∂t 3

(7.6.1)

◦

where Ric is the trace-free part of Ric . Then Rmin (t), the minimum of the scalar curvature at the time t, satisfies 2 2 d Rmin (t) ≥ Rmin (t) dt 3


469

for t ∈ (tk−1 , tk ), for each k = 1, 2, . . .. Since our surgery procedure had removed all components with nonnegative scalar curvature, the minimum Rmin (t) is negative for all t ∈ [0, +∞). Also recall that the cutoff surgeries were performed only on δ-necks. Thus the surgeries do not occur at the parts where Rmin (t) are achieved. So the differential inequality 2 2 d Rmin (t) ≥ Rmin (t) dt 3 holds for all t ≥ 0, and then by normalization, Rmin (0) ≥ −1, we have (7.6.2)

3 1 Rmin (t) ≥ − · , for all t ≥ 0. 2 t + 23

Meanwhile, on each time interval (tk−1 , tk ), the volume satisfies the evolution equation Z d V = − RdV dt and then by (7.6.2), d 3 1 V ≤ · V. dt 2 (t + 32 ) Since the cutoff surgeries do not increase volume, we thus have − 32 ! 3 d ≤0 (7.6.3) log V (t) t + dt 2 3

for all t ≥ 0. Equivalently, the function V (t)(t + 32 )− 2 is nonincreasing on [0, +∞). 3 We can now use the monotonicity of the function V (t)(t + 32 )− 2 to extract the information of the solution at large times. On each time interval (tk−1 , tk ), we have − 23 ! d 3 log V (t) t + dt 2 ! Z 1 3 + = − Rmin (t) + (Rmin (t) − R)dV. V M 2 t + 32

Then by noting that the cutoff surgeries do not increase volume, we get ( Z t V (0) 3 V (t) (7.6.4) − Rmin (t) + dt 3 ≤ 3 exp 2(t + 23 ) (t + 32 ) 2 ( 23 ) 2 0 ) Z t Z 1 − (R − Rmin (t))dV dt 0 V M

for all t > 0. Now by this inequality and the equation (7.6.1), we obtain the following consequence. Lemma 7.6.1. Let gij (t) be a solution to the Ricci flow with surgery, constructed by Theorem 7.4.3 with normalized initial data. If for a fixed 0 < r < 1 and a sequence of times tα → ∞, the rescalings of the solution on the parabolic neighborhoods

470


√ √ P (xα , tα , r tα , −r2 tα ) = {(x, t) | x ∈ Bt (xα , r tα ), t ∈ [tα − r2 tα , tα ]}, with factor (tα )−1 and shifting the times tα to 1, converge in the C ∞ topology to some smooth limiting solution, defined in an abstract parabolic neighborhood P (¯ x, 1, r, −r2 ), then this limiting solution has constant sectional curvature −1/4t at any time t ∈ [1 − r2 , 1]. In the previous section we obtained several curvature estimates for the solutions to the Ricci flow with surgery. Now we combine the curvature estimates with the above lemma to derive the following asymptotic result for the curvature. Lemma 7.6.2 (Perelman [104]). For any ε > 0, let gij (t), 0 ≤ t < +∞, be a solution to the Ricci flow with surgery, constructed by Theorem 7.4.3 with normalized initial data. (i) Given w > 0, r > 0, ξ > 0, √ one can find T = T (w, r, ξ, ε) < +∞ such that if the geodesic ball Bt0 (x0 , r t0 ) at some time t0 ≥ T has volume at least 3

wr3 t02 and the sectional curvature at least −r−2 t−1 0 , then the curvature at x0 at time t = t0 satisfies (7.6.5)

|2tRij + gij | < ξ.

(ii) Given in addition 1 ≤ A < ∞ and √ allowing T to depend on A, we can ensure (7.6.5) for all points in Bt0 (x0 , Ar t0 ). (iii) The same √ is true for all points in the forward parabolic neighborhood √ P (x0 , t0 , Ar t0 , Ar2 t0 ) , {(x, t) | x ∈ Bt (x0 , Ar t0 ), t ∈ [t0 , t0 + Ar2 t0 ]}. Proof. (i) By the assumptions and the standard volume comparison, we have Vol t0 (Bt0 (x0 , ρ)) ≥ cwρ3

√ for all 0 < ρ ≤ r t0 , where c is a universal positive constant. Let r¯ = r¯(cw, √ ε) be the positive constant in Theorem 7.5.2 and set r = min{r, r ¯ }. On B (x , r t0 )(⊂ 0 t 0 0 0 √ Bt0 (x0 , r t0 )), we have √ Rm ≥ −(r0 t0 )−2 (7.6.6) √ √ and Vol t0 (Bt0 (x0 , r0 t0 )) ≥ cw(r0 t0 )3 . √ √ Obviously, there holds θ−1 h ≤ r0 t0 ≤ r¯ t0 when t0 is large enough, where θ = θ(cw, ε) is the positive constant in Theorem 7.5.2 and h is the maximal cutoff radius for surgeries in [ t20 , t0 ] (if there is no surgery in the time interval [ t20 , t0 ], we take h = 0). Then it follows from Theorem 7.5.2 that the solution is defined and satisfies √ R < K(r0 t0 )−2 √ √ on whole parabolic neighborhood P (x0 , t0 , r0 4 t0 , −τ (r0 t0 )2 ). Here τ = τ (cw, ε) and K = K(cw, ε) are the positive constants in Theorem 7.5.2. By combining with the pinching assumption we have

Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R √ ≥ −const. K(r0 t0 )−2

√ √ in the region P (x0 , t0 , r0 4 t0 , −τ (r0 t0 )2 ). Thus we get the estimate √ (7.6.7) |Rm| ≤ K ′ (r0 t0 )−2


471

√ √ on P (x0 , t0 , r0 4 t0 , −τ (r0 t0 )2 ), for some positive constant K ′ = K ′ (w, ε) depending only on w and ε. The curvature estimate (7.6.7) and the volume estimate (7.6.6) ensure that as t0 → +∞ we can take smooth (subsequent) limits for the rescalings of the solution √ √ r0 t0 −1 with factor (t0 ) on parabolic neighborhoods P (x0 , t0 , 4 , −τ (r0 t0 )2 ). Then by applying Lemma 7.6.1, we can find T = T (w, r, ξ, ε) < +∞ such that when t0 ≥ T , there holds

|2tRij + gij | < ξ,

(7.6.8)

√ on P (x0 , t0 , r0 4 t0 , −τ (r0 t0 )2 ), in particular, √

|2tRij + gij |(x0 , t0 ) < ξ. This proves the assertion (i). (ii) In view √ of the above argument, to get the estimate (7.6.5) for all points bound for the scalar curvature on in Bt0 (x0 , Ar t0 ), the key point is to get √ a upper √ the parabolic neighborhood P (x0 , t0 , Ar t0 , −τ (r0 t0 )2 ). After having the estimates (7.6.6) and (7.6.7), one would like to use Theorem 7.5.1(iii) to obtain the desired scalar curvature estimate. Unfortunately it does not work since our r0 may be much larger than the constant r¯(A, ε) there when A is very large. In the following we will use Theorem 7.5.1(ii) to overcome the difficulty. Given 1 ≤ A < +∞, based on (7.6.6) and (7.6.7), we can use Theorem 7.5.1(ii) √ to find a positive constant K1 = K1 (w, r, A, ε) such that each point in B (x , 2Ar t0 ) t0 0 √ with its scalar curvature at least K1 (r t0 )−2 has a canonical neighborhood. We claim that there exists T = T (w, r, A, ε) < +∞ so that when t0 ≥ T , we have √ √ (7.6.9) R < K1 (r t0 )−2 , on Bt0 (x0 , 2Ar t0 ). Argue by contradiction. Suppose not; then there exist a sequence of times tα 0 → pα α α α α α +∞ and sequences of points x0 , x with x ∈ Bt0 (x0 , 2Ar t0 ) and R(xα , tα ) = 0 p −2 ) . Since there exist canonical neighborhoods (ε-necks or ε-caps) around K1 (r tα 0 the points (xα , tα 0 ), there exist positive constants c1 , C2 depending only on ε such that − 12 p α 3 −1 p Vol tα0 (Btα0 (xα , K1 2 (r tα 0 ))) ≥ c1 (K1 (r t0 ))

and

C2−1 K1 (r −1

on Btα0 (xα , K1 2 (r have

−1

p

on Btα0 (xα , K1 2 (r proved that

p

−2 tα ≤ R(x, tα 0 ) ≤ C2 K1 (r 0)

p

−2 tα , 0)

tα 0 )), for all α. By combining with the pinching assumption we

p

Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R p −2 , ≥ −const. C2 K1 (r tα 0)

tα 0 )), for all α. It then follows from the assertion (i) we just lim |2tRij + gij |(xα , tα 0 ) = 0.

α→+∞

472


In particular, we have α α tα 0 R(x , t0 ) < −1

for αp sufficiently large. This contradicts our assumption that R(xα , tα 0) = −2 α K1 (r t0 ) . So we have proved assertion (7.6.9). Now by combining (7.6.9) with the pinching assumption as before, we have (7.6.10)

√ Rm ≥ −K2 (r t0 )−2

√ on Bt0 (x0 , 2Ar t0 ), where K2 = K2 (w, r, A, ε) is some positive constant depending only on w, r, A and ε. Thus by (7.6.9) and (7.6.10) we have (7.6.11)

√ √ |Rm| ≤ K1′ (r t0 )−2 , on Bt0 (x0 , 2Ar t0 ),

for some positive constant K1′ = K1′ (w, r, A, ε) depending only on w, r, A and ε. This √ gives us the curvature estimate on Bt0 (x0 , 2Ar t0 ) for all t0 ≥ T (w, r, A, ε). From the arguments in proving the above assertion (i), we have the estimates (7.6.6) and (7.6.7) and the solution is well-defined on the whole parabolic neighbor√ √ hood P (x0 , t0 , r0 4 t0 , −τ (r0 t0 )2 ) for all t0 ≥ T (w, r, A, ε). Clearly we may assume √ 1 that (K1′ )− 2 r < min{ r40 , τ r0 }. Thus by combining with the curvature estimate (7.6.11), we can apply Theorem 7.5.1(i) to get the following volume control (7.6.12)

1 √ 1 √ Vol t0 (Bt0 (x, (K1′ )− 2 r t0 )) ≥ κ((K1′ )− 2 r t0 )3

√ for any x ∈ Bt0 (x0 , Ar t0 ), where κ = κ(w, r, A, ε) is some positive constant depending only on w, r, A and ε. So by using the assertion (i), we see that for t0 ≥ T with T = T (w, r,√ξ, A, ε) large enough, the curvature estimate (7.6.5) holds for all points in Bt0 (x0 , Ar t0 ). (iii) We next want to extend the curvature √ estimate (7.6.5) to all points in the forward parabolic neighborhood P (x0 , t0 , Ar t0 , Ar2 t0 ). Consider the time interval [t0 , t0 + Ar2 t0 ] in the parabolic neighborhood. In assertion (ii), we have obtained the desired estimate (7.6.5) at t = t0 . Suppose estimate (7.6.5) holds on a maximal time interval [t0 , t′ ) with t′ ≤ t0 + Ar2 t0 . This says that we have (7.6.13)

|2tRij + gij | < ξ

√ √ on P (x0 , t0 , Ar t0 , t′ − t0 ) , {(x, t) | x ∈√Bt (x0 , Ar t0 ), t ∈ [t0 , t′ )} so that either there exists a surgery in the√ball Bt′ (x0 , Ar t0 ) at t = t′ , or there holds |2tRij +gij | = ξ somewhere in Bt′ (x0 , Ar t0 ) at t = t′ . Since the Ricci curvature is near − 2t1′ in the geodesic ball, the surgeries cannot occur there. Thus we only need to consider the latter possibility. Recall that the evolution of the length of a curve γ and the volume of a domain Ω are given by Z d Lt (γ) = − Ric (γ, ˙ γ)ds ˙ t dt γ Z d and V olt (Ω) = − RdVt . dt Ω


473

By substituting the curvature estimate (7.6.13) into the above two evolution equations and using the volume lower bound (7.6.6), it is not hard to see √ 3 (7.6.14) Vol t′ (Bt′ (x0 , t′ )) ≥ κ′ (t′ ) 2 for some positive constant κ′ = κ′ (w, r, ξ, A, ε) depending only on w, r, ξ, A and ε. Then by the above assertion (ii), the combination of the curvature estimate (7.6.13) and the volume lower bound (7.6.14) √ implies that the curvature estimate (7.6.5) still holds for all points in Bt′ (x0 , Ar t0 ) provided T = T (w, r, ξ, A, ε) is chosen large enough. This is a contradiction. Therefore we have proved assertion (iii). We now state and prove the important Thick-thin decomposition theorem. A more general version (without the restriction on ε) was implicitly claimed by Perelman in [103] and [104]. Theorem 7.6.3 (The Thick-thin decomposition theorem). For any w > 0 and 0 < ε ≤ 21 w, there exists a positive constant ρ = ρ(w, ε) ≤ 1 with the following property. Suppose gij (t) (t ∈ [0, +∞)) is a solution, constructed by Theorem 7.4.3 with e and re(t), to the Ricci flow with the nonincreasing (continuous) positive functions δ(t) surgery and with a compact orientable normalized three-manifold as initial data, where e re(2t)}. Then for any arbitrarily each δ-cutoff at a time t has δ = δ(t) ≤ min{δ(t), fixed ξ > 0, for t large enough, the manifold Mt at time t admits a decomposition Mt = Mthin (w, t) ∪ Mthick(w, t) with the following properties: √ (a) For √ every x ∈ Mthin(w, t), there exists some r = r(x, t) > 0, with 0 < r t < ρ t, such that √ √ Rm ≥ −(r t)−2 on Bt (x, r t), and √ √ Vol t (Bt (x, r t)) < w(r t)3 . (b) For every x ∈ Mthick (w, t), we have

√ |2tRij + gij | < ξ on Bt (x, ρ t), and √ √ 1 w(ρ t)3 . Vol t (Bt (x, ρ t)) ≥ 10

Moreover, if we take any sequence of points xα ∈ Mthick (w, tα ), tα → +∞, then the scalings of gij (tα ) around xα with factor (tα )−1 converge smoothly, along a subsequence of α → +∞, to a complete hyperbolic manifold of finite volume with constant sectional curvature − 41 . Proof. Let r¯ = r¯(w, ε), θ = θ(w, ε) and h be the positive constants obtained in Corollary 7.5.3. We may assume ρ ≤ r¯ ≤ e−3 . For any point x ∈ Mt , there are two cases: either √ √ (i) min{Rm | Bt (x, ρ t)} ≥ −(ρ t)−2 , or (ii)

√ √ min{Rm | Bt (x, ρ t)} < −(ρ t)−2 .

474


√ Let us first consider Case (i). If Vol t (Bt (x, ρ t)) < choose r slightly less than ρ so that √ √ Rm ≥ −(ρ t)−2 ≥ −(r t)−2 √ √ on Bt (x, r t)(⊂ Bt (x, ρ t)), and

1 10 w(ρ

√

t)3 , then we can

√ √ √ 1 w(ρ t)3 < w(r t)3 ; Vol t (Bt (x, r t)) < 10 √ √ 1 w(ρ t)3 , we can apply Lemma 7.6.2(ii) thus x ∈ Mthin (w, t). If Vol t (Bt (x, ρ t)) ≥ 10 to conclude that for t large enough, √ |2tRij + gij | < ξ on Bt (x, ρ t); thus x ∈ Mthick (w, t). Next we consider Case (ii). By continuity, there exists 0 < r = r(x, t) < ρ such that √ √ (7.6.15) min{Rm | Bt (x, r t)} = −(r t)−2 . √ √ If θ−1 h ≤ r t (≤ r¯ t), we can apply Corollary 7.5.3 to conclude √ √ Vol t (Bt (x, r t)) < w(r t)3 ; thus x ∈ Mthin (w, t). √ We now consider the difficult subcase r t < θ−1 h. By the pinching assumption, we have √ √ R ≥ (r t)−2 (log[(r t)−2 (1 + t)] − 3) √ ≥ (log r−2 − 3)(r t)−2 √ ≥ 2(r t)−2 ≥ 2θ2 h−2

√ somewhere in Bt (x, r t). Since h is the maximal cutoff radius for surgeries in [ 2t , t], by the design of the δ-cutoff surgery, we have t ,t h ≤ sup δ 2 (s)e r (s) | s ∈ 2 t t re(t)e r . ≤ δe 2 2

e t ) → 0 as t → +∞. Thus from the canonical neighborhood assumption, Note also δ( 2 √ we see that for t large enough, there exists a point in the ball Bt (x, r t) which has a canonical neighborhood. We claim that for t sufficiently large, the point x satisfies R(x, t) ≥

1 √ −2 (r t) , 2

and then the above argument shows that the point x also has a canonical ε-neck√or ε-cap neighborhood. Otherwise, by continuity, we can choose a point x∗ ∈ Bt (x, r t)


475

√ with R(x∗ , t) = 12 (r t)−2 . Clearly the new point x∗ has a canonical neighborhood B ∗ by the above argument. In particular, there holds C2−1 (ε)R ≤

1 √ −2 (r t) ≤ C2 (ε)R 2

on the canonical neighborhood B ∗ . By the definition of canonical neighborhood assumption, we have Bt (x∗ , σ ∗ ) ⊂ B ∗ ⊂ Bt (x∗ , 2σ ∗ ) 1

for some σ ∗ ∈ (0, C1 (ε)R− 2 (x∗ , t)). Clearly, without loss of generality, we may assume 1 (in the definition of canonical neighborhood assumption) that σ ∗ > 2R− 2 (x∗ , t). Then 1 −1 C (ε)r−2 2 2 1 ≥ C2−1 (ε)ρ−2 2

R(1 + t) ≥

√ on Bt (x∗ , 2r t). Thus when we choose ρ = ρ(w, ε) small enough, it follows from the pinching assumption that Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R 1 √ ≥ − (r t)−2 , 2 √ on Bt (x∗ , 2r t). This is a contradiction with (7.6.15). We have seen that tR(x, t) ≥ 21 r−2 (≥ 21 ρ−2 ). Since r−2 > θ2 h−2 t in this subcase, we conclude that for arbitrarily given A < +∞, (7.6.16)

tR(x, t) > A2 ρ−2

as long as t is large enough. Let B, with Bt (x, σ) ⊂ B ⊂ Bt (x, 2σ), be the canonical ε-neck or ε-cap neighborhood of (x, t). By the definition of the canonical neighborhood assumption, we have 1

0 < σ < C1 (ε)R− 2 (x, t), C2−1 (ε)R ≤ R(x, t) ≤ C2 (ε)R, on B, and (7.6.17)

Vol t (B) ≤ εσ 3 ≤

1 3 wσ . 2

1

Choose 0 < A < C1 (ε) so that σ = AR− 2 (x, t). For sufficiently large t, since R(1 + t) ≥ C2−1 (ε)(tR(x, t)) 1 ≥ C2−1 (ε)ρ−2 , 2

476


on B, we can require ρ = ρ(w, ε) to be smaller still, and use the pinching assumption to conclude (7.6.18)

Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R 1

≥ −(AR− 2 (x, t))−2 = −σ −2 ,

on B. For sufficiently large t, we adjust √ (7.6.19) r = σ( t)−1

√ 1 = (AR− 2 (x, t))( t)−1 < ρ,

by (7.6.16). Then the combination of (7.6.17), (7.6.18) and (7.6.19) implies that x ∈ Mthin (w, t). The last statement in (b) follows directly from Lemma 7.6.2. (Here we also used Bishop-Gromov volume comparison, Theorem 7.5.2 and Hamilton’s compactness theorem to take a subsequent limit.) Therefore we have completed the proof of the theorem. To state the long-time behavior of a solution to the Ricci flow with surgery, we first recall some basic terminology in three-dimensional topology. A three-manifold M is called irreducible if every smooth two-sphere embedded in M bounds a threeball in M . If we have a solution (Mt , gij (t)) obtained by Theorem 7.4.3 with a compact, orientable and irreducible three-manifold (M, gij ) as initial data, then at each time t > 0, by the cutoff surgery procedure, the solution manifold Mt consists of a finite number of components where one of the components, called the essential (1) component and denoted by Mt , is diffeomorphic to the initial manifold M while the rest are diffeomorphic to the three-sphere S3 . The main result of this section is the following generalization of Theorem 5.3.4. A more general version of the result (without the restriction on ε) was implicitly claimed by Perelman in [104]. Theorem 7.6.4 (Long-time behavior of the Ricci flow with surgery). Let w > 0 and 0 < ε ≤ 21 w be any small positive constants and let (Mt , gij (t)), 0 < t < +∞, be a solution to the Ricci flow with surgery, constructed by Theorem 7.4.3 with e and re(t) and with a compact, the nonincreasing (continuous) positive functions δ(t) orientable, irreducible and normalized three-manifold M as initial data, where each e re(2t)}. Then one of the following holds: δ-cutoff at a time t has δ = δ(t) ≤ min{δ(t), either (i) for all sufficiently large t, we have Mt = Mthin(w, t); or (ii) there exists a sequence of times tα → +∞ such that the scalings of gij (tα ) on (1) the essential component Mtα , with factor (tα )−1 , converge in the C ∞ topology to a hyperbolic metric on the initial compact manifold M with constant sectional curvature − 41 ; or (iii) we can find a finite collection of complete noncompact hyperbolic threemanifolds H1 , . . . , Hm , with finite volume, and compact subsets K1 , . . . , Km of H1 , . . . , Hm respectively obtained by truncating each cusp of the hyperbolic manifolds along constant mean curvature torus of small area, and for all t beyond some time T < +∞ we can find diffeomorphisms ϕl , 1 ≤ l ≤ m, of Kl


477

into Mt so that as long as t is sufficiently large, the metric t−1 ϕ∗l (t)gij (t) is as close to the hyperbolic metric as we like on the compact sets K1 , . . . , Km ; moreover, the complement Mt \(ϕ1 (K1 ) ∪ · · · ∪ ϕm (Km )) is contained in the thin part Mthin (w, t), and the boundary tori of each Kl are incompressible in the sense that each ϕl injects π1 (∂Kl ) into π1 (Mt ). Proof. The proof of the theorem follows, with some modifications, essentially from the same argument of Hamilton as in the proof of Theorem 5.3.4. Clearly we may assume that the thick part Mthick (w, t) is not empty for a sequence tα → +∞, since otherwise we have case (i). If we take a sequence of points xα ∈ Mthick (w, tα ), then by Theorem 7.6.3(b) the scalings of gij (tα ) around xα with factor (tα )−1 converge smoothly, along a subsequence of α → +∞, to a complete hyperbolic manifold of finite volume with constant sectional curvature − 14 . The limits may be different for different choices of (xα , tα ). If a limit is compact, we have case (ii). Thus we assume that all limits are noncompact. Consider all the possible hyperbolic limits of the solution, and among them choose one such complete noncompact hyperbolic three-manifold H with the least possible number of cusps. Denote by hij the hyperbolic metric of H. For all small a > 0 we can truncate each cusp of H along a constant mean curvature torus of area a which is uniquely determined; we denote the remainder by Ha . Fix a > 0 so small that Lemma 5.3.7 is applicable for the compact set K = Ha . Pick an integer l0 sufficiently large and an ǫ0 sufficiently small to guarantee from Lemma 5.3.8 that the identity map Id is the only harmonic map F from Ha to itself with taking ∂Ha to itself, with the normal derivative of F at the boundary of the domain normal to the boundary of the target, and with dC l0 (Ha ) (F, Id) < ǫ0 . Then choose a positive integer q0 and a small number δ0 > 0 from Lemma 5.3.7 such that if Fe is a diffeomorphism of Ha e e into another complete noncompact hyperbolic three-manifold (H, hij ) with no fewer cusps (than H), of finite volume and satisfying kFe ∗e hij − hij kC q0 (Ha ) ≤ δ0 ,

e such that then there exists an isometry I of H to H

dC l0 (Ha ) (Fe , I) < ǫ0 .

By Lemma 5.3.8 we further require q0 and δ0 to guarantee the existence of a harmonic diffeomorphism from (Ha , geij ) to (Ha , hij ) for any metric e gij on Ha with ke gij − hij kC q0 (Ha ) ≤ δ0 . Let xα ∈ Mthick (w, tα ), tα → +∞, be a sequence of points such that the scalings of gij (tα ) around xα with factor (tα )−1 converge to hij . Then there exist a marked point x∞ ∈ Ha and a sequence of diffeomorphisms Fα from Ha into Mtα such that Fα (x∞ ) = xα and k(tα )−1 Fα∗ gij (tα ) − hij kC m (Ha ) → 0 as α → ∞ for all positive integers m. By applying Lemma 5.3.8 and the implicit function theorem, we can change Fα by an amount which goes to zero as α → ∞ so as to make Fα a harmonic diffeomorphism taking ∂Ha to a constant mean curvature hypersurface Fα (∂Ha ) of (Mtα , (tα )−1 gij (tα )) with the area a and satisfying the free boundary condition that the normal derivative of Fα at the boundary of the domain is normal to the boundary of the target; and by combining with Lemma 7.6.2 (iii),

478


we can smoothly continue each harmonic diffeomorphism Fα forward in time a little to a family of harmonic diffeomorphisms Fα (t) from Ha into Mt with the metric t−1 gij (t), with Fα (tα ) = Fα and with the time t slightly larger than tα , where Fα (t) takes ∂Ha into a constant mean curvature hypersurface of (Mt , t−1 gij (t)) with the area a and also satisfies the free boundary condition. Moreover, since the surgeries do not take place at the points where the scalar curvature is negative, by the same argument as in Theorem 5.3.4 for an arbitrarily given positive integer q ≥ q0 , positive number δ < δ0 , and sufficiently large α, we can ensure the extension Fα (t) satisfies kt−1 Fα∗ (t)gij (t) − hij kC q (Ha ) ≤ δ on a maximal time interval tα ≤ t ≤ ω α (or tα ≤ t < ω α when ω α = +∞), and with k(ω α )−1 Fα∗ (ω α )gij (ω α ) − hij kC q (Ha ) = δ, when ω α < +∞. Here we have implicitly used the fact that Fα (ω α )(∂Ha ) is still strictly concave to ensure the map Fα (ω α ) is diffeomorphic. We further claim that there must be some α such that ω α = +∞; in other words, at least one hyperbolic piece persists. Indeed, suppose that for each large enough α we can only continue the family Fα (t) on a finite interval tα ≤ t ≤ ω α < +∞ with k(ω α )−1 Fα∗ (ω α )gij (ω α ) − hij kC q (Ha ) = δ. Consider the new sequence of manifolds (Mωα , gij (ω α )). Clearly by Lemma 7.6.1, the scalings of gij (ω α ) around the new origins Fα (ω α )(x∞ ) with factor (ω α )−1 converge smoothly (by passing to a subsequence) to a complete noncompact hyperbolic threee with the metric e manifold H hij and the origin x e∞ and with finite volume. By the e choice of the old limit H, the new limit H has at least as many cusps as H. By the eα exhausting definition of convergence, we can find a sequence of compact subsets U ∞ e e eα H and containing x e , and a sequence of diffeomorphisms Fα of neighborhood of U ∞ α ∞ e e e into Mωα with Fα (e x ) = Fα (ω )(x ) such that for each compact subset U of H and each integer m, k(ω α )−1 Feα∗ (gij (ω α )) − e hij kC m (Ue ) → 0

as α → +∞. Thus for sufficiently large α, we have the map such that

e Gα = Feα−1 ◦ Fα (ω α ) : Ha → H kG∗α e hij − hij kC q (Ha ) < δe

for any fixed δe > δ. Then a subsequence of Gα converges at least in the C q−1 (Ha ) e which is a harmonic map from Ha into H e and topology to a map G∞ of Ha into H e e takes ∂Ha to a constant mean curvature hypersurface G∞ (∂Ha ) of (H, hij ) with the area a, as well as satisfies the free boundary condition. Clearly, G∞ is at least a local diffeomorphism. Since G∞ is the limit of diffeomorphisms, the only possibility of overlap is at the boundary. Note that G∞ (∂Ha ) is still strictly concave. So G∞ is still a diffeomorphism. Moreover by using the standard regularity result of elliptic partial differential equations (see for example [48]), we also have (7.6.20)

kG∗∞e hij − hij kC q (Ha ) = δ.

e with Now by Lemma 5.3.7 we deduce that there exists an isometry I of H to H dC l0 (Ha ) (G∞ , I) < ǫ0 .


479

Thus I −1 ◦ G∞ is a harmonic diffeomorphism of Ha to itself which satisfies the free boundary condition and dC l0 (Ha ) (I −1 ◦ G∞ , Id) < ǫ0 . However the uniqueness in Lemma 5.3.8 concludes that I −1 ◦ G∞ = Id which contradicts (7.6.20). So we have shown that at least one hyperbolic piece persists and the metric t−1 Fα∗ (t)gij (t), for ω α ≤ t < ∞, is as close to the hyperbolic metric hij as we like. We can continue to form other persistent hyperbolic pieces in the same way as long as there is a sequence of points y β ∈ Mthick(w, tβ ), tβ → +∞, lying outside the chosen 3 pieces. Note that V (t)(t + 23 )− 2 is nonincreasing on [0, +∞). Therefore by combining with Margulis lemma (see for example [55] or [76]), we have proved that there exists a finite collection of complete noncompact hyperbolic three-manifolds H1 , . . . , Hm with finite volume, a small number a > 0 and a time T < +∞ such that for all t beyond T we can find diffeomorphisms ϕl (t) of (Hl )a into Mt , 1 ≤ l ≤ m, so that as long as t is sufficiently large, the metric t−1 ϕ∗l (t)gij (t) is as close to the hyperbolic metrics as we like and the complement Mt \(ϕ1 (t)((H1 )a ) ∪ · · · ∪ ϕm (t)((Hm )a )) is contained in the thin part Mthin (w, t). It remains to show the boundary tori of any persistent hyperbolic piece are incompressible. Let B be a small positive number and assume the above positive number a is much smaller than B. Let Ma (t) = ϕl (t)((Hl )a ) (1 ≤ l ≤ m) be such a persistent hyperbolic piece of the manifold Mt truncated by boundary tori of area at ◦

with constant mean curvature, and denote by Mac (t) = Mt \ M a (t) the part of Mt exterior to Ma (t). Thus there exists a family of subsets MB (t) ⊂ Ma (t) which is a persistent hyperbolic piece of the manifold Mt truncated by boundary tori of area Bt ◦

with constant mean curvature. We also denote by MBc (t) = Mt \ M B (t). By Van Kampen’s Theorem, if π1 (∂MB (t)) injects into π1 (MBc (t)) then it injects into π1 (Mt ) also. Thus we only need to show π1 (∂MB (t)) injects into π1 (MBc (t)). As before we will use a contradiction argument to show π1 (∂MB (t)) injects into π1 (MBc (t)). Let T be a torus in ∂MB (t) and suppose π1 (T ) does not inject into π1 (MBc (t)). By Dehn’s Lemma we know that the kernel is a cyclic subgroup of π1 (T ) generated by a primitive element. Consider the normalized metric geij (t) = t−1 gij (t) on Mt . Then by the work of Meeks-Yau [86] or Meeks-Simon-Yau [87], we know that among all disks in MBc (t) whose boundary curve lies in T and generates the kernel of π1 (T ), there is a smooth embedded disk normal to the boundary which has the least possible area (with respect to the normalized metric e gij (t)). Denote by D the e = A(t) e its area. We will show that A(t) e decreases at a certain minimal disk and A rate which will arrive at a contradiction. We first consider the case that there exist no surgeries at the time t. Exactly as e comes from the in Part III of the proof of Theorem 5.3.4, the change of the area A(t) change in the metric and the change in the boundary. For the change in the metric, we choose an orthonormal frame X, Y, Z at a point x in the disk D so that X and Y are tangent to the disk D while Z is normal. Since the normalized metric e gij evolves by ∂ eij ), geij = −t−1 (e gij + 2R ∂t

480


the (normalized) area element de σ of the disk D around x satisfies ∂ g (X, X) + Ric g (Y, Y ))de de σ = −t−1 (1 + Ric σ. ∂t

eij is very small for For the change in the boundary, we notice that the tensor geij + 2R the persistent hyperbolic piece. Then by using the Gauss-Bonnet theorem as before, we obtain the rate of change of the area ! Z Z e e 1 R 1 2π 1 e dA e ≤− + de σ+ kde s− +o L, (7.6.21) dt t 2t t ∂D t t D

e is the length of the boundary where e k is the geodesic curvature of the boundary and L e ≥ −3t/2(t + 3 ) for curve ∂D (with respect to the normalized metric geij (t)). Since R 2 all t ≥ 0 by (7.6.2), the first term on the RHS of (7.6.21) is bounded above by ! Z e R 1 1 1 e + de σ≤− − o(1) A; − t 2t t 4 D while the second term on the RHS of (7.6.21) can be estimated exactly as before by Z 1 1 1 e e kde s≤ + o(1) L. t ∂D t 4

Thus we obtain (7.6.22)

e dA 1 ≤ dt t

1 1 e e + o(1) L − − o(1) A − 2π . 4 4

Next we show that these arguments also work for the case that there exist surgeries at the time t. To this end, we only need to check that the embedded minimal disk D lies in the region which is unaffected by surgery. Our surgeries for the irreducible three-manifold took place on δ-necks in ε-horns, where the scalar curvatures are at least δ −2 (e r (t))−1 , and the components with nonnegative scalar curvature have been removed. So the hyperbolic piece is not affected by the surgeries. In particular, the boundary ∂D is unaffected by the surgeries. Thus if surgeries occur on the minimal disk, the minimal disk has to pass through a long thin neck before it reaches the surgery regions. Look at the intersections of the embedding minimal disk with a generic center two-sphere S2 of the long thin neck; these are circles. Since the twosphere S2 is simply connected, we can replace the components of the minimal disk D outside the center two-sphere S2 by some corresponding components on the center two-sphere S2 to form a new disk which also has ∂D as its boundary. Since the metric on the long thin neck is nearly a product metric, we could choose the generic center two-sphere S2 properly so that the area of the new disk is strictly less than the area of the original disk D. This contradiction proves the minimal disk lies entirely in the region unaffected by surgery. Since a is much smaller than B, the region within a long distance from ∂MB (t) into MBc (t) will look nearly like a hyperbolic cusplike collar and is unaffected by the surgeries. So we can repeat the arguments in the last part of the proof of Theorem e by the area A e and to conclude 5.3.4 to bound the length L e π dA ≤− dt t


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for all sufficiently large times t, which is impossible because the RHS is not integrable. This proves that the boundary tori of any persistent hyperbolic piece are incompressible. Therefore we have proved the theorem. 7.7. Geometrization of Three-manifolds. In the late 70’s and early 80’s, Thurston [122], [123] [124] proved a number of remarkable results on the existence of geometric structures on a class of three-manifolds: Haken manifolds (i.e. each of them contains an incompressible surface of genus ≥ 1). These results motivated him to formulate a profound conjecture which roughly says every compact three-manifold admits a canonical decomposition into domains, each of which has a canonical geometric structure. To give a detailed description of the conjecture, we recall some terminology as follows. An n-dimensional complete Riemannian manifold (M, g) is called a homogeneous manifold if its group of isometries acts transitively on the manifold. This means that the homogeneous manifold looks the same metrically at everypoint. For example, the round n-sphere Sn , the Euclidean space Rn and the standard hyperbolic space Hn are homogeneous manifolds. A Riemannian manifold is said to be modeled on a given homogeneous manifold (M, g) if every point of the manifold has a neighborhood isometric to an open set of (M, g). And an n-dimensional Riemannian manifold is called a locally homogeneous manifold if it is complete and is modeled on a homogeneous manifold. By a theorem of Singer [119], the universal cover of a locally homogeneous manifold (with the pull-back metric) is a homogeneous manifold. In dimension three, every locally homogeneous manifold with finite volume is modeled on one of the following eight homogeneous manifolds (see for example Theorem 3.8.4 of [125]): (1) S3 , the round three-sphere; (2) R3 , the Euclidean space ; (3) H3 , the standard hyperbolic space; (4) S2 × R; (5) H2 × R; (6) N il, the three-dimensional nilpotent Heisenberg group (consisting of upper triangular 3 × 3 matrices with diagonal entries 1); ] (7) P SL(2, R), the universal cover of the unit sphere bundle of H2 ; (8) Sol, the three-dimensional solvable Lie group. A three-manifold M is called prime if it is not diffeomorphic to S3 and if every (topological) S2 ⊂ M , which separates M into two pieces, has the property that one of the two pieces is diffeomorphic to a three-ball. Recall that a three-manifold is irreducible if every embedded two-sphere bounds a three-ball in the manifold. Clearly an irreducible three-manifold is either prime or is diffeomorphic to S3 . Conversely, an orientable prime three-manifold is either irreducible or is diffeomorphic to S2 × S1 (see for example [69]). One of the first results in three-manifold topology is the following prime decomposition theorem obtained by Kneser [79] in 1929 (see also Theorem 3.15 of [69]). Prime Decomposition Theorem. Every compact orientable three-manifold admits a decomposition as a finite connected sum of orientable prime three-manifolds. In [90], Milnor showed that the factors involved in the above Prime Decomposition are unique. Based on the prime decomposition, the question about topology

482


of compact orientable three-manifolds is reduced to the question about prime threemanifolds. Thurston’s Geometrization Conjecture is about prime three-manifolds. Thurston’s Geometrization Conjecture. Let M be a compact, orientable and prime three-manifold. Then there is an embedding ` of a finite number of disjoint unions, possibly empty, of incompressible two-tori i Ti2 ⊂ M such that every component of the complement admits a locally homogeneous Riemannian metric of finite volume. We remark that the existence of a torus decomposition, also called JSJdecomposition, was already obtained by Jaco-Shalen [74] and Johannsen [75]. The JSJ-decomposition states that any compact, orientable, and prime three-manifold has a finite collection, possibly empty, of disjoint incompressible embedding two-tori {Ti2} which separate the manifold into a finite collection of compact three-manifolds (with toral boundary), each of which is either a graph manifold or is atoroidal in the sense that any subgroup of its fundamental group isomorphic to Z × Z is conjugate into the fundamental group of some component of its boundary. A compact three-manifold X, possibly with boundary, is called a graph manifold if there is S a finite collection of disjoint embedded tori Ti ⊂ X such that each component of X \ Ti is an S1 bundle over a surface. Thus the point of the conjecture is that the components should all be geometric. The geometrization conjecture for a general compact orientable 3-manifold is the statement that each of its prime factors satisfies the above conjecture. We say a compact orientable three-manifold is geometrizable if it satisfies the geometric conjecture. We also remark that the Poincaré conjecture can be deduced from Thurston’s geometrization conjecture. Indeed, suppose that we have a compact simply connected three-manifold that satisfies the conclusion of the geometrization conjecture. If it were not diffeomorphic to the three-sphere S3 , there would be a prime factor in the prime decomposition of the manifold. Since the prime factor still has vanishing fundamental group, the (torus) decomposition of the prime factor in the geometrization conjecture must be trivial. Thus the prime factor is a compact homogeneous manifold model. From the list of above eight models, we see that the only compact three-dimensional model is S3 . This is a contradiction. Consequently, the compact simply connected three-manifold is diffeomorphic to S3 . Now we apply the Ricci flow to discuss Thurston’s geometrization conjecture. Let M be a compact, orientable and prime three-manifold. Since a prime orientable three-manifold is either irreducible or is diffeomorphic to S2 × S1 , we may thus assume the manifold M is irreducible also. Arbitrarily given a (normalized) Riemannian metric for the manifold M , we use it as initial data to evolve the metric by the Ricci flow with surgery. From Theorem 7.4.3, we know that the Ricci flow with surgery has a long-time solution on a maximal time interval [0, T ) which satisfies the a priori assumptions and has a finite number of surgeries on each finite time interval. Furthermore, from the long-time behavior theorem (Theorem 7.6.4), we have wellunderstood geometric structures on the thick part. Whereas, to understand the thin part, Perelman announced the following assertion in [104]. α Perelman’s Claim ([104]). Suppose (M α , gij ) is a sequence of compact orientable three-manifolds, closed or with convex boundary, and wα → 0. Assume that (1) for each point x ∈ M α there exists a radius ρ = ρα (x), 0 < ρ < 1, not exceeding the diameter of the manifold, such that the ball B(x, ρ) in the


483

α metric gij has volume at most wα ρ3 and sectional curvatures at least −ρ−2 ; (2) each component of the boundary of M α has diameter at most wα , and has a (topologically trivial) collar of length one, where the sectional curvatures are between −1/4 − ǫ and −1/4 + ǫ. Then M α for sufficiently large α are diffeomorphic to graph manifolds.

The topology of graph manifolds is well understood; in particular, every graph manifold is geometrizable (see [126]). The proof of Perelman’s Claim promised in [104] is still not available in literature. Nevertheless, recently in [118], Shioya and Yamaguchi provided a proof of Perelman’s α Claim for the special case that all the manifolds (M α , gij ) are closed. That is, they proposed a proof for the following weaker assertion. Weaker Assertion (Theorem 8.1 of Shioya-Yamaguchi [118]). Suppose α (M α , gij ) is a sequence of compact orientable three-manifolds without boundary, and wα → 0. Assume that for each point x ∈ M α there exists a radius ρ = ρα (x), not exceeding the diameter of the manifold, such that the ball B(x, ρ) in the metric α gij has volume at most wα ρ3 and sectional curvatures at least −ρ−2 . Then M α for sufficiently large α are diffeomorphic to graph manifolds. Based on the the long-time behavior theorem (Theorem 7.6.4) and assuming the above Weaker Assertion, we can now give a proof for Thurston’s geometrization conjecture . We remark that if we assume the above Perelman’s Claim, then we does not need to use Thurston’s theorem for Haken manifolds in the proof of Theorem 7.7.1. Theorem 7.7.1. Thurston’s geometrization conjecture is true. Proof. Let M be a compact, orientable, and prime three-manifold (without boundary). Without loss of generality, we may assume that the manifold M is irreducible also. Recall that the theorem of Thurston (see for example Theorem A and Theorem B in Section 3 of [94], see also [85] and [102]) says that any compact, orientable, and irreducible Haken three-manifold (with or without boundary) is geometrizable. Thus in the following, we may assume that the compact three-manifold M (without boundary) is atoroidal, and then the fundamental group π1 (M ) contains no noncyclic, abelian subgroup. Arbitrarily given a (normalized) Riemannian metric on the manifold M , we use it as initial data for the Ricci flow. Arbitrarily take a sequence of small positive constants wα → 0 as α → +∞. For each fixed α, we set ε = wα /2 > 0. Then by α Theorem 7.4.3, the Ricci flow with surgery has a long-time solution (Mtα , gij (t)) on α a maximal time interval [0, T ), which satisfies the a priori assumptions (with the accuracy parameter ε = wα /2) and has a finite number of surgeries on each finite time interval. Since the initial manifold is irreducible, by the surgery procedure, we know that for each α and each t > 0 the solution manifold Mtα consists of a finite number of components where the essential component (Mtα )(1) is diffeomorphic to the initial manifold M and the others are diffeomorphic to the three-sphere S3 . α0 If for some α = α0 the maximal time T α0 is finite, then the solution (Mtα0 , gij (t)) α0 becomes extinct at T and the (irreducible) initial manifold M is diffeomorphic to S3 /Γ (the metric quotients of round three-sphere); in particular, the manifold M is geometrizable. Thus we may assume that the maximal time T α = +∞ for all α. We now apply the long-time behavior theorem (Theorem 7.6.4). If there is some α such that case (ii) of Theorem 7.6.4 occurs, then for some sufficiently large time

484


t, the essential component (Mtα )(1) of the solution manifold Mtα is diffeomorphic to a compact hyperbolic space, so the initial manifold M is geometrizable. Whereas if there is some sufficiently large α such that case (iii) of Theorem 7.6.4 occurs, then it follows that for all sufficiently large t, there is an embedding of a (nonempty) finite ` number of disjoint unions of incompressible two-tori i Ti2 in the essential component (Mtα )(1) of Mtα . This is a contradiction since we have assumed the initial manifold M is atoroidal. It remains to deal with the situation that there is a sequence of positive αk → αk +∞ such that the solutions (Mtαk , gij (t)) always satisfy case (i) of Theorem 7.6.4. αk That is, for each αk , Mt = Mthin (wαk , t) when the time t is sufficiently large. By the Thick-thin decomposition theorem (Theorem 7.6.3), there is a positive constant, αk 0 < ρ(wαk ) ≤ 1, such that as long as t is sufficiently √ large,α for √ every x ∈ Mt = αk k Mthin (w , t), we have some r = r(x, t), with 0 < r t < ρ(w ) t, such that √ √ (7.7.1) Rm ≥ −(r t)−2 on Bt (x, r t), and (7.7.2)

√ √ Vol t (Bt (x, r t)) < wαk (r t)3 .

Clearly we only need to consider the essential component (Mtαk )(1) . We divide the discussion into the following two cases: (1) there is a positive constant 1 < C < +∞ such that for each αk there is a sufficiently large time tk > 0 such that √ (7.7.3) r(x, tk ) tk < C · diam (Mtαkk )(1)

for all x ∈ (Mtαkk )(1) ⊂ Mthin (wαk , tk ); (2) there are a subsequence αk (still denoted by αk ), and sequences of positive constants Ck → +∞ and times Tk < +∞ such that for each t ≥ Tk , we have √ (7.7.4) r(x(t), t) t ≥ Ck · diam (Mtαk )(1)

for some x(t) ∈ (Mtαk )(1) , k = 1, 2, . . . . Here we denote by diam ((Mtα )(1) ) the diamα eter of the essential component (Mtα )(1) with the metric gij (t) at the time t. αk (1) Let us first consider case (1). For each point x ∈ (M ) ⊂ Mthin (wαk , tk ), we tk √ −1 denote by ρk (x) = C r(x, tk ) tk . Then by (7.7.1), (7.7.2) and (7.7.3), we have ρk (x) < diam (Mtαkk )(1) , √ Vol tk (Btk (x, ρk (x))) ≤ Vol tk (Btk (x, r(x, tk ) tk )) < C 3 wαk (ρk (x))3 , and √ Rm ≥ −(r(x, tk ) tk )−2 ≥ −(ρk (x))−2

on Btk (x, ρk (x)). Then it follows from the above Weaker Assertion that (Mtαkk )(1) , for sufficiently large k, are diffeomorphic to graph manifolds. This implies that the (irreducible) initial manifold M is diffeomorphic to a graph manifold. So the manifold M is geometrizable in case (1).


485

We next consider case (2). Clearly, for each αk and the chosen Tk , we may assume that the estimates (7.7.1) and (7.7.2) hold for all t ≥ Tk and x ∈ (Mtαk )(1) . The combination of (7.7.1) and (7.7.4) gives (7.7.5)

Rm ≥ −Ck−2 (diam ((Mtαk )(1) ))−2

on (Mtαk )(1) ,

for all t ≥ Tk . If there are a subsequence αk (still denoted by αk ) and a sequence of times tk ∈ (Tk , +∞) such that (7.7.6)

Vol tk ((Mtαkk )(1) ) < wk′ (diam ((Mtαkk )(1) ))3

for some sequence wk′ → 0, then it follows from the Weaker Assertion that (Mtαkk )(1) , for sufficiently large k, are diffeomorphic to graph manifolds which implies the initial manifold M is geometrizable. Thus we may assume that there is a positive constant w′ such that (7.7.7)

Vol t ((Mtαk )(1) ) ≥ w′ (diam ((Mtαk )(1) ))3

for each k and all t ≥ Tk . In view of the estimates (7.7.5) and (7.7.7), we now want to use Theorem 7.5.2 to get a uniform upper bound for the curvatures of the essential components αk ((Mtαk )(1) , gij (t)) with sufficiently large time t. Note that the estimate in Theorem 7.5.2 depends on the parameter ε and our ε’s depend on wαk with 0 < ε = wαk /2; so it does not work in the present situation. Fortunately we notice that the curvature estimate for smooth solutions in Corollary 7.2.3 is independent of ε. In the following we try to use Corollary 7.2.3 to obtain the desired curvature estimate. We first claim that for each k, there is a sufficiently large Tk′ ∈ (Tk , +∞) such αk that the solution, when restricted to the essential component ((Mtαk )(1) , gij (t)), has ′ no surgery for all t ≥ Tk . Indeed, for each fixed k, if there is a δ(t)-cutoff surgery αk at a sufficiently large time t, then the manifold ((Mtαk )(1) , gij (t)) would contain a −1 αk (1) − 21 δ(t)-neck Bt (y, δ(t) R(y, t) ) for some y ∈ (Mt ) with the volume ratio (7.7.8)

Vol t (Bt (y, δ(t)

−1

1

R(y, t)− 2 )) − 21

(δ(t)−1 R(y, t)

)3

2

≤ 8πδ(t) .

On the other hand, by (7.7.5) and (7.7.7), the standard Bishop-Gromov volume comparison implies that Vol t (Bt (y, δ(t) (δ(t)

−1

−1

1

R(y, t)− 2 )) 1

R(y, t)− 2 )3

≥ c(w′ )

for some positive constant c(w′ ) depending only on w′ . Since δ(t) is very small when t is large, this arrives at a contradiction with (7.7.8). So for each k, the essential αk component ((Mtαk )(1) , gij (t)) has no surgery for all sufficiently large t. αk For each k, we consider any fixed time t˜k > 3Tk′ . Let us scale the solution gij (t) αk (1) on the essential component (Mt ) by αk αk g˜ij (·, s) = (t˜k )−1 gij (·, t˜k s).

Note that (Mtαk )(1) is diffeomorphic to M for all t. By the above claim, we see that αk the rescaled solution (M, g˜ij (·, s)) is a smooth solution to the Ricci flow on the time 1 interval s ∈ [ 2 , 1]. Set p −1 t˜k diam (Mt˜αk )(1) . r˜k = k

486


Then by (7.7.4), (7.7.5) and (7.7.7), we have r˜k ≤ Ck−1 → 0,

and

g ≥ −C −2 (˜ Rm rk )−2 , k

as k → +∞, on B1 (x(t˜k ), r˜k ),

Vol 1 (B1 (x(t˜k ), r˜k )) ≥ w′ (˜ rk )3 , g is the rescaled curvature, x(t˜k ) is the point given by (7.7.4) and where Rm B1 (x(t˜k ), r˜k ) is the geodesic ball of rescaled solution at the time s = 1. Moreover, the αk closure of B1 (x(t˜k ), r˜k ) is the whole manifold (M, g˜ij (·, 1)). Note that in Theorem 7.2.1, Theorem 7.2.2 and Corollary 7.2.3, the condition about normalized initial metrics is just to ensure that the solutions satisfy the αk Hamilton-Ivey pinching estimate. Since our solutions (Mtαk , gij (t)) have already satisfied the pinching assumption, we can then apply Corollary 7.2.3 to conclude g |Rm(x, s)| ≤ K(w′ )(˜ rk )−2 ,

αk whenever s ∈ [1 − τ (w′ )(˜ rk )2 , 1], x ∈ (M, g˜ij (·, s)) and k is sufficiently large. Here ′ ′ K(w ) and τ (w ) are positive constants depending only on w′ . Equivalently, we have the curvature estimates

(7.7.9)

|Rm(·, t)| ≤ K(w′ )(diam ((Mt˜αk )(1) ))−2 ,

on M,

k

whenever t ∈ [t˜k − τ (w′ )(diam ((Mt˜αk )(1) ))2 , t˜k ] and k is sufficiently large. k

αk For each k, let us scale ((Mtαk )(1) , gij (t)) with the factor (diam((Mt˜αk )(1) ))−2 and shift k the time t˜k to the new time zero. By the curvature estimate (7.7.9) and Hamilton’s compactness theorem (Theorem 4.1.5), we can take a subsequential limit (in the C ∞ topology) and get a smooth solution to the Ricci flow on M × (−τ (w′ ), 0]. Moreover, by (7.7.5), the limit has nonnegative sectional curvature on M × (−τ (w′ ), 0]. Recall that we have removed all compact components with nonnegative scalar curvature. By combining this with the strong maximum principle, we conclude that the limit is a flat metric. Hence in case (2), M is diffeomorphic to a flat manifold and then it is also geometrizable. Therefore we have completed the proof of the theorem.

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491

INDEX Ti converges to a tensor T , 267 δ-cutoff surgery, 424 κ-noncollapsed, 255, 433 λ-remote, 360 L-Jacobian, 252 L-Jacobian field, 248 L-distance function, 245 L-exponential map with parameter τ¯, 252 L-geodesic, 244 curve, 244 equation, 244 L-length, 244 ε-cap, 414 ε-horn, 414 ε-neck, 414 strong, 416 ε-neck of radius r, 357 ε-tube, 414 k-jet distance, 343 space, 343 (almost) maximum points, 291 a priori assumptions (with accuracyε), 416 admissible curve, 435 ancient κ-solution, 357 solution, 233 asymptotic scalar curvature ratio, 303, 362 asymptotic volume ratio, 373 atoroidal, 482 barely admissible curve, 435 be modeled, 481 Berger’s rigidity theorem, 321 breather, 199 expanding, 199 shrinking, 199 steady, 199 canonical neighborhood assumption (with accuracy ε), 416 canonical neighborhood theorem, 396 capped ε-horn, 414 center of an evolving ε-neck, 394

Cheeger’s lemma, 288 classical sphere theorems, 321 classification of three-dimensional shrinking solitons, 384 collapsed, 338 compactness of ancient κ0 -solutions, 393 conjugate heat equation, 235 converges to a marked manifold, 267 converges to an evolving marked manifold, 282 curvature β-bump, 360 degree, 308 double ε-horn, 414 Einstein manifold, 174 metric, 174 elliptic type estimate, 391 essential component, 476 evolving ε-cap, 396 evolving ε-neck, 396 exceptional part, 338 finite bump theorem, 360 free boundary condition, 345 geometrizable, 482 gradient shrinking Ricci soliton, 384 graph manifold, 482 Gromoll-Meyer injectivity radius estimate, 286 Haken, 481 Hamilton’s advanced maximum principle, 218 Hamilton’s compactness theorem, 285 Hamilton’s strong maximum principle, 213 Hamilton-Ivey pinching estimate, 224, 336 homogeneous manifold, 481 locally, 481 incompressible, 338 injectivity radius, 286 condition, 291

492


irreducible, 476 Jacobian comparison theorem, 253 justification of the canonical neighborhood assumption, 433 justification of the pinching assumption, 424 Kähler-Ricci flow, 176 Kähler-Ricci soliton expanding, 176 shrinking, 176 steady, 176 Klingenberg’s lemma, 286 Li-Yau-Hamilton estimate, 226, 230 Li-Yau-Hamilton quadratic, 230 Li-Yau-Perelman distance, 238, 250 Little Loop Lemma, 290 marked Riemannian manifold, 267 marking, 267 maximal solution, 291 Mostow type rigidity, 343 no local collapsing theorem I, 255, 256 no local collapsing theorem I′ , 259 no local collapsing theorem II, 263 noncollapsing limit, 338 normalized, 398 normalized Ricci flow, 307 origin, 267 Perelman’s claim, 482 Perelman’s reduced volume, 243, 252 element, 254 pinching assumption, 416 Poincaré conjecture, 452 prime, 481 decomposition theorem, 481 regular, 411 Ricci flow, 173 Ricci flow with surgery, 416 Ricci soliton expanding, 175 shrinking, 175 gradient , 175 steady , 175 Shi’s derivative estimate, 192

singularity model, 292 singularity structure theorem, 399 soliton cigar, 177 steady, 175 solution ancient, 233 nonsingular, 336 standard, 420 solution becomes extinct, 417 solution develops a singularity, 267 standard capped infinite cylinder, 420 support function, 220 surgery times, 416 surgically modified solution, 416 tangent cone, 219 thick-thin decomposition theorem, 473 Thurston’s geometrization conjecture, 482, 483 type I, 291–293 type II, 293 (a), 291, 294 (b), 292, 295 type III, 293 (a), 292, 296 (b), 292 universal noncollapsing, 388 Weaker Assertion, 483

arXiv:math/0605667v3 [math.DG] 10 Oct 2008

NOTES ON PERELMAN’S PAPERS BRUCE KLEINER AND JOHN LOTT

1. Introduction These are notes on Perelman’s papers “The Entropy Formula for the Ricci Flow and its Geometric Applications” [51] and “Ricci Flow with Surgery on Three-Manifolds’ [52]. In these two remarkable preprints, which were posted on the ArXiv in 2002 and 2003, Grisha Perelman announced a proof of the Poincaré Conjecture, and more generally Thurston’s Geometrization Conjecture, using the Ricci flow approach of Hamilton. Perelman’s proofs are concise and, at times, sketchy. The purpose of these notes is to provide the details that are missing in [51] and [52], which contain Perelman’s arguments for the Geometrization Conjecture. Among other things, we cover the construction of the Ricci flow with surgery of [52]. We also discuss the long-time behavior of the Ricci flow with surgery, which is needed for the full Geometrization Conjecture. The papers [24, 53], which are not covered in these notes, each provide a shortcut in the case of the Poincaré Conjecture. Namely, these papers show that if the initial manifold is simply-connected then the Ricci flow with surgery becomes extinct in a finite time, thereby removing the issue of the long-time behavior. Combining this claim with the proof of existence of Ricci flow with surgery gives the shortened proof in the simply-connected case. These notes are intended for readers with a solid background in geometric analysis. Good sources for background material on Ricci flow are [22, 23, 33, 66]. The notes are selfcontained but are designed to be read along with [51, 52]. For the most part we follow the format of [51, 52] and use the section numbers of [51, 52] to label our sections. We have done this in order to respect the structure of [51, 52] and to facilitate the use of the present notes as a companion to [51, 52]. In some places we have rearranged Perelman’s arguments or provided alternative arguments, but we have refrained from an overall reorganization. Besides providing details for Perelman’s proofs, we have included some expository material in the form of overviews and appendices. Section 3 contains an overview of the Ricci flow approach to geometrization of 3-manifolds. Sections 4 and 57 contain overviews of [51] and [52], respectively. The appendices discuss some background material and techniques that are used throughout the notes. Regarding the proofs, the papers [51, 52] contain some incorrect statements and incomplete arguments, which we have attempted to point out to the reader. (Some of the mistakes in [51] were corrected in [52].) We did not find any serious problems, meaning problems that cannot be corrected using the methods introduced by Perelman. Date: September 19, 2008. Research supported by NSF grants DMS-0306242 and DMS-0204506, and the Clay Mathematics Institute. 1

2

BRUCE KLEINER AND JOHN LOTT

We will refer to Section X.Y of [51] as I.X.Y, and Section X.Y of [52] as II.X.Y. A reader may wish to start with the overviews, which explain the logical structure of the arguments and the interrelations between the sections. It may also be helpful to browse through the appendices before delving into the main body of the material. These notes have gone through various versions, which were posted at [39]. An initial version with notes on [51] was posted in June 2003. A version covering [51, 52] was posted in September 2004. After the May 2006 version of these notes was posted on the ArXiv, expositions of Perelman’s work appeared in [15] and [45]. Acknowledgements. In the preparation of the September 2004 version of our notes, we benefited from a workshop on Perelman’s surgery procedure that was held in August 2004, at Princeton. We thank the participants of that meeting, as well as the Clay Mathematics Institute for supporting it. We are grateful for comments and corrections that we have received regarding earlier versions of these notes. We especially thank Gérard Besson for comments on Theorem 13.3 and Lemma 80.4, Peng Lu for comments on Lemma 74.1 and Bernhard Leeb for discussions on the issue of uniqueness for the standard solution. Along with these people we thank Mike Anderson, Albert Chau, Ben Chow, Xianzhe Dai, Jonathan Dinkelbach, Sylvain Maillot, John Morgan, Joan Porti, Gang Tian, Peter Topping, Bing Wang, Guofang Wei, Hartmut Weiss, Jon Wolfson, Rugang Ye and Maciej Zworski. We thank the referees for their detailed and helpful comments on this paper. We thank the Clay Mathematics Institute for supporting the writing of the notes on [52]. Contents 1. Introduction

1

2. A reading guide

6

3. An overview of the Ricci flow approach to 3-manifold geometrization

7

3.1. The definition of Ricci flow, and some basic properties

7

3.2. A rough outline of the Ricci flow proof of the Poincaré Conjecture

8

3.3. Outline of the proof of the Geometrization Conjecture

9

3.4. Claim 3.2 and the structure of singularities

11

3.5. The proof of Claims 3.3 and 3.4

13

4. Overview of The Entropy Formula for the Ricci Flow and its Geometric Applications [51]

13

4.1. I.1-I.6

14

4.2. I.7-I.10

15

4.3. I.11

16

4.4. I.12-I.13

17

NOTES ON PERELMAN’S PAPERS

5. I.1.1. The F -functional and its monotonicity

3

18

6. Basic example for I.1

21

7. I.2.2. The λ-invariant and its applications

22

8. I.2.3. The rescaled λ-invariant

24

9. I.2.4. Gradient steady solitons on compact manifolds

25

10. Ricci flow as a gradient flow

26

11. The W-functional

27

12. I.3.1. Monotonicity of the W-functional

28

13. I.4. The no local collapsing theorem I

30

14. I.5. The W-functional as a time derivative

32

15. I.7. Overview of reduced length and reduced volume

33

16. Basic example for I.7

34

17. Remarks about L-Geodesics and L exp

34

18. I.(7.3)-(7.6). First derivatives of L

37

19. I.(7.7). Second variation of L

39

20. I.(7.8)-(7.9). Hessian bound for L

41

21. I.(7.10). The Laplacian of L

42

22. I.(7.11). Estimates on L-Jacobi fields 23. Monotonicity of the reduced volume Ve

42 43

24. I.(7.15). A differential inequality for L

44

25. I.7.2. Estimates on the reduced length

46

26. I.7.3. The no local collapsing theorem II

47

27. I.8.3. Length distortion estimates

50

28. I.8.2. No local collapsing propagates forward in time and to larger scales

52

29. I.9. Perelman’s differential Harnack inequality

54

30. The statement of the pseudolocality theorem

57

31. Claim 1 of I.10.1. A point selection argument

58

32. Claim 2 of I.10.1. Getting parabolic regions

59

33. Claim 3 of I.10.1. An upper bound on the integral of v

60

34. Theorem I.10.1. Proof of the pseudolocality theorem

62

35. I.10.2. The volumes of future balls

65

36. I.10.4. κ-noncollapsing at future times

66

37. I.10.5. Diffeomorphism finiteness

66

4


38. I.11.1. κ-solutions

67

39. I.11.2. Asymptotic solitons

68

40. I.11.3. Two dimensional κ-solutions

70

41. I.11.4. Asymptotic scalar curvature and asymptotic volume ratio

70

42. In a κ-solution, the curvature and the normalized volume control each other

74

43. An alternate proof of Corollary 40.1 using Proposition 41.13 and Corollary 42.1 76 44. I.11.5. A volume bound

77

45. I.11.6. Curvature bounds for Ricci flow solutions with nonnegative curvature operator, assuming a lower volume bound

78

46. I.11.7. Compactness of the space of three-dimensional κ-solutions

80

47. I.11.8. Necklike behavior at infinity of a three-dimensional κ-solution - weak version

82

48. I.11.8. Necklike behavior at infinity of a three-dimensional κ-solution - strong version

83

49. More properties of κ-solutions

85

50. I.11.9. Getting a uniform value of κ

85

51. II.1.2. Three-dimensional noncompact κ-noncollapsed gradient shrinkers are standard

86

52. I.12.1. Canonical neighborhood theorem

89

53. I.12.2. Later scalar curvature bounds on bigger balls from curvature and volume bounds 96 54. I.12.3. Earlier scalar curvature bounds on smaller balls from lower curvature bounds and volume bounds

99

55. I.12.4. Small balls with strongly negative curvature are volume-collapsed

101

56. I.13.1. Thick-thin decomposition for nonsingular flows

102

57. Overview of Ricci Flow with Surgery on Three-Manifolds [52]

103

57.1. II.1-II.3

104

57.2. II.4-II.5

104

57.3. II.6-II.8

107

58. II. Notation and terminology

108

59. II.1. Three-dimensional κ-solutions

110

60. II.2. Standard solutions

113

61. Claim 2 of II.2. The blow-up time for a standard solution is ≤ 1

115

62. Claim 4 of II.2. The blow-up time of a standard solution is 1

115

63. Claim 5 of II.2. Canonical neighborhood property for standard solutions

116


5

64. Compactness of the space of standard solutions

118

65. Claim 1 of II.2. Rotational symmetry of standard solutions

118

66. Claim 3 of II.2. Uniqueness of the standard solution

120

67. II.3. Structure at the first singularity time

122

68. Ricci flow with surgery: the general setting

127

69. II.4.1. A priori assumptions

130

70. II.4.2. Curvature bounds from the a priori assumptions

131

71. II.4.3. δ-necks in ǫ-horns

132

72. Surgery and the pinching condition

134

73. II.4.4. Performing surgery and continuing flows

138

74. II.4.5. Evolution of a surgery cap

142

75. II.4.6. Curves that penetrate the surgery region

144

76. II.4.7. A technical estimate

145

77. II.5. Statement of the the existence theorem for Ricci flow with surgery

146

78. The L-function of I.7 and Ricci flows with surgery

148

79. Establishing noncollapsing in the presence of surgery

152

80. Construction of the Ricci flow with surgery

157

81. II.6. Double sided curvature bound in the thick part

160

82. II.6.5. Earlier scalar curvature bounds on smaller balls from lower curvature bounds and a later volume bound

161

83. II.6.6. Locating small balls whose subballs have almost Euclidean volume

163

84. II.6.8. Proof of the double sided curvature bound in the thick part, modulo two propositions 164 85. II.6.3. Canonical neighborhoods and later curvature bounds on bigger balls from curvature and volume bounds

165

86. II.6.4. Earlier scalar curvature bounds on smaller balls from lower curvature bounds and volume bounds, in the presence of possible surgeries

169

87. II.7.1. Noncollapsed pointed limits are hyperbolic

171

88. II.7.2. Noncollapsed regions with a lower curvature bound are almost hyperbolic on a large scale 173 89. II.7.3. Thick-thin decomposition

175

90. Hyperbolic rigidity and stabilization of the thick part

177

91. Incompressibility of cuspidal tori

182

92. II.7.4. The thin part is a graph manifold

186

93. II.8. Alternative proof of cusp incompressibility

191

6


93.1. The approach using the σ-invariant

191

93.2. The approach using the λ-invariant

193

Appendix A. Maximum principles

199

Appendix B. φ-almost nonnegative curvature

201

Appendix C. Ricci solitons

202

Appendix D. Local derivative estimates

205

Appendix E. Convergent subsequences of Ricci flow solutions

205

Appendix F. Harnack inequalities for Ricci flow

206

Appendix G. Alexandrov spaces

208

Appendix H. Finding balls with controlled curvature

209

Appendix I. Statement of the geometrization conjecture

209

References

210

2. A reading guide Perelman’s papers contain a wealth of results about Ricci flow. We cover all of these results, whether or not they are directly relevant to the Poincaré and Geometrization Conjectures. Some readers may wish to take an abbreviated route that focuses on the proof of the Poincaré Conjecture or the Geometrization Conjecture. Such readers can try the following itinerary. Begin with the overviews in Sections 3 and 4. Then review Hamilton’s compactness theorem and its variants, as described in Appendix E; an exposition is in [66, Chapter 7]. Next, read I.7 (Sections 15-26), followed by I.8.3(b) (Section 27). After reviewing the theory of Riemannian manifolds and Alexandrov spaces of nonnegative sectional curvature (Appendix G and references therein), proceed to I.11 (Sections 38-50), followed by II.1.2 and I.12.1 (Sections 51-52). At this point, the reader should be ready for the overview of Perelman’s second paper in Section 57, and can proceed with II.1-II.5 (Sections 58-80). In conjunction with one of the finite extinction time results [24, 25, 53], this completes the proof of the Poincaré Conjecture. To proceed with the rest of the proof of the Geometrization Conjecture, the reader can begin with the large-time estimates for nonsingular Ricci flows, which appear in I.12.2-I.12.4 (Sections 53-55). The reader can then go to II.6 and II.7 (Sections 81-92). The main topics that are missed by such an abbreviated route are the F and W functionals (Sections 5-14), Perelman’s differential Harnack inequality (Section 29), pseudolocality (Sections 30-37) and Perelman’s alternative proof of cusp incompressibility (Section 93).


7

3. An overview of the Ricci flow approach to 3-manifold geometrization This section is an overview of the Ricci flow approach to 3-manifold geometrization. We make no attempt to present the history of the ideas that go into the argument. We caution the reader that for the sake of readability, in many places we have suppressed technical points and deliberately oversimplified the story. The overview will introduce the argument in three passes, with successively greater precision and detail : we start with a very crude sketch, then expand this to a step-by-step outline of the strategy, and then move on to more detailed commentary on specific points. Other overviews may be found in [14, 44]. The primary objective of our exposition is to prepare the reader for a more detailed study of Perelman’s work. We refer the reader to Appendix I for the statement of the geometrization conjecture. By convention, all manifolds and Riemannian metrics in this section will be smooth. We follow the notation of [51, 52] for pointwise quantities: R denotes the scalar curvature, Ric the Ricci curvature, and | Rm | the largest absolute value of the sectional curvatures. An inequality such as Rm ≥ C means that all of the sectional curvatures at a point or in a region, depending on the context, are bounded below by C. In this section, we will specialize to three dimensions. 3.1. The definition of Ricci flow, and some basic properties. Let M be a compact 3-manifold and let {g(t)}t∈[a,b] be a smoothly varying family of Riemannian metrics on M. Then g(·) satisfies the Ricci flow equation if ∂g (t) = − 2 Ric(g(t)) ∂t holds for every t ∈ [a, b]. Hamilton showed in [35] that for any Riemannian metric g0 on M, there is a T ∈ (0, ∞] with the property that there is a (unique) solution g(·) to the Ricci flow equation defined on the time interval [0, T ) with g(0) = g0 , so that if T < ∞ then the curvature of g(t) becomes unbounded as t → T . We refer to this maximal solution as the Ricci flow with initial condition g0 . If T < ∞ then we call T the blow-up time. A basic example is the shrinking round 3-sphere, with g0 = r02 gS 3 and g(t) = (r02 − 4t) gS 3 , in which r2 case T = 40 . (3.1)

Suppose that M is simply-connected. Based on the round 3-sphere example, one could hope that every Ricci flow on M blows up in finite time and becomes round while shrinking to a point, as t approaches the blow-up time T . If so, then by rescaling and taking a limit as t → T , one would show that M admits a metric of constant positive sectional curvature and therefore, by a classical theorem, is diffeomorphic to S 3 . The analogous argument does work in two dimensions [22, Chapter 5]. Furthermore, if the initial metric g0 has positive Ricci curvature then Hamilton showed in [33] that this is the correct picture: the manifold shrinks to a point in finite time and becomes round as it shrinks. One is then led to ask what can happen if M is simply-connected but g0 does not have positive Ricci curvature. Here a new phenomenon can occur — the Ricci flow solution may become singular before it has time to shrink to a point, due to a possible neckpinch. A neckpinch is modeled by a product region (−c, c) × S 2 in which one or many S 2 -fibers

8


separately shrink to a point at time T , due to the positive curvature of S 2 . The formation of neckpinch (and other) singularities prevents one from continuing the Ricci flow. In order to continue the evolution some intervention is required, and this is the role of surgery. Roughly speaking, the idea of surgery is to remove a neighborhood diffeomorphic to (−c′ , c′ ) × S 2 containing the shrinking 2-spheres, and cap off the resulting boundary components by gluing in 3-balls. Of course the topology of the manifold changes during surgery – for instance it may become disconnected – but it changes in a controlled way. The postsurgery Riemannian manifold is smooth, so one may restart the Ricci flow using it as an initial condition. When continuing the flow one may encounter further neckpinches, which give rise to further surgeries, etc. One hopes that eventually all of the connected components shrink to points while becoming round, i.e. that the Ricci flow solution has a finite extinction time. 3.2. A rough outline of the Ricci flow proof of the Poincar´ e Conjecture. We now give a step-by-step glimpse of the proof, stating the needed steps as claims. One starts with a compact orientable 3-manifold M with an arbitrary metric g0 . For the moment we do not assume that M is simply-connected. Let g(·) be the Ricci flow with initial condition g0 , defined on [0, T ). Suppose that T < ∞. Let Ω ⊂ M be the set of points x ∈ M for which limt→T − R(x, t) exists and is finite. Then M − Ω is the part of M that is going singular. (For example, in the case of a single standard neckpinch, M − Ω is a 2-sphere.) The first claim says what M looks like near this singularity set. Claim 3.2. [52] The set Ω is open and as t → T , the evolving metric g(·) converges smoothly on compact subsets of Ω to a Riemannian metric g. There is a geometrically defined neighborhood U of M − Ω such that each connected component of U is either A. Compact and diffeomorphic to S 1 × S 2 , S 1 ×Z2 S 2 or S 3 /Γ, where Γ is a finite subgroup of SO(4) that acts freely and isometrically on the round S 3 . (In writing S 1 ×Z2 S 2 , the generator of Z2 acts on S 1 by complex conjugation and on S 2 by the antipodal map. Then S 1 ×Z2 S 2 is diffeomorphic to RP 3 #RP 3 .) or

B. Noncompact and diffeomorphic to R×S 2 , R3 or the twisted line bundle R×Z2 S 2 over RP 2 . In Case B, the connected component meets Ω in geometrically controlled collar regions diffeomorphic to R × S 2 . Thus Claim 3.2 provides a topological description of a neighborhood U of the region M − Ω where the Ricci flow is going singular, along with some geometric control on U. Claim 3.3. [52] There is a well-defined way to perform surgery on M, which yields a smooth post-surgery manifold M ′ with a Riemannian metric g ′ . Claim 3.3 means that there is a well-defined procedure for specifying the part of M that will be removed, and for gluing caps on the resulting manifold with boundary. The discarded part corresponds to the neighborhood U in Claim 3.2. The procedure is required to satisfy a number of additional conditions which we do not mention here.


9

Undoing the surgery, i.e. going from the postsurgery manifold to the presurgery manifold, amounts to restoring some discarded components (as in Case A of Claim 3.2) and performing connected sums of some of the components of the postsurgery manifold, along with some possible connected sums with a finite number of new S 1 × S 2 and RP 3 factors. The S 1 × S 2 comes from the case when a surgery does not disconnect the connected component where it is performed. The RP 3 factors arise from the twisted line bundle components in Case B of Claim 3.2. After performing a surgery one lets the new manifold evolve under the Ricci flow until one encounters the next blowup time (if there is one). One then performs further surgery, lets the new manifold evolve, and repeats the process. Claim 3.4. [52] One can arrange the surgery procedure so that the surgery times do not accumulate. If the surgery times were to accumulate, then one would have trouble continuing the flow further, effectively killing the whole program. Claim 3.4 implies that by alternating Ricci flow and surgery, one obtains an evolutionary process that is defined for all time (though the manifold may become the empty set from some time onward). We call this Ricci flow with surgery. Claim 3.5. [24, 25, 53] If the original manifold M is simply-connected then any Ricci flow with surgery on M becomes extinct in finite time. Having a finite extinction time means that from some time onwards, the manifold is the empty set. More generally, the same proof shows that if the prime decomposition of the original manifold M has no aspherical factors, then every Ricci flow with surgery on M becomes extinct in finite time. (Recall that a connected manifold X is aspherical if πk (X) = 0 for all k > 1 or, equivalently, if its universal cover is contractible.) The Poincaré Conjecture follows immediately from the above claims. From Claim 3.5, after some finite time the manifold is the empty set. From Claims 3.2, 3.3 and 3.4, the original manifold M is diffeomorphic to a connected sum of factors that are each S 1 × S 2 or a standard quotient S 3 /Γ of S 3 . As we are assuming that M is simply-connected, van Kampen’s theorem implies that M is diffeomorphic to a connected sum of S 3 ’s, and hence is diffeomorphic to S 3 . 3.3. Outline of the proof of the Geometrization Conjecture. We now drop the assumption that M is simply-connected. The main difference is that Claim 3.5 no longer applies, so the Ricci flow with surgery may go on forever in a nontrivial way. (We remark that Claim 3.5 is needed only for a shortened proof of the Poincaré Conjecture; the proof in the general case is logically independent of Claim 3.5 and also implies the Poincaré Conjecture.) The possibility that there are infinitely many surgery times is not excluded, although it is not known whether this can actually happen. A simple example of a Ricci flow that does not become extinct is when M = H 3 /Γ, where Γ is a freely-acting cocompact discrete subgroup of the orientation-preserving isometries of hyperbolic space H 3 . If ghyp denotes the metric on M of constant sectional curvature −1

10


and g0 = r02 ghyp then g(t) = (r02 + 4t) ghyp . Putting b g (t) = 1t g(t), one finds that limt→∞ b g (t) 1 is the metric on M of constant sectional curvature − 4 , independent of r0 .

Returning to the general case, let Mt denote the time-t manifold in a Ricci flow with surgery. (If t is a surgery time then we consider the postsurgery manifold.) If for some t a component of Mt admits a metric with nonnegative scalar curvature then one can show that the component becomes extinct or admits a flat metric; either possiblity is good enough when we are trying to prove the Geometrization Conjecture for the initial manifold M. So we will assume that for every t, each component of Mt has a point with strictly negative scalar curvature.

Motivated by the hyperbolic example, we consider the metric gb(t) = 1t g(t) on Mt . Given x ∈ Mt , define the intrinsic scale ρ(x, t) to be the radius ρ such that inf B(x,ρ) Rm = − ρ−2 , where Rm denotes the sectional curvature of b g (t); this is well-defined because the scalar curvature is negative somewhere in the connected component of Mt containing x. Given w > 0, define the w-thick part of Mt by (3.6)

M + (w, t) = {x ∈ Mt : vol(B(x, ρ(x, t))) > w ρ(x, t)3 }.

It is not excluded that M + (w, t) = Mt or M + (w, t) = ∅. The next claim says that for any w > 0, as time goes on, M + (w, t) approaches the w-thick part of a manifold of constant sectional curvature − 14 . Claim 3.7. [52] There is a finite collection {(Hi, xi )}ki=1 of complete pointed finite-volume 3-manifolds with constant sectional curvature − 41 and, for large t, a decreasing function α(t) tending to zero and a family of maps (3.8)

ft :

k G

i=1

Hi ⊃

k G

i=1

1 → Mt , B xi , α(t)

such that 1. ft is α(t)-close to being an isometry. 2. The image of ft contains M + (α(t), t). 3. The image under ft of a cuspidal torus of {Hi }ki=1 is incompressible in Mt . The proof of Claim 3.7 uses earlier work by Hamilton [34]. S Claim 3.9. [52, 64] Let Yt be the truncation of ki=1 Hi obtained by removing horoballs at 1 distance approximately 2α(t) from the basepoints xi . Then for large t, Mt − ft (Yt ) is a graph manifold. Claim 3.9 reduces to a statement in Riemannian geometry about 3-manifolds that are locally volume-collapsed with a lower bound on sectional curvature. Claims 3.7 and 3.9, along with Claims 3.2-3.4, imply the geometrization conjecture, cf. Appendix I. In the remainder of this section, we will discuss some of the claims in more detail.


11

3.4. Claim 3.2 and the structure of singularities. Claim 3.2 is derived from a more localized statement, which says that near points of large scalar curvature, the Ricci flow looks very special : it is well-approximated by a special kind of model Ricci flow, called a κ-solution. Claim 3.10. Suppose that we have a given Ricci flow solution on a finite time interval. If x ∈ M and the scalar curvature R(x, t) is large then in the time-t slice, there is a ball 1 centered at x of radius comparable to R(x, t)− 2 in which the geometry of the Ricci flow is close to that of a ball in a κ-solution. 1

The quantity R(x, t)− 2 is sometimes called the curvature scale at (x, t), because it scales like a distance. We will define κ-solutions below, but mention here that they are Ricci flows with nonnegative sectional curvature, and they are ancient, i.e. defined on a time interval of the form (−∞, a). The strength of Claim 3.10 comes from the fact that there is a good description of κsolutions. Claim 3.11. [52] Any three-dimensional oriented κ-solution (M∞ , g∞ (·)) falls into one of the following types : (a) A finite isometric quotient of the round shrinking 3-sphere. (b) A Ricci flow on a manifold diffeomorphic to S 3 or RP 3 . (c) A standard shrinking round neck on R × S 2 (d) A Ricci flow on a manifold diffeomorphic to R3 , each time slice of which is asymptotically necklike at infinity. (e) The Z2 -quotient R ×Z2 S 2 of a shrinking round neck. Together, Claims 3.10 and 3.11 say that where the scalar curvature is large, there is a region of diameter comparable to the curvature scale where one sees either a closed manifold of known topology (cases (a) and (b)), a neck region (case (c)), a neck region capped off by a 3-ball (case (d)), or a neck region capped off by a twisted line bundle over RP 2 (case (e)). Applying this statement to every point of large scalar curvature at a time t just prior to the blow-up time T , one obtains a cover of M by regions with special geometry and topology. Any overlaps occur in neck-like regions, permitting one to splice them together to form the connected components with known topology whose existence is asserted in Claim 3.2. Claim 3.10 is proved using a rescaling (or blow-up) argument. This is a standard technique in geometric analysis and PDE’s for treating scale-invariant equations, such as the Ricci flow equation. The claim is equivalent to the statement that if {(xi , ti )}∞ i=1 is a sequence of spacetime points for which limi→∞ R(xi , ti ) = ∞, then by rescaling the Ricci flow and passing to a subsequence, one obtains a new sequence of Ricci flows which converges to a κ-solution. More precisely, view (xi , ti ) as a new spacetime basepoint and spatially expand 1 the solution around (xi , ti ) by R(xi , ti ) 2 . For dimensional reasons, in order for rescaling to produce a new Ricci flow solution one must also expand the time factor by R(xi , ti ). The new Ricci flow solution, with time parameter s, is given by (3.12) g i (s) = R(xi , ti ) g R(xi , ti )−1 s + ti . The new time interval for s is [− R(xi , ti ) ti , 0]. One would then hope to take an appropriate limit (M∞ , g ∞ ) of a subsequence of these rescaled solutions {(M, g i (·))}∞ i=1 . (Technically

12


speaking, one uses smooth convergence of sequences of Ricci flows with basepoints; this notion of convergence allows us to focus on what happens near the spacetime points (xi , ti ).) Any such limit solution (M∞ , g ∞ ) will be an ancient solution, since limi→∞ −R(xi , ti ) ti = −∞. Furthermore, from a 3-dimensional result of Hamilton and Ivey (see Appendix B), any limit solution will have nonnegative sectional curvature. Although this sounds promising, a major problem was to show that a limit solution actually exists. To prove this, one would like to invoke Hamilton’s compactness theorem [32]. In the present situation, the compactness theorem says that the sequence of rescaled Ricci flows {(M, g i (·))}∞ i=1 has a smoothly convergent subsequence provided two conditions are met: A. For every r > 0 and sufficiently large i, the sectional curvature of gi is bounded uniformly independent of i at each point (x, s) in spacetime such that x lies in the gi -ball B0 (xi , r) and s ∈ [−r 2 , 0] and B. The injectivity radii inj(xi , 0) in the time-0 slices of the gi ’s have a uniform positive lower bound. For the moment, we ignore the issue of verifying condition A, and simply assume that it holds for the sequence {(M, g i (·))}∞ i=1 . In the presence of the sectional curvature bounds in condition A, a lower bound on the injectivity radius is known to be equivalent to a lower bound on the volume of metric balls. In terms of the original Ricci flow solution, this becomes the condition that (3.13)

r −3 vol(Bt (x, r)) ≥ κ > 0,

where Bt (x, r) is an arbitrary metric r-ball in a time-t slice, and the curvature bound | Rm | ≤ r12 holds in Bt (x, r). The number κ could depend on the given Ricci flow solution, but the bound (3.13) should hold for all t ∈ [0, T ) and all r < ρ, where ρ is a relevant scale.

One of the outstanding achievements of [51] is to prove that for an arbitrary Ricci flow defined on a finite time interval, equation (3.13) does hold with appropriate values of κ and ρ. In fact, the proof works in arbitrary dimension. This result is called a “no local collapsing theorem” because it excludes the phenomenon of Cheeger-Gromov collapse, in which a sequence of Riemannian manifolds has uniformly bounded curvature, but fails to converge because the injectivity radii tend to zero.

One can then apply the no local collapsing theorem to the preceding rescaling argument, provided that one has the needed sectional curvature bounds, in order to construct the ancient solution (M∞ , g ∞ ). In the blowup limit the condition that r < ρ goes away, and so we can say that (M∞ , g ∞ ) is κ-noncollapsed (i.e. satisfies (3.13)) at all scales. In addition, in the three-dimensional case one can show that (M∞ , g ∞ ) has bounded sectional curvature. To summarize, (M∞ , g ∞ ) is a κ-solution, meaning that it is an ancient Ricci flow solution with nonnegative curvature operator on each time slice and bounded sectional curvature on compact time intervals, which is κ-noncollapsed at all scales. With the no local collapsing theorem in place, most of the proof of Claim 3.10 is concerned with showing that in the rescaling argument, we effectively have the needed curvature bounds


13

of condition A. The argument is a tour-de-force with many ingredients, including earlier work of Hamilton and the theory of Riemannian manifolds with nonnegative sectional curvature. 3.5. The proof of Claims 3.3 and 3.4. Claim 3.2 allows one to take the limit of the evolving metric g(·) as t → T , on the open set Ω where the metric is not becoming singular. It also provides geometrically defined regions – the connected components of the open set U – which one removes during surgery. Each boundary component of the resulting manifold is a nearly round 2-sphere with a nearly cylindrical collar, because the collar regions in Case B of Claim 3.2 have a neck-like geometry. This enables one to glue in 3-balls with a standard metric, using a partition of unity construction. The Ricci flow starting with the postsurgery metric may also go singular after a finite time. If so, one can appeal to Claim 3.2 again to perform surgery. However, the elapsed time between successive surgeries will depend on the scales at which surgeries are performed. Unless one performs the surgeries very carefully, the surgery times may accumulate. The way to rule out an accumulation of surgery times is to arrange the surgery procedure so that a surgery at time t removes a definite amount of volume v(t). That is, a surgery at time t should be performed at a definite scale h(t). In order to guarantee that this is possible, one needs to establish a quantitative version of Claim 3.2 for a Ricci flow with surgery, which applies not just at the first surgery time T but also at a later surgery time T ′ . The output of this quantitative version can depend on the surgery time T ′ and the time-zero metric, but it should be independent of whether or when surgeries occur before time T ′ . The general idea of the proof is similar to that of Claim 3.2, except that one has to carefully prescribe the surgery procedure in order to control the effect of the earlier surgeries. In particular, one of Perelman’s remarkable achievements is a version of the no local collapsing theorem for Ricci flows with surgery. We refer the reader to Section 57 for a more detailed overview of the proof of Claim 3.4, and for further discussion of Claims 3.7 and 3.9. 4. Overview of The Entropy Formula for the Ricci Flow and its Geometric Applications [51] The paper [51] deals with nonsingular Ricci flows; the surgery process is considered in [52]. In particular, the final conclusion of [51] concerns Ricci flows that are singularity-free and exist for all positive time. It does not apply to compact 3-manifolds with finite fundamental group or positive scalar curvature. The purpose of the present overview is not to give a comprehensive summary of the results of [51]. Rather we indicate its organization and the interdependence of its sections, for the convenience of the reader. Some of the remarks in the overview may only become clear once the reader has absorbed a portion of the detailed notes. Sections I.1-I.10, along with the first part of I.11, deal with Ricci flow on n-dimensional manifolds. The second part of I.11, and Sections I.12-I.13, deal more specifically with Ricci flow on 3-dimensional manifolds. The main result is that geometrization holds if a

14


compact 3-manifold admits a Riemannian metric which is the initial metric of a smooth Ricci flow. This was previously shown in [34] under the additional assumption that the sectional curvatures in the Ricci flow are O(t−1 ) as t → ∞. The paper [51] can be divided into four main parts.

Sections I.1-I.6 construct certain entropy-type functionals F and W that are monotonic under Ricci flow. The functional W is used to prove a no local collapsing theorem. Sections I.7-I.10 introduce and apply another monotonic quantity, the reduced volume V˜ . It is also used to prove a no local collapsing theorem. The construction of V˜ uses a modified notion of a geodesic, called an L-geodesic. For technical reasons the reduced volume V˜ seems to be easier to work with than the W-functional, and is used in most of the sequel. A reader who wants to focus on the Poincaré Conjecture or the Geometrization Conjecture could in principle start with I.7. Section I.11 is concerned with κ-solutions, meaning nonflat ancient solutions that are κ-noncollapsed at all scales (for some κ > 0) and have bounded nonnegative curvature operator on each time slice. In three dimensions, a blowup limit of a finite-time singularity will be a κ-solution. Sections I.12-I.13 are about three-dimensional Ricci flow solutions. It is shown that highscalar-curvature regions are modeled by rescalings of κ-solutions. A decomposition of the time-t manifold into “thick” and “thin” pieces is described. It is stated that as t → ∞, the thick piece becomes more and more hyperbolic, with incompressible cuspidal tori, and the thin piece is a graph manifold. More details of these assertions appear in [52], which also deals with the necessary modifications if the solution has singularities. We now describe each of these four parts in a bit more detail. 4.1. I.1-I.6. In these sections M is assumed to be a closed n-dimensional manifold. A functional F (g) of Riemannian metrics g is said to be monotonic under Ricci flow if F (g(t)) is nondecreasing in t whenever g(·) is a Ricci flow solution. Monotonic quantities are an important tool for understanding Ricci flow. One wants to have useful monotonic quantities, in particular with a characterization of the Ricci flows for which F (g(t)) is constant in t. Formally thinking of Ricci flow as a flow on the space of metrics, one way to get a monotonic quantity would be if the Ricci flow were the gradient flow of a functional F . In Sections I.1-I.2, a functional F is introduced whose gradient flow is not quite Ricci flow, but only differs from the Ricci flow by the action of diffeomorphisms. (If one formally considers the Ricci flow as a flow on the space of metrics modulo diffeomorphisms then it turns out to be the gradient flow of a functional λ1 .) The functional F actually depends on a Riemannian metric g and a function f . If g(·) satisfies the Ricci flow equation and e−f (·) satisfies a conjugate or “backward” heat equation, in terms of g(·), then F (g(t), f (t)) is nondecreasing in t. Furthermore, it is constant in t if and only if g(·) is a gradient steady R −f soliton with associated function f (·). Minimizing F (g, f ) over all functions f with e dV = 1 gives the monotonic quantity λ1 (g), which turns out to be the lowest M eigenvalue of − 4 △ + R.


15

In Section I.3 a modified “entropy” functional W(g, f, τ ) is introduced. It is nondecreasing n in t provided that g(·) is a Ricci flow, τ = t0 − t and (4πτ )− 2 e−f satisfies the conjugate heat equation. The functional W is constant on a Ricci flow if and only if the flow is a gradient shrinking soliton that terminates at time t0 . Because of this, W is more suitable than F when one wants information that is localized in spacetime. In Section I.4 the entropy functional W is used to prove a no local collapsing theorem. The statement is that if g(·) is a given Ricci flow on a finite time interval [0, T ) then for any (scale) ρ > 0, there is a number κ > 0 so that if Bt (x, r) is a time-t ball with radius r less than ρ, on which | Rm | ≤ r12 , then vol(Bt (x, r)) ≥ κ rn . The method of proof is to show that if r −n vol(Bt (x, r)) is very small then the evaluation of W at time t is very negative, which contradicts the monotonicity of W. The significance of a no local collapsing theorem is that it allows one to use Hamilton’s compactness theorem to construct blowup limits of finite time singularities, and more generally to understand high curvature regions.

Section I.5 and I.6 are not needed in the sequel. Section I.5 gives some thermodynamiclike equations in which W appears as an entropy. Section I.6 motivates the construction of the reduced volume of Section I.7. 4.2. I.7-I.10. A new monotonic quantity, the reduced volume V˜ , is introduced in I.7. It is defined in terms of so-called L-geodesics. Let (p, t0 ) be a fixed spacetime point. Define backward time by τ = t0 − t. Given a curve γ(τ ) in M defined for 0 ≤ τ ≤ τ (i.e. going backward in real time) with γ(0) = p, its L-length is Z τ √ τ |γ(τ ˙ )|2 + R(γ(τ ), t0 − τ ) dτ. (4.1) L(γ) = 0

One can compute the first and second variations of L, in analogy to what is done in Riemannian geometry.

= p and γ(τ ) = q. Put l(q, τ ) = Let L(q, τ ) be the infimum of L(γ) over curves γ with R γ(0) −n −l(q,τ ) ˜ 2 e dvol(q). The remarkable The reduced volume is defined by V (τ ) = M τ ˜ fact is that if g(·) is a Ricci flow solution then V is nonincreasing in τ , i.e. nondecreasing in real time t. Furthermore, it is constant if and only if g(·) is a gradient shrinking soliton that terminates at time t0 . The proof of monotonicity uses a subtle cancellation between the τ derivative of l(γ(τ ), τ ) along an L-geodesic and the Jacobian of the so-called L-exponential map. L(q,τ ) √ . 2 τ

Using a differential inequality, it is shown that for each τ there is some point q(τ ) ∈ M so that l(q(τ ), τ ) ≤ n2 . This is then used to prove a no local collapsing theorem : Given a Ricci flow solution g(·) defined on a finite time interval [0, t0 ] and a scale ρ > 0 there is a number κ > 0 with the following property. For r < ρ, suppose that | Rm | ≤ r12 on the “parabolic” ball {(x, t) : distt0 (x, p) ≤ r, t0 − r 2 ≤ t ≤ t0 }. Then vol(Bt0 (p, r)) ≥ κ rn . The number κ can be chosen to depend on ρ, n, t0 and bounds on the geometry of the initial metric g(0). The hypotheses of the no local collapsing theorem proved using V˜ are more stringent than those of the no local collapsing theorem proved using W, but the consequences turn out to

16


be the same. The no local collapsing theorem is used extensively in Sections I.11 and I.12 when extracting a convergent subsequence from a sequence of Ricci flow solutions. Theorem I.8.2 of Section I.8 says that under appropriate assumptions, local κ-noncollapsing extends forwards in time to larger distances. This will be used in I.12 to analyze long-time behavior. The statement is that for any A < ∞ there is a κ = κ(A) > 0 with the following property. Given r0 > 0 and x0 ∈ M, suppose that g(·) is defined for t ∈ [0, r02], with | Rm(x, t)| ≤ r0−2 for all (x, t) satisfying dist0 (x, x0 ) < r0 , and the volume of the time-zero ball B0 (x0 , r0 ) is at least A−1 r0n . Then the metric g(t) cannot be κ-collapsed on scales less than r0 at a point (x, r02 ) with distr02 (x, x0 ) ≤ Ar0 . A localized version of the W-functional appears in I.9. Section I.10, which is not needed for the sequel but is of independent interest, shows if a point in a time slice lies in a ball with quantitatively bounded geometry then at nearby later times, the curvature at the point is quantitatively bounded. That is, there is a damping effect with regard to exterior high-curvature regions. The “bounded geometry” assumptions on the initial ball are a lower bound on its scalar curvature and an assumption that the isoperimetric constants of subballs are close to the Euclidean value.

4.3. I.11. Section I.11 contains an analysis of κ-solutions. As mentioned before, in three dimensions κ-solutions arise as blowup limits of finite-time singularities and, more generally, as limits of rescalings of high-scalar-curvature regions. In addition to the no local collapsing theorem, some of the tools used to analyze κ-solutions are Hamilton’s Harnack inequality for Ricci flows with nonnegative curvature operator, and the comparison geometry of nonnegatively curved manifolds. The first result is that any time slice of a κ-solution has vanishing asymptotic volume ratio limr→∞ r −n vol(Bt (p, r)). This apparently technical result is used to show that if a κ-solution (M, g(·)) has scalar curvature normalized to equal one at some spacetime point (p, t) then there is an a priori upper bound on the scalar curvature R(q, t) at other points q in terms of distt (p, q). Using the curvature bound, it is shown that a sequence {(Mi , pi , gi(·))}∞ i=1 of pointed n-dimensional κ-solutions, normalized so that R(pi , 0) = 1 for each i, has a convergent subsequence whose limit satisfies all of the requirements to be a κ-solution, except possibly the condition of having bounded sectional curvature on each time slice. In three dimensions this statement is improved by showing that the sectional curvature will be bounded on each compact time interval, so the space of pointed 3-dimensional κsolutions (M, p, g(·)) with R(p, 0) = 1 is in fact compact. This is used to draw conclusions about the global geometry of 3-dimensional κ-solutions. If M is a compact 3-dimensional κ-solution then Hamilton’s theorem about compact 3manifolds with nonnegative Ricci curvature implies that M is diffeomorphic to a spherical space form. If M is noncompact then assuming that M is oriented, it follows easily that M is diffeomorphic to R3 , or isometric to the round shrinking cylinder R ×S 2 or its Z2 -quotient R ×Z2 S 2 . In the noncompact case it is shown that after rescaling, each time slice is neck-like at infinity. More precisely, considering a given time-t slice, for each ǫ > 0 there is a compact


17

subset Mǫ ⊂ M so that / Mǫ then the pointed manifold (M, y, R(y, t)g(t)) is ǫ-close to if 1y ∈ 1 the standard cylinder − ǫ , ǫ × S 2 of scalar curvature one. 4.4. I.12-I.13. Sections I.12 and I.13 deal with three-dimensional Ricci flows. Theorem I.12.1 uses the results of Section I.11 to model the high-scalar-curvature regions of a Ricci flow. Let us assume a pinching condition of the form Rm ≥ − Φ(R) for an appropriate function Φ with lims→∞ Φ(s) = 0. (This will eventually follow from Hamiltons Ivey pinching, cf. Appendix B.) Theorem I.12.1 says that given numbers ǫ, κ > 0, one can find r0 > 0 with the following property. Suppose that g(·) is a Ricci flow solution defined on some time interval [0, T ] that satisfies the pinching condition and is κ-noncollapsed at scales less than one. Then for any point (x0 , t0 ) with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , after scaling by the factor Q, the solution in the region {(x, t) : dist2t0 (x, x0 ) ≤ (ǫQ)−1 , t0 − (ǫQ)−1 ≤ t ≤ t0 } is ǫ-close to the corresponding subset of a κ-solution. Theorem I.12.1 says in particular that near a first singularity, the geometry is modeled by a κ-solution, for some κ. This fact is used in [52]. Although Theorem I.12.1 is not used directly in [51], its method of proof is used in Theorem I.12.2.

The method of proof of Theorem I.12.1 is by contradiction. If it were not true then there (i) would be a sequence r0 → 0 and a sequence(Mi , gi (·)) of Ricci flow solutions that satisfy (i) (i) the assumptions, each with a spacetime point x0 , t0 that does not satisfy the conclusion. (i) (i) To consider first a special case, suppose that each point x0 , t0 is the first point at which (i) (i) a certain curvature threshold Ri is achieved, i.e. R(y, t) ≤ R x0 , t0 for each y ∈ Mi h i (i) (i) (i) and t ∈ 0, t0 . Then after rescaling the Ricci flow gi (·) by Qi = R x0 , t0 and shifting (i)

the time parameter, one has the curvature bounds on the time interval [−Qi t0 , 0] that form part of the hypotheses of Hamilton’s compactness theorem. Furthermore, the no local collapsing theorem gives the lower injectivity radius bound needed to apply Hamilton’s theorem and take a convergent subsequence of the pointed rescaled solutions. The limit will be a κ-solution, giving the contradiction. In the general case, one effectively proceedsby induction on the size of the scalar curvature. (i) (i) By modifying the choice of points x0 , t0 , one can assume that the conclusion of the (i) (i) theorem holds for all of the points (y, t) in a large spacetime neighborhood of x0 , t0 that have R(y, t) > 2Qi . One then shows that one has the curvature bounds needed to form the time-zero slice of the putative κ-solution. One shows that this “time-zero” metric can be extended backward in time to form a κ-solution, thereby giving the contradiction. The rest of Section I.12 begins the analysis of the long-time behaviour of a nonsingular 3-dimensional Ricci flow. There are two main results, Theorems I.12.2 and I.12.3. They extend curvature bounds forward and backward in time, respectively. Theorem I.12.2 roughly says that if one has | Rm | ≤ r0−2 on a spacetime region of spatial size r0 and temporal size r02 , and if one has a lower bound on the volume of the initial time face of the region, then one gets scalar curvature bounds on much larger spatial balls at

18


the final time. More precisely, for any A < ∞ there are numbers K = K(A) < ∞ and ρ = ρ(A) < ∞ so that with the hypotheses and notation of Theorem I.8.2, if in addition r02 Φ(r0−2 ) < ρ then R(x, r02 ) ≤ Kr0−2 for points x lying in the ball of radius Ar0 around x0 at time r02 . Theorem I.12.3 says that if one has a lower bound on volume and sectional curvature on a ball at a certain time then one obtains an upper scalar curvature bound on a smaller ball at an earlier time. More precisely, given w > 0 there exist τ = τ (w) > 0, ρ = ρ(w) > 0 and K = K(w) < ∞ with the following property. Suppose that a ball B(x0 , r0 ) at time t0 has volume bounded below by wr03 and sectional curvature bounded below by − r0−2. Then R(x, t) < Kr0−2 for t ∈ [t0 − τ r02 , t0 ] and distt (x, x0 ) < 41 r0 , provided that φ(r0−2 ) < ρ.

Applying a back-and-forth argument using Theorems I.12.2 and I.12.3, along with the pinching condition, one concludes, roughly speaking, that if a metric ball of small radius r has infimal sectional curvature exactly equal to − r −2 then the ball has a small volume compared to r 3 . Such a ball can be said to be locally volume-collapsed with respect to a lower sectional curvature bound. Section I.13 defines the thick-thin decomposition of a large-time slice of a nonsingular Ricci flow and shows the geometrization. Rescaling the metric to b g (t) = t−1 g(t), there is a universal function Φ so that for large t, the metric b g (t) satisfies the Φ-pinching condition. In terms of the original unscaled metric, given x ∈ M let rb(x, t) > 0 be the unique number such that inf Rm Bt (x,br) = − rb−2 .

Given w > 0, define the w-thin part Mthin (w, t) of the time-t slice to be the points x ∈ M so that vol(Bt (x, rb(x, t))) < w rb(x, t)3 . That is, a point in Mthin (w, t) lies in a ball that is locally volume-collapsed with respect to a lower sectional curvature bound. Put Mthick (w, t) = M − Mthin (w, t). One shows that for large t, the subset Mthick (w, t) has bounded geometry in the sense that >0√ and K = K(w) < ∞ √ there are numbers √ ρ = ρ(w) 1 w (ρ t))3 , whenever x ∈ so that | Rm | ≤ Kt−1 on B(x, ρ t) and vol(B(x, ρ t)) ≥ 10 Mthick (w, t).

Invoking arguments of Hamilton (that are written out in more detail in [52]) one can take a sequence t → ∞ and w → 0 so that Mthick (w, t) converges to a complete finite-volume manifold with constant sectional curvature − 14 , whose cuspidal tori are incompressible in M. On the other hand, a result from Riemannian geometry implies that for large t and small w, Mthin (w, t) is homeomorphic to a graph manifold; again a more precise statement appears in [52]. The conclusion is that M satisfies the geometrization conjecture. Again, one is assuming in I.13 that the Ricci flow is nonsingular for all times. 5. I.1.1. The F -functional and its monotonicity The goal of this section is to show that in an appropriate sense, Ricci flow is a gradient flow on the space of metrics. We introduce the entropy functional F . We compute its formal variation and show that the corresponding gradient flow is a modified Ricci flow. In Sections 5 through 14 of these notes, M is a closed manifold. We will use the Einstein summation convention freely. We also follow Perelman’s convention that a condition like a > 0 means that a should be considered to be a small parameter, while a condition like


19

A < ∞ means that A should be considered to be a large parameter. This convention is only for pedagogical purposes and may be ignored by a logically minded reader. Let M denote the space of smooth Riemannian metrics g on M. We think of M formally as an infinite-dimensional manifold. The tangent space Tg M consists of the symmetric covariant 2-tensors vij on M. Similarly, C ∞ (M) is an infinite-dimensional manifold with Tf C ∞ (M) = C ∞ (M). The diffeomorphism group Diff(M) acts on M and C ∞ (M) by pullback. Let dV denote the Riemannian volume density associated to a metric g. We use the convention that △ = div grad. Definition 5.1. The F -functional F : M × C ∞ (M) → R is given by Z (5.2) F (g, f ) = R + |∇f |2 e−f dV. M

Given vij ∈ Tg M and h ∈ Tf C ∞ (M), the evaluation of the differential dF on (vij , h) is written as δF (vij , h). Put v = g ij vij . Proposition 5.3. (cf. I.1.1) We have Z i h v 2 −f − h (2△f − |∇f | + R) dV. (5.4) δF (vij , h) = e − vij (Rij + ∇i ∇j f ) + 2 M Proof. From a standard formula, δR = − △v + ∇i ∇j vij − Rij vij .

(5.5) As

|∇f |2 = g ij ∇i f ∇j f,

(5.6) we have (5.7)

δ|∇f |2 = − v ij ∇i f ∇j f + 2 h∇f, ∇hi.

p

det(g) dx1 . . . dxn , we have δ(dV ) = v2 dV , so v (5.8) δ e−f dV = − h e−f dV. 2 Putting this together gives Z (5.9) δF = e−f [− △v + ∇i ∇j vij − Rij vij − vij ∇i f ∇j f + M v i 2 h∇f, ∇hi + (R + |∇f |2 ) − h dV. 2

As dV =

The goal now is to rewrite the right-hand side of (5.9) so that vij and h appear algebraically, i.e. without derivatives. As △e−f = (|∇f |2 − △f ) e−f ,

(5.10) we have (5.11)

Z

M

−f

e

[− △v] dV = −

Z

−f

(△e M

) v dV =

Z

M

e−f (△f − |∇f |2 ) v dV.

20

Next, (5.12)


Z

−f

e

M

∇i ∇j vij dV = =

Z

−f

ZM M

Finally, (5.13)

2

Z

−f

e

M

(∇i ∇j e

) vij dV = −

Z

M

∇i (e−f ∇j f ) vij dV

e−f (∇i f ∇j f − ∇i ∇j f ) vij dV. Z

−f

Z

h∇f, ∇hi dV = − 2 h∇e , ∇hi dV = 2 (△e−f ) h dV M Z M = 2 e−f (|∇f |2 − △f ) h dV. M

Then

Z

− h (2△f − 2|∇f |2) − vij (Rij + ∇i ∇j f ) 2 M v i 2 + − h (R + |∇f | ) dV 2 Z i h v 2 −f − h (2△f − |∇f | + R) dV. = e − vij (Rij + ∇i ∇j f ) + 2 M This proves the proposition. We would like to get rid of the v2 − h (2△f − |∇f |2 + R) term in (5.14). We can do this by restricting our variations so that v2 − h = 0. From (5.8), this amounts to assuming that assuming e−f dV is fixed. We now fix a smooth measure dm on M and relate f to g by requiring that e−f dV = dm. Equivalently, we define a section s : M → M × C ∞ (M) dV by s(g) = g, ln dm . Then the composition F m = F ◦ s is a function on M and its differential is given by Z m (5.15) dF (vij ) = e−f [− vij (Rij + ∇i ∇j f )] dV. (5.14)

δF =

e−f

h v

M

Defining a formal Riemannian metric on M by Z 1 (5.16) hvij , vij ig = v ij vij dm, 2 M the gradient flow of F m on M is given by

(5.17)

(gij )t = −2 (Rij + ∇i ∇j f ).

The induced flow equation for f is dV 1 ∂ ln = g ij (gij )t = − △f − R. (5.18) ft = ∂t dm 2 As with any gradient flow, the function F m is nondecreasing along the flow line with its derivative being given by the length squared of the gradient, i.e. Z m (5.19) Ft = 2 |Rij + ∇i ∇j f |2 dm, M

as follows from (5.14) and (5.17)


21

We now perform time-dependent diffeomorphisms to transform (5.17) into the Ricci flow equation. If V (t) is the time-dependent generating vector field of the diffeomorphisms then the new equations for g and f become (5.20)

(gij )t = −2 (Rij + ∇i ∇j f ) + LV g, ft = − △f − R + LV f.

Taking V = ∇f gives (5.21)

(gij )t = −2 Rij ,

ft = − △f − R + |∇f |2.

As F (g, f ) is unchanged by a simultaneous pullback of g and f , and the right-hand side of (5.19) is also unchanged under a simultaneous pullback, it follows that under the new evolution equations (5.21) we still have Z d (5.22) F (g(t), f (t)) = 2 |Rij + ∇i ∇j f |2 e−f dV dt M (This can also be checked directly). Because of the diffeomorphisms that we applied, g and f are no longer related by e−f dV = R −f dm. We do have that M e dV is constant in t, as e−f dV is related to dm by a diffeomorphism.

The relation between g and f is as follows: we solve (or assume that we have a solution for) the first equation in (5.21), with some initial metric. Then given the solution g(t), we require that f satisfy the second equation in (5.21) (which is in terms of g(t)). The second equation in (5.21) can be written as

∂ −f e = − △e−f + R e−f . ∂t As this is a backward heat equation, we cannot solve for f forward in time starting with an arbitrary smooth function. Instead, (5.21) will be applied by starting with a solution for (gij )t = −2 Rij on some time interval [t1 , t2 ] and then solving (5.23) backwards in time on [t1 , t2 ] (which can always be done) starting with some initial f (t2 ). Having done this, the solution (g(t), f (t)) on [t1 , t2 ] will satisfy (5.22). (5.23)

6. Basic example for I.1 In this section we compute F in a Euclidean example.

Consider Rn with the standard metric, constant in time. Fix t0 > 0. Put τ = t0 − t and (6.1)

f (t, x) =

|x|2 n + ln(4πτ ), 4τ 2

so (6.2)

e−f = (4πτ )−n/2 e−

|x|2 4τ

.

22


This is the standard heat kernel when considered for τ going from 0 to t0 , i.e. for t going from t0 to 0. One can check that (g, f ) solves (5.21). As Z |x|2 (6.3) e− 4τ dV = (4πτ )n/2 , Rn

2

x and |∇f |2 = |x| . Differentiating (6.3) with f is properly normalized. Then ∇f = 2τ 4τ 2 respect to τ gives Z |x|2 − |x|2 n (6.4) e 4τ dV = (4πτ )n/2 , 2 2τ Rn 4τ so Z n . (6.5) |∇f |2 e−f dV = 2τ Rn

Then F (t) =

n 2τ

=

n . 2(t0 −t)

In particular, this is nondecreasing as a function of t ∈ [0, t0 ).

7. I.2.2. The λ-invariant and its applications In this section we define λ(g) and show that it is nondecreasing under Ricci flow. We use this to show that a steady breather on a compact manifold is a gradient steady soliton. Proposition Given a metric g, there is a unique minimizer f of F (g, f ) under the R 7.1. −f constraint M e dV = 1.

Proof. Write

F =

(7.2) −f /2

Putting Φ = e (7.3)

,

F =

Z

2

Z

M

Re−f + 4|∇e−f /2 |2 dV. 2

Z

4|∇Φ| + R Φ dV = Φ (− 4△Φ + RΦ) dV. M R The constraint equation becomes M Φ2 dV = 1. Then λ is the smallest eigenvalue of −4△ + R and e−f /2 is a corresponding normalized eigenvector. As the operator is a Schrödinger operator, there is a unique normalized positive eigenvector [55, Chapter XIII.12]. M

Definition 7.4. The λ-functional is given by λ(g) = F (g, f). If g(t) is a smooth family of metrics then it follows from eigenvalue perturbation theory that λ(g(t)) and f(t) are smooth in t [55, Chapter XII]. Proposition 7.5. (cf. I.2.2) If g(·) is a Ricci flow solution then λ(g(t)) is nondecreasing in t. Proof. Consider a time interval [t1 , t2 ], and the minimizer f (t2 ). In particular, λ(t2 ) = F (g(t2), f (t2 )). Put u(t2 ) = e−f (t2 ) . Solve the backward heat equation (7.6)

∂u = − △u + Ru ∂t


23

backward on [t1 , t2 ]. We claim that u(x′ , t′ ) > 0 for all x′ ∈ M and t′ ∈ [t1 , t2 ]. To see this, we take t′ ∈ = △h on (t′ , t2 ] with [t1 , t2 ), and let h be the solution to the forward heat equation ∂h ∂t limt→t′ h(t) = δx′ . We have Z Z d (7.7) u(t) h(t) dV = [(∂t u + △u − Ru) v + u (∂t h − △h)] dV = 0. dt M M One knows that h(t) > 0 for all t ∈ (t′ , t2 ]. Then Z Z Z ′ ′ ′ (7.8) u(x , t ) = u(x, t ) δx′ (x) dV (x) = lim′ u(t) h(t) dV = u(t2 ) h(t2 ) dV > 0. t→t

M

M

M

For t ∈ [t1 , t2 ], define f (t) by u(t) = e−f (t) . By (5.22), F (g(t1), f (t1 )) ≤ F (g(t2), f (t2 )). By the of λ, λ(tR1 ) = F (g(t1 ), f(t1 )) ≤ F (g(t1), f (t1 )). (We are using the fact R definition −f (t1 ) that M e dV (t1 ) = M e−f (t2 ) dV (t2 ) = 1.) Thus λ(t1 ) ≤ λ(t2 ).

Definition 7.9. A steady breather is a Ricci flow solution on an interval [t1 , t2 ] that satisfies the equation g(t2 ) = φ∗ g(t1 ) for some φ ∈ Diff(M). Steady soliton solutions are steady breathers.

Again, we are assuming that M is compact. The next result is not essential for the sequel, but gives a good illustration of how a monotonicity formula is used. Proposition 7.10. (cf. I.2.2) A steady breather is a gradient steady soliton. Proof. We have λ(g(t2 )) = λ(φ∗ g(t1 )) = λ(g(t1)). Thus we have equality in Proposition 7.5. Tracing through the proof, F (g(t), f (t)) must be constant in t. From (5.22), Rij + ∇i ∇j f = 0. Then R + △f = 0 and so (5.21) becomes (C.5). One can sharpen Proposition 7.5. Lemma 7.11. (cf. Proposition I.1.2) 2 2 dλ ≥ λ (t). dt n

(7.12)

Proof. Given a time interval [t1 , t2 ], with the notation of the proof of Proposition 7.5 we have Z t2 Z (7.13) λ(t1 ) ≤ F (g(t1 ), f (t1 )) = F (g(t2), f (t2 )) − 2 |Rij + ∇i ∇j f |2 e−f dV dt = λ(t2 ) − 2 Then (7.14)

Z

t2

t1

Z

t1

M

M

|Rij + ∇i ∇j f |2 e−f dV dt.

λ(t2 ) − λ(t1 ) dλ = lim− ≥ 2 dt t=t2 t2 − t1 t1 →t2

where the right-hand side is evaluated at time t2 .

Z

M

|Rij + ∇i ∇j f |2 e−f dV,

24


Hence for all t, dλ ≥ 2 dt

(7.15) and so

Z

M

dλ 2 ≥ dt n

(7.16)

|Rij + ∇i ∇j f |2 e−f dV Z

M

(R + △f)2 e−f dV.

R From the Cauchy-Schwarz inequality and the fact that M e−f dV = 1, Z 2 Z −f ≤ (R + △f)2 e−f dV. (7.17) (R + △f ) e dV M

M

Finally, (5.10) gives Z Z −f (7.18) (R + △f) e dV = (R + |∇f |2 ) e−f dV = F (g(t), f(t)) = λ(t). M

M

This proves the lemma.

8. I.2.3. The rescaled λ-invariant In this section we show the monotonicity of a scale-invariant version of λ. This will be used in Section 93. We then show that an expanding breather on a compact manifold is a gradient expanding soliton. 2

Put λ(g) = λ(g)V (g) n . As λ is scale-invariant, it is constant in t along a steady, shrinking or expanding soliton solution. Proposition 8.1. (cf. Claim of I.2.3) If g(·) is a Ricci flow solution and λ(g(t)) ≤ 0 for some t then dtd λ(g(t)) ≥ 0. Proof. We have 2−n 2 dλ 2 dλ = V (t) n − V (t) n λ(t) dt dt n

(8.2) From (7.15), (8.3)

2 dλ ≥ V (t) n dt

Z

R dV.

M

Z Z 2 −1 2 −f R dV . 2 |Rij + ∇i ∇j f | e dV − V (t) λ(t) n M M

Using the spatially-constant function ln V (t) as a test function for F gives Z −1 (8.4) λ(t) ≤ V (t) R dV. M

The assumption that λ(t) ≤ 0 gives (8.5)

2

− λ(t) ≤ − V (t)

and so (8.6)

2 dλ ≥ V (t) n dt

−1

λ(t)

Z

R dV

M

Z 2 2 −f 2 2 |Rij + ∇i ∇j f| e dV − λ(t) . n M


Next, (8.7)

|Rij

25

2 1 1 + ∇i ∇j f | ≥ Rij + ∇i ∇j f − (R + △f) gij + (R + △f)2 . n n 2

Using (7.18), one obtains (8.8) "Z 2 Z −f 2 1 dλ 1 Rij + ∇i ∇j f − (R + △f) gij e dV + ≥ 2 V (t) n (R + △f )2 e−f dV dt n n M M Z 2 # 1 − (R + △f) e−f dV . n M R As M e−f dV = 1, the Cauchy-Schwarz inequality implies that the right-hand side of (8.8) is nonnegative. Corollary 8.9. If λ is a constant nonpositive number on an interval [t1 , t2 ] then the Ricci flow solution is a gradient soliton. Proof. From equation (8.8), we obtain that R + △f = α(t) for some function α that is spatially constant, and Rij + ∇i ∇j f = α(t) gij . Thus g evolves by diffeomorphisms and n dilations. After a shift of the time parameter, α(t) is proportionate to t cf. [22, Lemma 2.4]. This is the gradient soliton equation of Appendix C. Definition 8.10. An expanding breather is a Ricci flow solution on [t1 , t2 ] that satisfies g(t2 ) = c φ∗ g(t1) for some c > 1 and φ ∈ Diff(M). Expanding soliton solutions are expanding breathers. Again, we are assuming that M is compact. Proposition 8.11. An expanding breather is a gradient expanding soliton. Proof. First, λ(t2 ) = λ(t1 ). As V (t2 ) > V (t1 ), we must have dV > 0 for some t ∈ [t1 , t2 ]. dt R dV From (8.4), dt = − M R dV ≤ − λ(t) V (t), so λ(t) must be negative for some t ∈ [t1 , t2 ]. Proposition 8.1 implies that λ(t1 ) < 0. Then as λ(t2 ) = λ(t1 ), it follows that λ is a negative constant on [t1 , t2 ]. From Corollary 8.9, the solution is a gradient expanding soliton. 9. I.2.4. Gradient steady solitons on compact manifolds It was shown in Section 7 that a steady breather on a compact manifold is a gradient steady soliton. We now show that it is in fact Ricci flat. This was previously shown in [33, Theorem 20.1]. Proposition 9.1. A gradient steady solution on a compact manifold is Ricci flat. Proof. As we are in the equality case of Proposition 7.5, the function f (t) must be the minimizer of F (g(t), ·) for all t. That is, (9.2)

f

f

(− 4△ + R) e− 2 = λ e− 2

26


for all t, where λ is constant in t. Equivalently, 2△f −|∇f |2 +RR= λ. As R+△f = R0, we have △f −|∇f |2 = λ. Then △e−f = −λe−f . Integrating gives 0 = M △e−f dV = −λ M e−f dV , 2 R R so λ = 0. Then 0 = − M e−f △e−f dV = M ∇e−f dV , so f is constant and g is Ricci flat. A similar argument shows that a gradient expanding soliton on a compact manifold comes from an Einstein metric with negative Ricci curvature. 10. Ricci flow as a gradient flow We have shown in Section 5 that the modified Ricci flow is the gradient flow for the functional F m on the space of metrics M. One can ask if the unmodified Ricci flow is a gradient flow. This turns out to be true provided that one considers it as a flow on the space M/ Diff(M).

As mentioned in II.8, Ricci flow is the gradient flow for the function λ. More precisely, this statement is valid on M/ Diff(M), with the latter being equipped with an appropriate metric. To see this, we first consider λ as a function on the space of metrics M. Here the formal Riemannian metric on M comes from saying that for vij ∈ Tg M, Z 1 (10.1) hvij , vij i = v ij vij Φ2 dV (g), 2 M where Φ = Φ(g) is the unique normalized positive eigenvector corresponding to λ(g). Lemma 10.2. The formal gradient flow of λ is ∂gij = − 2 (Rij − 2 ∇i ∇j ln Φ) . (10.3) ∂t Proof. We set (10.4)

λ(g) =

inf R

f ∈C ∞ (M ) :

M

e−f dV = 1

F (g, f ).

To calculate the variation in λ due to a variation δgij = vij , we let h = δf be the variation induced by letting f be the minimizer in (10.4). Then Z Z v −f (10.5) 0=δ e dV = − h e−f dV. M M 2 Now equation (5.14) gives

(10.6)

δλ(vij ) =

f

As Φ = e− 2 satisfies (10.7) it follows that (10.8)

Z

e−f [− vij (Rij + ∇i ∇j f ) + M i v 2 − h (2△f − |∇f | + R) dV. 2 − 4 △Φ + RΦ = λΦ, 2 △f − |∇f |2 + R = λ.


27

Hence the last term in (10.6) vanishes, and (10.6) becomes Z (10.9) δλ(vij ) = − e−f vij (Rij + ∇i ∇j f ) dV. M

The corresponding gradient flow is ∂gij (10.10) = − 2 (Rij + ∇i ∇j f ) = − 2 (Rij − 2 ∇i ∇j ln Φ) . ∂t We note that it follows from (10.8) that (10.11)

∇j (Rij + ∇i ∇j f ) e−f

= 0.

This implies that the gradient vector field of λ is perpendicular to the infinitesimal diffeomorphisms at g, as one would expect. In the sense of [10], the quotient space M/ Diff(M) is a stratified infinite-dimensional Riemannian manifold, with the strata corresponding to the possible isometry groups Isom(M, g). We give it the quotient Riemannian metric coming from (10.1). The modified Ricci flow (10.10) on M projects to a flow on M/ Diff(M) that coincides with the projection of the unmodified Ricci flow dg = − 2 Ric. The upshot is that the Ricci flow, as a flow on dt M/ Diff(M), is the gradient flow of λ, the latter now being considered as a function on M/ Diff(M). One sees an intuitive explanation for Proposition 7.10. If a gradient flow on a finitedimensional manifold has a periodic orbit then it must be a fixed-point. Applying this principle formally to the Ricci flow on M/ Diff(M), one infers that a steady breather only evolves by diffeomorphisms. 11. The W-functional Definition 11.1. The W-functional W : M × C ∞ (M) × R+ → R is given by Z n (11.2) W(g, f, τ ) = τ |∇f |2 + R + f − n (4πτ )− 2 e−f dV. M

The W-functional is a scale-invariant variant of F . It has the symmetries W(φ∗ g, φ∗f, τ ) = W(g, f, τ ) for φ ∈ Diff(M), and W(cg, f, cτ ) = W(g, f, τ ) for c > 0. Hence it is constant in t = − τ along a gradient shrinking soliton defined for t ∈ (−∞, 0), as in Appendix C. In this sense, W is constant on gradient shrinking solitons just as F is constant on gradient steady solitons. As an example of a gradient shrinking soliton, consider Rn with the flat metric, constant in time t ∈ (−∞, 0). Put τ = − t and (11.3)

f (t, x) =

|x|2 , 4τ

so (11.4)

e−f = e−

|x|2 4τ

.

28


One can check that (g(t), f (t), τ (t)) satisfies (12.12) and (12.13). Now |x|2 |x|2 |x|2 + − n = − n. 4τ 2 4τ 2τ It follows from (6.3) and (6.4) that W(t) = 0 for all t. τ (|∇f |2 + R) + f − n = τ ·

(11.5)

12. I.3.1. Monotonicity of the W-functional In this section we compute the variation of W, in analogy with the computation in Section 5 of the variation of F . We then show that a shrinking breather on a compact manifold is a gradient shrinking soliton. As in Section 5, we write δgij = vij and δf = h. Put σ = δτ . Proposition 12.1. We have (12.2) Z δW(vij , h, σ) = σ(R + |∇f |2 ) − τ vij (Rij + ∇i ∇j f ) + h + M v nσ i 2 τ (2△f − |∇f | + R) + f − n − h − (4πτ )−n/2 e−f dV. 2 2τ Proof. One finds

δ (4πτ )−n/2 e−f dV

(12.3) Writing

W =

(12.4)

Z

M

we can use (5.14) to obtain

=

v

2

− h −

nσ (4πτ )−n/2 e−f dV. 2τ

τ (R + |∇f |2 ) + f − n (4πτ )−n/2 e−f dV

(12.5)

Z h v 2 δW = σ(R + |∇f | ) + τ − h (2△f − 2|∇f |2) − τ vij (Rij + ∇i ∇j f ) + 2 M v nσ i (12.6) − h − (4πτ )−n/2 e−f dV. h + τ (R + |∇f |2) + f − n 2 2τ Then (5.10) gives (12.7)

Z

σ(R + |∇f |2) − τ vij (Rij + ∇i ∇j f ) + h + M v nσ i τ (2△f − |∇f |2 + R) + f − n − h − (4πτ )−n/2 e−f dV. 2 2τ This proves the proposition. δW =

We now fix a smooth measure dm on M with mass 1 and relate f to g and τ by requiring that (4πτ )−n/2 e−f dV = dm. Then v2 − h − nσ = 0 and 2τ Z (12.8) δW = σ(R + |∇f |2 ) − τ vij (Rij + ∇i ∇j f ) + h (4πτ )−n/2 e−f dV. M


We now consider

dW dt

when (gij )t = − 2 (Rij + ∇i ∇j f ), n ft = − △f − R + , 2τ τt = −1.

(12.9)

To apply (12.8), we put

vij = − 2 (Rij + ∇i ∇j f ), n h = − △f − R + , 2τ σ = −1.

(12.10)

v 2

− h − nσ = 0. Then from (12.8), 2τ Z dW = − (R + |∇f |2) + 2 τ |Rij + ∇i ∇j f |2 dt M ni − △f − R + (4πτ )−n/2 e−f dV 2τ Z h ni (4πτ )−n/2 e−f dV = − 2(R + △f ) + 2 τ |Rij + ∇i ∇j f |2 + 2τ ZM 1 gij |2 (4πτ )−n/2 e−f dV. 2 τ |Rij + ∇i ∇j f − = 2τ M

We do have (12.11)

29

Adding a Lie derivative to the right-hand side of (12.9) gives the new flow equations (12.12)

(gij )t = − 2 Rij ,

ft = − △f + |∇f |2 − R +

n , 2τ

τt = −1,

with (12.11) still holding. We no longer have (4πτ )−n/2 e−f dV = dm, but we do have Z (12.13) (4πτ )−n/2 e−f dV = 1. M

We now wantR to look at the variational problem of minimizing W(g, f, τ ) under the constraint that M (4πτ )−n/2 e−f dV = 1. We write Z (12.14) µ(g, τ ) = inf {W(g, f, τ ) : (4πτ )−n/2 e−f dV = 1}. f

M

− f2

Making the change of variable Φ = e , we are minimizing Z −n/2 (12.15) (4πτ ) τ (4|∇Φ|2 + RΦ2 ) − 2Φ2 log Φ − n Φ2 dV M

R under the constraint (4πτ ) Φ2 dV = 1. From [56, Section 1] the infimum is finite M and there is a positive continuous minimizer Φ. It will be a weak solution of the variational equation −n/2

(12.16)

τ (−4△ + R)Φ = 2Φ log Φ + (µ(g, τ ) + n)Φ.

30


From elliptic theory, Φ is smooth. Then f = − 2 log Φ is also smooth.

As in Section 7, it follows that µ(g(t), t0 −t) is nondecreasing in t for a Ricci flow solution, where t0 is any fixed number and t < t0 . If it is constant in t then the solution must be a gradient shrinking soliton that goes singular at time t0 . Definition 12.17. A shrinking breather is a Ricci flow solution on [t1 , t2 ] that satisfies g(t2 ) = c φ∗ g(t1) for some c < 1 and φ ∈ Diff(M). Gradient shrinking soliton solutions are shrinking breathers. Again, we are assuming that M is compact. Proposition 12.18. A shrinking breather is a gradient shrinking soliton. t2 −ct1 . 1−c

Then if τ1 = t0 − t1 and τ2 = t0 − t2 , we have τ2 = cτ1 . Hence τ2 ∗ φ g(t1 ), τ2 = µ (φ∗ g(t1 ), τ1 ) = µ (g(t1 ), τ1 ) . (12.19) µ(g(t2), τ2 ) = µ τ1 It follows that the solution is a gradient shrinking soliton. Proof. Put t0 =

13. I.4. The no local collapsing theorem I In this section we prove the no local collapsing theorem. Definition 13.1. A smooth Ricci flow solution g(·) on a time interval [0, T ) is said to be locally collapsing at T if there is a sequence of times tk → T and a sequence of metric balls Bk = B(pk , rk ) at times tk such that rk2 /tk is bounded, | Rm |(g(tk )) ≤ rk−2 in Bk and limk→∞ rk−n vol(Bk ) = 0. Remark 13.2. In the definition of noncollapsing, T could be infinite. This is why it is written that rk2 /tk stays bounded, while if T < ∞ then this is obviously the same as saying that rk stays bounded. Theorem 13.3. (cf. Theorem I.4.1) If M is closed and T < ∞ then g(·) is not locally collapsing at T . Proof. We first sketch the idea of the proof. In Section 11 we showed that in the case of flat n

−f

−

|x|2 4τ

rk2

−fk

−

|x|2 4r 2 k

R , taking e (x) = e , , we get W(g, f, τ ) = 0. So putting τ = and e (x) = e we have W(g, fk , rk2 ) = 0. In the collapsing case, the idea is to use a test function fk so that (13.4)

−

e−fk (x) ∼ e−ck e

distt (x,pk )2 k 4r 2 k

,

where ck is determined by the normalization condition Z (13.5) (4πrk2 )−n/2 e−fk dV = 1. M

The main difference between computing (13.5) in M and in Rn comes from the difference in 1 volumes, which means that e−ck ∼ r−n vol(B . In particular, as k → ∞, we have ck → −∞. ) k

k


31

Now that fk is normalized correctly, the main difference between computing W(g(tk ), fk , rk2 ) in M, and the analogous computation for the Gaussian in Rn , comes from the f term in the integrand of W. Since fk ∼ ck , this will drive W(g(tk ), fk , rk2 ) to −∞ as k → ∞, so µ(g(tk ), rk2 ) → −∞; by the monotonicity of µ(g(t), t0 − t) it follows that µ(g(0), tk + rk2 ) → −∞ as k → ∞. This contradicts the fact that µ(g(0), τ ) is a continuous function of τ . To write this out precisely, let us put Φ = e−f /2 , so that Z −n/2 (13.6) W(g, f, τ ) = (4πτ ) 4τ |∇Φ|2 + (τ R − 2 ln Φ − n) Φ2 dV. M

For the argument, it is enough to obtain small values of W for positive Φ. Since lims→0 (−2 ln s)s2 = 0, by an approximation it is enough to obtain small values of W for nonnegative Φ, where the integrand is declared to be 4τ |∇Φ|2 at points where Φ vanishes. Take Φk (x) = e−ck /2 φ(disttk (x, pk )/rk ),

(13.7)

where φ : [0, ∞) → [0, 1] is a monotonically nonincreasing function such that φ(s) = 1 if s ∈ [0, 1/2], φ(s) = 0 if s ≥ 1 and |φ′ (s)| ≤ 10 for s ∈ [1/2, 1]. The function Φk is a priori only Lipschitz, but by smoothing it slightly we can use Φk in the variational formula to bound W from above. The constant ck is determined by Z ck (13.8) e = (4πrk2 )−n/2 φ2 (disttk (x, pk )/rk ) dV ≤ (4πrk2 )−n/2 vol(Bk ). M

Thus ck → −∞. Next, (13.9)

W(g(tk ), fk , rk2 )

=

(4πrk2 )−n/2

Z

M

4rk2 |∇Φk |2 + (rk2 R − 2 ln Φk − n) Φ2k dV.

Let Ak (s) be the mass of the distance sphere S(pk , rk s) around pk . Put Z 2 −1 (13.10) Rk (s) = rk Ak (s) R d area . S(pk ,rk s)

We can compute the integral in (13.9) radially to get (13.11) R1 ′ 4(φ (s))2 + (Rk (s) + ck − 2 ln φ(s) − n) φ2 (s) Ak (s) ds 2 0 . W(g(tk ), fk , rk ) = R1 2 (s) A (s) ds φ k 0

The expression 4(φ′ (s))2 − 2 ln φ(s) φ2 (s) vanishes if s ∈ / [1/2, 1], and is bounded above by −1 400 + e if s ∈ [1/2, 1]. Then the lower bound on the Ricci curvature and the BishopGromov inequality give R1 [4(φ′ (s))2 − 2 ln φ(s) φ2 (s)] Ak (s) ds vol(B(pk , rk )) − vol(B(pk , rk /2)) ≤ 401 (13.12) 0 R1 vol(B(pk , rk /2)) φ2 (s) Ak (s) ds 0 ! R1 n−1 sinh (s) ds 0 − 1 . ≤ 401 R 1/2 n−1 sinh (s) ds 0 Next, from the upper bound on scalar curvature, Rk (s) ≤ n(n − 1) for s ∈ [0, 1]. Putting this together gives W(g(tk ), fk , rk2 ) ≤ const. + ck and so W(g(tk ), fk , rk2 ) → −∞ as k → ∞.

32


Thus µ(g(tk ), rk2 ) → −∞. For any t0 > t, µ(g(t), t0 − t) is nondecreasing in t. Hence µ(g(0), tk + rk2 ) ≤ µ(g(tk ), rk2 ), so µ(g(0), tk + rk2 ) → −∞. Since T is finite, tk and rk2 are uniformly bounded, and tk uniformly positive, which contradicts the fact that µ(g(0), τ ) is a continuous function of τ . Remark 13.13. In the preceding argument we only used the upper bound on scalar curvature and the lower bound on Ricci curvature, i.e. in the definition of local collapsing one could have assumed that R(gij (tk )) ≤ n(n − 1) rk−2 in Bk and Ric(gij (tk )) ≥ − (n − 1) rk−2 in Bk . In fact, one can also remove the lower bound on Ricci curvature (observation of Perelman, communicated by Gang Tian). The necessary ingredients of the preceding argument were that 1. rk−n vol(B(pk , rk )) → 0, 2. rk2 R is uniformly bounded above on B(pk , rk ) and vol(B(pk ,rk )) is uniformly bounded above. 3. vol(B(p k ,rk /2)) Suppose only that rk−n vol(B(pk , rk )) → 0 and for all k, rk2 R ≤ n(n − 1) on B(pk , rk ). If vol(B(pk ,rk )) < 3n for all k then we are done. If not, suppose that for a given k, vol(B(p ≥ k ,rk /2)) −n ′ ′ −n ′ 3 . Putting rk = rk /2, we have that (rk ) vol(B(pk , rk )) ≤ rk vol(B(pk , rk )) and vol(B(pk ,rk )) (rk′ )2 R ≤ n(n − 1) on B(pk , rk′ ). We replace rk by rk′ . If now vol(B(p < 3n then we k ,rk /2)) stop. If not then we repeat the process and replace rk by rk /2. Eventually we will achieve vol(B(pk ,rk )) < 3n . Then we can apply the preceding argument to this new sequence that vol(B(p k ,rk /2)) of pairs {(pk , rk )}∞ k=1 . vol(B(pk ,rk )) vol(B(pk ,rk /2)) n

Definition 13.14. (cf. Definition I.4.2) We say that a metric g is κ-noncollapsed on the scale ρ if every metric ball B of radius r < ρ, which satisfies | Rm(x)| ≤ r −2 for every x ∈ B, has volume at least κrn . Remark 13.15. We caution the reader that this definition differs slightly from the definition of noncollapsing that is used from section I.7 onwards. Note that except for the overall scale ρ, the κ-noncollapsed condition is scale-invariant. From the proof of Theorem 13.3 we extract the following statement. Given a Ricci flow defined on an interval [0, T ), with T < ∞, and a scale ρ, there is some number κ = κ(g(0), T, ρ) so that the solution is κ-noncollapsed on the scale ρ for all t ∈ [0, T ). We note that the estimate on κ deteriorates as T → ∞, as there are Ricci flow solutions that collapse at long time.

14. I.5. The W-functional as a time derivative We will only discuss one formula from I.5, showing that along a Ricci flow, W is itself the time-derivative of an integral expression.


33

Again, we put τ = − t. Consider the evolution equations (12.9), with (4πτ )−n/2 e−f dV = dm. Then Z Z Z d n n n (14.1) τ f − dm = f − dm + τ △f + R − dm dτ 2 2 2τ M M M Z Z n n dm + τ |∇f |2 + R − dm = f − 2 2τ M M = W(g(t), f (t), τ ). With respect to the evolution equations (12.12) obtained by performing diffeomorphisms, we get Z d n −n/2 −f (14.2) τ f − (4πτ ) e dV = W(g(t), f (t), τ ). dτ 2 M Similarly, with respect to (5.21), Z d −f − f e dV = F (g(t), f (t)). (14.3) dt M 15. I.7. Overview of reduced length and reduced volume We first give a brief summary of I.7. In I.7, the variable τ = t0 − t is used and so the corresponding Ricci flow equation is (gij )τ = 2Rij . The goal is to prove a no local collapsing theorem by means of the L-lengths of curves γ : [τ1 , τ2 ] → M, defined by Z τ2 2 √ (15.1) L(γ) = τ R(γ(τ )) + γ(τ ˙ ) dτ, τ1

where the scalar curvature R(γ(τ )) and the norm |γ(τ ˙ )| are evaluated using the metric at dγ time t0 − τ . Here τ1 ≥ 0. With X = dτ , the corresponding L-geodesic equation is

1 1 ∇R + X + 2 Ric(X, ·) = 0, 2 2τ where again the connection and curvature are taken at the corresponding time, and the 1-form Ric(X, ·) has been identified with the corresponding dual vector field. √ Fix p ∈ M. Taking τ1 = 0 and γ(0) = p, the vector v = limτ →0 τ X(τ ) is welldefined in Tp M and is called the initial vector of the geodesic. The L-exponential map Lexpτ : Tp M → M sends v to γ(τ ). (15.2)

∇X X −

The function L(q, τ ) is the infimal L-length of curves γ with γ(0) = p and γ(τ ) = q. Defining the reduced length by (15.3)

l(q, τ ) =

and the reduced volume by (15.4)

V˜ (τ ) =

Z

M

L(q, τ ) √ 2 τ

n

τ − 2 e−l(q,τ ) dq,

34


the goal is to show that V˜ (τ ) is nonincreasing in τ , i.e. nondecreasing in t. To do this, one uses the L-exponential map to write V˜ (τ ) as an integral over Tp M: Z n ˜ (15.5) V (τ ) = τ − 2 e−l(Lexpτ (v),τ ) J (v, τ ) χτ (v) dv, Tp M

where J (v, τ ) = det d (Lexpτ )v is the Jacobian factor in the change of variable and χτ is a cutoff function related to the L-cut locus of p. To show that V˜ (τ ) is nonincreasing in τ n it suffices to show that τ − 2 e−l(Lexpτ (v),τ ) J (v, τ ) is nonincreasing in τ or, equivalently, that − n2 ln(τ ) − l(Lexpτ (v), τ ) + ln J (v, τ ) is nonincreasing in τ . Hence it is necessary to (v,τ ) . The computation of the latter will involve the L-Jacobi compute dl(Lexpdττ (v),τ ) and dJdτ fields. The fact that V˜ (τ ) is nonincreasing in τ is then used to show that the Ricci flow solution cannot be collapsed near p. 16. Basic example for I.7 In this section we say what the various expressions of I.7 become in the model case of a flat Euclidean Ricci solution. If M is flat Rn and p = ~0 then the unique L-geodesic γ with γ(0) = ~0 and γ(τ ) = ~q is τ 21 1 (16.1) γ(τ ) = ~q = 2 τ 2 ~v . τ The function L is given by 1 1 (16.2) L(q, τ ) = τ − 2 |q|2 2 and the reduced length (15.3) is given by (16.3)

l(q, τ ) =

|q|2 . 4τ

1

The function L(q, τ ) = 2 τ 2 L(q, τ ) is L(q, τ ) = |q|2.

(16.4) Then (16.5)

V˜ (τ ) =

Z

n

|q|2

n

τ − 2 e− 4τ dn q = (4π) 2

Rn

is constant in τ .

17. Remarks about L-Geodesics and L exp In this section we discuss the variational equation corresponding to (15.1). To derive the L-geodesic equation, as in Riemannian geometry we consider a 1-parameter family of curves γs : [τ1 , τ2 ] → M, parametrized by s ∈ (−ǫ, ǫ). Equivalently, we have a map γ γ and Y = ∂˜ , we have [X, Y ] = 0. γ˜ (s, τ ) with s ∈ (−ǫ, ǫ) and t ∈ [τ1 , τ2 ]. Putting X = ∂˜ ∂τ ∂s


35

Then ∇X Y = ∇Y X. Restricting to the curve γ(τ ) = γ˜ (0, τ ) and writing δY as shorthand d for ds , we have (δY γ)(τ ) = Y (τ ) and (δY X)(τ ) = (∇X Y )(τ ). Then s=0 Z τ2 √ τ (hY, ∇Ri + 2 h∇X Y, Xi) dτ. (17.1) δY L = τ1

Using the fact that

dgij dτ

(17.2)

dhY, Xi = h∇X Y, Xi + hY, ∇X Xi + 2 Ric(Y, X). dτ

Then

Z

= 2Rij , we have

τ2

√ τ (hY, ∇Ri + 2 h∇X Y, Xi) dτ = τ1 Z τ2 √ d τ hY, ∇Ri + 2 hY, Xi − 2hY, ∇X Xi − 4 Ric(Y, X) dτ = dτ τ1 Z τ2 τ2 √ √ 1 τ Y, ∇R − 2∇X X − 4 Ric(X, ·) − X dτ. 2 τ hX, Y i τ1 + τ τ1 Hence the L-geodesic equation is 1 1 (17.4) ∇X X − ∇R + X + 2 Ric(X, ·) = 0. 2 2τ

(17.3)

We now discuss some technical issues about L-geodesics and the L-exponential map. We are assuming that (M, g(·)) is a Ricci flow, where the curvature operator of M is uniformly bounded on a τ -interval [τ1 , τ2 ], and each τ -slice (M, g(τ )) is complete for τ ∈ [τ1 , τ2 ]. By Appendix D, for every τ ′ < τ2 there is a constant D < ∞ such that D (17.5) |∇R(x, τ )| < √ τ2 − τ for all x ∈ M, τ ∈ [τ1 , τ2 ). √ Making the change of variable s = τ in the formula for L-length, we get Z s2 2 1 dγ 2 (17.6) L(γ) = 2 + s R(γ(s)) ds. 4 ds s1 The Euler-Lagrange equation becomes ˆ − 2s2 ∇R + 4s Ric(X, ˆ ·) = 0, (17.7) ∇Xˆ X ˆ = dγ = 2sX. Putting s1 = √τ1 , it follows from standard existence theory for where X ds ODE’s that for each p ∈ M and v ∈ Tp M, there is a unique solution γ(s) to (17.7), defined on an interval [s1 , s1 + ǫ), with γ(s1 ) = p and √ dγ 1 ′ γ (s1 ) = lim τ = v. (17.8) τ →τ1 2 dτ If γ(s) is defined for s ∈ [s1 , s′ ] then d ˆ 2 d ˆ ˆ ˆ X) ˆ + 2h∇ ˆ X, ˆ Xi ˆ (17.9) |X| = hX, Xi = 4s Ric(X, X ds ds ˆ X) ˆ + 4s2 h∇R, Xi ˆ = −4s Ric(X,

36


ˆ and so if X(s) 6= 0 then (17.10)

1 d ˆ 2 d ˆ ˆ Ric |X| = |X| = −2s|X| ˆ ds ds 2|X|

ˆ X ˆ X , ˆ |X| ˆ |X|

!

+ 2s2

*

ˆ X ∇R, ˆ |X|

+

.

By (17.5), d ˆ ˆ + √ C2 |X| ≤ C1 |X| ds s2 − s √ for appropriate constants C1 and C2 , where s2 = τ2 . Since the metrics g(τ ) are uniformly comparable for τ ∈ [τ1 , τ2 ], we conclude (by a continuity argument in s ) that the L-geodesic γv with 12 γv′ (s1 ) = v is defined on the whole interval [s1 , s2 ]. In particular, in terms of the original variable τ , for each τ ∈ [τ1 , τ2 ] and each p ∈ M, we get a globally defined and smooth L-exponential √ dγ map L expτ : Tp M → M which takes each v ∈ Tp M to γ(τ ), where v = limτ ′ →τ1 τ ′ dτ ′ . Note that unlike in the case of Riemannian geometry, L expτ (0) may not be p, because of the ∇R term in (17.4). (17.11)

We now fix p ∈ M, take τ1 = 0, and let L(q, τ¯) be the minimizer function as in Section 15. We can imitate the traditional Riemannian geometry proof that geodesics minimize √ for a short time. Using the change of variable s = τ and the implicit function theorem, there is an r = r(p) > 0 (which varies continuously with p) such that for every q ∈ M with d(q, p) ≤ 10r at τ = 0, and every 0 < τ¯ ≤ r 2 , there is a unique L-geodesic γ(q,¯τ ) : [0, τ¯] → M, starting at p and ending at q, which remains within the ball B(p, 100r) (in the τ = 0 slice (M, g(0))), and γ(q,¯τ ) varies smoothly with (q, τ¯). Thus, the L-length of γ(q,¯τ ) varies smoothly ˆ τ¯) near (p, 0). We claim that L ˆ = L near (p, 0). with (q, τ¯), and defines a function L(q, Suppose that q ∈ B(p, r) and let α : [0, τ¯] → M be a smooth curve whose L-length is close to L(q, τ¯). If r is small, relative to the assumed curvature bound, then α must stay within B(p, 10r). Equations (18.2) and (18.6) below imply that (17.12) dα 2 √ √ √ d ˆ d dα 2 L(α(τ ), τ ) = h2 τ X, i + τ (R−|X| ) ≤ τ R + = Llength(α [0,τ ] ) . dτ dτ dτ dτ Thus γ(q,¯τ ) minimizes when (q, τ¯) is close to (p, 0).

We can now deduce that for all (q, τ¯), there is an L-geodesic γ : [0, τ¯] → M which has infimal L-length among all piecewise smooth curves starting at p and ending at q (with domain [0, τ¯]). This can be done by imitating the usual broken geodesic argument, using the fact that for x, y in a given small ball of M and for sufficiently small time intervals R τ ′ +ǫ √ [τ ′ , τ ′ + ǫ] ⊂ [0, τ ], there is a unique minimizer γ for τ ′ τ R(γ(τ )) + |γ(τ ˙ )|2 dτ with √ γ(τ ′ ) = x and γ(τ ′ + ǫ) = y. Alternatively, using the change of variable s = τ , one can take a minimizer of L among H 1,2-regular curves.

Another technical issue is the justification of the change of variables from M to Tp M in the proof of monotonicity of reduced volume. Fix p ∈ M and τ > 0, and let L expτ : Tp M → M be the√map which takes v ∈ Tp M to γv (τ ), where γv : [0, τ ] → M is the unique L-geodesic v with τ ′ dγ → v as τ ′ → 0. Let B ⊂ M be the set of points which are either endpoints of dτ ′ more than one minimizing L-geodesic, or which are the endpoint of a minimizing geodesic γv : [0, τ ] → M where v ∈ Tp M is a critical point of L expτ . We will call B the time-τ L-cut


37

locus of p. It is a closed subset of M. Let G ⊂ M be the complement of B and let Ωτ ⊂ Tp M be the corresponding set of initial conditions for minimizing L-geodesics. Then Ωτ is an open set, and L exp maps it diffeomorphically onto G. We claim that B has measure zero. By Sard’s theorem, to prove this it suffices to prove that the set B′ of points q ∈ B which are regular values of L expτ , has measure zero. Pick q ∈ B′ , and distinct points v1 , v2 ∈ Tp M such that γvi : [0, τ ] → M are both minimizing geodesics ending at q. Then L expτ is a local diffeomorphism near each vi . The first variation formula and the implicit function theorem then show that there are neighborhoods Ui of vi , and a smooth hypersurface H passing through q, such that if we have points wi ∈ Ui with (17.13)

q ′ = L expτ (w1 ) = L expτ (w2 ) and L length(γw1 ) = L length(γw2 ),

then q ′ lies on H. Thus B′ is contained in a countable union of hypersurfaces, and hence has measure zero. Therefore one may compute the integral of any integrable function on M by pulling it back to Ωτ ⊂ Tp M and using the change of variables formula. Note that if τ ≤ τ ′ then Ωτ ′ ⊂ Ωτ . 18. I.(7.3)-(7.6). First derivatives of L In this section we do some preliminary calculations leading up to the computation of the second variation of L. A remark about the notation : L is a function of a point q and a time τ . The notation Lτ refers to the partial derivative with respect to τ , i.e. differentiation while keeping q fixed. The notation dτd refers to differentiation along an L-geodesic, i.e. simultaneously varying both the point and the time. If q is not in the time-τ L-cut locus of p, let γ : [0, τ ] → M be the unique minimizing Lgeodesic from p to q, with length L(q, τ ). If c : (−ǫ, ǫ) → M is a short curve with c(0) = q, consider the 1-parameter family of minimizing L-geodesics γ˜ (s, τ ) with γ˜ (s, 0) = p and ∂˜ γ (s,τ ) γ˜ (s, τ ) = c(s). Putting Y (τ ) = ∂s , equation (17.3) gives s=0

(18.1)

Hence (18.2) and (18.3)

h∇L, c′ (0)i =

√ dL(c(s), τ ) = 2 τ hX(τ), Y (τ )i. ds s=0

(∇L)(q, τ ) = 2

√ τ X(τ )

|∇L|2 (q, τ ) = 4 τ |X(τ )|2 = − 4 τ R(q) + 4 τ R(q) + |X(τ )|2 .

If we simply extend the L-geodesic γ in τ , we obtain √ dL(γ(τ ), τ ) (18.4) = τ R(γ(τ )) + |X(τ )|2 . dτ As dL(γ(τ ), τ ) (18.5) = Lτ (q, τ ) + h(∇L)(q, τ ), X(τ)i, dτ

38


equations (18.2) and (18.4) give √ Lτ (q, τ ) = τ R(q) + |X(τ )|2 − h(∇L)(q, τ ), X(τ )i (18.6) √ √ = 2 τ R(q) − τ R(q) + |X(τ)|2 .

When computing in I.(7.3), (18.7)

dl(γ(τ ),τ ) , dτ

2 it will be useful to have a formula for R(γ(τ )) + X(τ ) . As

2 d R(γ(τ )) + X(τ ) = Rτ + h∇R, Xi + 2h∇X X, Xi + 2 Ric(X, X). dτ

Using the L-geodesic equation (17.4) gives

(18.8) 2 1 1 d R(γ(τ )) + X(τ ) = Rτ + R + 2h∇R, Xi − 2 Ric(X, X) − (R + |X|2 ) dτ τ τ 1 = − H(X) − (R + |X|2), τ where (18.9)

H(X) = − Rτ −

1 R − 2h∇R, Xi + 2 Ric(X, X) τ 3

is the expression of (F.9) after the change τ = −t and X → −X. Multiplying (18.8) by τ 2 and integrating gives Z τ 2 3 d R(γ(τ )) + X(τ ) dτ = − K − L(q, τ ), (18.10) τ2 dτ 0 where

(18.11)

K =

Z

τ

3

τ 2 H(X(τ )) dτ.

0

Then integrating the left-hand side of (18.10) by parts gives (18.12)

τ

3 2

2 1 R(γ(τ )) + X(τ ) = − K + L(q, τ ). 2

Plugging this back into (18.6) and (18.3) gives (18.13)

√ 1 1 Lτ (q, τ ) = 2 τ R(q) − L(q, τ ) + K 2τ τ

and (18.14)

2 4 |∇L|2 (q, τ ) = − 4 τ R(q) + √ L(q, τ ) − √ K. τ τ


39

19. I.(7.7). Second variation of L In this section we compute the second variation of L. We use it to compute the Hessian of L on M. To compute the second variation δY2 L, we start with the first variation equation Z τ √ τ (hY, ∇Ri + 2 h∇Y X, Xi) dτ. (19.1) δY L = 0

Recalling that δY γ(τ ) = Y (τ ) and δY X(τ ) = (∇Y X)(τ ), the second variation is (19.2) δY2

L = = =

Z

Z

Z

τ

0 τ

0 τ 0

√

τ (Y · Y · R + 2 h∇Y ∇Y X, Xi + 2 h∇Y X, ∇Y Xi) dτ

2 √ τ Y · Y · R + 2 h∇Y ∇X Y, Xi + 2 ∇X Y dτ

2 √ τ Y · Y · R + 2 h∇X ∇Y Y, Xi + 2 hR(Y, X)Y, Xi + 2 ∇X Y dτ,

where the notation Z · u refers to the directional derivative, i.e. Z · u = iZ du. In order to deal with the h∇X ∇Y Y, Xi term, we have to compute dτd h∇Y Y, Xi.

From the general equation for the Levi-Civita connection in terms of the metric [17, dg ˙ = d∇ , then (1.29)], if g(τ ) is a 1-parameter family of metrics, with g˙ = dτ and ∇ dτ

(19.3)

˙ X Y, Zi = (∇X g)(Y, 2h∇ ˙ Z) + (∇Y g)(Z, ˙ X) − (∇Z g)(X, ˙ Y ).

In our case g˙ = 2 Ric and so (19.4)

d ˙ Y Y, Xi h∇Y Y, Xi = h∇X ∇Y Y, Xi + h∇Y Y, ∇X Xi + 2 Ric(∇Y Y, X) + h∇ dτ = h∇X ∇Y Y, Xi + h∇Y Y, ∇X Xi + 2 Ric(∇Y Y, X) + 2(∇Y Ric)(Y, X) − (∇X Ric)(Y, Y ).

(Although we will not need it, we can write (19.5) 2Y · Ric(Y, X) − X · Ric(Y, Y ) = 2(∇Y Ric)(Y, X) + 2 Ric(∇Y Y, X) + 2 Ric(Y, ∇Y X) − (∇X Ric)(Y, Y ) − 2 Ric(∇X Y, Y )

= 2 Ric(∇Y Y, X) + 2(∇Y Ric)(Y, X)

− (∇X Ric)(Y, Y ) − 2 Ric([X, Y ], Y ). We are assuming that the variation field Y satisfies [X, Y ] = 0 (this was used in deriving the L-geodesic equation). Hence one obtains the formula (19.6) of I.7.)

d h∇Y Y, Xi = h∇X ∇Y Y, Xi + h∇Y Y, ∇X Xi + 2Y · Ric(Y, X) − X · Ric(Y, Y ) dτ

40


Next, using (19.4), (19.7) Z τ √ d √ 2 τ h∇Y Y, Xi = 2 τ h∇Y Y, Xi dτ 0 dτ Z τ √ 1 d = τ h∇Y Y, Xi + 2 h∇Y Y, Xi dτ τ dτ 0 Z τ √ 1 τ = h∇Y Y, Xi + 2h∇X ∇Y Y, Xi + 2h∇Y Y, ∇X Xi + τ 0 4 Ric(∇Y Y, X) + 4(∇Y Ric)(Y, X) − 2(∇X Ric)(Y, Y )] dτ Z τ √ τ [2h∇X ∇Y Y, Xi + (∇Y Y )R + = 0

−

Z

4(∇Y Ric)(Y, X) − 2(∇X Ric)(Y, Y )] dτ τ √ 1 τ ∇Y Y, ∇R − 2∇X X − 4 Ric(X, ·) − X dτ. τ 0

(Of course the last term vanishes if γ is an L-geodesic, but we do not need to assume this here.) The quadratic form Q representing the Hessian of L on the path space is given by (19.8)

Q(Y, Y ) = δY2 L − δ∇Y Y L √ = δY2 L − 2 τ h∇Y Y, Xi − Z τ √ 1 τ ∇Y Y, ∇R − 2∇X X − 4 Ric(X, ·) − X . τ 0

It follows that (19.9)

Z

τ

√ τ [Y · Y · R − (∇Y Y )R + 2hR(Y, X)Y, Xi + 0 2 |∇X Y |2 − 4(∇Y Ric)(Y, X) + 2(∇X Ric)(Y, Y ) dτ Z τ √ τ HessR (Y, Y ) + 2hR(Y, X)Y, Xi + 2 |∇X Y |2 =

Q(Y, Y ) =

0

− 4(∇Y Ric)(Y, X) + 2(∇X Ric)(Y, Y )] dτ.

There √ is an associated second-order differential operator T on√vector fields Y given by saying that 2 τ hY, T Y i equals the integrand of (19.9) minus 2 dτd ( τ h∇X Y, Y i). Explicitly, (19.10) T Y = − ∇X ∇X Y −

1 1 ∇X Y + HessR (Y, ·) − 2 (∇Y Ric)(X, ·) − 2 Ric(∇Y X, ·). 2τ 2

Then (19.11)

Q(Y, Y ) = 2

Z

0

τ

√

τ hY, T Y i dτ + 2

√ τ h∇X Y (τ ), Y (τ )i.

An L-Jacobi field along an L-geodesic is a field Y (τ ) that is annihilated by T .


41

The Hessian of the function L(·, τ ) can be computed as follows. Assume that q ∈ M is outside of the time-τ L-cut locus. Let γ : [0, τ ] → M be the minimizing L-geodesic with γ(0) = p and γ(τ ) = q. Given w ∈ Tq M, take a short geodesic c : (−ǫ, ǫ) → M with c(0) = q and c′ (0) = w. Form the 1-parameter family of L-geodesics γ˜ (s, τ ) with (s,τ ) is an L-Jacobi field Y along γ γ˜ (s, 0) = p and γ˜ (s, τ ) = c(s). Then Y (τ ) = ∂˜γ∂s s=0

with Y (0) = 0 and Y (τ ) = w. We have √ d2 L(c(s)) τ h∇X Y (τ ), Y (τ )i. (19.12) HessL (w, w) = = Q(Y, Y ) = 2 ds2 s=0

From (19.11), a minimizer of Q(Y, Y ), among fields Y with given values at the endpoints, is an L-Jacobi field. It follows that HessL (w, w) ≤ Q(Y, Y ) for any field Y along γ satisfying Y (0) = 0 and Y (τ ) = w. 20. I.(7.8)-(7.9). Hessian bound for L In this section we use a test variation field in order to estimate the Hessian of L. If Y (τ ) is a unit vector at γ(τ ), solve for Ye (τ ) in the equation

1 e ∇X Ye = − Ric(Ye , ·) + Y, 2τ on the interval 0 < τ ≤ τ with the endpoint condition Ye (τ ) = Y (τ ). (For this section, we change notation from the Y in I.7 to Ye .) Then (20.1)

d e e 1 hY , Y i = 2 Ric(Ye , Ye ) + 2h∇X Ye , Ye i = hYe , Ye i, dτ τ so hYe (τ ), Ye (τ )i = ττ . Thus we can extend Ye continuously to the interval [0, τ ] by putting Ye (0) = 0. Substituting into (19.9) gives (20.2)

(20.3)

"

2 1 τ HessR (Ye , Ye ) + 2hR(Ye , X)Ye , Xi + 2 − Ric(Ye , ·) + Q(Ye , Ye ) = Ye 2τ 0 i − 4(∇Ye Ric)(Ye , X) + 2(∇X Ric)(Ye , Ye ) dτ Z τ √ h = τ HessR (Ye , Ye ) + 2hR(Ye , X)Ye , Xi + 2(∇X Ric)(Ye , Ye ) 0 1 2 2 dτ. − 4(∇Ye Ric)(Ye , X) + 2 | Ric(Ye , ·)| − Ric(Ye , Ye ) + τ 2τ τ Z

τ

√

From (20.4) d Ric(Ye (τ ), Ye (τ )) = Ricτ (Ye , Ye ) + (∇X Ric)(Ye , Ye ) + 2 Ric(∇X Ye , Ye ) dτ 1 = Ricτ (Ye , Ye ) + (∇X Ric)(Ye , Ye ) + Ric(Ye , Ye ) − 2| Ric(Ye , ·)|2, τ

42


one obtains (20.5) Z τ √ d √ τ Ric(Ye , Ye ) dτ = − 2 τ Ric(Y (τ ), Y (τ )) = − 2 0 dτ Z τ √ 1 2 2 − τ Ric(Ye , Ye ) + 2 Ricτ (Ye , Ye ) + 2(∇X Ric)(Ye , Ye ) + Ric(Ye , Ye ) − 4| Ric(Ye , ·)| . τ τ 0

Combining (20.3) and (20.5) gives

Z τ √ √ 1 e e √ (20.6) HessL (Y (τ ), Y (τ )) ≤ Q(Y , Y ) = τ H(X, Ye ) dτ, − 2 τ Ric(Y (τ ), Y (τ )) − τ 0 where (20.7) e e e e e e e e H(X, Y ) = − HessR (Y , Y ) − 2hR(Y , X)Y , Xi − 4 ∇X Ric(Y , Y ) − ∇Ye Ric(Y , X) 2 1 Ric(Ye , Ye ). − 2 Ricτ (Ye , Ye ) + 2 Ric(Ye , ·) − τ is the expression appearing in (F.4), after the change τ = −t and X → −X. Note that H(X, Ye ) is a quadratic form in Ye . For its relation to the expression H(X) from (18.9), see Appendix F. 21. I.(7.10). The Laplacian of L In this section we estimate △L.

Let {Yi (τ )}ni=1 be an orthonormal basis of Tγ(τ ) M. Solve for Yei (τ ) from (20.1). Putting 1/2 ei (τ ), the vectors {ei (τ )}ni=1 form an orthonormal basis of Tγ(τ ) M. SubstiYei (τ ) = ττ tuting into (20.6) and summing over i gives Z τ X √ n 1 (21.1) △L ≤ √ − 2 τ R − τ 3/2 H(X, ei ) dτ. τ 0 τ i Then from (F.8), (21.2)

Z τ √ n 1 △L ≤ √ − 2 τ R − τ 3/2 H(X) dτ τ τ 0 √ 1 n = √ − 2 τ R − K. τ τ 22. I.(7.11). Estimates on L-Jacobi fields

In this section we estimate the growth rate of an L-Jacobi field.

Given an L-Jacobi field Y , we have (22.1)

d |Y |2 = 2 Ric(Y, Y ) + 2h∇X Y, Y i = 2 Ric(Y, Y ) + 2h∇Y X, Y i. dτ


43

Thus (22.2)

d|Y |2 1 = 2 Ric(Y (τ ), Y (τ )) + √ HessL (Y (τ ), Y (τ )), dτ τ =τ τ

where we have used (19.12). Let Y˜ be a field along γ as in Section 20, satisfying (20.1) with Y˜ (τ ) = Y (τ ) and |Y (τ )| = 1. Then from (20.6), Z τ √ 1 1 1 √ HessL (Y (τ ), Y (τ )) ≤ (22.3) τ H(X, Y˜ ) dτ. − 2 Ric(Y (τ ), Y (τ )) − √ τ τ τ 0 Thus

(22.4)

Z τ √ d|Y |2 1 1 τ H(X, Y˜ ) dτ. ≤ − √ dτ τ =τ τ τ 0

The inequality is sharp if and only if the first inequality in (20.6) is an equality. This is the case if and only if Y˜ is actually the L-Jacobi field Y , in which case 1 1 d|Y |2 d|Y˜ |2 (22.5) = = 2 Ric(Y (τ ), Y (τ )) + √ HessL (Y (τ ), Y (τ )). = τ dτ τ =τ dτ τ =τ τ

τ.

23. Monotonicity of the reduced volume Ve

In this section we show that the reduced volume Ve (τ ) is monotonically nonincreasing in

Fix p ∈ M. Define l(q, τ ) as in (15.3). In order to show that V˜ (τ ) is well-defined in the noncompact case, we will need a lower bound on l(q, τ ). For later use, we prove something slightly more general. Recall that we are assuming that we have bounded curvature on compact time intervals, and that time slices are complete.

Lemma 23.1. Given 0 < τ 1 ≤ τ 2 , there constants C1 , C2 > 0 so that for all τ ∈ [τ 1 , τ 2 ] and all q ∈ M, we have (23.2)

l(q, τ ) ≥ C1 d(p, q)2 − C2 .

Proof. We write L in the form (17.6). Given an L-geodesic γ with γ(0) = p and γ(τ ) = q, we obtain Z √τ 2 dγ 1 ds − const. (23.3) L(γ) ≥ 2 0 ds

As the multiplicative change in the metric between times τ 1 and τ 2 is bounded by a factor econst. (τ 2 −τ 1 ) , it follows that L(q, τ ) ≥ const. d(p, q)2 − const., where the distance is measured at time τ . The lemma follows. Define V˜ (τ ) as in (15.4). As the volume of time-τ balls in M increases at most exponentially fast in the radius, it follows that V˜ (τ ) is well-defined. From the discussion in Section

44


17, we can write (23.4)

V˜ (τ ) =

Z

n

Tp M

τ − 2 e−l(Lexpτ (v),τ ) J (v, τ ) χτ (v) dv,

where J (v, τ ) = det d (Lexpτ )v is the Jacobian factor in the change of variable and χτ is the characteristic function of the time-τ domain Ωτ of Section 17. We first show that for each v, the expression − n2 ln(τ ) − l(Lexpτ (v), τ ) + ln J (v, τ ) is nonincreasing in τ . Let γ be the L-geodesic with initial vector v ∈ Tp M. From (18.4) and (18.12), 3 dl(γ(τ ), τ ) 1 1 1 (23.5) = − l(γ(τ )) + R(γ(τ )) + |X(τ )|2 = − τ − 2 K. dτ τ =τ 2τ 2 2

Next, let {Yi }ni=1 be a basis for the Jacobi fields along γ that vanish at τ = 0. We can write (23.6)

ln J (v, τ )2 = ln det ((d (Lexpτ )v )∗ d (Lexpτ )v ) = ln det(S(τ )) + const.,

where S is the matrix (23.7) Then (23.8)

Sij (τ ) = hYi (τ ), Yj (τ )i. d ln J (v, τ ) 1 −1 dS = Tr S . dτ 2 dτ

To compute the derivative at τ = τ , we can choose a basis so that S(τ ) = In , i.e. hYi (τ ), Yj (τ )i = δij . Then using (22.4) and computing as in Section 21, n 3 1 d ln J (v, τ ) 1 X d|Yi|2 n − τ − 2 K. (23.9) = ≤ dτ 2 i=1 dτ τ =τ 2τ 2 τ =τ If we have equality then (22.5) holds for each Yi , i.e. 2 Ric +

√1 τ

HessL =

g τ

at γ(τ ).

n

From (23.5) and (23.9), we deduce that τ − 2 e−l(Lexpτ (v),τ ) J (v, τ ) is nonincreasing in τ . Finally, recall that if τ ≤ τ ′ then Ωτ ′ ⊂ Ωτ , so χτ (v) is nonincreasing in τ . Hence V˜ (τ ) is nonincreasing in τ . If it is not strictly decreasing then we must have (23.10)

g(τ ) 1 . 2 Ric(τ ) + √ HessL(τ ) = τ τ

on all of M. Hence we have a gradient shrinking soliton solution. 24. I.(7.15). A differential inequality for L In this section we discuss an important differential inequality concerning the reduced length l. We use the differential inequality to estimate min l(·, τ ) from above. We then give a lower bound on l. √ With L(q, τ ) = 2 τ L(q, τ ), equations (18.13) and (21.2) imply that (24.1)

Lτ + △L ≤ 2n


45

away from the time-τ L-cut locus of p. We will eventually apply the maximum principle to this differential inequality. However to do so, we must discuss several senses in which the inequality can be made global on M, i.e. how it can be interpreted on the cut locus. The first sense is that of a barrier differential inequality. Given f ∈ C(M) and a function g on M, one says that △f ≤ g in the sense of barriers if for all q ∈ M and ǫ > 0, there is a neighborhood V of q and some uǫ ∈ C 2 (V ) so that uǫ (q) = f (q), uǫ ≥ f on V and ≤ g [26]. △uǫ ≤ g + ǫ on V [13]. There is a similar spacetime definition for △f − ∂f ∂t The point of barrier differential inequalities is that they allow one to apply the maximum principle just as with smooth solutions. We illustrate this by constructing a barrier function for L in (24.1). Given the spacetime point (q, τ ), let γ : [0, τ ] → M be a minimizing L-geodesic with γ(0) = p and γ(τ ) = q. Given a small ǫ > 0, let uǫ (q ′ , τ ′ ) be the minimum of Z τ′ Z ǫ 2 2 √ √ (24.2) τ R(γ2 (τ )) + γ˙2 (τ ) dτ + τ R(γ(τ )) + γ(τ ˙ ) dτ ǫ

0

′

among curves γ2 : [ǫ, τ ] → M with γ2 (ǫ) = γ(ǫ) and γ2 (τ ′ ) = q ′ . Because the new basepoint (γ(ǫ), ǫ) is moved in along γ from p, the minimizer γ2 will be unique and will vary smoothly with q ′ , when q ′ is close to q; otherwise a second minimizer or a “conjugate point” would imply that γ was not minimizing. Thus √ the function uǫ is smooth in a spacetime ′ ′ neighborhood V of (q, τ ). Put Uǫ (q , τ ) = 2 τ ′ uǫ (q ′ , τ ′ ). By construction, Uǫ ≥ L in V and Uǫ (q, τ ) = L(q, τ ). For small ǫ, (Uǫ )τ ′ + △Uǫ will be bounded above on V by something close to 2n. Hence L satisfies (24.1) globally on M in the barrier sense. As we are assuming bounded curvature on compact time intervals, we can now apply the maximum principle of Appendix A to conclude that the minimum of L(·, τ ) − 2nτ is nonincreasing in τ . (Note that from Lemma 23.1, the minimum of L(·, τ ) − 2nτ exists.) Lemma 24.3. For small positive τ , we have min L(·, τ) − 2nτ < 0. Proof. Consider the static curve at the point p. Then for small τ , we have L(·, τ ) ≤ const. τ 2 , from which the claim follows. (Being a bit more careful with the estimates in the proof of Lemma 23.1, one sees that limτ →0 min L(·, τ) = 0.) Then for τ > 0, we must have min L(·, τ ) ≤ 2nτ , so min l(·, τ ) ≤ n2 . The other sense of a differential inequality is the distributional sense, i.e. △f ≤ g if for every nonnegative compactly-supported smooth function φ on M, Z Z (24.4) (△φ) f dV ≤ φ g dV. M

M

A general fact is that a barrier differential inequality implies a distributional differential inequality [37, 67]. We illustrate this by giving an alternative proof that V˜ (τ ) is nonincreasing in τ . From (18.13), (18.14) and (21.2), one finds that in the barrier sense (and hence in the distributional sense as well) n (24.5) lτ − △l + |∇l|2 − R + ≥ 0 2τ

46


or, equivalently, that n

(∂τ − △) τ − 2 e−l dV

(24.6)

≤ 0.

Then for all nonnegative φ ∈ Cc∞ (M) and 0 < τ 1 ≤ τ 2 , one obtains (24.7) Z

−n τ2 2

φ M Z τ2 Z

Z

τ1 τ2 τ1

Z

−l(·,τ 2 )

e

n

M

n

−n 2

e−l(·,τ 1 ) dV (τ 1 ) = M Z τ2 Z n −l(·,τ ) e dV (τ ) dτ + (△φ) τ − 2 e−l(·,τ ) dV (τ ) dτ ≤

dV (τ 2 ) −

φ (∂τ − △) τ − 2

Z

φ τ1

τ1

M

(△φ) τ − 2 e−l(·,τ ) dV (τ ) dτ. M

We can find a sequence {φi }∞ i=1 of such functions φ with range [0, 1] so that φi is one on B(p, i), vanishes outside of B(p, i2 ), and supM |△φi | ≤ i−1 , uniformly in τ ∈ [τ 1 , τ 2 ]. Then to finish the argument it suffices to have a good upper bound on e−l(·,τ ) in terms of d(p, ·), uniformly in τ ∈ [τ 1 , τ 2 ]. This is given by Lemma 23.1. The monotonicity of V˜ follows. We also note the equation

(24.8)

2△l − |∇l|2 + R +

l−n ≤ 0, τ

which follows from (18.14) and (21.2). Finally, suppose that the Ricci flow exists on a time interval τ ∈ [0, τ0 ]. From the maximum principle of Appendix A, R(·, τ ) ≥ − 2(τ0n−τ ) . Then one obtains a lower bound on l as before, using the better lower bound for R. 25. I.7.2. Estimates on the reduced length In this section we suppose that our solution has nonnegative curvature operator. We use this to derive estimates on the reduced length l. We refer to Appendix F for Hamilton’s differential Harnack inequality. We consider a Ricci flow defined on a time interval t ∈ [0, τ0 ], with bounded nonnegative curvature operator, and put τ = τ0 − t. The differential Harnack inequality gives the nonnegativity of the expression in (F.4). Comparing this with the formula for H(X, Y ) in (20.7), we can write the nonnegativity as Ric(Y, Y ) Ric(Y, Y ) + ≥ 0, (25.1) H(X, Y ) + τ τ0 − τ or 1 1 (25.2) H(X, Y ) ≥ − Ric(Y, Y ) + . τ τ0 − τ

Then

(25.3)

H(X) ≥ − R

1 1 + τ τ0 − τ

.


47

As long as τ ≤ (1 − c) τ0 , equations (18.3) and (18.12) give Z τ 4 2 4τ |∇l| = − 4τ R + 4l − √ (25.4) τ˜3/2 H(X) d˜ τ τ 0 Z τ 4 1 1 3/2 ≤ − 4τ R + 4l + √ τ˜ R + d˜ τ τ 0 τ˜ τ0 − τ˜ Z τ√ τ0 4 τ˜R = − 4τ R + 4l + √ d˜ τ τ 0 τ0 − τ˜ Z τ√ 4 ≤ − 4τ R + 4l + √ τ˜R d˜ τ c τ 0 8l ≤ − 4τ R + 4l + , c where the last line uses (15.1). Thus Cl (25.5) |∇l|2 + R ≤ τ for a constant C = C(c). One shows similarly that 1 d ln |Y |2 ≤ (Cl + 1) (25.6) dτ τ for an L-Jacobi field Y , using (22.4). 26. I.7.3. The no local collapsing theorem II In this section we use the reduced volume to prove a no-local-collapsing theorem for a Ricci flow on a finite time interval. Definition 26.1. We now say that a Ricci flow solution g(·) defined on a time interval [0, T ) is κ-noncollapsed on the scale ρ if for each r < ρ and all (x0 , t0 ) ∈ M × [0, T ) with t0 ≥ r 2 , whenever it is true that | Rm(x, t)| ≤ r −2 for every x ∈ Bt0 (x0 , r) and t ∈ [t0 − r 2 , t0 ], then we also have vol(Bt0 (x0 , r)) ≥ κrn . Definition 26.1 differs from Definition 13.14 by the requirement that the curvature bound holds in the entire parabolic region Bt0 (x0 , r) × [t0 − r 2 , t0 ] instead of just on the ball Bt0 (x0 , r) in the final time slice. Therefore a Ricci flow which is κ-noncollapsed in the sense of Definition 13.14 is also κ-noncollapsed in the sense of Definition 26.1. Theorem 26.2. Given numbers n ∈ Z+ , T < ∞ and ρ, K, c > 0, there is a number κ = κ(n, K, c, ρ, T ) > 0 with the following property. Let (M n , g(·)) be a Ricci flow solution defined on a time interval [0, T ) with T < ∞, such that the curvature | Rm | is bounded on every compact subinterval [0, T ′ ] ⊂ [0, T ). Suppose that (M, g(0)) is a complete Riemannian manifold with | Rm | ≤ K and inj(M, g(0)) ≥ c > 0. Then the Ricci flow solution is κ-noncollapsed on the scale ρ, in the sense of Definition 26.1. Furthermore, with the other constants fixed, we can take κ to be nonincreasing in T . Proof. We first observe that the existence of L-geodesics and the monotonicity of the reduced volume are valid in this setting; see Section 17.

48


Suppose that the theorem were false. Then for given T < ∞ and ρ, K, c > 0, there are : 1. A sequence {(Mk , gk (·))}∞ k=1 of Ricci flow solutions, each defined on the time interval [0, T ), with | Rm | ≤ K on (Mk , gk (0)) and inj(Mk , gk (0)) ≥ c, 2. Spacetime points (pk , tk ) ∈ Mk × [0, T ) and 3. Numbers rk ∈ (0, ρ) having the following property : tk ≥ rk2 and if we put Bk = Btk (pk , rk ) ⊂ Mk then 1 | Rm |(x, t) ≤ rk−2 whenever x ∈ Bk and t ∈ [tk − rk2 , tk ], but ǫk = rk−1 vol(Bk ) n → 0 as k → ∞. From short-time curvature estimates along with the assumed bounded geometry at time zero, there is some t > 0 so that we have uniformly bounded geometry on the time interval [0, t]. In particular, we may assume that each tk is greater than t. We define V˜k using curves going backward in real time from the basepoint (pk , tk ), i.e. forward in τ -time from τ = 0. The first step is to show that V˜k (ǫk rk2 ) is small. Note that τ = ǫk rk2 corresponds to a real time of tk − ǫk rk2 , which is very close to tk . Given an L-geodesic γ(τ ) with γ(0) = pk and velocity vector X(τ ) = dγ , its initial dτ √ −1/2 vector is v = limτ →0 τ X(τ ) ∈ Tpk Mk . We first want to show that if |v| ≤ .1 ǫk then 2 γ does not escape from Bk in time ǫk rk . We have (26.3)

d hX(τ ), X(τ )i = 2 Ric(X, X) + 2hX, ∇X Xi dτ = 2 Ric(X, X) + hX, ∇R − = −

1 X − 4 Ric(X, ·)i τ

|X|2 − 2 Ric(X, X) + hX, ∇Ri, τ

so d τ |X|2 = − 2 τ Ric(X, X) + τ hX, ∇Ri. dτ Letting C denote a generic n-dependent constant, for x ∈ B(pk , rk /2) and t ∈ [tk − rk2 /2, tk ], the fact that gk satisfies the Ricci flow gives an estimate |∇R|(x, t) ≤ Crk−3 , as follows from the case l = 0, m = 1 of Appendix D. Then in terms of dimensionless variables, d 2 τ |X| ≤ C τ |X|2 + C (τ /rk2 )1/2 (τ |X|2 )1/2 . (26.5) 2 d(τ /rk )

(26.4)

Equivalently,

(26.6)

1/2 √ √ d τ |X| . ≤ C τ |X| + C τ /rk2 2 d(τ /rk )

Let us rewrite this as 1/2 d 1√ 1√ τ 2 2 2 ǫ τ |X| ≤ C ǫk ǫk τ |X| + C ǫk . (26.7) d( ǫkτr2 ) k ǫk rk2 k

We are interested in the time range when ǫkτr2 ∈ [0, 1] and the initial condition satisfies k 1 √ 2 limτ →0 ǫk τ |X|(τ ) ≤ .1. Then because of the ǫk -factors on the right-hand side, it follows


49

1 √ from (26.7) that for large k, we will have ǫk2 τ |X|(τ ) ≤ .11 for all τ ∈ [0, ǫk rk2 ]. Next, Z ǫk r 2 Z ǫk r 2 1 Z ǫk r 2 k k k dτ √ dτ − 12 − 12 2 √ = .22 rk . (26.8) |X(τ )| dτ = ǫk ǫk τ |X|(τ ) √ ≤ .11 ǫk τ τ 0 0 0

From the Ricci flow equation gτ = 2 Ric, it follows that the metrics g(τ ) between τ = 0 and τ = ǫk rk2 are eCǫk -biLipschitz close to each other. Then for ǫk small, the length of γ, as measured with the metric at time tk , will be at most .3 rk . This shows that γ does not leave Bk within time ǫk rk2 . 1

˜k (ǫk r 2 ) coming from vectors v ∈ Tp Mk with |v| ≤ .1 ǫ− 2 is at Hence the contribution to V k k k R n 2 most Bk (ǫk rk2 )− 2 e− l(q,ǫk rk ) dq. We now want to give a lower bound on l(q, ǫk rk2 ) for q ∈ Bk . Given the L-geodesic γ : [0, ǫk rk2 ] → Mk with γ(0) = pk and γ(ǫk rk2 ) = q, we have Z ǫk rk2 Z ǫk rk2 3 √ √ 2 (26.9) L(γ) ≥ τ R(γ(τ )) dτ ≥ − τ n(n − 1) rk−2 dτ = − n(n − 1) ǫk2 rk . 3 0 0

Then

l(q, ǫk rk2 ) ≥ −

(26.10)

1 n(n − 1) ǫk . 3 1

− Thus the contribution to V˜k (ǫk rk2 ) coming from vectors v ∈ Tpk Mk with |v| ≤ .1 ǫk 2 is at most

(26.11)

e

1 3

n(n−1) ǫk

n (ǫk rk2 )− 2

vol

tk −ǫk rk2

n

(Bk ) ≤ e

1 3

n(n−1) ǫk

const.

e

1 r2 k

ǫk rk2

n

ǫk2 ,

which is less than 2ǫk2 for large k. 1

− To estimate the contribution to V˜k (ǫk rk2 ) coming from vectors v ∈ Tpk Mk with |v| > .1ǫk 2 , we can use the previously-shown monotonicity of the integrand in τ . As τ → 0, the Euclidean 2 calculation of Section 16 shows that τ −n/2 e− l(Lexpτ (v),τ ) J (v, τ ) → 2n e− |v| . Then for all τ > 0 and all v ∈ Ωτ , 2

τ −n/2 e− l(Lτ (v),τ ) J (v, τ ) ≤ 2n e− |v| ,

(26.12) giving (26.13) Z

τ −1/2

Tpk Mk −B(0,.1 ǫk

)

−n/2 − l(Lτ (v),τ )

e

n

J(v, τ ) χτ d v ≤ 2

n

Z

2

−1/2

Tpk Mk −B(0,.1 ǫk

)

−

e− |v| dn v ≤ e

1 10ǫk

for k large. The conclusion is that limk→∞ V˜k (ǫk rk2 ) = 0. We now claim that there is a uniform positive lower bound on V˜k (tk ). To estimate V˜k (tk ) (where τ = tk corresponds to t = 0), we choose a point qk at time t = 2t , i.e. at τ = tk − 2t , for which l(qk , tk − t/2) ≤ n2 ; see Section 24. Then we consider (k) (k) the concatenation of a fixed curve γ1 : [0, tk − t/2] → Mk , having γ1 (0) = pk and (k) (k) (k) γ1 (tk − t/2) = qk , with a fan of curves γ2 : [tk − t/2, tk ] → Mk having γ2 (tk − t/2) = qk . Because of the uniformly bounded geometry in the spacetime region with t ∈ [0, t/2], we

50


can get an upper bound in this way for l(·, tk ) in a region around qk . Integrating e−l(·,tk ) , we get a positive lower bound on V˜k (tk ) that is uniform in k. As ǫk rk2 → 0, the monotonicity of V˜ implies that V˜k (tk ) ≤ V˜k (ǫk rk2 ) for large k, which is a contradiction. 27. I.8.3. Length distortion estimates The distortion of distances under Ricci flow can be estimated in terms of the Ricci tensor. We first mention a crude estimate. Lemma 27.1. If Ric ≤ (n − 1) K then for t1 > t0 , (27.2)

distt1 (x0 , x1 ) ≥ e−(n−1)K(t1 −t0 ) . distt0 (x0 , x1 )

Proof. For any curve γ : [0, a] → M, we have Z a s Z a d dγ dγ d dγ dγ ds dγ ≥ − (n − 1) K L(γ). (27.3) L(γ) = , ds = − Ric , dt dt 0 ds ds ds ds 0 ds

Integrating gives (27.4)

L(γ) t1 ≥ e−(n−1)K(t1 −t0 ) . L(γ) t0

The lemma follows by taking γ to be a minimal geodesic at time t1 between x0 and x1 . Remark 27.5. By a similar argument, if Ric ≥ − (n − 1) K then for t1 > t0 , (27.6)

distt1 (x0 , x1 ) ≤ e(n−1)K(t1 −t0 ) . distt0 (x0 , x1 )

We can write the conclusion of Lemma 27.1 as (27.7)

d distt (x0 , x1 ) ≥ − (n − 1)K distt (x0 , x1 ), dt

where the derivative is interpreted in the sense of forward difference quotients. The estimate in Lemma 27.1 is multiplicative. We now give an estimate that is additive in the distance. Lemma 27.8. (cf. Lemma I.8.3(b)) Suppose distt0 (x0 , x1 ) ≥ 2r0 , and Ric(x, t0 ) ≤ (n−1)K for all x ∈ Bt0 (x0 , r0 ) ∪ Bt0 (x1 , r0 ). Then 2 d −1 distt (x0 , x1 ) ≥ − 2 (n − 1) K r0 + r 0 (27.9) dt 3 at time t = t0 .


51

Proof. If γ is a normalized minimal geodesic from x0 to x1 with velocity field X(s) = dγ ds then for any piecewise-smooth normal vector field V along γ that vanishes at the endpoints, the second variation formula gives Z

(27.10)

d(x0 ,x1 )

0

2 ∇X V + hR(V, X)V, Xi ds ≥ 0.

Let {ei (s)}n−1 i=1 be a parallel orthonormal frame along γ that is perpendicular to X. Put Vi (s) = f (s) ei (s), where  s   r0 f (s) = 1   d(x0 ,x1 )−s

(27.11)

r0

Then ∇X Vi = |f ′ (s)| and Z

(27.12)

d(x0 ,x1 )

0

Next,

(27.13)

Z

if 0 ≤ s ≤ r0 , if r0 ≤ s ≤ d(x0 , x1 ) − r0 , if d(x0 , x1 ) − r0 ≤ s ≤ d(x0 , x1 ).

Z 2 ∇X Vi ds = 2

0

d(x0 ,x1 )

hR(Vi , X)Vi , Xi ds =

0

r0

Z

r0

0

Z

2 1 ds = . 2 r0 r0

s2 hR(ei , X)ei , Xi ds + r02

d(x0 ,x1 )−r0

r0 Z d(x0 ,x1 )

d(x0 ,x1 )−r0

hR(ei , X)ei, Xi ds + (d(x0 , x1 ) − s)2 hR(ei , X)ei , Xi ds. r02

Then

(27.14)

0 ≤

n−1 Z X i=1

d(x0 ,x1 ) 0

2 ∇X Vi + hR(Vi , X)Vi, Xi ds

Z d(x0 ,x1 ) Z r0 2(n − 1) s2 − Ric(X, X) ds + 1 − 2 Ric(X, X) ds + = r0 r0 0 0 Z d(x0 ,x1 ) (d(x0 , x1 ) − s)2 1− Ric(X, X) ds. r02 d(x0 ,x1 )−r0

52


This gives (27.15)

d distt (x0 , x1 ) = − dt

Z

d(x0 ,x1 )

Ric(X, X) ds Z r0 2(n − 1) s2 ≥ − − 1 − 2 Ric(X, X) ds r0 r0 0 Z d(x0 ,x1) (d(x0 , x1 ) − s)2 Ric(X, X) ds − 1− r02 d(x0 ,x1 )−r0

≥ −

0

2 2(n − 1) − 2(n − 1)K · r0 , r0 3

which proves the lemma.

We now give an additive version of Lemma 27.1. Corollary 27.16. [33, Theorem 17.2] If Ric ≤ K with K > 0 then for all x0 , x1 ∈ M, (27.17)

d distt (x0 , x1 ) ≥ − const.(n) K 1/2 . dt

Proof. Put r0 = K −1/2 . If distt (x0 , x1 ) ≤ 2r0 then the corollary follows from (27.7). If distt (x0 , x1 ) > 2r0 then it follows from Lemma 27.8. The proof of the next lemma is similar to that of Lemma 27.8 and is given in I.8. Lemma 27.18. (cf. Lemma I.8.3(a)) Suppose that Ric(x, t0 ) ≤ (n − 1) K on Bt0 (x0 , r0 ). Then the distance function d(x, t) = distt (x, x0 ) satisfies 2 −1 (27.19) dt − △d ≥ − (n − 1) K r0 + r 0 3

at time t = t0 , outside of Bt0 (x0 , r0 ). The inequality must be understood in the barrier sense (see Section 24) if necessary. 28. I.8.2. No local collapsing propagates forward in time and to larger scales This section is concerned with a localized version of the no-local-collapsing theorem. The main result, Theorem 28.2, says that noncollapsing propagates forward in time and to a larger distance scale. We first give a local version of Definition 26.1. Definition 28.1. (cf. Definition of I.8.1) A Ricci flow solution is said to be κ-collapsed at (x0 , t0 ), on the scale r > 0, if | Rm |(x, t) ≤ r −2 for all (x, t) ∈ Bt0 (x0 , r) × [t0 − r 2 , t0 ], but vol(Bt0 (x0 , r 2 )) ≤ κrn . Theorem 28.2. (cf. Theorem I.8.2) For any 0 < A < ∞, there is some κ = κ(A) > 0 with the following property. Let g(·) be a Ricci flow solution defined for t ∈ [0, r02], having complete time slices and uniformly bounded sectional curvature.. Suppose that vol(B0 (x0 , r0 )) ≥


53

A−1 r0n and that | Rm |(x, t) ≤ nr1 2 for all (x, t) ∈ B0 (x0 , r0 )×[0, r02 ]. Then the solution cannot 0 be κ-collapsed on a scale less than r0 at any point (x, r02 ) with x ∈ Br02 (x0 , Ar0 ). Remark 28.3. In [51, Theorem I.8.2] the assumption is that | Rm |(x, t) ≤ r0−2 . We make the slightly stronger assumption that | Rm |(x, t) ≤ nr1 2 . The extra factor of n is needed in order 0 1 to assert in the proof that the region {(y, t) : dist 1 (y, x0 ) ≤ 10 , t ∈ [0, 12 ]} has bounded 2 geometry; see below. Clearly this change of hypothesis does not make any substantial difference in the sequel. Proof. We follow the lines of the proof of Theorem 26.2. By scaling, we can take r0 = 1. Choose x ∈ M with dist1 (x, x0 ) < A. Define the reduced volume V˜ (τ ) by means of curves starting at (x, 1). An effective lower bound on V˜ (1) would imply that the solution is not κ-collapsed at (x, 1), on a scale less than 1, for an appropriate κ > 0. 1 We first note that the geometry of the region {(y, t) : dist 1 (y, x0 ) ≤ 10 , t ∈ [0, 12 ]} is 2 uniformly bounded. To see this, the upper sectional curvature bound implies that Ric ≤ 1, 1 so the distance distortion estimate of Section 27 implies that B 1 (x0 , 10 ) ⊂ B0 (x0 , 1). In 2 1 1 and particular, | Rm |(y, t) ≤ n on the region. By Remark 27.5, if dist 1 (y, x0 ) ≤ 10 2 1 1 1 t ∈ [0, 2 ] then B0 (y, 1000 ) ⊂ Bt (y, 100 ). The Bishop-Gromov inequality gives a lower bound 1 1 for the time-zero volume of B0 (y, 1000 ), of the form vol0 (B0 (y, 1000 )) ≥ C1 (n, A). The Ricci 1 flow equation then gives a lower bound for the time-t volume of B0 (y, 1000 ), of the form 1 1 1 volt (B0 (y, 1000 )) ≥ C2 (n, A). Thus the time-t volume of Bt (y, 100 ) satisfies vol(Bt (y, 100 )) ≥ C2 (n, A). This, along with the uniform sectional curvature bound, implies that the region has uniformly bounded geometry.

If we have an effective upper bound on miny l(y, 12 ), where y ranges over points that 1 , then we obtain a lower bound on V˜ (1). Thus it suffices to obtain satisfy dist 1 (y, x0 ) ≤ 10 2 an effective upper bound on miny l(y, 21 ) or, equivalently, on miny L(y, 12 ) (as defined using 1 L-geodesics from (x, 1)) for y satisfying dist 1 (y, x0) ≤ 10 . Applying the maximum principle 2 to (24.1) gave an upper bound on inf M L. The idea is to spatially localize this estimate near x0 , by means of a radial function φ. 1 1 Let φ = φ(u) be a smooth function that equals 1 on (−∞, 20 ), equals infinity on ( 10 , ∞) 1 1 and is increasing on ( 20 , 10 ), with

2(φ′ )2 /φ − φ′′ ≥ (2A + 100n)φ′ − C(A)φ

(28.4)

for some constant C(A) < ∞. To satisfy (28.4), it suffices to take φ(u) =

for u near

1 . 10

1

1

e(2A+100n)( 10 −u) −1

We claim that L + 2n + 1 ≥ 1 for t ≥ 12 . To see this, from the end of Section 24, n (28.5) R(·, τ ) ≥ − . 2(1 − τ )

Then for τ ∈ [0, 12 ], (28.6)

L(q, τ ) ≥ −

Z

0

τ

√

n τ dτ ≥ − n 2(1 − τ )

Z

0

τ

√

τ dτ = −

2n 3/2 τ . 3

54


Hence (28.7)

L(q, τ ) = 2

√ 4n n τ L(q, τ ) ≥ − τ 2 ≥ − , 3 3

which proves the claim. Now put h(y, t) = φ(d(y, t) − A(2t − 1)) (L(y, 1 − t) + 2n + 1),

(28.8)

where d(y, t) = distt (y, x0 ). It follows from the above claim that h(y, t) ≥ 0 if t ≥ 21 . Also, min h(y, 1) ≤ h(x, 1) = φ(dist1 (x, x0 ) − A) · (2n + 1) = 2n + 1.

(28.9)

y

1 As φ is infinite on ( 10 , ∞) and L(·, 21 ) + 2n + 1 ≥ 1, the minimum of h(·, 12 ) is achieved at 1 . some y satisfying d(y, 21 ) ≤ 10

The calculations in I.8 give

h ≥ −(2n + C(A))h

(28.10)

at a minimum point of h, where = ∂t − △. Then dtd hmin (t) ≥ −(2n + C(A)) hmin (t), so C(A) C(A) 1 ≤ en+ 2 hmin (1) ≤ (2n + 1) en+ 2 . (28.11) hmin 2 It follows that C(A) 1 L(y, ) + 2n + 1 ≤ (2n + 1) en+ 2 . (28.12) min 1 1 2 y : d(y, 2 )≤ 10 This implies the theorem.

29. I.9. Perelman’s differential Harnack inequality This section is concerned with a localized version of the W-functional. It is mainly used in I.10. Let g(·) be a Ricci flow solution on a manifold M, defined for t ∈ (a, b). Put = ∂t − △. For f1 , f2 ∈ Cc∞ ((a, b) × M), we have Z b Z d (29.1) f1 (t, x)f2 (t, x) dV dt 0 = a dt M Z bZ Z bZ = ((∂t − △)f1 ) f2 dV + f1 (∂t + △ − R)f2 dV a M a M Z bZ Z bZ = (f1 ) f2 dV − f1 ∗ f2 dV, a

∗

M

a

M ∗

where = − ∂t − △ + R. In this sense, is the formal adjoint to . Now suppose that the Ricci flow is defined for t ∈ [0, T ). Suppose that

(29.2)

satisfies ∗ u = 0. Put (29.3)

u = (4π(T − t))−

n 2

e−f

v = [(T − t)(2△f − |∇f |2 + R) + f − n] u.


If M is compact then using (5.10), W(gij , f, T − t) =

(29.4)

Z

55

v dV.

M

Proposition 29.5. (cf. Proposition I.9.1) (29.6) ∗ v = − 2(T − t) Rij + ∇i ∇j f −

gij 2 u. 2(T − t)

Proof. We note that the right-hand side of I.(9.1) should be multiplied by u. To prove the proposition, we first claim that d△ = 2 Rij ∇i ∇j . (29.7) dt To see this, for f1 , f2 ∈ Cc∞ (M), we have Z Z (29.8) f1 △f2 dV = − hdf1 , df2 i dV. M

M

Differentiating with respect to t gives Z Z Z Z d△ f2 dV − f1 △f2 R dV = − 2 Ric(df1 , df2) dV + hdf1 , df2 i R dV, (29.9) f1 dt M M M M so d△ (29.10) f2 − R △f2 = 2 ∇i (Rij ∇j f2 ) − ∇i (R ∇i f2 ). dt Then (29.7) follows from the traced second Bianchi identity. Next, one can check that ∗ u = 0 is equivalent to n 1 + |∇f |2 − R. (29.11) (∂t + △) f = 2 T −t Then one obtains (29.12) u−1 ∗ v = − (∂t + △) (T − t) (2△f − |∇f |2 + R) + f − 2h∇ (T − t) (2△f − |∇f |2 + R) + f , u−1 ∇ui

= 2△f − |∇f |2 + R − (T − t) (∂t + △) (2△f − |∇f |2 + R)

− (∂t + △) f + 2 (T − t) h∇(2△f − |∇f |2 + R), ∇f i + 2 |∇f |2.

Now (29.13) (∂t + △) (2△f − |∇f |2 + R) = 2(∂t △)f + 2△ (∂t + △) f

− (∂t + △) |∇f |2 + (∂t + △) R

= 4 Rij ∇i ∇j f + 2 △ (|∇f |2 − R) − 2 Ric(df, df )

− 2h∇ft , ∇f i − △|∇f |2 + △R + 2 | Ric |2 + △R

= 4 Rij ∇i ∇j f + 2 △ |∇f |2 − 2 Ric(df, df )

− 2h∇(− △f + |∇f |2 − R), ∇f i − △|∇f |2 + 2 | Ric |2 .

56


Hence the term in u−1 ∗ v proportionate to (T − t)−1 is n 1 (29.14) − . 2 T −t The term proportionate to (T − t)0 is

(29.15)

2△f − |∇f |2 + R − |∇f |2 + R + 2|∇f |2 = 2(△f + R).

The term proportionate to (T − t) is (T − t) times (29.16)

− 4 Rij ∇i ∇j f − 2 △ |∇f |2 + 2 Ric(df, df ) +

2h∇(− △f + |∇f |2 − R), ∇f i + △|∇f |2 − 2 | Ric |2 +

2 h∇(2△f − |∇f |2 + R), ∇f i =

− 4 Rij ∇i ∇j f − △ |∇f |2 + 2 Ric(df, df ) + 2h∇△f, ∇f i − 2 | Ric |2 = − 4 Rij ∇i ∇j f − 2 | Hess(f )|2 − 2 | Ric |2 .

Putting this together gives (29.17)

v = − 2 (T − t) Rij + ∇i ∇j f − ∗

This proves the proposition.

As a consequence of Proposition 29.5, Z d d (29.18) W(gij , f, T − t) = v dV dt dt M Z = 2 (T − t)

2 1 gij u. 2(T − t)

Z

= (∂t + △ − R)v dV M 2 1 gij u dV. Rij + ∇i ∇j f − 2(T − t) M

In this sense, Proposition 29.5 is a local version of the monotonicity of W.

Corollary 29.19. (cf. Corollary I.9.2) If M is closed, or whenever the maximum principle holds, then max v/u is nondecreasing in t. Proof. We note that the statement of Corollary I.9.2 should have max v/u instead of min v/u. To prove the corollary, we have v∗ u − u∗ v 2 D vE v − ∇u, ∇ . (29.20) (∂t + △) = u u2 u u As ∗ u = 0 and ∗ v ≤ 0, the corollary now follows from the maximum principle.

We now assume that the Ricci flow solution is defined on the closed interval [0, T ]. Corollary 29.21. (cf. Corollary I.9.3) Under the same assumptions, if the solution is defined for t ∈ [0, T ] and u tends to a δ-function as t → T then v ≤ 0 for all t < T . Proof. Suppose that h is a positive solution of h = 0. Then Z Z Z d ∗ (29.22) hv dV = ((h) v − h v) dV = − h ∗ v dV ≥ 0. dt M M M


57

R As t → T , the computation of M hv approaches the flat-space calculation, which one finds to be zero; see [50] for details. (Strictly speaking, the paper [50] deals with the case when M R is closed. It is indicated that the proof should extend to the noncompact setting.) Thus h(t0 )v(t0 ) dV is nonpositive for all t0 < T . As h(t0 ) can be taken to be an arbitrary M positive function, and then flowed forward to a positive solution of h = 0, it follows that v(t0 ) ≤ 0 for all t0 < T . The next result compares the function f used in the W-functional and the function l used in the reduced volume. Corollary 29.23. (cf. Corollary I.9.5) Under the assumptions of the previous corollary, let p ∈ M be the point where the limit δ-function is concentrated. Then f (q, t) ≤ l(q, T − t), where l is the reduced distance defined using curves starting from (p, T ). Proof. Equation (24.6) implies that ∗ (4πτ )−n/2 e−l ≤ 0. (This corrects the statement at ∗ −n/2 −f the top of page 23 of I.) From this and the fact that (4πτ ) e = 0, the argument of the proof of Corollary 29.19 gives that max ef −l is nondecreasing in t, so max(f − l) is nondecreasing in t. As t → T one obtains the flat-space result, namely that f − l vanishes. Thus f (t) ≤ l(T − t) for all t ∈ [0, T ). Remark 29.24. To give an alternative proof of Corollary 29.23, putting τ = T − t, Corollary I.9.4 of [51] says that for any smooth curve γ, 2 d 1 1 f (γ(τ ), τ ) ≤ f (γ(τ ), τ ), (29.25) R(γ(τ ), τ ) + γ(t) ˙ − dτ 2 2τ or 2 1 1/2 d 1/2 τ f (γ(τ ), τ ) ≤ τ R(γ(τ ), τ ) + γ(t) ˙ . (29.26) dτ 2 Take γ to be a curve emanating from (p, T ). For small τ , (29.27)

f (γ(τ ), τ ) ∼ d(p, γ(τ ))2 /4τ = O(τ 0 ).

Then integration gives τ 1/2 f ≤

1 2

L, or f ≤ l.

30. The statement of the pseudolocality theorem The next theorem says that, in a localized sense, if the initial data of a Ricci flow solution has a lower bound on the scalar curvature and satisfies an isoperimetric inequality close to that of Euclidean space then there is a sectional curvature bound in a forward region. The result is not used in the sequel. Theorem 30.1. (cf. Theorem I.10.1) For every α > 0 there exist δ, ǫ > 0 with the following property. Suppose that we have a smooth pointed Ricci flow solution (M, (x0 , 0), g(·)) defined for t ∈ [0, (ǫr0 )2 ], such that each time slice is complete. Suppose that for any x ∈ B0 (x0 , r0 ) and Ω ⊂ B0 (x0 , r0 ), we have R(x, 0) ≥ −r0−2 and vol(∂Ω)n ≥ (1 − δ) cn vol(Ω)n−1 , where cn is the Euclidean isoperimetric constant. Then | Rm |(x, t) < αt−1 + (ǫr0 )−2 whenever 0 < t ≤ (ǫr0 )2 and d(x, t) = distt (x, x0 ) ≤ ǫr0 .

58


The sectional curvature bound | Rm |(x, t) < αt−1 + (ǫr0 )−2 necessarily blows up as t → 0, as nothing was assumed about the sectional curvature at t = 0. We first sketch the idea of the proof of Theorem 30.1. It is an argument by contradiction. One takes a Ricci flow solution that satisfies the assumptions and picks a point (x, t) where the desired curvature bound does not hold. One can assume, roughly speaking, that (x, t) is the first point in the given solution where the bound does not hold. (This will give the curvature bound needed for taking a limit in a sequence of counterexamples.) One now considers the solution u to the conjugate heat equation, starting as a δ-function at (x, t), and the corresponding function v. We know that v ≤ 0. The first goal is to get a negative upper bound for the integral of v over an appropriate ball B at a time e t near t; see Section 33. The argument to get such a bound is by contradiction. If there were R not such a bound then one could consider a rescaled sequence of counterexamples with B v dV → 0, and try to take a limit. If one has the R injectivity radius bounds needed to take a limit then one obtains a limit solution with B v dV = 0, which implies that the limit solution is a gradient shrinking soliton, which violates curvature assumptions. If one doesn’tRhave the injectivity radius bounds then one can do a further rescaling to see that in fact B v dV → −∞ for some subsequence, which is a contradiction. R If M is compact then M v dV is monotonically nondecreasing in t. As (29.6) is a localized version of this statement, whether M is compact or noncompact we can use a cutoff R function Rh and equation (29.6) to get a negative upper bound on M hv dV at time t = 0. Finally, M v dV is the expression that appears in the logarithmic Sobolev inequality. If the isoperimetric constant is sufficiently close to the Euclidean value cn then one concludes R that M hv dV must be bounded below by a constant close to zero, which contradicts the R negative upper bound on M hv dV . 31. Claim 1 of I.10.1. A point selection argument

In Theorem 30.1, we can assume that r0 = 1 and α < | Rm(x, t)| ≥ αt−1 }.

1 . 100n

Fix α and put Mα = {(x, t) :

The next lemma says that if we have a point (x, t) where the conclusion of Theorem 30.1 −1 does not hold then there is another point (x, t) with | Rm(x, t)| large (relative to t ) so that any other such point (x′ , t′ ) either has t′ > t or is much farther from x0 than x is. Lemma 31.1. (cf. Claim 1 of I.10.1) For any A > 0, if g(·) is a Ricci flow solution for 1 t ∈ [0, ǫ2 ], with Aǫ < 100n , and | Rm |(x, t) ≥ αt−1 + ǫ−2 for some (x, t) satisfying t ∈ (0, ǫ2 ] and d(x, t) ≤ ǫ, then one can find (x, t) ∈ Mα with t ∈ (0, ǫ2 ] and d(x, t) < (2A + 1)ǫ, such that | Rm(x′ , t′ )| ≤ 4 | Rm(x, t)|

(31.2) whenever (31.3)

(x′ , t′ ) ∈ Mα ,

t′ ∈ (0, t],

1

d(x′ , t′ ) ≤ d(x, t) + A| Rm |− 2 (x, t).

Proof. The proof is by a point selection argument as in Appendix H. By assumption, there is a point (x, t) satisfying t ∈ (0, ǫ2 ], d(x, t) ≤ ǫ and | Rm(x, t)| ≥ αt−1 + ǫ−2 . Clearly (x, t) ∈ Mα . Define points (xk , tk ) inductively as follows. First, (x1 , t1 ) = (x, t). Next,


59

suppose that (xk , tk ) is constructed but cannot be taken for (x, t). Then there is some point 1 (xk+1 , tk+1 ) ∈ Mα such that 0 < tk+1 ≤ tk , d(xk+1 , tk+1) ≤ d(xk , tk ) + A| Rm |− 2 (xk , tk ) and | Rm |(xk+1 , tk+1 ) > 4| Rm |(xk , tk ). Continuing in this way, the point (xk , tk ) constructed has | Rm |(xk , tk ) ≥ 4k−1 | Rm |(x1 , t1 ) ≥ 4k−1 ǫ−2 . Then (31.4)

1

1

d(xk , tk ) ≤ d(x1 , t1 ) + A| Rm |− 2 (x1 , t1 ) + . . . + A| Rm |− 2 (xk−1 , tk−1 ) 1

≤ ǫ + 2A| Rm |− 2 (x1 , t1 ) ≤ (2A + 1)ǫ.

As the solution is smooth, the induction process must terminate after a finite number of steps and the last value (xk , tk ) can be taken for (x, t). 32. Claim 2 of I.10.1. Getting parabolic regions In Lemma 31.1, we know that (31.2) is satisfied under the condition (31.3). The spacetime region described in (31.3) is not a product region, due to the fact that d(x, t) is timedependent. The next goal is to obtain the estimate (31.2) on a product region in spacetime; this will be necessary when taking limits of Ricci flow solutions. To get the estimate on a product region, one needs to bound how fast distances are changing with respect to t. Lemma 32.1. (cf. Claim 2 of I.10.1) For the point (x, t) constructed in Lemma 31.1, | Rm(x′ , t′ )| ≤ 4 | Rm(x, t)|

(32.2) holds whenever (32.3)

t −

1 αQ−1 ≤ t′ ≤ t, 2

distt (x′ , x) ≤

1 1 AQ− 2 , 10

where Q = | Rm(x, t)|. Proof. We first claim that if (x′ , t′ ) satisfies t− 21 αQ−1 ≤ t′ ≤ t and d(x′ , t′ ) ≤ d(x, t)+AQ−1/2 then | Rm |(x′ , t′ ) ≤ 4Q. To see this, if (x′ , t′ ) ∈ Mα then it is true by Lemma 31.1. If −1 (x′ , t′ ) ∈ / Mα then | Rm |(x′ , t′ ) < α(t′ )−1 . As (x, t) ∈ Mα , we know that Q ≥ αt . Then −1 t′ ≥ t − 21 αQ−1 ≥ 12 t and so | Rm |(x′ , t′ ) < 2 αt ≤ 2Q.

Thus we have a uniform curvature bound on the time-t′ distance ball B(x0 , d(x, t) + AQ−1/2 ), provided that t − 12 αQ−1 ≤ t′ ≤ t. We now claim that the time-t ball 1 B(x0 , d(x, t) + 10 AQ−1/2 ) lies in the time-t′ distance ball B(x0 , d(x, t) + AQ−1/2 ). To see 1 this, applying Lemma 27.8 with r0 = 100 AQ−1/2 and the above curvature bound, if x′ is in 1 AQ−1/2 ) then the time-t ball B(x0 , d(x, t) + 10 (32.4) 2 1 1 ′ −1 −1/2 −1 1/2 ′ distt′ (x0 , x ) − distt (x0 , x ) ≤ αQ · 2(n − 1) · 4Q( AQ ) + 100 A Q . 2 3 100

Assuming that A is sufficiently large (we’ll take A → ∞ later) and using the fact that 1 1 α < 100n , it follows that d(x′ , t′ ) ≤ d(x′ , t) + 12 AQ−1/2 ≤ d(x, t) + AQ− 2 , which is what we want to show. We note that the argument also shows that is indeed self-consistent to use the curvature bounds in the application of Lemma 27.8.

60


Now suppose that (x′ , t′ ) satisfies (32.3). By the triangle inequality, x′ lies in the time-t 1 distance ball B(x0 , d(x, t) + 10 AQ−1/2 ). Then x′ is in the time-t′ distance ball B(x0 , d(x, t) + AQ−1/2 ) and so | Rm(x′ , t′ )| ≤ 4Q, which proves the lemma. 33. Claim 3 of I.10.1. An upper bound on the integral of v We first make some remarks about the fundamental solution to the backward heat equation. Let (M, (x, b), g(·)) be a smooth one-parameter family of complete pointed Riemannian manifolds, parametrized by t ∈ (a, b]. The fundamental solution u of the backward heat equation is a positive solution of ∗ u = 0 on M ×(a, b) such that u(·, t) converges to δx in the distributional sense, as t → b− . It is constructed as follows (cf. [26, Section 3]). Let {Di }∞ i=1 be an exhaustion of M by an increasing sequence of smooth compact codimension-zero submanifolds-with-boundary containing x in the interior. Let u(i) be the unique solution of ∗ u(i) = 0 on Di × (a, b) with limt→b− u(i) (x, t) = δx (x), as constructed using Dirichlet boundary conditions on Di . If Di ⊂ Dj then u(i) ≤ u(j) on Di . Then the fundamental solution is defined to be the limit u = limi→∞ u(i) , with smooth convergence on compact subsets of M × (a, b). The function Ru is independent of the choice of exhaustion sequence R ∞ {Di }i=1 . For any t ∈ (a, b), we have M u(x, t) dV (x) ≤ 1. If M u(x, t) dV (x) = 1 for all t then we say that (M, (x, b), g(·)) is stochastically complete for ∗ . This will be the case if one has bounded curvature on compact time intervals, but need not be the case in general. Lemma 33.1. Let {(Mk , (xk , b), gk (·))}∞ k=1 be a sequence of manifolds as above, each defined on the time interval (a, b]. Suppose that limk→∞ (Mk , (xk , b), gk (·)) = (M∞ , (x∞ , b), g∞ (·)) in the pointed smooth topology, and that (M∞ , (x∞ , b), g∞ (·)) is stochastically complete for ∗ . Then after passing to a subsequence, the fundamental solutions {uk }∞ k=1 converge smoothly on compact subsets of M∞ × (a, b) to the fundamental solution u∞ . (Of course, we use the pointed diffeomorphisms inherent in the statement of pointed convergence in order to compare the uk ’s with u∞ .) Proof. From the uniform upper L1 -bound on {uk (·, t)}∞ k=1 and parabolic regularity, after ∞ passing to a subsequence we can assume that {uk }k=1 converges smoothly on compact subsets of M∞ × (a, b) to some function U. From the construction of u∞ , it follows easily that u∞ ≤ U. For any t ∈ (a, b), we have (33.2) Z

M∞

Z

lim inf (uk (x, t) − u∞ (x, t)) dvol(x) k Z ≤ lim inf (uk (x, t) − u∞ (x, t)) dvol(x) ≤ 0,

(U(x, t) − u∞ (x, t)) dvol(x) =

M∞

k

so U = u∞ .

M∞

Starting the proof of Theorem 30.1, we suppose that the theorem is not true. Then there are sequences ǫk → 0 and δk → 0, and pointed Ricci flow solutions (Mk , (x0,k , 0), gk (·)) which satisfy the hypotheses of the theorem but for which there is a point (xk , tk ) with 0 < tk ≤ ǫ2k , −2 d(xk , tk ) ≤ ǫk and | Rm |(xk , tk ) ≥ αt−1 k + ǫk . Given the flow (Mk , gk (·)), we reduce ǫk as


61

much as possible so that there is still such a point (xk , tk ). Then −2 | Rm |(x, t) < αt−1 k + 2ǫk

(33.3)

1 whenever 0 < t ≤ ǫ2k and d(x, t) ≤ ǫk . Put Ak = 100nǫ . Construct points (xk , tk ) as k n in Lemma 31.1. Consider fundamental solutions uk = (4π(tk − t))− 2 e−fk of ∗ uk = 0 satisfying limt→t− u(x, t) = δxk (x). Construct the corresponding functions vk from (29.3). k

Lemma 33.4. (cf. Claim 3 of I.10.1) There Ris some β > 0 so that for all sufficiently large k, there is some e tk ∈ [tk − 21 αQ−1 k , tk ] with Bk vk dVk ≤ −β, where Qk = | Rm |(xk , tk ) p tk centered at xk . and Bk is the time-e tk ball of radius tk − e

Proof. Suppose that the claim is notR true. After passing to a subsequence, we can assume that for any choice of e tk , lim inf k→∞ Bk vk dVk ≥ 0.

Consider the pointed solution (Mk , (xk , tk ), gk (·)) parabolically rescaled by Qk . Suppose first that there is a subsequence so that the injectivity radii of the scaled metrics at (xk , tk ) are bounded away from zero. Since Ak → ∞, we can use Lemma 32.1 and Appendix E to take a subsequence that converges to a complete Ricci flow solution (M∞ , (x∞ , t∞ ), g∞ (·)) on a time interval (t∞ − 12 α, t∞ ], with | Rm | ≤ 4 and | Rm |(x∞ , t∞ ) = 1. Consider the fundamental solution u∞ of ∗ on M∞ with limt→t−∞ u∞ (x∞ , t) = δx∞ (x∞ ). As before, let uk be the fundamental solution of ∗ on Mk with limt→t− uk (xk , t) = δxk (xk ). In view of the k pointed convergence of the rescalings of (Mk , (xk , tk ), gk (·)) to (M∞ , (x∞ , t∞ ), g∞ (·)), Lemma 33.1 implies that after passing to a further subsequence we can ensure that limk→∞ uk = u∞ , with smooth convergence on compact subsets of M∞ ×(t∞ − 21 α, t∞ ). (The curvature bounds on (M∞ , (x∞ , t∞ ), g∞ (·)) ensure that it is stochastically complete for ∗ .) From Corollary 29.21, v∞ ≤ 0. Note that we are applying Corollary 29.21 on M∞ × (t∞ − 21 α, t∞ ), where we have the curvature bounds needed to use the maximum principle. p t∞ ball of radius t∞ − e t∞ centered at x∞ . Given e t∞ ∈ (t∞ − 12 α, t∞ ), let B∞ be the time-e In view of the smooth Rconvergence limk→∞ uk = u∞ on compact subsets of M∞ × (t∞ − 1 α, t∞ ), it follows that B∞ v∞ dV∞ = 0 at time e t∞ , so v∞ vanishes on B∞ at time e t∞ . Let 2 tR∞ ) a nonnegative nonzero function h be a solution to h = 0 on M∞ × [e t∞ , t∞ ) with h(·, e supported in B∞ . As in the proof of Corollary 29.21, M∞ hv∞ dV∞ is nondecreasing in t R and vanishes for t = e t∞ and t → t∞ . Thus M∞ hv∞ dV∞ vanishes for all t ∈ [e t∞ , t∞ ). However, for t ∈ (e t∞ , t∞ ), h is strictly positive and v∞ is nonpositive. Thus v∞ vanishes on M∞ for all t ∈ (e t∞ , t∞ ), and so 1 (33.5) Ric(g∞ ) + Hess f∞ − g∞ = 0. 2(t − t)

on this interval. We know that | Rm | ≤ 4 on M∞ × (t∞ − 21 α, t∞ ]. From the evolution equation, 1 dg∞ = − 2 Ric(g∞ ) = 2 Hess f∞ − g∞ . (33.6) dt t−t It follows that the supremal and infimal sectional curvatures of g∞ (·, t) go like (t∞ − t)−1 . Hence g∞ is flat, which contradicts the fact that | Rm |(x∞ , t∞ ) = 1.

62


Suppose now that there is a subsequence so that the injectivity radii of the scaled metrics at (xk , tk ) tend to zero. Parabolically rescale (Mk , (xk , tk ), gk (·)) further so that the injectivity radius becomes one. After passing to a subsequence we will have convergence to a flat Ricci flow solution (−∞, 0] × L. The complete flat manifold L can be described as the total space of a flat orthogonal Rm -bundle over a flat compact manifold C. After separating variables, the fundamental solution u∞ on L will be Gaussian in the fiber directions and will decay exponentially fast to a constant in the base directions, i.e. |x|2

1 u(x, τ ) ∼ (4πτ )−m/2 e− 4τ vol(C) , where |x| is the fiber norm. With this for u∞ , one finds √ of radius τ , that v∞ = (m − n) 1 + 12 ln(4πτ ) u∞ . With Bτ the ball around a basepoint R the integral of u over Bτ has a positive limit as τ → ∞, and so limτ →∞ Bτ v∞ dV∞ = − ∞. R , t ] so that lim v dVk = −∞, which is a Then there are times e tk ∈ [tk − 21 αQ−1 k k→∞ k Bk k contradiction.

34. Theorem I.10.1. Proof of the pseudolocality theorem Continuing with the proof of Theorem 30.1, we now use Lemma 33.4 to get a contradiction to a log Sobolev inequality. For simplicity of notation, we drop the subscript k and deal with a particular (Mk , (xk , tk ), gk (·)) for k large. Define a smooth function φ on R which is one on (−∞, 1], decreasing on [1, 2] and zero on [2, ∞], with φ′′ ≥ −10φ′ and (φ′ )2 ≤ 10φ. To construct φ we can take the function which is 1 on (−∞, 1], 1 − 2(x − 1)2 on [1, 3/2], 2(x − 2)2 on [3/2, 2] and 0 on [2, ∞), and smooth it slightly. √ e t) = d(y, t) + 200n t. We claim that if 10Aǫ ≤ d(y, e t) ≤ 20Aǫ then dt (y, t) − Put d(y, 100n 2 e t) ≤ 20Aǫ and A is △d(y, t) + √ ≥ 0. To see this, recalling that t ∈ [0, ǫ ], if 10Aǫ ≤ d(y, t

sufficiently large then 9Aǫ ≤ √ d(y, t) ≤ 21Aǫ. We apply Lemma 27.18 with the parameter r0 of Lemma 27.18 equal to t. As r0 ≤ ǫ, we have y ∈ / B(x0 , r0 ). From (33.3), on B(x0 , r0 ) −1 −2 we have | Rm |(·, t) ≤ α t + 2 ǫ . Then from Lemma 27.18, at (y, t) we have

(34.1)

2 dt − △d ≥ − (n − 1) (α t−1 + 2 ǫ−2 )t1/2 + t−1/2 3 2 4 −2 = − (n − 1) 1 + α + ǫ t t−1/2 . 3 3

√ It follows that dt − △d + 100n ≥ 0. t e . Then h = Now put h(y, t) = φ d(y,t) 10Aǫ e

1 10Aǫ

dt − △d +

100n √ t

φ′ −

1 φ′′ , (10Aǫ)2

where

√ the arguments of φ′ and φ′′ are d(y,t) . Where φ′ 6= 0, we have dt − △d + 100n ≥ 0. The 10Aǫ t −n −f (x,t) ∗ fundamental solution u(x, t) = (4π(t − t)) 2 e of is positive for t ∈ [0, t) and we R have M u dV ≤ 1 for all t. (Recall that we are not assuming stochastic completeness.)


63

Then (34.2) Z

hu dV M

t

Z

Z

1 = ((h)u − h u) dV = (h)u dV ≤ − (10Aǫ)2 M M Z Z 10 10 10 φu dV ≤ u dV ≤ . ≤ 2 2 (10Aǫ) M (10Aǫ) M (10Aǫ)2 ∗

Z

φ′′ u dV

M

Hence Z

(34.3)

M

hu dV

t=0

≥

Z

M

hu dV

t=t

−

t ≥ 1 − A−2 . 2 (Aǫ)

Similarly, using Proposition 29.5 and Corollary 29.21, (34.4) Z Z Z ∗ − hv dV = − ((h)v − h v) dV ≤ − (h)v dV M M M t Z Z Z 1 10 10 ′′ ≤ φ v dV ≤ − φv dV = − hv dV. (10Aǫ)2 M (10Aǫ)2 M (10Aǫ)2 M

p t ∈ [t/2, t]. p Then t − e t ≤ 2−1/2 ǫ and Consider the time e t of Lemma 33.4. As (x, t) ∈ Mα , e so for large A, h will be one on the ball B at time e t of radius t − e t centered at x. Then at time e t, −

(34.5)

Z

M

hv dV ≥ −

Z

B

v dV ≥ β.

Thus (34.6)

−

Z

M

hv dV

t=0

−

≥ βe

e t (Aǫ)2

−

≥ βe

t (Aǫ)2

≥ β 1−

t (Aǫ)2

≥ β(1 − A−2 ).

Working at time 0, put u e = hu and fe = f − log h. In what follows we implicitly integrate over supp(h). We have (34.7)

−2

β(1 − A ) ≤ −

Z

hv dV = M

Z

M

[(−2△f + |∇f |2 − R)t − f + n]hu dV.

We claim that Z Z −f |∇h|2 2 2 e he−f dV. (34.8) −2△f + |∇f | he dV = − |∇f | + 2 h M M

64


This follows from Z Z −f 2 (34.9) −2△f + |∇f | he dV = 2h∇f, ∇(he−f )i + |∇f |2 he−f dV M ZM ∇h 2 = 2h∇f, − ∇f i + |∇f | he−f dV h ZM ∇h − ∇f i he−f dV = h∇f, 2 h ZM ∇h ∇h e he−f dV = h∇fe + , − ∇fi h h ZM 2 |∇h| 2 e + = − |∇f| he−f dV. 2 h M Then

Z

(34.10)

[(−2△f + |∇f |2 − R)t − f + n]hu dV = Z ZM 2 e e u dV + [t(|∇h|2 /h − Rh) − h log h]u dV. [−t|∇f | − f + n]e M

M

Next, Then

|∇h|2 h

≤

Z

(34.11) Also, (34.12)

10 (10Aǫ)2

−

Z

M

t M

and − Rh ≤ 1 (from the assumed lower bound on R at time zero).

|∇h|2 10 2 − Rh u dV ≤ ǫ + 1 ≤ A−2 + ǫ2 . h (10Aǫ)2

uh log h dV = −

Z

B(x0 ,20Aǫ)−B(x0 ,10Aǫ)

≤ 1 − Putting h(y) = φ (34.13)

d(y) 5Aǫ

Z

uh log h dV ≤

Z

u dV

M −B(x0 ,10Aǫ)

u dV. B(x0 ,10Aǫ)

, a result similar to (34.3) shows that Z Z u dV ≥ hu dV ≥ 1 − cA−2 B(x0 ,10Aǫ)

M

for an appropriate constant c. Putting this together gives Z −2 e 2 − fe + n u (34.14) β(1 − A ) ≤ −t|∇f| e dV + (1 + c)A−2 + ǫ2 . M

n n b b = (2t) 2 u e and define fb by u b = (2π)− 2 e−f . From (34.3) and (34.14), if we Put b g = 2t1 g, u R restore the subscript k then limk→∞ M u bk dVbk = 1 and for large k, Z 1 1 2 b b (34.15) β ≤ − |∇fk | − fk + n u bk dVbk . 2 2 Mk


If we normalize u bk by putting Uk = for large k, we also have

1 β ≤ 2

(34.16)

R

Mk

u bk , u bk dVbk

65 n

and define Fk by Uk = (2π)− 2 e− Fk , then

1 2 − |∇Fk | − Fk + n Uk dVbk . 2 Mk

Z

On the other hand, the logarithmic Sobolev inequality for Rn [8, I.(8)] says that Z 1 2 (34.17) − |∇F | − F + n U dV ≤ 0, 2 Rn R provided that the compactly-supported function U = (2π)−n/2 e−F satisfies Rn U dV = 1. As was mentioned to us by Peter Topping, one can get a sharper inequality by applying (34.17) to the rescaled function Uc (x) = cn U(cx) and optimizing with respect to c. The result is Z R 2 (34.18) |∇F |2 U dV ≥ n e1 − n Rn F U dV . Rn

Given this inequality on Rn , one can use a symmetrization argument to prove the same inequality for a compactly-supported function on any complete Riemannian manifold, provided that the Euclidean isoperimetric inequality holds for domains in the support of U. See, for example, [48, Proposition 4.1] which gives the symmetrization argument for (34.17), attributing it to Perelman. Again using the inequality for Rn , if instead we have vol(∂Ω)n ≥ (1 − δk ) cn vol(Ω)n−1 for domains Ω ⊂ supp(Uk ) then the symmetrization argument gives Z R 2 1− 2 F U dVb (34.19) |∇Fk |2 Uk dVbk ≥ (1 − δk ) n n e n Mk k k k . Mk

Equations (34.16) and (34.19) imply that Z R 2 bk 2 2 β n 1− n F U d V k k Mk (1 − δk ) n e − 1 − 1− Fk Uk dVbk ≤ − . (34.20) 2 n Mk 2

However, (34.21)

2 x n lim inf (1 − δk ) e − 1 − x = 0.

k→∞ x∈R

This is a contradiction.

35. I.10.2. The volumes of future balls The next result gives a lower bound on the volumes of future balls. Corollary 35.1. (cf. Corollary √ I.10.2)n Under the assumptions of Theorem 30.1, for 0 < t ≤ (ǫr0 )2 we have vol(Bt (x, t)) ≥ ct 2 for x ∈ B0 (x0 , ǫr0 ), where c = c(n) is a universal constant.

66


Proof. (Sketch) If the corollary were not true then taking√a sequence of counterexamples, we can center ourselves around the collapsing balls B(x, t) to obtain functions f as in Section 34. As in the proof of Theorem 13.3, the volume condition with the fact that R R along 1 −n/2 −f 2 (2π) e dV → 1 means that f → −∞, which implies that − |∇f | − f + n udV → M M 2 ∞. This contradicts the logarithmic Sobolev inequality. 36. I.10.4. κ-noncollapsing at future times The next result gives κ-noncollapsing at future times. Corollary 36.1. (cf. Corollary I.10.4) There are δ, ǫ > 0 such that for any A > 0 there exists κ = κ(A) > 0 with the following property. Suppose that we have a Ricci flow solution g(·) defined for t ∈ [0, (ǫr0 )2 ] which has bounded | Rm | and complete time slices. Suppose that for any x ∈ B(x0 , r0 ) and Ω ⊂ B(x0 , r0 ), we have R(x, 0) ≥ −r0−2 and vol(∂Ω)n ≥ (1 − δ) cn vol(Ω)n−1 , where cn is the Euclidean isoperimetric constant. If (x, t) satisfies A−1 (ǫr0 )2 √ ≤ t ≤ (ǫr0 )2 and distt (x, x0 ) ≤ Ar0 then g(·) is not κ-collapsed at (x, t) on scales less than t. Proof. Using Theorem 30.1 and Corollary 35.1, we can apply Theorem 28.2 starting at time A−1 (ǫr0 )2 . 37. I.10.5. Diffeomorphism finiteness In this section we prove the diffeomorphism finiteness of Riemannian manifolds with local isoperimetric inequalities, a lower bound on scalar curvature and an upper bound on volume. Theorem 37.1. Given n ∈ Z+ , there is a δ > 0 with the following property. For any r0 , V > 0, there are finitely many diffeomorphism types of compact n-dimensional Riemannian manifolds (M, g0 ) satisfying 1. R ≥ −r0−2 . 2. vol(M, g0 ) ≤ V . 3. Any domain Ω ⊂ M contained in a metric r0 -ball satisfies vol(∂Ω)n ≥ (1−δ)cn vol(Ω)n−1 , where cn is the Euclidean isoperimetric constant. Proof. Choose α > 0. Let δ and ǫ be the parameters of Theorem 30.1. Consider Ricci flow g(·) starting from (M, g0 ). Let T > 0 be the maximal number so that a smooth flow exists for t ∈ [0, T ). If T < ∞ then limt→T − supx∈M | Rm(x, t)| = ∞. It follows from Theorem 30.1 that T > (ǫr0 )2 . Put b g = g((ǫr0 )2 ). Theorem 30.1 gives a uniform double-sided sectional curvature bound on (M, b g ). Corollary 35.1 gives a uniform lower bound on the volumes of (ǫr0 )-balls in (M, b g ). Let {xi }N b). i=1 be a maximal (2ǫr0 )-separated net in (M, g

From the lower bound R ≥ −r0−2 on (M, g0 ) and the maximum principle, we have R(x, t) ≥ −r0−2 for t ∈ [0, (ǫr0 )2 ]. Then the Ricci flow equation gives a uniform upper bound on vol(M, b g ). This implies a uniform upper bound on N or, equivalently, a uniform upper bound on diam(M, gb). The theorem now follows from the diffeomorphism finiteness of n-dimensional Riemannian manifolds with double-sided sectional curvature bounds, upper bounds on diameter and lower bounds on volume.


67

38. I.11.1. κ-solutions Definition 38.1. Given κ > 0, a κ-solution is a Ricci flow solution (M, g(·)) that is defined on a time interval of the form (−∞, C) (or (−∞, C]) such that • The curvature | Rm | is bounded on each compact time interval [t1 , t2 ] ⊂ (−∞, C) (or (−∞, C]), and each time slice (M, g(t)) is complete. • The curvature operator is nonnegative and the scalar curvature is everywhere positive. • The Ricci flow is κ-noncollapsed at all scales.

By abuse of terminology, we may sometimes write that “(M, g(·)) is a κ-solution” if it is a κ-solution for some κ > 0. From Appendix F, Rt ≥ 0 for an ancient solution. This implies the essential equivalence of the notions of κ-noncollapsing in Definitions 13.14 and 26.1 , when restricted to ancient solutions. Namely, if a solution is κ-collapsed in the sense of Definition 26.1 then it is automatically κ-collapsed in the sense of Definition 13.14. Conversely, if a time-t0 slice of an ancient solution is collapsed in the sense of Definition 13.14 then the fact that Rt ≥ 0, together with bounds on distance distortion, implies that it is collapsed in the sense of Definition 26.1 (possibly for a different value of κ). The relevance of κ-solutions is that a blowup limit of a finite-time singularity on a compact manifold will be a κ-solution. For examples of κ-solutions, if n ≥ 3 then there is a κ-solution on the cylinder R×S n−1(r), where the radius satisfies r 2 (t) = r02 − 2(n−2)t. There is also a κ-solution on the Z2 -quotient R ×Z2 S n−1 (r), where the generator of Z2 acts by reflection on R and by the antipodal map on S n−1 . On the other hand, the quotient solution on S 1 × S n−1 (r) is not κ-noncollapsed for any κ > 0, as can be seen by looking at large negative time. Bryant’s gradient steady soliton is a three-dimensional κ-solution given by g(t) = φ∗t g0 , where g0 = dr 2 + µ(r) dΘ2 is a certain rotationally symmetric metric on R3 . It has sectional curvatures that go like r −1 , and µ(r) ∼ r. The gradient function f satisfies Rij + ∇i ∇j f = 0, with f (r) ∼ −2r. Then for r and r − 2t large, φt (r, Θ) ∼ (r − 2t, Θ). In particular, if R0 ∈ C ∞ (R3 ) is the scalar curvature function of g0 then R(t, r, Θ) ∼ R0 (r − 2t, Θ). To check the conclusion of Corollary 47.2 in this case, given a point (r0 , Θ) ∈ R3 at time 0, the scalar curvature goes like r0−1 . Multiplying the soliton metric by r0−1 and sending t → r0 t gives the asymptotic metric (38.2)

√ r − 2r0 t dΘ2 . d(r/ r0 )2 + r0

√ Putting u = (r − r0 )/ r0 , the rescaled metric is approximately u 2 (38.3) du + 1 + √ − 2t dΘ2 . r0 Given ǫ > 0, this will be ǫ-biLipschitz close to the evolving cylinder du2 + (1 − 2t) dΘ2 √ provided that |u| ≤ ǫ r0 , i.e. |r − r0 | ≤ ǫr0 . To have an ǫ-neck, we want this to hold

68


whenever |r − r0 |2 ≤ (ǫr0−1 )−1 . This will be the case if r0 ≥ ǫ−3 . Thus Mǫ is approximately

(38.4)

{(r, Θ) ∈ R3 : r ≤ ǫ−3 }

and Q = R(x0 , 0) ∼ ǫ3 . Then diam(Mǫ ) ∼ ǫ−3 and at the origin 0 ∈ Mǫ , R(0, 0) ∼ ǫ0 . It follows that for the value of κ corresponding to this solution, C(ǫ, κ) must grow at least as fast as ǫ−3 as ǫ → 0. 39. I.11.2. Asymptotic solitons This section shows that every κ-solution has a gradient shrinking soliton buried inside of it, in an asymptotic sense as t → −∞. Such a soliton will be called an asymptotic soliton.

Heuristically, the existence of an asymptotic soliton is a consequence of the compactness results and the monotonicity of the reduced volume. Taking an appropriate sequence of spacetime points going backward in time, one constructs a limiting rescaled solution. As the limit reduced volume is constant in time, the monotonicity formula implies that this limit solution is a gradient shrinking soliton. This is the basic idea but the rigorous argument is a bit more subtle. Pick an arbitrary point (p, t0 ) in the κ-solution (M, g(·)). Define the reduced volume Ve (τ ) and the reduced length l(q, τ ) as in Section 15, by means of curves starting from (p, t0 ), with τ = t0 − t. From Section 24, for each τ > 0 there is some q(τ ) ∈ M such that l(q(τ ), τ ) ≤ n2 . (Note that l ≥ 0 from the curvature assumption.)

Proposition 39.1. (cf. Proposition I.11.2) There is a sequence τ i → ∞ so that if we consider the solution g(·) on the time interval [t0 − τ i , t0 − 12 τ i ] and parabolically rescale it at the point (q(τ i ), t0 − τ i ) by the factor τi −1 then as i → ∞, the rescaled solutions converge to a nonflat gradient shrinking soliton (restricted to [−1, − 21 ]). Proof. Equation (25.5) implies that |∇l1/2 |2 ≤

C , 4τ

r

and so

C distt0 −τ (q, q(τ )). 4τ We apply this estimate initially at some fixed time τ = τ , to obtain r r !2 C n distt0 −τ (q, q(τ )) + . (39.3) l(q, τ ) ≤ 4τ 2 (39.2)

|l1/2 (q, τ ) − l1/2 (q(τ ), τ )| ≤

From (18.13), (18.14) and (25.5), R |∇l|2 l (1 + C)l ∂τ l = − − ≥ − . 2 2 2τ 2τ This implies that for τ ∈ 12 τ , τ , r r !2 1+C 2 τ C n (39.5) l(q, τ ) ≤ distt0 −τ (q, q(τ )) + . τ 4τ 2 (39.4)

Also from (25.5), we have τ R ≤ Cl. Then we can plug in the previous bound on l to get 1 an upper bound on τ R for τ ∈ 2 τ , τ . The upshot is that for any ǫ > 0, one can find


69

δ > 0 so that both l(q, τ ) and τ R(q, t0 − τ ) do not exceed δ −1 whenever 12 τ ≤ τ ≤ τ and dist2t0 −τ (q, q(τ )) ≤ ǫ−1 τ .

Varying τ , as the rescaled solutions (with basepoints at (q(τ ), t0 − τ )) are uniformly noncollapsing and have uniform curvature bounds on balls, Appendix E implies that we can take a sequence τ i → ∞ to get a pointed limit (M , q, g(·)) that is a complete Ricci flow solution (in the backward parameter τ ) for 12 < τ < 1. We may assume that we have locally Lipschitz convergence of l to a limit function l. We define the reduced volume V˜ (τ ) for the limit solution using the limit function l. We 1 claim that for any τ ∈ 2 , 1 , if we put τi = τ τ i then the number V˜ (τ ) for the limit solution is the limit of numbers V˜ (τi ) for the original solution. One wishes to apply dominated R −n/2 −n/2 convergence to the integrals M e−l(q,τi ) τi dvol(q, t0 −τi ). (Note that τi dvol(q, t0 −τi ) = −n/2 −n/2 −n/2 τi dvol(q, t0 −τi ) and τ i dvol(q, t0 −τi ) is the volume form for the rescaled metric τ −1 τ i g(t0 − τi ).) However, to do so one needs uniform lower bounds on l(q, τ ′ ) for the original solution in terms of dt0 −τ ′ (q, q(τ ′ )), for τ ′ ∈ (−∞, 0). By an argument of Perelman, written in detail in [67], one does indeed have a lower bound of the form dt0 −τ ′ (q, q(τ ′ ))2 . τ′ The nonnegative curvature gives polynomial volume growth for distance balls, so using (39.6) R −n/2 one can apply dominated convergence to the integrals M e−l(q,τi ) τi dvol(q, t0 − τi ). Thus ˜ ˜ limi→∞ V (τi ) = V (τ ). (39.6)

l(q, τ ′ ) ≥ − l(q(τ ′ )) − 1 + C(n)

As (39.5) gives a uniform upper bound on l on an appropriate ball around q(τi ), and there is a lower volume bound on the ball, it follows that as i → ∞, V˜ (τi ) is uniformly bounded away from zero. From this argument and the monotonicity of V˜ , V˜ (τ ) is a positive constant c as a function of τ , namely the limit of the reduced volume of the original solution as real time goes to −∞. As the original solution is nonflat, the constant c is strictly less than the n limit of the reduced volume of the original solution as real time goes to zero, which is (4π) 2 . Next, we will apply (24.6) and (24.8). As (24.6) holds distributionally for each rescaled solution, it follows that it holds distributionally for l. In particular, the nonpositivity implies that the left-hand side of (24.6), when computed for the limit solution, is actually a nonpositive measure. If the left-hand side of (24.6) (for the limit solution) were not ˜ strictly zero then using (24.7) we would conclude that ddτV is somewhere negative, which is a contradiction. (We use the fact that (39.6) passes to the limit to give a similar lower bound on l.) Thus we must have equality in (24.6) for the limit solution. This implies equality in (21.2), which implies equality in (24.8). Writing (24.8) as l l − n −l (39.7) (4△ − R) e− 2 = e 2, τ elliptic theory gives smoothness of l. In I.11.2 it is said that equality in (24.8) implies equality in (23.9), which implies that one has a gradient shrinking soliton. There is a problem with this argument, as the use of (23.9) implicitly assumes that the solution is defined for all τ ≥ 0, which we do not know. (The function l is only defined by a limiting procedure, and not in terms of L-geodesics on some

70


Ricci flow solution.) However, one can instead use Proposition 29.5, with f = l. Equality in (24.8) implies that v = 0, so (29.6) directly gives the gradient shrinking soliton equation. (The problem with the argument using (23.9), and its resolution using (29.6), were pointed out by the UCSB group.) If the gradient shrinking soliton g(·) is flat then, as it will be κ-noncollapsed at all scales, g n it must be Rn . From the soliton equation, ∂i ∂j l = 2τij and △l = 2τ . Putting this into the equality (24.8) gives |∇l|2 = τl . It follows that the level sets of l are distance spheres. Then 2 (24.6) implies that with an appropriate choice of origin, l = |x| . The reduced volume V˜ (τ ) 4τ n for the limit solution is now computed to be (4π) 2 , which is a contradiction. Therefore the gradient shrinking soliton is not flat. We remark that the gradient soliton constructed here does not, a priori, have bounded curvature on compact time intervals, i.e. it may not be a κ-solution. In the 2 and 3dimensional cases one can prove this using additional reasoning. See Section 43 where it is shown that 2-dimensional κ-solutions are round spheres, and Section 46 where the 3-dimensional case is discussed. 40. I.11.3. Two dimensional κ-solutions The next result is a classification of two-dimensional κ-solutions. It is important when doing dimensional reduction. Corollary 40.1. (cf. Corollary I.11.3) The only oriented two-dimensional κ-solution is the shrinking round 2-sphere. Proof. First, the only nonflat oriented nonnegatively curved gradient shrinking 2-D soliton is the round S 2 . The reference [30] given in I.11.3 for this fact does not actually cover it, as the reference only deals with compact solitons. A proof using Proposition 39.1 to rule out the noncompact case appears in [68]. Given this, the limit solution in Proposition 39.1 is a shrinking round 2-sphere. Thus the rescalings τ −1 i g(t0 − τ i ) converge to a round 2-sphere as i → ∞. However, by [30] the Ricci flow makes an almost-round 2-sphere become more round. Thus any given time slice of the original κ-solution must be a round 2-sphere. Remark 40.2. One can employ a somewhat different line of reasoning to prove Corollary 40.1; see Section 43. 41. I.11.4. Asymptotic scalar curvature and asymptotic volume ratio In this section we first show that the asymptotic scalar curvature ratio of a κ-solution is infinite. We then show that the asymptotic volume ratio vanishes. The proofs are somewhat rearranged from those in I.11.4. They are logically independent of Section 40, i.e. also cover the case n = 2. We will use results from Appendices F and G, in particular (F.14).


71

Definition 41.1. If M is a complete connected Riemannian manifold then its asymptotic scalar curvature ratio is R = lim supx→∞ R(x) d(x, p)2 . It is independent of the choice of basepoint p. Theorem 41.2. Let (M, g(·)) be a noncompact κ-solution. Then the asymptotic scalar curvature ratio R is infinite for each time slice. Proof. Suppose that M is n-dimensional, with n ≥ 2. Pick p ∈ M and consider a time-t0 slice (M, g(t0 )). We deal with the cases R ∈ (0, ∞) and R = 0 separately and show that they lead to contradictions. Case 1: 0 < R < ∞. We choose a sequence xk ∈ M such that dt0 (xk , p) → ∞ and R(xk , t0 )d2 (xk , p) → R. Consider the rescaled pointed solution (M, xk , gk (t)) with gk (t) = R(xk , t0 )g(t0 + R(xkt ,t0 ) ) and t ∈ (−∞, 0]. We have Rk (xk , 0) = 1, and for all b > 0, for sufficiently large k, we have Rk (x, t) ≤ Rk (x, 0) ≤ d22R for all x such that dk (x, p) > b. k (x,p) √ Fix numbers b, B > 0 so that b < R < B. The κ-noncollapsing assumption gives a uniform positive lower bound on the injectivity radius of gk (0) at xk , and so by Appendix E we may extract a pointed limit solution (M∞ , x∞ , g∞ (·)), defined on a time interval (−∞, 0] from the sequence (Mk , xk , gk (·)) where Mk = {x ∈ M | b < dk (x, p) < B}. Note that g∞ has nonnegative curvature operator and the time slice (M∞ , g∞ (0)) is locally isometric to an annular portion of a nonflat metric cone, since (Mk , p, gk (0)) Gromov-Hausdorff converges to the Tits cone CT (M, g(t0 )). (We use the word “locally” because the annulus in CT (M, g(t0 )) need not be geodesically convex in CT (M, g(t0 )), so we are only saying that the distance functions in small balls match up.) When n = 2 this contradicts the fact that R∞ (x∞ , 0) = 1. When n ≥ 3, we will derive a contradiction from Hamilton’s curvature evolution equation (41.3)

Rmt = ∆ Rm +Q(Rm).

Let dv : CT (M, g(t0 )) → R be the distance function from the vertex and let ρ : M∞ → R be the pullback of dv under the inclusion of the annulus M∞ in CT (M, g(t0 )). Lemma 41.4. The metric cone structure on (M∞ , g∞ (0)) is smooth, i.e. ρ is a smooth function. Proof. Consider a unit speed geodesic segment γ in the Tits cone CT (M, g(t0 )), such that γ is disjoint from the vertex v ∈ CT (M, g(t0 )). Note that since CT (M, g(t0 )) is a Euclidean cone over the Tits boundary ∂T (M, g(t0 )), the geodesic γ lies in the cone over a geodesic segment γˆ ⊂ ∂T (M, g(t0 )). Thus γ lies in a 2-dimensional locally convex flat subspace of CT (M, g(t0 )). Also, as in a flat 2-dimensional cone, the second derivative of the composite function d2v ◦ γ is identically 2.

Since ρ is obtained from dv by composition with a locally isometric embedding (M∞ , g∞ (0)) → CT (M, g(t0 )), the composition of ρ2 with any unit speed geodesic segment in M∞ also has second derivative identically equal to 2. Since ρ2 is Lipschitz, Rademacher’s theorem implies that ρ2 is differentiable almost everywhere. Let y ∈ M∞ be a point of differentiability of ρ2 . If the injectivity radius of g∞ (0)

72


at y is > a, then the function hy given by the composition expy

ρ2

Ty M∞ ⊃ B(0, a) −→ M∞ −→ R

has the property that its second radial derivative is identically 2, and it is differentiable at the origin 0 ∈ Ty M∞ . Therefore hy is a second order polynomial with Hessian identically 2, and is smooth. As the injectivity radius is a continuous function, this implies that ρ2 is smooth everywhere in M∞ . p Since ρ2 is strictly positive on M∞ , it follows that ρ = ρ2 is smooth as well.

By the lemma, we may choose a smooth local orthonormal frame e1 , . . . , en for (M∞ , g∞ (0)) near x∞ such that e1 points radially outward (with respect to the cone structure), and e2 , e3 span a 2-plane at x∞ with strictly positive curvature; such a 2-plane exists because R∞ (x∞ , 1) = 1. Put P = e1 ∧ e2 . In terms of the curvature operator, the fact that Rm∞ (e1 , e2 , e2 , e1 ) = 0 is equivalent to hP, Rm∞ P i = 0. As the curvature operator is nonnegative, it follows that Rm∞ P = 0. (In fact, this is true for any metric cone.) Differentiating gives (41.5)

(∇ei Rm∞ ) P + Rm∞ (∇ei P ) = 0

and (41.6)

(△ Rm∞ ) P + 2

X i

(∇ei Rm∞ ) ∇ei P + Rm∞ (△P ) = 0.

Taking the inner product of (41.6) with P gives X (41.7) 0 = hP, (△ Rm∞ ) P i + 2 hP, (∇ei Rm∞ ) ∇ei P i i

X = hP, (△ Rm∞ ) P i + 2 h∇ei P, (∇ei Rm∞ ) P i. i

Then (41.5) gives (41.8)

hP, (△ Rm∞ )P i = 2

X i

h∇ei P, Rm∞ (∇ei P )i.

As the sphere of distance r from the vertex in a metric cone has principal curvatures r1 , we have ∇e3 e1 = − 1r e3 . Then

1 (e2 ∧ e3 ) + e1 ∧ ∇e3 e2 . r This shows that ∇e3 P has a nonradial component 1r e2 ∧e3 . Thus (∆ Rm∞ )(e1 , e2 , e2 , e1 ) > 0. The zeroth order quadratic term Q(Rm) appearing in (41.3) is nonnegative when Rm is nonnegative, so we conclude that ∂t Rm∞ (e1 , e2 , e2 , e1 ) > 0 at t = 0. This means that Rm∞ (−ǫ)(e1 , e2 , e2 , e1 ) < 0 for ǫ > 0 sufficiently small, which is impossible. (41.9)

∇e3 (e1 ∧ e2 ) = (∇e3 e1 ) ∧ e2 + e1 ∧ ∇e3 e2 =

Case 2: R = 0. Let us take any sequence xk ∈ M with dt0 (xk , p) → ∞. Set rk = dt0 (xk , p), put (41.10)

gk (t) = rk−2 g(t0 + rk2 t)


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for t ∈ (−∞, 0], and let dk (·, ·) be the distance function associated to gk (0). For any 0 < b < B, put (41.11)

Mk (b, B) = {x ∈ M | 0 < b < dk (x, p) < B}.

Since R = 0, we get that supx∈Mk (b,B) | Rmk (x, 0)| → 0 as k → ∞. Invoking the κnoncollapsed assumption as in the previous case, we may assume that (M, p, gk (0)) GromovHausdorff converges to a metric cone (M∞ , p∞ , g∞ ) (the Tits cone CT (M, g(t0 ))) which is flat and smooth away from the vertex p∞ , and the convergence is smooth away from p∞ . The “unit sphere” in CT (M, g(t0 )) defines a compact smooth hypersurface S∞ in (M∞ − {p∞ }, g∞ (0)) whose principal curvatures are identically 1. If n ≥ 3 then S∞ must be a quotient of the standard (n − 1)-sphere by the free action of a finite group of isometries. We have a sequence Sk ⊂ Mk of approximating smooth hypersurfaces whose principal curvatures (with respect to gk (0)) go to 1 as k → ∞. In view of the convergence to (M∞ , p∞ , g∞ ), for sufficiently large k, the inward principal curvatures of Sk with respect to gk (0) are close to 1. As M has nonnegative curvature, Sk is diffeomorphic to a sphere [28, Theorem A]. Thus S∞ is isometric to the standard (n − 1)-sphere, and so CT (M, g(t0 )) is isometric to n-dimensional Euclidean space. Then (M, g(t0 )) is isometric to Rn , which contradicts the definition of a κ-solution. In the case n = 2 we know that S∞ is diffeomorphic to a circle but we do not know a priori that it has length 2π. To handle the case n = 2, we use the fact that gk (t) is a Ricci flow solution, to extract a limiting smooth incomplete time-independent Ricci flow solution (M∞ \p∞ , g∞ (t)) for t ∈ [−1, 0]. Note that this solution is unpointed. In view of the convergence to the limiting solution, for sufficiently large k, the inward principal curvatures of Sk with respect to gk (t) are close to 1 for all t ∈ [−1, 0]. This implies that Sk bounds a domain Bk ⊂ M whose diameter with respect to gk (t) is uniformly bounded above, say by 10 (see Appendix G). Applying the Harnack inequality (F.14) with yk ∈ Sk (at time 0) and x ∈ Bk (at time −1), we see that supx∈Bk | Rmk (x, −1)| → 0 as k → ∞. Thus (Bk , p, gk (−1)) Gromov-Hausdorff converges to a flat manifold (B∞ , p¯∞ , g∞ (−1)) with convex boundary. As all of the principal curvatures of ∂B∞ are 1, B∞ must be isometric to a Euclidean unit ball. This implies that S∞ is isometric to the standard S 1 of length 2π, and we obtain a contradiction as before. Definition 41.12. If M is a complete n-dimensional Riemannian manifold with nonnegative Ricci curvature then its asymptotic volume ratio is V = limr→∞ r −n vol(B(p, r)). It is independent of the choice of basepoint x0 . Proposition 41.13. (cf. Proposition I.11.4) Let (M, g(·)) be a noncompact κ-solution. Then the asymptotic volume ratio V vanishes for each time slice (M, g(t0)). Moreover, there is a sequence of points xk ∈ M going to infinity such that the pointed sequence {(M, (xk , t0 ), g(·))}∞ k=1 converges, modulo rescaling by R(xk , t0 ), to a κ-solution which isometrically splits off an R-factor. Proof. Consider the time-t0 slice. Suppose that V > 0. As R = ∞, there are sequences sk xk ∈ M and sk > 0 such that dt0 (xk , p) → ∞, dt (x → 0, R(xk , t0 )s2k → ∞, and k ,p) 0 R(x, t0 ) ≤ 2R(xk , t0 ) for all x ∈ Bt0 (xk , sk ) [33, Lemma 22.2]. Consider the rescaled pointed

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solution (M, xk , gk (t)) with gk (t) = R(xk , t0 ) g(t0 + R(xtk ,t0 ) ) and t ∈ (−∞, 0]. As Rt ≥ 0, we have Rk (x, t) ≤ 2 whenever t ≤ 0 and dk (x, xk ) ≤ R(xk , t0 )1/2 sk , where dk is the distance function for gk (0) and Rk (·, ·) is the scalar curvature of gk (·). The κ-noncollapsing assumption gives a uniform positive lower bound on the injectivity radius of gk (0) at xk , so by Appendix E we may extract a complete pointed limit solution (M∞ , x∞ , g∞ (t)), t ∈ (−∞, 0], of a subsequence of the sequence of pointed Ricci flows. By relative volume comparison, (M∞ , x∞ , g∞ (0)) has positive asymptotic volume ratio. By Appendix G, the Riemannian manifold (M∞ , x∞ , g∞ (0)) is isometric to an Alexandrov space which splits off a line, which means that it is a Riemannian product R × N. This implies a product structure for earlier times; see Appendix A. Now when n = 2, we have a contradiction, since R(x∞ , 0) = 1 but (M∞ , g∞ (0)) is a product surface, and must therefore be flat. When n > 2 we obtain a κ-solution on an (n − 1)-manifold with positive asymptotic volume ratio at time zero, and by induction this is impossible. 42. In a κ-solution, the curvature and the normalized volume control each other In this section we show that, roughly speaking, in a κ-solution the curvature and the normalized volume control each other. Corollary 42.1. 1. If B(x0 , r0 ) is a ball in a time slice of a κ-solution, then the normalized volume r0−n vol(B(x0 , r0 )) is controlled (i.e. bounded away from zero) ⇐⇒ the normalized scalar curvature r02 R(x0 ) is controlled (i.e. bounded above). 2. If B(x0 , r0 ) is a ball in a time slice of a κ-solution, then the normalized volume vol(B(x0 , r0 )) is almost maximal ⇐⇒ the normalized scalar curvature r02 R(x0 ) is almost zero. r0−n

3. (Precompactness) If (Mk , (xk , tk ), gk (·)) is a sequence of pointed κ-solutions (without the assumption that R(xk , tk ) = 1) and for some r > 0, the r-balls B(xk , r) ⊂ (Mk , gk (tk )) have controlled normalized volume, then a subsequence converges to an ancient solution (M∞ , (x∞ , 0), g∞(·)) which has nonnegative curvature operator, and is κ-noncollapsed (though a priori the curvature may be unbounded on a given time slice). 4. There is a constant η = η(n, κ) such that for every n-dimensional κ-solution (M, g(·)), 3 and all x ∈ M, we have |∇R|(x, t) ≤ ηR 2 (x, t) and |Rt |(x, t) ≤ ηR2 (x, t). More generally, there are scale invariant bounds on all derivatives of the curvature tensor, that only depend on n and κ. That is, for each ρ, k, l < ∞ there is a constant C = C(n, ρ, k, l, κ) < ∞ such l 1 ∂k l that ∂tk ∇ Rm (y, t) ≤ C R(x, t)(k+ 2 +1) for any y ∈ Bt (x, ρR(x, t)− 2 ).

5. There is a function α : [0, ∞) → [0, ∞) depending only on κ such that lims→∞ α(s) = ∞, and for every κ-solution (M, g(·)) and x, y ∈ M, we have R(y)d2(x, y) ≥ α(R(x)d2 (x, y)).

Proof. Assertion 1, =⇒. Suppose we have a sequence of κ-solutions (Mk , gk (·)), and sequences tk ∈ (−∞, 0], xk ∈ Mk , rk > 0, such that at time tk , the normalized volume of B(xk , rk ) is ≥ c > 0, and R(xk , tk )rk2 → ∞. By Appendix H, for each k, we can find yk ∈ B(xk , 5rk ), r¯k ≤ rk , such that R(yk , tk )¯ rk2 ≥ R(xk , tk )rk2 , and R(z, tk ) ≤ 2R(yk , tk ) for


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all z ∈ B(yk , r¯k ). Note that by relative volume comparison, whenever rek ≤ rk we have (42.2)

vol(B(yk , r¯k )) vol(B(yk , 10rk )) vol(B(xk , rk )) c vol(B(yk , rek )) ≥ ≥ ≥ ≥ . rekn r¯kn (10rk )n (10rk )n 10n

Rescaling the sequence of pointed solutions (Mk , (yk , tk ), gk (·)) by R(yk , tk ), we get a sequence satisfying the hypotheses of Appendix E (we use here the fact that Rt ≥ 0 for an ancient solution), so it accumulates on a limit flow (M∞ , (y∞ , 0), g∞(·)) which is a κ-solution. By (42.2), the asymptotic volume ratio of (M, g∞ (0)) is ≥ 10cn > 0. This contradicts Proposition 41.13. Assertion 3. By relative volume comparison, it follows that every r-ball in (Mk , gk (tk )) has normalized volume bounded below by a (k-independent) function of its distance to xk . By 1, this implies that the curvature of (Mk , gk (tk )) is bounded by a k-independent function of the distance to xk , and hence we can apply Appendix E to extract a smoothly converging subsequence. Assertion 1, ⇐=. Suppose we have a sequence (Mk , gk (·)) of κ-solutions, and sequences xk ∈ Mk , rk > 0, such that R(xk , tk )rk2 < c for all k, but rk−n vol(B(xk , rk )) → 0. For large k, we can choose r¯k ∈ (0, rk ) such that r¯k−n vol(B(xk , r¯k )) = 21 cn where cn is the volume of the unit Euclidean n-ball. By relative volume comparison, rr¯kk → 0. Applying 3, we see that the pointed sequence (Mk , (xk , tk ), gk (·)), rescaled by the factor r¯k−2 , accumulates on a pointed ancient solution (M∞ , (x∞ , 0), g∞(·)), such that the ball B(x∞ , 1) ⊂ (M∞ , g∞ ) has normalized volume 12 cn at t = 0. Suppose the ball B(x∞ , 1) ⊂ (M∞ , g∞ (0)) were flat. Then by the Harnack inequality (F.14) (applied to the approximators) we would have R∞ (x, t) = 0 for all x ∈ M∞ , t ≤ 0, i.e. (M∞ , g∞ (t)) would be a time-independent flat manifold. It cannot be Rn since vol(B(x∞ , 1)) = 12 cn . But flat manifolds other than Euclidean space have zero asymptotic volume ratio (as follows from the Bieberbach theorem that if N = Rn /Γ is a flat manifold and Γ is nontrivial then there is a Γ-invariant affine subspace A ⊂ Rn of dimension at least 1 on which Γ acts cocompactly). This contradicts the assumption that the sequence (Mk , gk (·)) is κ-noncollapsed. Thus B(x∞ , 1) ⊂ (M∞ , g∞ (0)) is not flat, which means, by the Harnack inequality, that the scalar curvature of g∞ (0) is strictly positive everywhere. Therefore, with respect to gk , we have 2 2 rk rk 2 2 (42.3) lim inf R(xk , tk )rk = lim inf (R(xk , tk ))¯ rk ) ≥ const. lim inf = ∞, k→∞ k→∞ k→∞ r¯k r¯k which is a contradiction. Assertion 2, =⇒. Apply 1, the precompactness criterion, and the fact that a nonnegativelycurved manifold whose balls have normalized volume cn must be flat. Assertion 2, ⇐=. Apply 1, the precompactness criterion, and the Harnack inequality (F.14) (to the approximators). Assertion 4. This follows by rescaling g so that R(x, t) = 1, and applying 1 and 3.

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Assertion 5. The quantity R(z)d2 (u, v) is scale invariant. If the assertion failed then we would have sequences (Mk , gk (·)), xk , yk ∈ Mk , such that R(yk ) = 1 and d(xk , yk ) remains bounded, but the curvature at xk blows up. This contradicts 1 and 3. 43. An alternate proof of Corollary 40.1 using Proposition 41.13 and Corollary 42.1 In this section we give an alternate proof of Corollary 40.1. It uses Proposition 41.13 and Corollary 42.1 To clarify the chain of logical dependence, we remark that this section is concerned with 2-dimensional κ-solutions, and does not use anything from Sections 39 or 40. It does use Proposition 41.13. However, we avoid circularity here because the proof of Proposition 41.13 given in Section 41, unlike the proof in [51], does not use Corollary 40.1. Lemma 43.1. There is a constant v = v(κ) > 0 such that if (M, g(·)) is a 2-dimensional κ-solution (a priori either compact or noncompact), x, y ∈ M and r = d(x, y) then (43.2)

vol(Bt (x, r)) ≥ vr 2 .

Proof. If the lemma were not true then there would be a sequence (Mk , gk (·)) of 2-dimensional κ-solutions, and sequences xk , yk ∈ Mk , tk ∈ R such that rk−2 vol(Btk (xk , rk )) → 0, where rk = d(xk , yk ). Let zk be the midpoint of a shortest segment from xk to yk in the tk -time slice (Mk , gk (tk )). For large k, choose r¯k ∈ (0, rk /2) such that π (43.3) r¯k−2 vol(Btk (zk , r¯k )) = , 2 2 i.e. half the area of the unit disk in R . As π = r¯k−2 vol(Btk (zk , r¯k )) ≤ r¯k−2 vol(Btk (xk , rk )) = (¯ rk /rk )−2 rk−2 vol(Btk (xk , rk )), (43.4) 2 it follows that limk→∞ rr¯kk = 0. Then by part 3 of Corollary 42.1, the sequence of pointed Ricci flows (Mk , (zk , tk ), gk (·)), when rescaled by r¯k−2 , accumulates on a complete Ricci flow (M∞ , (z∞ , 0), g∞ (·)). The segments from zk to xk and yk accumulate on a line in (M∞ , g∞ (0)), and hence (M∞ , g∞ (0)) splits off a line. By (43.3), (M∞ , g∞ (0)) cannot be isometric to R2 , and hence must be a cylinder. Considering the approximating Ricci flows, we get a contradiction to the κ-noncollapsing assumption. Lemma 43.1 implies that the asymptotic volume ratio of any noncompact 2-dimensional κ-solution is at least v > 0. By Proposition 41.13 we therefore conclude that every 2dimensional κ-solution is compact. (This was implicitly assumed in the proof of Corollary I.11.3 in [51], as its reference [30] is about compact surfaces.) Consider the family F of 2-dimensional κ-solutions (M, (x, 0), g(·)) with diam(M, g(0)) = 1. By Lemma 43.1, there is uniform lower bound on the volume of the t = 0 time slices of κ-solutions in F . Thus F is compact in the smooth topology by part 3 of Corollary 42.1 (the precompactness leads to compactness in view of the diameter bound). This implies (recall that R > 0) that there is a constant K ≥ 1 such that every time slice of every 2-dimensional κ-solution has K-pinched curvature.


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Hamilton has shown that volume-normalized Ricci flow on compact surfaces with positively pinched initial data converges exponentially fast to a constant curvature metric [30]. His argument shows that there is a small ǫ > 0, depending continuously on the initial data, so that when the volume of the (unnormalized) solution has gone down by a factor of at least ǫ−1 , the pinching is at most the square root of the initial pinching. By the compactness of the family F , this ǫ can be chosen uniformly when we take the initial data to be the t = 0 time slice of a κ-solution in F .

Now let K0 be the worst pinching of a 2-dimensional κ-solution, and let (M, g(·)) be a κ-solution where the curvature pinching of (M, g(0)) is K0 . Choosing t < 0 such that ǫ vol(M, g(t)) = vol(M, g(0)), the previous paragraph implies the curvature pinching of (M, g(t)) is at least K02 . This would contradict the fact that K0 is the upper bound on the pinching for all κ-solutions, unless K0 = 1.

44. I.11.5. A volume bound In this section we give a consequence of Proposition 41.13 concerning the volumes of metric balls in Ricci flow solutions with nonnegative curvature operator. Corollary 44.1. (cf. Corollary I.11.5) For every ǫ > 0, there is an A < ∞ with the following property. Suppose that we have a sequence of (not necessarily complete) Ricci flow solutions gk (·) with nonnegative curvature operator, defined on Mk × [tk , 0], such that 1. For each k, the time-zero ball B(xk , rk ) has compact closure in Mk . 2. For all (x, t) ∈ B(xk , rk ) × [tk , 0], 21 R(x, t) ≤ R(xk , 0) = Qk . 3. limk→∞ tk Qk = −∞. 4. limk→∞ rk2 Qk = ∞. −

1

−

1

Then for large k, vol(B(xk , AQk 2 )) ≤ ǫ(AQk 2 )n at time zero.

Proof. Given ǫ > 0, suppose that the corollary is not true. Then there is a sequence of such −1 −1 Ricci flow solutions with vol(B(xk , Ak Qk 2 )) > ǫ(Ak Qk 2 )n at time zero, where Ak → ∞. −

1

−

n

By Bishop-Gromov, vol(B(xk , Qk 2 )) > ǫQk 2 at time zero, so we can parabolically rescale by Qk and take a convergent subsequence. The limit (M∞ , g∞ (·)) will be a nonflat complete ancient solution with nonnegative curvature operator, bounded curvature and V(0) > 0. By Proposition 41.13, it cannot be κ-noncollapsed for any κ. Thus for each κ > 0, there are a point (xκ , tκ ) ∈ M∞ × (−∞, 0] and a radius rκ so that | Rm(xκ , tκ )| ≤ rκ−2 on the time-tκ ball B(xκ , rκ ), but vol(B(xκ , rκ )) < κ rκn . From the Bishop-Gromov inequality, V(tκ ) < κ for the limit solution. R ) = U R dV ≥ 0 We claim that V(t) is nonincreasing in t. To see this, we have dvol(U dt for any domain U ⊂ M∞ . Also, as R ≤ 2 on M∞ × (−∞, 0], Corollary 27.16 gives that distances on M∞ decrease at most linearly in t, which implies the claim. Thus V(0) = 0 for the limit solution, which is a contradiction.

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45. I.11.6. Curvature bounds for Ricci flow solutions with nonnegative curvature operator, assuming a lower volume bound In this section we show that for a Ricci flow solution with nonnegative curvature operator, a lower bound on the volume of a ball implies an earlier upper curvature bound on a slightly smaller ball. This will be used in Section 54. Corollary 45.1. (cf. Corollary I.11.6) For every w > 0, there are B = B(w) < ∞, C = C(w) < ∞ and τ0 = τ0 (w) > 0 with the following properties. (a) Take t0 ∈ [−r02 , 0). Suppose that we have a (not necessarily complete) Ricci flow solution (M, g(·)), defined for t ∈ [t0 , 0], so that at time zero the metric ball B(x0 , r0 ) has compact closure. Suppose that for each t ∈ [t0 , 0], g(t) has nonnegative curvature operator and vol(Bt (x0 , r0 )) ≥ wr0n . Then (45.2)

R(x, t) ≤ Cr0−2 + B(t − t0 )−1

whenever distt (x, x0 ) ≤ 41 r0 . (b) Suppose that we have a (not necessarily complete) Ricci flow solution (M, g(·)), defined for t ∈ [−τ0 r02 , 0], so that at time zero the metric ball B(x0 , r0 ) has compact closure. Suppose that for each t ∈ [−τ0 r02 , 0], g(t) has nonnegative curvature operator. If we assume a timezero volume bound vol(B0 (x0 , r0 )) ≥ wr0n then (45.3)

R(x, t) ≤ Cr0−2 + B(t + τ0 r02 )−1

whenever t ∈ [−τ0 r02 , 0] and distt (x, x0 ) ≤ 14 r0 . Remark 45.4. The statement in [51, Corollary 11.6(a)] does not have any constraint on t0 . In our proof we seem to need that −t0 ≤ cr02 for some arbitrary but fixed constant c < ∞. (The statement R(x, t) > C + B(t − t0 )−1 in [51, Proof of Corollary 11.6(a)] is the issue.) For simplicity we take −t0 ≤ r02 . This point does not affect the proof of Corollary 45.1(b), which is what ends up getting used. Proof. For part (a), we can assume that r0 = 1. Given B, C > 0, suppose that g(·) is a Ricci flow solution for t ∈ [t0 , 0] that satisfies the hypotheses of the corollary, with R(x, t) > C + B(t − t0 )−1 for some (x, t) satisfying distt (x, x0 ) ≤ 14 . Following the notation b put A b = λC 21 of the proof of Theorem 30.1, except changing the A of Theorem 30.1 to A, 1 and α = min(λ2 C 2 , B), where we will take λ to be a sufficiently small number that only depends on n. Put (45.5) Clearly (x, t) ∈ Mα .

Mα = {(x′ , t′ ) : R(x′ , t′ ) ≥ α(t′ − t0 )−1 }.

We first go through the analog of the proof of Lemma 31.1. We claim that there is some (x, t) ∈ Mα , with t ∈ (t0 , 0] and distt (x, x0 ) ≤ 13 , such that R(x′ , t′ ) ≤ 2Q = 2R(x, t) b − 12 . Put (x1 , t1 ) = whenever (x′ , t′ ) ∈ Mα , t′ ∈ (t0 , t] and distt′ (x′ , x0 ) ≤ distt (x, x0 ) + AQ (x, t). Inductively, if we cannot take (xk , tk ) for (x, t) then there is some (xk+1 , tk+1) ∈ Mα with tk+1 ∈ (t0 , tk ], R(xk+1 , tk+1 ) > 2R(xk , tk ) and disttk+1 (xk+1 , x0 ) ≤ disttk (xk , x0 ) +


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−1 b AR(x k , tk ) 2 . As the process must terminate, we end up with (x, t) satisfying 1 1 b R(x, t)− 12 ≤ 1 p + A (45.6) distt (x, x0 ) ≤ 4 3 1 − 1/2

if λ is sufficiently small.

Next, we go through the analog of the proof of Lemma 32.1. As in the proof of Lemma 32.1, b − 12 . R(x′ , t′ ) ≤ 2R(x, t) whenever t − 21 αQ−1 ≤ t′ ≤ t and distt′ (x′ , x0 ) ≤ distt (x, x0 ) + AQ 1 b −1/2 We claim that the time-t ball B(x0 , distt (x, x0 ) + 10 AQ ) is contained in the time-t′ ball b − 12 ). To see this, we apply Lemma 27.8 with r0 = 1 Q−1/2 to give B(x0 , distt (x, x0 ) + AQ 2 (45.7)

b −1/2 . distt (x0 , x) − distt (x0 , x) ≤ const.(n) αQ−1/2 ≤ λ const.(n)AQ

If λ is sufficiently small then the claim follows. The argument also shows that it is consistent to use the curvature bound when applying Lemma 27.8. Hence R(x′ , t′ ) ≤ 2R(x, t) whenever t − 21 αQ−1 ≤ t′ ≤ t and distt (x′ , x0 ) ≤ distt (x, x0 ) + 1 b −1/2 AQ . It follows that R(x′ , t′ ) ≤ 2R(x, t) whenever t− 12 αQ−1 ≤ t′ ≤ t and distt (x′ , x) ≤ 10 1 b −1/2 AQ . This shows that there is an A′ = A′ (B, C), which goes to infinity as B, C → ∞, 10

so that R(x′ , t′ ) ≤ 2R(x, t) whenever t − A′ Q−1 ≤ t′ ≤ t and distt (x′ , x) ≤ A′ Q−1/2 .

Now suppose that Corollary 45.1(a) is not true. Fixing w > 0, for any sequences {Bk }∞ k=1 and {Ck }∞ going to infinity and for each k, there is a Ricci flow solution g (·) which satisfies k k=1 the hypotheses of the corollary but for which R(xk , tk ) ≥ Ck + Bk (tk − t0,k )−1 for some point 1

(xk , tk ) satisfying disttk (xk , x0,k ) ≤ 41 . We can assume that λ2 Ck2 ≥ Bk . From the preceding discussion, there is a sequence A′k → ∞ and points (xk , tk ) with disttk (xk , x0,k ) ≤ 13 so that −1/2 ≤ t′k ≤ tk and disttk (x′k , xk ) ≤ A′k Qk , R(x′k , t′k ) ≤ 2R(xk , tk ) whenever tk − A′k Q−1 k where (45.8)

Qk = R(xk , tk ) ≥ Bk (tk − t0,k )−1 ≥ Bk .

By Corollary 44.1, for any ǫ > 0 there is some A = A(ǫ) < ∞ so that for large k, p p (45.9) vol(B(xk , A/ Qk )) ≤ ǫ(A/ Qk )n

at time zero. By the Bishop-Gromov inequality, vol(B(xk , 1)) ≤ ǫ for large k, since Qk → ∞. If we took ǫ sufficiently small from the beginning then we would get a contradiction to the fact that n n 2 2 2 ≥ vol(B(x0,k , 1)) ≥ w. (45.10) vol(B(xk , 1)) ≥ vol B x0,k , 3 3 3

For part (b), the idea is to choose the parameter τ0 sufficiently small so that we will still have the estimate vol(B(x0 , r0 )) ≥ 5−n w r0n for the time-t ball B(x0 , r0 ) when t ∈ [−τ0 r02 , 0], and so we can apply part (a) with w replaced by w5 . The value of τ0 will emerge from the proof. More precisely, putting r0 = 1 and with a given τ0 , let τ be the largest number in [0, τ0 ] so that the time-t ball B(x0 , 1) satisfies vol(B(x0 , 1)) ≥ 5−n w whenever t ∈ [−τ, 0]. If τ < τ0 then at time −τ , we have vol(B(x0 , 1)) = 5−n w. The conclusion of part (a) holds in the sense that (45.11)

R(x, t) ≤ C(5−n w) + B(5−n w)(t + τ )−1

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whenever t ∈ [−τ, 0] and distt (x, x0 ) ≤ 41 . Lemma 27.8, along with (45.11), implies that √ √ the time-(−τ ) ball B(x0 , 41 ) contains the time-0 ball B(x0 , 14 − 10(n − 1)(τ C + 2 Bτ )). From the nonnegative curvature, the time-(−τ ) volume of the first ball is at least as large as the time-0 volume of the second ball. Then 1 (45.12) 5−n w = vol(B(x0 , 1)) ≥ vol(B(x0 , )) 4 √ √ 1 ≥ vol(B(x0 , − 10(n − 1)(τ C + 2 Bτ ))) 4 √ √ 1 ≥ ( − 10(n − 1)(τ C + 2 Bτ ))n vol(B(x0 , 1)) 4 √ √ 1 ≥ ( − 10(n − 1)(τ C + 2 Bτ ))n w, 4 where the balls on the top at time-(−τ ), and the other balls are at time-0. √ line of√(45.12) are 1 1 Thus 4 − 10(n − 1)(τ C + 2 √Bτ ) ≤ 5 . This contradicts our assumption that τ < τ0 √ provided that 41 − 10(n − 1)(τ0 C + 2 Bτ0 ) = 15 . Finally, we give a version of Corollary 45.1(b) where instead of assuming a nonnegative curvature operator, we assume that the curvature operator in the time-dependent ball of radius r0 around x0 is bounded below by − r0−2 . Corollary 45.13. (cf. end of Section I.11.6) For every w > 0, there are B = B(w) < ∞, C = C(w) < ∞ and τ0 = τ0 (w) > 0 with the following property. Suppose that we have a (not necessarily complete) Ricci flow solution (M, g(·)), defined for t ∈ [−τ0 r02 , 0], so that at time zero the metric ball B(x0 , r0 ) has compact closure. Suppose that for each t ∈ [−τ0 r02 , 0], the curvature operator in the time-t ball B(x0 , r0 ) is bounded below by − r0−2 . If we assume a time-zero volume bound vol(B0 (x0 , r0 )) ≥ wr0n then (45.14)

R(x, t) ≤ Cr0−2 + B(t + τ0 r02 )−1

whenever t ∈ [−τ0 r02 , 0] and distt (x, x0 ) ≤ 41 r0 .

Proof. The blowup argument goes through as before. The only real difference is that the −2 be at least e− const. τ r0 times the volume of the volume of the time-(−τ ) ball B(x0√, 41 ) will √ time-0 ball B(x0 , 14 − 10(n − 1)(τ C + 2 Bτ )). 46. I.11.7. Compactness of the space of three-dimensional κ-solutions In this section we prove a compactness result for the space of three-dimensional κsolutions. The three-dimensionality assumption is used to show that the limit solution has bounded curvature. If a three-dimensional κ-solution M is compact then it is diffeomorphic to a quotient of S 3 or R × S 2 , as it has nonnegative curvature and is nonflat. If its asymptotic soliton (see Section 39) is also closed then M is a quotient of the round S 3 or R × S 2 . There are κ-solutions on S 3 and RP 3 with noncompact asymptotic soliton; see [52, Section 1.4]. They are not isometric to the round metric; this corrects the statement in the first paragraph of [51, Section 11.7].


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Theorem 46.1. (cf. Theorem I.11.7) Given κ > 0, the set of oriented three-dimensional κ-solutions is compact modulo scaling. That is, from any sequence of such solutions and points (xk , 0), after appropriate dilations we can extract a smoothly converging subsequence that satisfies the same conditions. Proof. If (Mk , (xk , 0), gk (·)) is a sequence of such κ-solutions with R(xk , 0) = 1 then parts 1 and 3 of Corollary 42.1 imply there is a subsequence that converges to an ancient solution (M∞ , (x∞ , 0), g∞(·)) which has nonnegative curvature operator and is κ-noncollapsed. The remaining issue is to show that it has bounded curvature. Note that Rt ≥ 0 since g∞ (·) is a limit of a sequence of Ricci flows satisfying Rt ≥ 0. Hence it is enough to show that (M∞ , g(0)) has bounded scalar curvature. If not, there is a sequence of points yi going to infinity in M∞ such that R(yi , 0) → ∞ 1 and R(y, 0) ≤ 2R(yi , 0) for y ∈ B(yi , Ai R(yi, 0)− 2 ), where Ai → ∞; compare [33, Lemma 22.2]. Using the κ-noncollapsing, a subsequence of the rescalings (M∞ , yi, R(yi, 0)g∞ ) will converge to a limit manifold N∞ . As in the proof of Proposition 41.13 from Appendix G, N∞ will split off a line. By Corollary 40.1 or Section 43, N∞ must be the standard solution on R × S 2 . Thus (M, g(0)) contains a sequence Di of neck regions, with their cross-sectional radii tending to zero as i → ∞.

Note that M has to be 1-ended. Otherwise, it would contain a line, and would therefore have to split off a line isometrically [18, Theorem 8.17]. But then M, the product of a line and a surface, could not have neck regions with cross-sections tending to zero.

From theStheory of nonnegatively curved manifolds [18, Chapter 8.5], there is an exhaustion M = t≥0 Ct by nonempty totally convex compact sets Ct so that (t1 ≤ t2 ) ⇒ (Ct1 ⊂ Ct2 ), and (46.2)

Ct1 = {q ∈ Ct2 : dist(q, ∂Ct2 ) ≥ t2 − t1 }.

Now consider a neck region D which is close to a cylinder. Note by triangle comparison – or simply because the distance function in D is close to that of a product metric – any minimizing geodesic segment γ ⊂ D of length large compared to cross-sectional radius of D must be nearly orthogonal to the cross-section. It follows from this and (46.2) that if t > 0 and ∂Ct contains a point p ∈ D such that d(p, ∂D) is large compared to the cross-section of D, then ∂Ct ∩ D is an approximate 2-sphere cross-section of D. Fix such a neck region D0 and let Ct0 be the corresponding convex set. As M has one end, ∂Ct0 has only one connected component, namely the approximate 2-sphere cross-section. For all t > t0 , there is a distance-nonincreasing retraction r : Ct → Ct0 which maps Ct − Ct0 onto ∂Ct0 [60]. Let D be a neck region with a very small cross-section and let Ct be a convex set so that ∂Ct intersects D in an approximate 2-sphere cross-section. Then ∂Ct consists entirely of this approximate cross-section. The restriction of r to ∂Ct is distance-nonincreasing, but will map the 2-sphere ∂Ct onto the 2-sphere ∂Ct0 . This is a contradiction. Remark 46.3. The statement of [51, Theorem 11.7] is about noncompact κ-solutions but the proof works whether the solutions are compact or noncompact.

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Remark 46.4. One may wonder where we have used the fact that we have a Ricci flow solution, i.e. whether the curvature is bounded for any κ-noncollapsed Riemannian 3manifold with nonnegative sectional curvature. Following the above argument, we could again split off a line in a rescaling around high-curvature points. However, we would not necessarily know that the ensuing nonnegatively-curved surface is compact. (A priori, it could be a smoothed-out cone, for example.) In the case of a Ricci flow, the compactness comes from Corollary 40.1 or Section 43. Corollary 46.5. Let (M, g(·)) be a 3-dimensional κ-solution. Then any asymptotic soliton constructed as in Section 39 is also a κ-solution. 47. I.11.8. Necklike behavior at infinity of a three-dimensional κ-solution - weak version The next corollary says that outside of a compact region, any oriented noncompact threedimensional κ-solution looks necklike (after rescaling). In this section we give a simple argument to prove the corollary, except for a diameter bound on the compact region. In the next section we give an argument that also proves the diameter bound. More information on three-dimensional κ-solutions is in Section 59. Definition 47.1. Fix ǫ > 0. Let (M, g(·)) be an oriented three-dimensional κ-solution. We say that a point x0 ∈ M is the center of an ǫ-neck if the solution g(·) in the set {(x, t) : −(ǫQ)−1 < t ≤ 0, dist0 (x, x0 )2 < (ǫQ)−1 }, where Q = R(x0 , 0), is, after scaling with the factor Q, ǫ-close in some fixed smooth topology to the corresponding subset of the evolving round cylinder (having scalar curvature one at time zero). (See Definition 58.1 below for a more precise statement.) We let Mǫ denote the points in M that are not centers of ǫ-necks. Corollary 47.2. (cf. Corollary I.11.8) For any ǫ > 0, there exists C = C(ǫ, κ) > 0 such that if (M, g(·)) is an oriented noncompact three-dimensional κ-solution then 1 1. Mǫ is compact with diam(Mǫ ) ≤ CQ− 2 and 2. C −1 Q ≤ R(x, 0) ≤ CQ whenever x ∈ Mǫ , where Q = R(x0 , 0) for some x0 ∈ ∂Mǫ . Proof. We prove here the claims of Corollary 47.2, except for the diameter bound. In the next section we give another argument which also proves the diameter bound. We claim first that Mǫ is compact. Suppose not. Then there is a sequence of points xk ∈ Mǫ going to infinity. Fix a basepoint x0 ∈ M. Then R(x0 ) dist20 (x0 , xk ) → ∞. By part 5 of Corollary 42.1, R(xk ) dist20 (x0 , xk ) → ∞. Rescaling around (xk , 0) to make its scalar curvature one, we can use Theorem 46.1 to extract a convergent subsequence (M∞ , x∞ ). As in the proof of Proposition 41.13, we can say that (M∞ , x∞ ) splits off a line. Hence for large k, xk is the center of an ǫ-neck, which is a contradiction. Next we claim that for any ǫ, there exists C = C(ǫ, κ) > 0 such that if gij (t) is a κ-solution then for any point x ∈ Mǫ , there is a point x0 ∈ ∂Mǫ such that dist0 (x, x0 ) ≤ CQ−1/2 and C −1 Q ≤ R(x, 0) ≤ CQ, where Q = R(x0 , 0).


83

If not then there is a sequence {Mi }∞ i=1 of κ-solutions along with points xi ∈ Mi,ǫ such that for each yi ∈ ∂Mi,ǫ , we have 1. dist20 (xi , yi) R(yi , 0) ≥ i or 2. R(yi , 0) ≥ i R(xi , 0) or 3. R(xi , 0) ≥ i R(yi , 0).

Rescale the metric on Mi so that R(xi , 0) = 1. From Theorem 46.1, a subsequence of the pointed spaces (Mi , xi ) will converge smoothly to a κ-solution (M∞ , x∞ ). Also, x∞ ∈ M∞,ǫ . Taking a subsequence, we can assume that 1. occurs for each i, or 2. occurs for each i, or 3. occurs for each i. If M∞ 6= M∞,ǫ , choose y∞ ∈ ∂M∞,ǫ . Then y∞ is the limit of a subsequence of points yi ∈ ∂Mi,ǫ .

If 1. occurs for each i then dist20 (x∞ , y∞ ) R(y∞ , 0) = ∞, which is impossible. If 2. occurs for each i then R(y∞ , 0) = ∞, which is impossible. If 3. occurs for each i then R(y∞ , 0) = 0. It follows from (F.14) that M∞ is flat, which is impossible, as R(x∞ , 0) = 1. Hence M∞ = M∞,ǫ , i.e. no point in the noncompact ancient solution M∞ is the center of an ǫ-neck. This contradicts the previous conclusion that M∞,ǫ is compact. 48. I.11.8. Necklike behavior at infinity of a three-dimensional κ-solution - strong version The following corollary is an application of the compactness result Theorem 46.1. it is a refinement of [51, Cor. I.11.8]. Corollary 48.1. For all κ > 0, there exists an ǫ0 > 0 such that for all 0 < ǫ < ǫ0 there exists an α = α(ǫ, κ) with the property that for any κ-solution (M, g(·)), and at any time t, precisely one of the following holds (Mǫ denotes the set of points which are not centers of ǫ-necks at time t): A. (M, g(·)) is round cylindrical flow, and so every point at every time is the center of an ǫ-neck for all ǫ > 0. B. M is noncompact, Mǫ 6= ∅, and for all x, y ∈ Mǫ , we have R(x)d2 (x, y) < α.

C. M is compact, and there is a pair of points x, y ∈ Mǫ such that R(x)d2 (x, y) > α, (48.2)

1

1

Mǫ ⊂ B(x, αR(x)− 2 ) ∪ B(y, αR(y)− 2 ),

and there is a minimizing geodesic xy such that every z ∈ M − Mǫ satisfies R(z)d2 (z, xy) < α. D. M is compact and there exists a point x ∈ Mǫ such that R(x)d2 (x, z) < α for all z ∈ M. Lemma 48.3. For all ǫ > 0, κ > 0, there exists α = α(ǫ, κ) with the following property. Suppose (M, g(·)) is any κ-solution, x, y, z ∈ M, and at time t we have x, y ∈ Mǫ and R(x)d2 (x, y) > α. Then at time t either R(x)d2 (z, x) < α or R(y)d2 (z, y) < α or (R(z)d2 (z, xy) < α and z ∈ / Mǫ ). Proof. Pick ǫ > 0, κ > 0, and suppose no such α exists. Then there is a sequence αk → ∞, a sequence of κ-solutions (Mk , gk (·)), and sequences xk , yk , zk ∈ Mk , tk ∈ R violating the

84


αk -version of the statement for all k. In particular, xk , yk ∈ (Mk )ǫ and (48.4) R(xk , tk )d2tk (xk , yk ) → ∞, R(xk , tk )d2tk (zk , xk ) → ∞, and R(yk , tk )d2tk (zk , yk ) → ∞. Let zk′ ∈ xk yk be a point in xk yk nearest zk in (Mk , gk (tk )).

We first show that R(xk , tk )d2tk (zk′ , xk ) → ∞. If not, we may pass to a subsequence on which R(xk , tk )d2tk (zk′ , xk ) remains bounded. Applying Theorem 46.1, we may pass to a subsequence and rescale by R(xk , tk ), to make the sequence (Mk , (xk , tk ), gk (·)) converge to a κ-solution (M∞ , (x∞ , 0), g∞(·)), the segments xk yk ⊂ (Mk , gk (tk )) converge to a ray x∞ ξ ⊂ ′ η. Recall that the comparison (M∞ , g∞ (0)), and the segments zk′ zk converge to a ray z∞ e z ′ (u, v) tends to the Tits angle ∂T (ξ, η) as u ∈ z ′ ξ, v ∈ z ′ η tend to infinity. Since angle ∠ ∞ ∞ ∞ ′ ξ d(zk , zk′ ) = d(zk , xk yk ) we must have ∂T (ξ, η) ≥ π2 . Now consider a sequence uk ∈ z∞ tending to infinity. By Theorem 46.1, part 5 of Corollary 42.1, and the remarks about Alexandrov spaces in Appendix G, if we rescale (M∞ , (uk , 0), g∞(·)) by R(uk , 0), we get round cylindrical flow as a limit. When k is sufficiently large, we may find an almost ′ η, and whose crossproduct region D ⊂ (M∞ , g∞ (·)) containing uk which is disjoint from z∞ ′ ξ transversely at a single point. section Σ × {0} ⊂ Σ × (−1, 1) ≃ D intersects the ray z∞ ′ ξ ∪ z ′ η from each other; hence M This implies that Σ × {0} separates the two ends of z∞ ∞ is ∞ two-ended, and (M∞ , g∞ (·)) is round cylindrical flow. This contradicts the assumption that xk is not the center of an ǫ-neck. Hence R(xk , tk )d2tk (zk′ , xk ) → ∞, and similar reasoning shows that R(yk , tk )d2tk (zk′ , yk ) → ∞.

By part 5 of Corollary 42.1, we therefore have R(zk′ , tk )d2tk (zk′ , xk ) → ∞ and R(zk′ , tk )d2tk (zk′ , yk ) → ∞. Rescaling the sequence (Mk , (zk′ , tk ), gk (·)) by R(zk′ , tk ), we get convergence to round cylindrical flow (since any limit flow contains a line), and zk′ zk subconverges to a segment orthogonal to the R-factor, which implies that R(zk′ , tk )d2tk (zk , zk′ ) is bounded and zk is the center of an ǫ-neck for large k. This contradicts our assumption that the αk -version of the lemma is violated for each k.

Proof of Corollary 48.1. Let (M, g(·)) be a κ-solution, and ǫ > 0. Case 1: Every x ∈ (M, g(t)) is the center of an ǫ-neck. In this case, if ǫ > 0 is sufficiently small, M fibers over a 1-manifold with fiber S 2 . If the 1-manifold is homeomorphic to R, then M has two ends, which implies that the flow (M, g(·)) is an evolving round cylinder. ˜ , g˜(t)) would split off a If the base of the fibration were a circle, then the universal cover (M line, which would imply that the universal covering flow would be a round cylindrical flow; but this would violate the κ-noncollapsed assumption at very negative times. Thus A holds in this case. Case 2: There exist x, y ∈Mǫ such that R(x)d2 (x, y) > α. By Lemma 48.3 and Corollary − 21 − 12 42.1 part 5, for all z ∈ M − B(x, αR(x) ) ∪ B(y, αR(y) ) , we have R(z)d2 (z, xy) < α and z ∈ / Mǫ . This implies (again by Corollary 42.1 part 5) that there exists a γ = γ(ǫ, κ) such that for every z ∈ M there is a z ′ ∈ xy for which R(z ′ )d2 (z ′ , z) < γ, which means that M must be compact, and C holds.


85

Case 3: Mǫ 6= ∅, and for all x, y ∈ Mǫ , we have R(x)d2 (x, y) < α. If M is noncompact then we are in case B and are done, so assume that M is compact. Pick x ∈ Mǫ , and suppose z ∈ M maximizes R(x)d2 (·, x). If R(x)d2 (z, x) ≥ α, then z is the center of an ǫ-neck, and we may look at the cross-section Σ of the neck region. If Σ separates M, then when ǫ > 0 is sufficiently small, we get a contradiction to the assumption that z maximizes R(x)d2 (x, ·). Hence Σ cannot separate M, and there is a loop passing through x which ˜ , g˜(·)) intersects Σ transversely at one point. It follows that the universal covering flow (M is cylindrical flow, a contradiction. Hence R(x)d2 (x, z) < α for all z ∈ M, so D holds. 49. More properties of κ-solutions In this section we prove some additional properties of κ-solutions. In particular, Corollary 49.2 implies that if z lies in a geodesic segment η in a κ-solution M and if the endpoints of 1 η are sufficiently far from z (relative to R(z)− 2 ) then z ∈ / Mǫ . The results of this section will be used in the proof of Theorem 52.7. Proposition 49.1. For all κ > 0, α > 0, θ > 0, there exists a β(κ, α, θ) < ∞ such that if (M, g(t)) is a time slice of a κ-solution, x, y1 , y2 ∈ M, R(x)d2 (x, yi ) > β for i = 1, 2, and e x (y1 , y2) ≥ θ, then (a) x is the center of an α-neck, and (b) ∠ e x (y1 , y2 ) ≥ π − α. ∠

Proof. The proof of this is similar to the first part of the proof of Lemma 48.3. Note that when α is small, then after enlarging β if necessary, the neck region around x will separate y1 from y2 ; this implies (b). Corollary 49.2. For all κ > 0, ǫ > 0, there exists a ρ = ρ(κ, ǫ) such that if (M, g(t)) is a time slice of a κ-solution, η ⊂ (M, g(t)) is a minimizing geodesic segment with endpoints y1 , y2 , z ∈ M, z ′ ∈ η is a point in η nearest z, and R(z ′ )d2 (z ′ , yi) > ρ for i = 1, 2, then z, z ′ are centers of ǫ-necks, and max(R(z)d2 (z, z ′ ), R(z ′ )d2 (z, z ′ )) < 4π 2 . Proof. Pick ǫ′ > 0. Under the assumptions, if (49.3)

min(R(z ′ )d2 (z ′ , y1 ), R(z ′ )d2 (z ′ , y2 ))

is sufficiently large, we can apply the preceding proposition to the triple z ′ , y1, y2 , to conclude that z ′ is the center of an ǫ′ -neck. Since the shortest segment from z to z ′ is orthogonal to η, when ǫ′ is small enough the segment zz ′ will lie close to an S 2 cross-section in the approximating round cylinder, which gives R(z ′ )d2 (z, z ′ ) . 2π 2 . 50. I.11.9. Getting a uniform value of κ Proposition 50.1. There is a κ0 > 0 so that if (M, g(·)) is an oriented three-dimensional κ-solution, for some κ > 0, then it is a κ0 -solution or it is a quotient of the round shrinking S 3. Proof. Let (M, g(·)) be a κ-solution. Suppose that for some κ′ > 0, the solution is κ′ collapsed at some scale. After rescaling, we can assume that there is a point (x0 , 0) so that | Rm(x, t)| ≤ 1 for all (x, t) satisfying dist0 (x, x0 ) < 1 and t ∈ [−1, 0], with vol(B0 (x0 , 1)) < κ′ . Let Ve (t) denote the reduced volume as a function of t ∈ (−∞, 0], as defined using curves

86


from (x0 , 0). It is nondecreasing in t. As in the proof of Theorem 26.2, there is an estimate Ve (−κ′ ) ≤ 3(κ′ )3/2 . Take a sequence of times ti → −∞. For each ti , choose qi ∈ M so that l(qi , ti ) ≤ 23 . From the proof of Proposition 39.1, for all ǫ > 0 there is a δ > 0 such that l(q, t) does not exceed δ −1 whenever t ∈ [ti , ti /2] and dist2ti (q, qi ) ≤ ǫ−1 ti . Given the monotonicity of Ve and the obtain an upper bound on the volume p upper bound on l(q, t), we 3/2 δ−1 of the time-ti ball B(qi , ti /ǫ) of the form const. ti e (κ′ )3/2 . On the other hand, from Proposition 39.1, a subsequence of the rescalings of the ancient solution around (qi , ti ) converges to a nonflat gradient shrinking soliton. If the gradient shrinking soliton is compact then it must be a quotient of the round shrinking S 3 [35]. Otherwise, Corollary 51.22 says that if the gradient shrinking soliton is noncompact then it must bep an evolving cylinder or its Z2 -quotient. Fixing ǫ, this gives a lower bound on vol(Bti (qi , ti /ǫ)) in terms of the noncollapsing constants of the evolving cylinder and its Z2 -quotient. Hence there is a universal constant κ0 so that if κ′ < κ0 then we obtain a contradiction to the assumption of κ′ -collapsing.

Remark 50.2. The hypotheses of Corollary 51.22 assume a global upper bound on the sectional curvature of any time slice, which in the n-dimensional case is not a priori true for the asymptotic soliton of Proposition 39.1. However, in our 3-dimensional case, the argument of Theorem 46.1 shows that there is such an upper bound. 51. II.1.2. Three-dimensional noncompact κ-noncollapsed gradient shrinkers are standard In this section we show that any complete oriented 3-dimensional noncompact κ-noncollapsed gradient shrinking soliton with bounded nonnegative curvature is either the evolving round cylinder R × S 2 or its Z2 -quotient.

The basic example of √ a gradient shrinking soliton is the metric on R × S 2 which gives the 2-sphere a radius of −2t at time t ∈ (−∞, 0). With coordinates (s, θ) on R × S 2 , the 2 function f is given by f (t, s, θ) = − s4t .

Lemma 51.1. (cf. Lemma of II.1.2) There is no complete oriented 3-dimensional noncompact κ-noncollapsed gradient shrinking soliton with bounded positive sectional curvature. Proof. The idea of the proof is to show that the soliton has the qualitative features of a shrinking cylinder, and then to get a contradiction to the assumption of positive sectional curvature. Applying ∇i to the gradient shrinker equation ∇i ∇j f + Rij +

(51.2)

1 gij = 0 2t

gives (51.3) As ∇i Rij =

(51.4)

1 2

△∇j f + ∇i Rij = 0.

∇j R and △∇j f = ∇j △f + Rjk ∇k f = ∇j −R − ∇i R = 2 Rij ∇j f.

n 2t

+ Rjk ∇k f , we obtain


87

Fix a basepoint x0 ∈ M and consider a normalized minimal geodesic γ : [0, s] → M in the time −1 slice with γ(0) = x0 . Put X(s) = dγ . As in the proof of Lemma 27.8, ds Rs Ric(X, X) ds ≤ const. for some constant independent of s. If {Yi }3i=1 are orthonormal 0 parallel vector fields along γ then 2 Z s Z s 3 Z s X 2 | Ric(X, Y1 )| ds ≤ s | Ric(X, Y1 )| ds ≤ s | Ric(X, Yi )|2 ds. (51.5) 0

0

P3

i=1

0

Thinking of Ric as a self-adjoint linear operator on T M, i=1 | Ric(X, Yi)|2 = hX, Ric2 Xi. In terms of a pointwise orthonormal frame {ei } of eigenvectors of Ric, with eigenvalues λi , P3 write X = i=1 Xi ei . Then (51.6)

Hence (51.7)

hX, Ric2 Xi = Z

3 X i=1

λ2i Xi2 ≤ (

s 0

| Ric(X, Y1 )| ds

2

3 X i=1

3 X λi ) ( λi Xi2 ) = R · Ric(X, X).

≤ (sup R) s M

i=1

Z

s

Ric(X, X) ds ≤ const. s.

0 2

(γ(s)) Multiplying (51.2) by X i X j and summing gives d fds + Ric(X, X) − 12 = 0. Then 2 Z s 1 1 df (γ(s)) df (γ(s)) + s − Ric(X, X) ds ≥ s − const. (51.8) = ds ds 2 2 s=s s=0 0 This implies that there is a compact subset of M outside of which f has no critical points.

If Y is a unit vector field perpendicular to X then multiplying (51.2) by X i Y j and d (Y · f )(γ(s)) + Ric(X, Y ) = 0. Then summing gives ds Z s (51.9) (Y · f )(γ(s)) = (Y · f )(γ(0)) − Ric(X, Y ) ds 0

and

(51.10)

√ |(Y · f )(γ(s))| ≤ const.( s + 1).

For large s, |(Y · f )(γ(s))| is small compared to (X · f )(γ(s)). This means that as one approaches infinity, the gradient of f becomes more and more parallel to the gradient of the distance function from x0 , where by the latter we mean the vectors X that are tangent to minimal geodesics. The gradient flow of f is given by the equation dx = (∇f )(x). (51.11) du Then along a flowline, equation (51.4) implies that dR(x) dx (51.12) = ∇R, = 2 Ric(∇f, ∇f ). du du

In particular, outside of a compact set, R is strictly increasing along the flowlines. Put R = lim supx→∞ R. Take points xα tending toward infinity, with R(xα ) → R. Putting

88


p α rα = → 0 and R(xα ) rα2 → ∞. Then the argument dist−1 (x0 , xα ), we have dist−1r(x 0 ,xα ) of the proof of Proposition 41.13 shows that any convergent subsequence of the rescalings around (xα , −1) splits off a line. Hence the limit is a shrinking round cylinder with scalar curvature R at time −1. Because our original solution exists up to time zero, we must have R ≤ 1. Now equation (C.13) says that the Ricci flow is given by g(−t) = −tφ∗t g(−1), where φt is the flow generated by ∇f . It follows that inf x∈M R(x, t) = Ct−1 for some C > 0. That is, the curvature blows up uniformly as t → 0. Comparing this with the singularity time of the shrinking round cylinder implies that R = 1. Performing a similar argument with any sequence of xα ’s tending toward infinity, with the property that R(xα ) has a limit, shows that limx→∞ R(x) = 1. Let N denote a (connected component of a) level surface of f . At a point of N, choose an orthonormal frame {e1 , e2 , e3 } with e3 = X normal to N. From the Gauss-Codazzi equation, (51.13)

RN = 2 K N (e1 , e2 ) = 2(K M (e1 , e2 ) + det(S)),

where S is the shape operator. As R = 2(K M (e1 , e2 ) + K M (e1 , e3 ) + K M (e2 , e3 )) and Ric(X, X) = K M (e1 , e3 ) + K M (e2 , e3 ), we obtain (51.14)

RN = R − 2 Ric(X, X) + 2 det(S).

|T N The shape operator is given by S = Hessf . From (51.2), Hess f = 12 − Ric. We can  |∇f |  r1 0 c1 diagonalize Ric to write Ric =  0 r2 c2 , where r3 = Ric(X, X). Then TN c1 c2 r3 1 1 1 = (51.15) det Hess f − r1 − r2 = (1 − r1 − r2 )2 − (r1 − r2 )2 2 2 4 TN 1 1 ≤ (1 − r1 − r2 )2 = (1 − R + Ric(X, X))2 . 4 4 This shows that the scalar curvature of N is bounded above by

(51.16)

R − 2 Ric(X, X) +

(1 − R + Ric(X, X))2 . 2|∇f |2

If |∇f | is large then 1 − R + Ric(X, X) < 2 |∇f |2 . As 1 − R + Ric(X, X) is positive when the distance from x to x0 is large enough, (51.17) and so (51.18) Hence (51.19)

(1 − R + Ric(X, X))2 < 2(1 − R + Ric(X, X))|∇f |2

≤ 2(1 − R + Ric(X, X))|∇f |2 + 2|∇f |2 Ric(X, X)

(1 − R + Ric(X, X))2 < 1 − R + 2 Ric(X, X). 2|∇f |2 R − 2 Ric(X, X) +

(1 − R + Ric(X, X))2 < 1. 2|∇f |2


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This shows that RN < 1 if N is sufficiently far from x0 . If Y is a unit vector that is tangential to N then from (51.2), 1 (51.20) ∇Y ∇Y f = − Ric(Y, Y ). 2 If {Y, Z, W } is an orthonormal basis then

1 R. 2 Hence ∇Y ∇Y f ≥ 12 (1 − R), which is positive if N is sufficiently far from x0 . Thus N is convex and so the area of the level set increases as the level increases. On the other hand, we can take points xα on the level sets going to infinity, apply the previous splitting argument and use the fact that grad f becomes almost parallel to grad d(·, x0 ). Within one of the approximate cylinders coming from the splitting argument, there is a projection π to its base S 2 . As a tangent plane of N is almost perpendicular to Ker(dπ), the restriction of π to N is an almost-isometry from N to S 2 . By the monotonicity of area(N), we conclude that area(N) ≤ 8π if N is sufficiently far from R x0 . However as N is a topologically a 2-sphere, the Gauss-Bonnet theorem says that N RN dA = 8π. This contradicts the facts that RN < 1 and area(N) ≤ 8π. (51.21) Ric(Y, Y ) = K M (Y, Z)+K M (Y, W ) ≤ K M (Y, Z)+K M (Y, W )+K M (Z, W ) =

Corollary 51.22. The only complete oriented 3-dimensional noncompact κ-noncollapsed gradient shrinking solitons with bounded nonnegative sectional curvature are the round evolving R × S 2 and its Z2 -quotient R ×Z2 S 2 . Proof. Let (M, g(·)) be a complete oriented 3-dimensional noncompact κ-noncollapsed gradient shrinking soliton with bounded nonnegative sectional curvature. By Lemma 51.1, M cannot have positive sectional curvature. From Theorem A.7, M must locally split off an R-factor. Then the universal cover splits off an R-factor and so, by Corollary 40.1 or Section 43, must be the standard R × S 2 . From the κ-noncollapsing, M must be R × S 2 or R ×Z2 S 2 . 52. I.12.1. Canonical neighborhood theorem In this section we show that a high-curvature region of a three-dimensional Ricci flow is modeled by part of a κ-solution. We first define the notion of Φ-almost nonnegative curvature. Definition 52.1. (cf. I.12) Let Φ ∈ C ∞ (R) be a positive nondecreasing function such that for positive s, Φ(s) is a decreasing function which tends to zero as s → ∞. A Ricci flow s solution is said to have Φ-almost nonnegative curvature if for all (x, t), we have (52.2)

Rm(x, t) ≥ − Φ(R(x, t)).

Remark 52.3. Note that Φ-almost nonnegative curvature implies that the scalar curvature is uniformly bounded below by − 6 Φ(0). The formulation of the pinching condition in [51, Section 12] is that there is a decreasing function φ, tending to zero at infinity, so that Rm(x, t) ≥ − φ(R(x, t)) R(x, t) for each (x, t). This formulation has a problem when R(x, t) < 0, if one takes φ to be defined on all of R. The condition in Definition 52.1 is what

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comes out of the three-dimensional Hamilton-Ivey pinching result (applied to the rescaled metric g˜(t) = g(t) ) if we assume normalized initial conditions; see Appendix B. t We note that since the sectional curvatures have to add up to R, the lower bound (52.2) implies a double-sided bound on the sectional curvatures. Namely, R n(n − 1) (52.4) −Φ(R) ≤ Rm ≤ + − 1 Φ(R). 2 2 The main use of the pinching condition is to show that blowup limits have nonnegative sectional curvature. Lemma 52.5. Let {(Mk , pk , gk )}∞ k=1 be a sequence of complete pointed Riemannian manifolds with Φ-almost nonnegative curvature. Given a sequence Qk → ∞, put g k = Qk gk and suppose that there is a pointed smooth limit (M∞ , p∞ , g ∞ ) = limk→∞ (Mk , pk , g k ). Then M∞ has nonnegative sectional curvature. Proof. First, the Φ-almost nonnegative curvature condition implies that the scalar curvature of Mk is bounded below uniformly in k. For m ∈ M∞ , let mk ∈ (Mk , g¯k ) be a sequence of approximants to m. Then limk→∞ Rm(mk ) = Rm∞ (m), where Rm(mk ) = Q−1 k Rm(mk ). There are two possibilities : either the numbers R(mk ) are uniformly bounded above or they are not. If they are uniformly bounded above then (52.4) implies that Rm(mk ) is uniformly bounded above and below, so Rm∞ (m) = 0. Suppose on the other hand that a subsequence of the numbers R(mk ) tends to infinity. We pass to this subsequence. Now R∞ (m) = limk→∞ R(mk ) exists by assumption and is nonnegative. Applying (52.2) gives that Rm∞ (m), the limit of Rm(mk ) (52.6) Q−1 , k Rm(mk ) = R(mk ) R(mk ) is nonnegative. We now prove the first version of the “canonical neighborhood” theorem. Theorem 52.7. (cf. Theorem I.12.1) Given ǫ, κ, σ > 0 and a function Φ as above, one can find r0 > 0 with the following property. Let g(·) be a Ricci flow solution on a threemanifold M, defined for 0 ≤ t ≤ T with T ≥ 1. We suppose that for each t, g(t) is complete, and the sectional curvature is bounded on compact time intervals. Suppose that the Ricci flow has Φ-almost nonnegative curvature and is κ-noncollapsed on scales less than σ. Then for any point (x0 , t0 ) with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , the solution in {(x, t) : dist2t0 (x, x0 ) < (ǫQ)−1 , t0 − (ǫQ)−1 ≤ t ≤ t0 } is, after scaling by the factor Q, ǫ-close to the corresponding subset of a κ-solution. Remark 52.8. Our statement of Theorem 52.7 differs slightly from that in [51, Theorem 12.1]. First, we allow M to be noncompact, provided that there is bounded sectional curvature on compact time intervals. This generalization will be useful for later work. More importantly, the statement in [51, Theorem 12.1] has noncollapsing at scales less than r0 , whereas we require noncollapsing at scales less than σ. See Remark 52.19 for further comment. In the phrase “t0 ≥ 1” there is an implied scale which comes from the Φ-almost nonnegativity assumption, and similarly for the statement “scales less than σ”.


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Proof. We give a proof which differs in some points from the proof in [51] but which has the same ingredients. We first outline the argument. Suppose that the theorem is false. Then for some ǫ, κ, σ > 0, we have a sequence of such Φ-nonnegatively curved 3-dimensional Ricci flows (Mk , gk (·)) defined on intervals [0, Tk ], and sequences rk → 0, xˆk ∈ Mk , tˆk ≥ 1 such that Mk is κ-noncollapsed on scales < σ and Qk = R(ˆ xk , tˆk ) ≥ rk−2 , but the Qk -rescaled solution in B(ǫQk )−1/2 (ˆ xk ) × [tˆk − (ǫQk )−1 , tˆk ] is not ǫ-close to the corresponding subset of any κ-solution. We note that if the statement were false for ǫ then it would also be false for any smaller ǫ. Because of this, somewhat paradoxically, we will begin the argument with a given ǫ but will allow ourselves to make ǫ small enough later so that the argument works. To be clear, we will eventually get a contradiction using a fixed (small) value of ǫ, but as the proof goes along we will impose some upper bounds on this value in order for the proof to work. (If we tried to list all of the constraints at the beginning of the argument then they would look unmotivated.) The goal is to get a contradiction based on the “bad” points (b xk , b tk ). In a sense, the method of proof of Theorem 52.7 is an induction on the curvature scale. For example, if we were to make the additional assumption in the theorem that R(x, t) ≤ R(x0 , t0 ) for all x ∈ M and t ≤ t0 then the theorem would be very easy to prove. We would just take a convergent subsequence of the rescaled solutions, based at (ˆ xk , tˆk ), to get a κ-solution; this would give a contradiction. This simple argument can be considered to be the first step in a proof by induction on curvature scale. In the proof of Theorem 52.7 one effectively proves the result at a given curvature scale inductively by assuming that the result is true at higher curvature scales. The actual proof consists of four steps. Step 1 consists of replacing the sequence (b xk , b tk ) by another sequence of “bad” points (xk , tk ) which have the property that points near (xk , tk ) with distinctly higher scalar curvature are “good” points. It then suffices to get a contradiction based on the existence of the sequence (xk , tk ). In steps 2-4 one uses the points (xk , tk ) to build up a κ-solution, whose existence then contradicts the “badness” of the points (xk , tk ). More precisely, let (Mk , (xk , tk ), g¯k (·)) be the result of rescaling gk (·) by R(xk , tk ). We will show that the sequence of pointed flows (Mk , (xk , tk ), g¯k (·)) accumulates on a κ-solution (M∞ , (x∞ , t0 ), g ∞ (·)), thereby obtaining a contradiction. In step 2 one takes a pointed limit of the manifolds (Mk , xk , g¯k (tk )) in order to construct what will become the final time slice of the κ-solution, (M∞ , x∞ , g ∞ (t0 )). In order to take this limit, it is necessary to show that the manifolds (Mk , xk , g¯k (tk )) have uniformly bounded curvature on distance balls of a fixed radius. If this were not true then for some radius, a subsequence of the manifolds (Mk , xk , g¯k (tk )) would have curvatures that asymptotically blowup on the ball of that radius. One shows that geometrically, the curvature blowup is due to the asymptotic formation of a cone-like point at the blowup radius. Doing a further rescaling at this cone-like point, one obtains a Ricci flow solution that ends on a part of a nonflat metric cone. This gives a contradiction as in the case 0 < R < ∞ of Theorem 41.2.

Thus one can construct the pointed limit (M∞ , x∞ , g ∞ (t0 )). The goal now is to show that (M∞ , x∞ , g ∞ (t0 )) is the final time slice of a κ-solution (M∞ , (x∞ , t0 ), g ∞ (·)). In step 3

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one shows that (M∞ , x∞ , g ∞ (t0 )) extends backward to a Ricci flow solution on some time interval [t0 − ∆, t0 ], and that the time slices have bounded nonnegative curvature. In step 4 one shows that the Ricci flow solution can be extended all the way to time (−∞, t0 ], thereby constructing a κ-solution Step 1: Adjusting the choice of basepoints. We first modify the points (b xk , b tk ) slightly in time to points (xk , tk ) so that in the given Ricci flow solution, there are no other “bad” points with much larger scalar curvature in a earlier time interval whose length is large compared to R(xk , tk )−1 . (The phrase “nearly the smallest curvature Q” in [51, Proof of Theorem 12.1] should read “nearly the largest curvature Q”. This is clear from the sentence in parentheses that follows.) The proof of the next lemma is by pointpicking, as in Appendix H. Lemma 52.9. We can find Hk → ∞, xk ∈ Mk , and tk ≥ 12 such that Qk = R(xk , tk ) → ∞, and for all k the conclusion of Theorem 52.7 fails at (xk , tk ), but holds for any (y, t) ∈ Mk × [tk − Hk Q−1 k , tk ] for which R(y, t) ≥ 2R(xk , tk ). 1 Proof. Choose Hk → ∞ such that Hk (R(b xk , b tk ))−1 ≤ 10 for all k. For each k, initially set ˆ (xk , tk ) = (ˆ xk , tk ). Put Qk = R(xk , tk ) and look for a point in Mk × [tk − Hk Q−1 k , tk ] at which Theorem 52.7 fails, and the scalar curvature is at least 2R(xk , tk ). If such a point exists, replace (xk , tk ) by this point; otherwise do nothing. Repeat this until the second alternative occurs. This process must terminate with a new choice of (xk , tk ) satisfying the lemma.

Hereafter we use this modified sequence (xk , tk ). Let (Mk , (xk , tk ), g¯k (·)) be the result of ¯ k to denote its scalar curvature; in particular, rescaling gk (·) by Qk = R(xk , tk ). We use R ¯ Rk (xk , tk ) = 1. Note that the rescaled time interval of Lemma 52.9 has duration Hk → ∞; this is what we want in order to try to extract an ancient solution. ¯ k is uniformly bounded on the ρ-balls Step 2: For every ρ < ∞, the scalar curvature R B(xk , ρ) ⊂ (Mk , g¯k (tk )) (the argument for this is essentially equivalent to [51, Pf. of Claim 2 of Theorem I.12.1]). Before proceeding, we need some bounds which come from our choice of basepoints, and the derivative bounds inherited (by approximation) from κ-solutions. Lemma 52.10. There is a constant C = C(κ) so that for any (x, t) in a Ricci flow solution, if R(x, t) > 0 and the solution in Bt (x, (ǫR(x, t))−1/2 ) × [t − (ǫR(x, t))−1 , t] is ǫ-close to a corresponding subset of a κ-solution then |∇R−1/2 |(x, t) ≤ C and |∂t R−1 |(x, t) ≤ C. Proof. This follows from the compactness in Theorem 46.1.

Note that the same value of C in Lemma 52.10 also works for smaller ǫ. Lemma 52.11. (cf. Claim 1 of I.12.1) For each (x, t) with tk − −1

1 2

Hk Q−1 ≤ t ≤ tk , k −1/2

we have Rk (x, t) ≤ 4Qk whenever t − c Qk ≤ t ≤ t and distt (x, x) ≤ c Qk Qk = Qk + |Rk (x, t)| and c = c(κ) > 0 is a small constant.

, where


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Proof. If Rk (x, t) ≤ 2Qk then there is nothing to show. If Rk (x, t) > 2Qk , consider a spacetime curve γ that goes linearly from (x, t) to (x, t), and then goes from (x, t) to (x, t) along a minimizing geodesic. If there is a point on γ with curvature 2Qk , let p be the nearest such point to (x, t). If not, put p = (x, t). From the conclusion of Lemma 52.9, we can apply Lemma 52.10 along γ from (x, t) to p. The claim follows from integrating the ensuing derivative bounds along γ. Lemma 52.12. In terms of the rescaled solution g k (·), for each (x, t) with tk − 21 Hk ≤ t ≤ tk , ek whenever t − c Q e−1 ≤ t ≤ t and distt (x, x) ≤ c Q e−1/2 , where we have Rk (x, t) ≤ 4Q k k ek = 1 + |Rk (x, t)|. Q Proof. This is just the rescaled version of Lemma 52.11.

For all ρ ≥ 0, put

(52.13)

¯ k (x, tk ) | k ≥ 1, x ∈ B(xk , ρ) ⊂ (Mk , g¯k (tk ))}, D(ρ) = sup{R

and let ρ0 be the supremum of the ρ’s for which D(ρ) < ∞. Note that ρ0 > 0, in view of Lemma 52.12 (taking (¯ x, t¯) = (xk , tk )). Suppose that ρ0 < ∞. After passing to a subsequence if necessary, we can find a sequence yk ∈ Mk with disttk (xk , yk ) → ρ0 and ¯ k , tk ) → ∞. Let ηk ⊂ (Mk , g¯k (tk )) be a minimizing geodesic segment from xk to yk . Let R(y ¯ k , tk ) = 2, and let γk be the subsegment zk ∈ ηk be the point on ηk closest to yk at which R(z of ηk running from yk to zk . By Lemma 52.12 the length of γk is bounded away from zero independent of k. Due to the Φ-pinching (see (52.4)), for all ρ < ρ0 , we have a uniform bound on | Rm | on the balls B(xk , ρ) ⊂ (Mk , g¯k (tk )). The injectivity radius is also controlled in B(xk , ρ), in view of the curvature bounds and the κ-noncollapsing. Therefore after passing to a subsequence, we can assume that the pointed sequence (B(xk , ρ0 ), g¯k (tk ), xk ) converges in the pointed Gromov-Hausdorff topology (i.e. for all ρ < ρ0 we have the usual GromovHausdorff convergence) to a pointed C 1 -Riemannian manifold (Z, g¯∞ , x∞ ), the segments ηk converge to a segment (missing an endpoint) η∞ ⊂ Z emanating from x∞ , and γk converges to γ∞ ⊂ η∞ . Let Z¯ denote the completion of (Z, g¯∞), and y∞ ∈ Z¯ the limit point of η∞ . Note that by Lemma 52.9 and part 4 of Corollary 42.1, the Riemannian structure near γ∞ may be chosen to be many times differentiable. (Alternatively, this follows from ¯∞ Lemma 52.12 and the Shi estimates of Appendix D.) In particular the scalar curvature R is defined, differentiable, and satisfies the bound in Lemma 52.10 near γ∞ . Lemma 52.14. 1. There is a function c : (0, ∞) → R depending only on κ, with limt→0 c(t) = ¯ ∞ (w) d(y∞ , w)2 > c(ǫ). ∞, such that if w ∈ γ∞ then R

2. There is a function ǫ′ : (0, ∞) → R ∪ {∞} depending only on κ, with limt→0 ǫ′ (t) = 0, ¯ ∞ (w)¯ such that if w ∈ γ∞ and d(y∞ , w) is sufficiently small then the pointed manifold (Z, w, R g∞ ) ′ 2 is 2ǫ (ǫ)-close to a round cylinder in the C topology. Proof. It follows from Lemma 52.9 that for all w ∈ γ∞ , the pointed Riemannian manifold ¯ ∞(w)¯ (Z, w, R g∞ ) is 2ǫ-close to (a time slice of) a pointed κ-solution. From the definition of pointed closeness, there is an embedded region around w, large on the scale defined by ¯ ∞ (w), which is close to the corresponding subset of a pointed κ-solution. This gives a lower R bound on the distance ρ0 − d(w, x∞ ) to the point of curvature blowup, thereby proving part 1 of the lemma.

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¯ ∞(w)¯ We know that (Z, w, R g∞ ) is 2ǫ-close to a pointed κ-solution (N, ⋆, h(t)) in the 2 ¯ ∞ (w) tends to ∞ as d(w, x∞) → ρ0 , for pointed C -topology. By Lemma 52.10, we know R 2 ¯ w ∈ γ∞ . In particular, R∞ (w)d (w, x∞ ) → ∞. From part 1 above, we can choose ǫ small ¯ ∞ (w)d2(w, y∞ ) large enough to apply Proposition 49.1. Hence the enough in order to make R pointed manifold (N, ⋆, R(⋆)h(t)) is ǫ′′ (ǫ)-close to a round cylinder, where ǫ′′ : (0, ∞) → R is a function with limt→0 ǫ′′ (t) = 0. The lemma follows. Note that the function ǫ′ in Lemma 52.14 is independent of the particular manifold Z that arises in our proof. From part 2 of Lemma 52.14, if ǫ is small and w ∈ γ∞ is sufficiently close to y∞ then (Z, w, R∞ (w)g ∞ ) is C 2 -close to a round cylinder. The cross-section of the cylinder has diam1 1 eter approximately π(R∞ (w)/2)− 2 . If we form the union of the balls B(w, 2π(R∞ (w)/2)− 2 ), as w ranges over such points in γ∞ , then we obtain a connected Riemannian manifold W . ¯ which is locally complete, and geodesic By adding in the point y∞ , we get a metric space W near y∞ . As the original manifolds Mk had Φ-almost nonnegative curvature, it follows from Lemma 52.5 that W is nonnegatively curved. Furthermore, y∞ cannot be an interior point ¯ , since such a geodesic would have to pass through a cylindrical of any geodesic segment in W region near y∞ twice. The usual proof of the Toponogov triangle comparison inequality now ¯. applies near y∞ since minimizers remain in the smooth nonnegatively curved part of W Then W has nonnegative curvature in the Alexandrov sense. ¯ , y∞ ) converge to the tangent cone Cy∞ W ¯ . As W is This implies that blowups of (W 2 ¯ three-dimensional, so is Cy∞ W [12, Corollary 7.11]. It will be C -smooth away from the ¯ such that d(z, y∞ ) = 1. vertex and nowhere flat, by part 2 of Lemma 52.14. Pick z ∈ Cy∞ W 1 ¯ Then the ball B(z, 2 ) ⊂ Cy∞ W is the Gromov-Hausdorff limit of a sequence of rescaled balls B(b zk , rbk ) ⊂ (Mk , g¯k (tk )) where rbk → 0, whose center points (b zk , tk ) satisfy the conclusions of Theorem 52.7. Applying Lemma 52.12 and Appendix B, we get the curvature bounds needed to extract a limiting Ricci flow solution whose time zero slice is isometric to B(z, 12 ). Now we can apply the reasoning from the 0 < R < ∞ case of Theorem 41.2 to get a contradiction. This completes step 2. Step 3: The sequence of pointed flows (Mk , g¯k (·), (xk , tk )) accumulates on a pointed Ricci flow (M∞ , g¯∞ (·), (x∞ , t0 )) which is defined on a time interval [t′ , t0 ] with t′ < t0 . By step 2, we know that the scalar curvature of (Mk , g¯k (tk )) at y ∈ Mk is bounded by a function of the distance from y to xk . Lemma 52.12 extends this curvature control to a backward parabolic neighborhood centered at y whose radius depends only on d(y, xk ). Thus we can conclude, using Φ-pinching (52.4) and Shi’s estimates (Appendix D), that all derivatives of the curvature (Mk , g¯k (tk )) are controlled as a function of the distance from xk , which means that the sequence of pointed manifolds (Mk , g¯k (tk ), xk ) accumulates to a smooth manifold (M∞ , g¯∞ ). From Lemma 52.5, M∞ has nonnegative sectional curvature. We claim that M∞ has bounded curvature. If not then there is a sequence of points qk ∈ M∞ so that limk→∞ R(qk ) = 1 ∞ and R(q) ≤ 2R(qk ) for q ∈ B(yk , Ak R(qk )− 2 ), where Ak → ∞; compare [33, Lemma 22.2]. Lemma 52.9 implies that for large k, a rescaled neighborhood of (M∞ , qk ) is ǫ-close to the corresponding subset of a time slice of a κ-solution. As in the proof of Theorem 46.1, we


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obtain a sequence of neck-like regions in M∞ with smaller and smaller cross-sections, which contradicts the existence of the Sharafutdinov retraction. By Lemma 52.12 again, we now get curvature control on (Mk , g¯k (·)) for a time interval [tk − ∆, tk ] for some ∆ > 0, and hence we can extract a subsequence which converges to a pointed Ricci flow (M∞ , (x∞ , t0 ), g¯∞ (·)) defined for t ∈ [t0 − ∆, t0 ], which has nonnegative curvature and bounded curvature on compact time intervals. Step 4: Getting an ancient solution. Let (t′ , t0 ] be the maximal time interval on which we can extract a limiting solution (M∞ , g¯∞ (·)) with bounded curvature on compact time intervals. Suppose that t′ > −∞. By Lemma 52.12 the maximum of the scalar curvature on the time slice (M∞ , g¯∞ (t)) must tend to infinity as t → t′ . From the trace Harnack R inequality, Rt + t−t ′ ≥ 0, and so ′

¯ ∞ (x, t) ≤ Q t0 − t , R t − t′ where Q is the maximum of the scalar curvature on (M∞ , g¯∞ (t0 )). Combining this with Corollary 27.16, we get r d t0 − t′ (52.16) dt (x, y) ≥ const. Q . dt t − t′ Since the right hand side is integrable on (t′ , t0 ], and using the fact that distances are nonincreasing in time (since Rm ≥ 0), it follows that there is a constant C such that (52.15)

(52.17)

for all x, y ∈ M∞ , t ∈ (t′ , t0 ].

|dt (x, y) − dt0 (x, y)| < C

If M∞ is compact then by (52.17) the diameter of (M∞ , g¯∞ (t)) is bounded independent of t ∈ (t′ , t0 ]. Since the minimum of the scalar curvature is increasing in time, it is also bounded independent of t. Now the argument in Step 2 shows that the curvature is bounded everywhere independent of t. (We can apply the argument of Step 2 to the time-t slice because the main ingredient was Lemma 52.9, which holds for rescaled time t.) We may therefore assume M∞ is noncompact. To be consistent with the notation of I.12.1, we now relabel the basepoint (x∞ , t0 ) as (x0 , t0 ). Since nonnegatively curved manifolds are asymptotically conical (see Appendix G), there is a constant D such that if y ∈ M∞ , and dt0 (y, x0 ) > D, then there is a point x ∈ M∞ such that 3 (52.18) dt0 (x, y) = dt0 (y, x0) and dt0 (x, x0 ) ≥ dt0 (y, x0); 2 ′ by (52.17) the same conditions hold at all times t ∈ (t , t0 ], up to error C. If for some such y, ¯ ∞ (y, t)¯ and some t ∈ (t′ , t0 ] the scalar curvature were large, then (M∞ , (y, t), R g∞(t)) would be 2ǫ-close to a κ-solution (N, h(·), (z, t0 )). When ǫ is small we could use Proposition 49.1 1 to see that y lies in a neck region U in (M∞ , g¯∞ (t)) of diameter ≈ R(y, t)− 2 ≪ 1. We claim that U separates x0 from x in the sense that x0 and x belong to disjoint components of M∞ − U, where x0 , y, and x satisfy (52.18). To see this, we recall that if M∞ has more than one end then it splits isometrically, in which case the claim is clear. If M∞ has one end then we consider an exhaustion of M∞ by totally convex compact sets Ct as

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in Section 46. One of the sets Ct will have boundary consisting of an approximate 2-sphere cross-section in the neck region U, giving the separation. e y (x0 , x) > π (For another argument, suppose that U does not separate x0 from x. Since ∠ 2 (from Proposition 49.1), the segments yx0 and yx must exit U by different ends. If x0 and x can be joined by a curve avoiding U then there is a nonzero element of H1 (M∞ , Z). The corresponding infinite cyclic cover of M∞ will then isometrically split off a line and its quotient M∞ will be compact, which is a contradiction. We thank one of the referees for this argument.) Obviously, at time t0 the set U still separates x0 from x. Since g¯∞ has nonnegative curvature, we have diamt0 (U) ≤ diamt (U) ≪ 1. Since (M∞ , g¯∞ (t0 )) has bounded geometry, there cannot be topologically separating subsets of arbitrarily small diameter. Thus there must be a uniform upper bound on R(y, t) and the curvature of (M∞ , g¯∞ ) is uniformly bounded (in space and time) outside a set of uniformly bounded diameter. Repeating the reasoning from Step 2, we get uniform bounds everywhere. This contradicts our assumption that the curvature blows up as t → t′ .

It remains to show that the ancient solution is a κ-solution. The only remaining point is to show that it is κ-noncollapsed at all scales. This follows from the fact that the original Ricci flow solutions (Mk , gk (·)) were κ-noncollapsed on scales less than the fixed number σ.

Remark 52.19. As mentioned in Remark 52.8, the statement of [51, Theorem 12.1] instead assumes noncollapsing at all scales less than r0 . Bing Wang pointed out that with this assumption, after constructing the ancient solution in Step 4 of the proof, one only gets that it is κ-collapsed at all scales less than one. Hence it may not be a κ-solution. The literal statement of [51, Theorem 12.1] is not used in the rest of [51, 52], but rather its method of proof. Because of this, the change of hypotheses does not seem to lead to any problems. The method of proof of Theorem 52.7 is used in two different ways. The first way is to construct the Ricci flow with surgery on a fixed finite time interval, as in Section 77. In this case the noncollapsing at a given scale σ comes from Theorem 26.2, and its extension when surgeries are allowed. The second way is to analyze the large-time behavior of the Ricci flow, as in the next few sections. 53. I.12.2. Later scalar curvature bounds on bigger balls from curvature and volume bounds The next theorem roughly says that if one has a sectional curvature bound on a ball, for a certain time interval, and a lower bound on the volume of the ball at the initial time, then one obtains an upper scalar curvature bound on a larger ball at the final time. We first write out the corrected version of the theorem (see II.6.2). Theorem 53.1. For any A < ∞, there exist K = K(A) < ∞ and ρ = ρ(A) > 0 with the following property. Suppose in dimension three we have a Ricci flow solution with Φalmost nonnegative curvature. Given x0 ∈ M and r0 > 0, suppose that r02 Φ(r0−2 ) < ρ, the solution is defined for 0 ≤ t ≤ r02 and it has | Rm |(x, t) ≤ 3r12 for all (x, t) satisfying 0


97

dist0 (x, x0 ) < r0 . Suppose in addition that the volume of the metric ball B(x0 , r0 ) at time zero is at least A−1 r03 . Then R(x, r02 ) ≤ Kr0−2 whenever distr02 (x, x0 ) < Ar0 . Remark 53.2. The added restriction that r02 Φ(r0−2 ) < ρ (see II.6.2) imposes an upper bound on r0 . This is necessary, as otherwise the conclusion would imply that neck pinches cannot occur. There is an apparent gap in the proof of [51, Theorem 12.2], in the sentence (There is a little subtlety...). We instead follow the proof of II.6.3(b,c) (see Proposition 84.1(b,c)), which proves the same statement in the presence of surgeries. The volume assumption in the theorem is used to guarantee noncollapsing, by means of Theorem 28.2. The reason for the “3” in the hypothesis | Rm |(x, t) ≤ 3r12 comes from 0 Remark 28.3. Proof. The proof is in two steps. In the first step one shows that if R(x, r02 ) is large then a parabolic neighborhood of (x, r02 ) is close to the corresponding subset of a κ-solution. In the second part one uses this to prove the theorem. The first step is the following lemma. Lemma 53.3. For any ǫ > 0 there exists K = K(A, ǫ) < ∞ so that for any r0 , whenever we have a solution as in the statement of the theorem and distr02 (x, x0 ) < Ar0 then (a) R(x, r02 ) < Kr0−2 or (b) The solution in {(x′ , t′ ) : distt (x′ , x) < (ǫQ)−1 , t − (ǫQ)−1 ≤ t′ ≤ t} is, after scaling by the factor Q, ǫ-close to the corresponding subset of a κ-solution. Here t = r02 and Q = R(x, t). Remark 53.4. One can think of this lemma as a localized analog of Theorem 52.7, where “localized” refers to the fact that both the hypotheses and the conclusion involve the point x0 . Proof. To prove the lemma, suppose that there is a sequence of such pointed solutions (Mk , x0,k , gk (·)), along with points xˆk ∈ Mk , so that distr02 (ˆ xk , x0,k ) < Ar0 and r02 R(ˆ xk , r02 ) → ∞, but (ˆ xk , r02 ) does not satisfy conclusion (b) of the lemma. As in the proof of Theorem 52.7, we will allow ourselves to make ǫ smaller during the course of the proof. We first show that there is a sequence Dk → ∞ and modified points (xk , tk ) with 43 r02 ≤ tk ≤ r02 , disttk (xk , x0,k ) < (A + 1)r0 and Qk = R(xk , tk ) → ∞, so that any point (x′k , t′k ) −1/2 ′ ′ ′ with R(x′k , t′k ) > 2Qk , tk − Dk2 Q−1 k ≤ tk ≤ tk and disttk (xk , x0,k ) < disttk (xk , x0,k ) + Dk Qk satisfies conclusion (b) of the lemma, but (xk , tk ) does not satisfy conclusion (b) of the lemma. (Of course, in saying “(x′k , t′k ) satisfies conclusion (b)” or “(xk , tk ) does not satisfy conclusion (b)”, we mean that the (x, t) in conclusion (b) is replaced by (x′k , t′k ) or (xk , tk ), respectively.) r R(ˆ x ,r 2 )1/2

k 0 . Start The construction of (xk , tk ) is by a pointpicking argument. Put Dk = 0 10 2 ′ ′ ′ ′ with (xk , tk ) = (ˆ xk , r0 ) and look if there is a point (xk , tk ) with R(xk , tk ) > 2R(xk , tk ), 2 −1 tk − Dk R(xk , tk ) ≤ t′k ≤ tk and distt′k (x′k , x0,k ) < disttk (xk , x0,k ) + Dk R(xk , tk )−1/2 , but

98


which does not have a neighborhood that is ǫ-close to the corresponding subset of a κsolution. If there is such a point, we replace (xk , tk ) by (x′k , t′k ) and repeat the process. The process must terminate after a finite number of steps to give a point (xk , tk ) with the desired property. −1/2

involves the metric (Note that the condition distt′k (x′k , x0,k ) < disttk (xk , x0,k ) + Dk Qk at time t′k . In order to construct an ancient solution, one of the issues will be to replace this by a condition that only involves the metric at time tk , i.e. that involves a parabolic neighborhood around (xk , tk ).) Let g k (·) denote the rescaling of the solution gk (·) by Qk . We normalize the time interval of the rescaled solution by fixing a number t∞ and saying that for all k, the time-tk slice of (Mk , gk ) corresponds to the time-t∞ slice of (Mk , g k ). Then the scalar curvature Rk of g k satisfies Rk (xk , t∞ ) = 1. By the argument of Step 2 of the proof of Theorem 52.7, a subsequence of the pointed spaces (Mk , xk , g k (t∞ )) will smoothly converge to a nonnegatively-curved pointed space (M∞ , x∞ , g ∞ ). By the pointpicking, if m ∈ M∞ has R(m) ≥ 3 then a parabolic neighborhood of m is ǫ-close to the corresponding region in a κ-solution. It follows, as in Step 3 of the proof of Theorem 52.7, that the sectional curvature of M∞ will be bounded above by some C < ∞. Using Lemma 52.12, the metric on M∞ is the time-t∞ slice of a nonnegativelycurved Ricci flow solution defined on some time interval [t∞ −c, t∞ ], with c > 0, and one has convergence of a subsequence g k (t) → g ∞ (t) for t ∈ [t∞ −c, t∞ ]. As Rt ≥ 0, the scalar curvature on this time interval will be uniformly bounded above by 6C and so from the Φ-almost nonnegative curvature (see (52.4)), the sectional curvature will be uniformly bounded above on the time interval. Hence we can apply Lemma 27.8 to get a uniform additive bound on the length distortion between times t∞ − c and t∞ (see Step 4 of the proof of Theorem 52.7). More precisely, in applying Lemma 27.8, we use the curvature bound coming from the hypotheses of the theorem near x0 , and the just-derived upper curvature bound near xk . ′ It follows that for a given A′ > 0, for large k, if t′k ∈ [tk − cQ−1 k /2, tk ] and disttk (xk , xk ) < −1/2 −1/2 A′ Qk then distt′k (x′k , x0,k ) < disttk (xk , x0,k )+Dk Qk . In particular, if a point (x′k , t′k ) lies ′ ′ −1/2 in the parabolic neighborhood given by t′k ∈ [tk − cQ−1 , k /2, tk ] and disttk (xk , xk ) < A Qk and has R(x′k , t′k ) > 2Qk , then it has a neighborhood that is ǫ-close to the corresponding subset of a κ-solution.

As in Step 4 of the proof of Theorem 52.7, we now extend (M∞ , g∞ , x∞ ) backward to an ancient solution g ∞ (·), defined for t ∈ (−∞, t∞ ]. To do so, we use the fact that if the solution is defined backward to a time-t slice then the length distortion bound, along with the pointpicking, implies that a point m in a time-t slice with R∞ (m) > 3 has a neighborhood that is ǫ-close to the corresponding subset of a κ-solution. The ancient solution is κ-noncollapsed at all scales since the original solution was κ-noncollapsed at some scale, by Theorem 28.2. Then we obtain smooth convergence of parabolic regions of the points (xk , tk ) to the κ-solution, which is a contradiction to the choice of the (xk , tk )’s. We now know that regions of high scalar curvature are modeled by corresponding regions in κ-solutions. To continue with the proof of the theorem, fix A large. Suppose that the


99

theorem is not true. Then there are 1. Numbers ρk → 0, −2 2 2. Numbers r0,k with r0,k Φ(r0,k ) ≤ ρk , 2 3. Solutions (Mk , gk (·)) defined for 0 ≤ t ≤ r0,k , 4. Points x0,k ∈ Mk and 5. Points xk ∈ Mk so that −2 2 a. | Rm |(x, t) ≤ r0,k for all (x, t) ∈ Mk × [0, r0,k ] satisfying dist0 (x, x0,k ) < r0,k , 3 b. The volume of the metric ball B(x0,k , r0,k ) at time zero is at least A−1 r0,k and 2 (xk , x0,k ) < Ar0,k , but c. distr0,k 2 2 d. r0,k R(xk , r0,k ) → ∞.

We now apply Step 2 of the proof of Theorem 52.7 to obtain a contradiction. That is, we −2 2 take a subsequence of {(Mk , x0,k , r0,k gk (r0,k ))}∞ k=1 that converges on a maximal ball. The only difference is that in Theorem 52.7, the nonnegative curvature of W came from the Φ-almost nonnegative curvature assumption on the original manifolds Mk along with the fact (with the notation of the proof of Theorem 52.7) that the numbers Qk = R(xk , tk ), which we used to rescale, go to infinity. In the present case the rescaled scalar curvatures 2 2 r0,k R(x0,k , r0,k ) at the basepoints x0,k stay bounded. However, if a point y ∈ W is a limit of points x ek ∈ Mk then the equations 2 2 Rm(e xk , r0,k ) ≥ − Φ(R(e xk , r0,k ))

(53.5)

in the form (53.6)

2 r0,k

2 Rm(e xk , r0,k )

−2 2 2 Φ r0,k R(e xk , r0,k ) · r0,k 2 2 r0,k R(e xk , r0,k ) ≥ − 2 R(e xk , r0,k )

2 2 pass to the limit to give Rm(y) ≥ 0 (using that y ∈ W , so r0,k Rm(e xk , r0,k ) → R(y) > 0). This is enough to carry out the argument.

54. I.12.3. Earlier scalar curvature bounds on smaller balls from lower curvature bounds and volume bounds The main result of this section says that if one has a lower bound on volume and sectional curvature on a ball at a certain time then one obtains an upper scalar curvature bound on a smaller ball at an earlier time. We first prove a result in Riemannian geometry saying that under certain hypotheses, metric balls have subballs of a controlled size with almost-Euclidean volume. Lemma 54.1. Given w ′ > 0 and n ∈ Z+ , there is a number c = c(w ′ , n) > 0 with the following property. Let B be a radius-r ball with compact closure in an n-dimensional Riemannian manifold. Suppose that the sectional curvatures of B are bounded below by − r −2 . Suppose that vol(B) ≥ w ′r n . Then there is a subball B ′ ⊂ B of radius r ′ ≥ cr so that vol(B ′ ) ≥ 21 ωn (r ′ )n , where ωn is the volume of the unit ball in Rn . Proof. Suppose that the lemma is not true. Rescale so that r = 1. Then there is a sequence of Riemannian manifolds {Mi }∞ i=1 with balls B(xi , 1) ⊂ Mi having compact closure

100


so that Rm

B(xi ,1)

≥ − 1 and vol(B(xi , 1)) ≥ w ′, but with the property that all balls

B(x′i , r ′) ⊂ B(xi , 1) with r ′ ≥ i−1 satisfy vol(B(x′i , r ′ )) < 12 ωn (r ′ )n . After taking a subsequence, we can assume that limi→∞ (B(xi , 1), xi ) = (X, x∞ ) in the pointed GromovHausdorff topology. From [12, Theorem 10.8], the Riemannian volume forms dvolMi converge weakly to the three-dimensional Hausdorff measure µ of X. From [12, Corollary 6.7 and Section 9], for any ǫ > 0, there are small balls B(x′∞ , r ′ ) ⊂ X with compact closure in X such that µ(B(x′∞ , r ′ )) ≥ (1 − ǫ) ωn (r ′)n . This gives a contradiction. Theorem 54.2. (cf. Theorem I.12.3) For any w > 0 there exist τ = τ (w) > 0, K = K(w) < ∞ and ρ = ρ(w) > 0 with the following property. Suppose that g(·) is a Ricci flow on a closed three-manifold M, defined for t ∈ [0, T ), with Φ-almost nonnegative curvature. Let (x0 , t0 ) be a spacetime point and let r0 > 0 be a radius with t0 ≥ 4τ r02 and r02 Φ(r0−2 ) < ρ. Suppose that volt0 (B(x0 , r0 )) ≥ wr03 and the time-t0 sectional curvatures on B(x0 , r0 ) are bounded below by − r0−2 . Then R(x, t) ≤ Kr0−2 whenever t ∈ [t0 − τ r02 , t0 ] and distt (x, x0 ) ≤ 1 r . 4 0 Proof. Let τ0 (w), B(w) and C(w) be the constants of Corollary 45.13. Put τ (w) = 12 τ0 (w) and K(w) = C(w) + 2 τB(w) . The function ρ(w) will be specified in the course of the proof. 0 (w) Suppose that the theorem is not true. Take a counterexample with a point (x0 , t0 ) and a radius r0 > 0 such that the time-t0 ball B(x0 , r0 ) satisfies the assumptions of the theorem, but the conclusion of the theorem fails. We claim there is a counterexample coming from a point (b x0 , b t0 ) and a radius rb0 > 0, with the additional property that for any (x′0 , t′0 ) and r0′ ′ having t0 ∈ [b t0 − 2τ rb02 , b t0 ] and r0′ ≤ 21 rb0 , if volt′0 (B(x′0 , r0′ )) ≥ w(r0′ )3 and the time-t′0 sectional curvatures on B(x′0 , r0′ ) are bounded below by − (r0′ )−2 then R(x, t) ≤ K(r0′ )−2 whenever t ∈ [t′0 − τ (r0′ )2 , t′0 ] and distt (x, x′0 ) ≤ 41 r0′ . This follows from a pointpicking argument suppose that it is not true for the original x0 , t0 , r0 . Then there are (x′0 , t′0 ) and r0′ with t′0 ∈ [t0 − 2τ r02 , t0 ] and r0′ ≤ 12 r0 , for which the assumptions of the theorem hold but the conclusion does not. If the triple (x′0 , t′0 , r0′ ) satisfies the claim then we stop, and otherwise we iterate the procedure. The iteration must terminate, which provides the desired triple (b x0 , b t0 , rb0 ). Note that b t0 > t0 − 4τ r02 ≥ 0. We relabel (b x0 , b t0 , rb0 ) as (x0 , t0 , r0 ). For simplicity, let us assume that the time-t0 sectional curvatures on B(x0 , r0 ) are strictly greater than − r0−2 ; the general case will follow from continuity. Let τ ′ > 0 be the largest number such that Rm(x, t) ≥ − r0−2 whenever t ∈ [t0 − τ ′ r02 , t0 ] and distt (x, x0 ) ≤ r0 . If τ ′ ≥ 2τ = τ0 (w) then Corollary 45.13 implies that R(x, t) ≤ Cr0−2 + B(t − t0 + 2τ r02 )−1 whenever t ∈ [t0 − 2τ r02 , t0 ] and distt (x, x0 ) ≤ 14 r0 . In particular, R(x, t) ≤ K whenever t ∈ [t0 − τ r02 , t0 ] and distt (x, x0 ) ≤ 41 r0 , which contradicts our assumption that the conclusion of the theorem fails. Now suppose that τ ′ < 2τ . Put t′ = t0 − τ ′ r02 . From estimates on the length and volume distortion under the Ricci flow, we know that there are numbers α = α(w) > 0 and w ′ = w ′ (w) > 0 so that the time-t′ ball B(x0 , αr0) has volume at least w ′ (αr0 )3 . From Lemma 54.1, there is a subball B(x′ , r ′ ) ⊂ B(x0 , αr0 ) with r ′ ≥ cαr0 and vol(B(x′ , r ′ )) ≥ 12 ω3 (r ′ )3 . From the preceding pointpicking argument, we have the estimate R(x, t) ≤ K(r ′ )−2 whenever t ∈ [t′ − τ (r ′ )2 , t′ ] and distt (x, x′ ) ≤ 41 r ′ . From the Φ-almost nonnegative curvature, we have


101

a bound | Rm |(x, t) ≤ const. K (r ′ )−2 + const. Φ(K(r ′ )−2 ) at such a point (see (52.4)). If ρ(w) is taken sufficiently small then we can ensure that r0 is small enough, and hence r ′ is small enough, to make K (r ′ )−2 + Φ(K(r ′ )−2 ) ≤ 2 K (r ′)−2 . Then we can apply Theorem 53.1 to a time interval ending at time t′ , after a redefinition of its constants, to obtain a bound of the form R(x, t′ ) ≤ K ′ (r ′ )−2 whenever distt (x, x′ ) ≤ 10r0 , where K ′ is related to the constant K of Theorem 53.1. (We also obtain a similar estimate at times slightly less than t′ .) Thus at such a point, Rm(x, t′ ) ≥ − Φ(K ′ (r ′ )−2 ). If we choose ρ(w) to be sufficiently small to force r ′ to be sufficiently small to force − Φ(K ′ (r ′ )−2 ) > − r0−2 then we have Rm > −r0−2 on Bt′ (x0 , r0 ) ⊂ Bt′ (x′ , 10r0 ), which contradicts the assumed maximality of τ ′ . We note that in the application of Theorem 53.1 at the end of the proof, we must take into account the extra hypothesis, in the notation of Theorem 53.1, that r02 Φ(r0−2 ) < ρ (see Remark 53.2). This will be satisfied if the r0 in Theorem 53.1 is small enough, which is ensured by taking the ρ of Theorem 54.2 small enough. 55. I.12.4. Small balls with strongly negative curvature are volume-collapsed In this section we show that under certain hypotheses, if the infimal sectional curvature on an r-ball is exactly − r −2 then the volume of the ball is small compared to r 3 . Corollary 55.1. (cf. Corollary I.12.4) For any w > 0, one can find ρ > 0 with the following property. Suppose that g(·) is a Φ-almost nonnegatively curved Ricci flow solution on a closed three-manifold M, defined for t ∈ [0, T ) with T ≥ 1. If B(x0 , r0 ) is a metric ball at time t0 ≥ 1 with r0 < ρ and if inf x∈B(x0 ,r0 ) Rm(x, t0 ) = −r0−2 then vol(B(x0 , r0 )) ≤ wr03. Proof. Fix w > 0. The number ρ will be specified in the course of the proof. Suppose that the corollary is not true, i.e. there is a Ricci flow solution as in the statement of the corollary along with a metric ball B(x0 , r0 ) at a time t0 ≥ 1 so that inf x∈B(x0 ,r0 ) Rm(x, t0 ) = −r0−2 and vol(B(x0 , r0 )) > wr0n . The idea is to use Theorem 54.2, along with the Φ-almost nonnegative curvature, to get a double-sided sectional curvature bound on a smaller ball at an earlier time. Then one goes forward in time using Theorem 53.1, along with the Φ-almost nonnegative curvature, to get a lower sectional curvature bound on the original ball, thereby obtaining a contradiction. 1

Looking at the hypotheses of Theorem 54.2, if we require r0 < (4τ )− 2 then 4τ r02 < 1 ≤ t0 . From Theorem 54.2, R(x, t) ≤ Kr0−2 whenever t ∈ [t0 − τ r02 , t0 ] and distt (x, x0 ) ≤ 14 r0 , provided that r0 is small enough that r02 Φ(r0−2 ) is less than the ρ of Theorem 54.2. If in addition r0 is sufficiently small then it follows that | Rm(x, t)| ≤ const. Φ(Kr0−2 ) ≤ r0−2 .

From the Bishop-Gromov inequality and the bounds on length and volume distortion under Ricci flow, there is a small number c so that we are ensured that | Rm(x, t)| ≤ (cr0 )−2 for all (x, t) satisfying distt0 −(cr0 )2 (x, x0 ) < cr0 and t ∈ [t0 − (cr0 )2 , t0 ], and in addition the volume of B(x0 , cr0 ) at time t0 − (cr0 )2 is at least c(cr0 )3 . Choosing the constant A of Theorem 53.1 appropriately in terms of c, we can apply Theorem 53.1 to the ball B(x0 , cr0 ) and the time interval [t0 − (cr0 )2 , t0 ] to conclude that at time t0 , R(·, t0 ) ≤ B(x0 ,r0 )

102


K(A) (cr0 )−2 , where K(A) is as in the statement of Theorem 53.1. From the Φ-almost nonnegative curvature condition, (55.2) Rm ≥ − Φ K(A) (cr0 )−2 . B(x0 ,r0 )

If r0 is sufficiently small then we contradict the assumption that inf Rm(x, t0 )

−

B(x0 ,r0 )

1 . r02

=

56. I.13.1. Thick-thin decomposition for nonsingular flows The main result of this section says that if g(·) is a Ricci flow solution on a closed oriented three-dimensional manifold M that exists for t ∈ [0, ∞) then for large t, (M, g(t)) has a thick-thin decomposition. A fuller description is in Sections 87-92. We assume that at time zero, the sectional curvatures are bounded below by −1. This can always be achieved by rescaling the initial metric. Then we have the Φ-almost nonnegative curvature result of (B.8). If the metric g(t) has nonnegative sectional curvature then it must be flat, as we are assuming that the Ricci flow exists for all time. Let us assume that g(t) is not flat, so it has some negative sectional curvature. Given x ∈ M, consider the time-t ball Bt (x, r). Clearly if r is sufficiently small then Rm > − r −2 , while if r is sufficiently large (maybe Bt (x,r) > − r −2 . Let rb(x, t) > 0 greater than the diameter of M) then it is not true that Rm Bt (x,r) −2 be the unique number such that inf Rm = − rb . Let Mthin (w, t) be the set of points x ∈ M for which (56.1)

Bt (x,b r)

vol(Bt (x, rb(x, t))) < w rb(x, t)3 .

Put Mthick (w, t) = M − Mthin (w, t).

As the statement of (B.8) is invariant under parabolic rescaling (although we must take t ≥ t0 for (B.8) to apply), if t ≥ t0 and we are interested in the Ricci flow at time t then we can apply Theorem 53.1, Theorem 54.2 and Corollary 55.1 to the rescaled flow g(t′ ) = t−1 g(tt′ ).√ From Corollary 55.1, for any w > 0 we can find ρb = ρb(w) > 0 so that if rb(x, t) < ρb t then x ∈ Mthin (w, t), provided that t is sufficiently large (depending on w). Equivalently, if t is sufficiently large (depending on w) and x ∈ Mthick (w, t) then √ rb(x, t) ≥ ρb t.

Theorem 56.2. (cf. I.13.1) There are numbers T = T (w) > 0, ρ = ρ(w) > 0 √ and −1 K = K(w) < ∞ so that if t ≥ T and x ∈ Mthick (w, t) then | Rm | ≤ Kt on Bt (x, ρ t), √ √ 3 1 and vol(Bt (x, ρ t)) ≥ 10 w ρ t .

Proof. The method of proof is the same as in Corollary 55.1. By assumption, Rm ≥ Bt (x,b r(x,t)) √ − rb(x, t)−2 and vol(Bt (x, rb(x, t))) ≥ w rb(x, t)3 . As rb(x, t) ≥ ρb t, for any c ∈ (0, 1) we have


Rm

√ Bt (x,cb ρ t)

(56.3)

103

≥ − (cb ρ)−2 t−1 . By the Bishop-Gromov inequality,

√ vol(Bt (x, cb ρ t)) ≥ ≥

R

√ cρ b t r b(x,t)

0

R1 0

sinh2 (u) du 2

sinh (u) du

3 1 w (cb ρ)3 t 2 . 10

w rb(x, t)3 ≥

3

R1 0

1

3

2

sinh (u) du

w (cb ρ )3 t 2

w Considering Theorem 54.2 with its w replaced by 10 , if c = c(w) is taken sufficiently small √ 2 (to ensure t ≥ 4τ (cb ρ t) ) and t is larger than a certain w-dependent constant (to ensure √ √ Φ((cb ρ t)−2 ) √ < ρ) then we can apply Theorem 54.2 with r = cb ρ t to obtain R(x′ , t′ ) ≤ 0 −2 (cb ρ t) √ K ′ (w)c−2 ρb−2 t−1 whenever t′ ∈ [t − τ c2 ρb2 t, t] and distt′ (x′ , x) ≤ 14 cb ρ t. From the Φ-almost nonnegative curvature (see (52.4)),

(56.4)

| Rm |(x′ , t′ ) ≤ const. K ′ c−2 ρb−2 t−1 + const. Φ(K ′ c−2 ρb−2 t−1 ),

which is bounded above by 2 const. K ′ c−2 ρb−2 t−1 if t is larger than a certain w-dependent constant. Then from length and volume distortion √ estimates for the Ricci flow, we obtain a √ lower volume bound vol(Bt′ (x, c′ ρb t)) ≥ w ′ (c′ ρb t)3 on a smaller ball of controlled radius, √ for ′ ′ ′′ some c = c√ (w). Using Theorem 53.1, we finally obtain an upper bound R ≤ K (w)(cb ρ t)−2 nonnegative curvature, an upper bound of the on Bt (x, cb ρ t) and hence, by the Φ-almost √ form | Rm | ≤ K(w) t−1 on Bt (x, cb ρ t), provided that t ≥ T for an appropriate T = T (w). ρ, the theorem follows. Taking ρ = cb Remark 56.5. The use of Theorem 53.1 in the √ proof of Theorem 56.2 also gives an upper bound | Rm |(x, t) ≤ K(A, w) t−1 on Bt (x, Aρ t) if x ∈ Mthick (w, t), for any A > 0. We now take w sufficiently small. Then for large t, Mthick (w, t) has a boundary consisting of tori that are incompressible in M and the interior of Mthick (w, t) admits a complete Riemannian metric with constant sectional curvature − 41 and finite volume; see Sections 90 and 91. In addition, Mthin (w, t) is a graph manifold; see Section 92. 57. Overview of Ricci Flow with Surgery on Three-Manifolds [52] The paper [52] is concerned with the Ricci flow on compact oriented 3-manifolds. The main difference with respect to [51] is that singularity formation is allowed, so the paper deals with a “Ricci flow with surgery”. The main part of the paper is concerned with setting up the surgery procedure and showing that it is well-defined, in the sense that surgery times do not accumulate. In addition, the long-time behavior of a Ricci flow with surgery is analyzed. The paper can be divided into three main parts. Sections II.1-II.3 contain preparatory material about ancient solutions, the so-called standard solution and the geometry at the first singular time. Sections II.4-II.5 set up the surgery procedure and prove that it is well-defined. Sections II.6-II.8 analyze the long-time behavior.

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57.1. II.1-II.3. Section II.1 continues the analysis of three-dimensional κ-solutions from I.11. From I.11, any κ-solution contains an “asymptotic soliton”, a gradient shrinking soliton that arises as a rescaled limit of the κ-solution as t → −∞. It is shown that any such gradient shrinking soliton must be a shrinking round cylinder R × S 2 , its Z2 -quotient R ×Z2 S 2 or a finite quotient of the round shrinking S 3 . Using this, one obtains a finer description of the κ-solutions. In particular, any compact κ-solution must be isometric to a finite quotient of the round shrinking S 3 , or diffeomorphic to S 3 or RP 3 . It is shown that there is a universal number κ0 > 0 so that any κ-solution is a finite quotient of the round shrinking S 3 or is a κ0 -solution. This implies universal derivative bounds on the scalar curvature of a κ-solution. Section II.2 defines and analyzes the Ricci flow of the so-called standard solution. This is a Ricci flow on R3 whose initial metric is a capped-off half cylinder. The surgery procedure will amount to gluing in a truncated copy of the time-zero slice of the standard solution. Hence one needs to understand the Ricci flow on the standard solution itself. It is shown that the Ricci flow of the standard solution exists on a maximal time interval [0, 1), and the solution goes singular everywhere as t → 1.

The geometry of the solution at the first singular time T (assuming that there is one) is considered in II.3. Put Ω = {x ∈ M : lim supt→T − | Rm(x, t)| < ∞}. Then Ω is an open subset of M, and x ∈ M − Ω if and only if limt→T − R(x, t) = ∞. If Ω = ∅ then for t slightly less than T , the manifold (M, g(t)) consists of nothing but high-scalar-curvature regions. Using Theorem I.12.1, one shows that M is diffeomorphic to S 1 × S 2 , RP 3 #RP 3 or a finite isometric quotient of S 3 . If Ω 6= ∅ then there is a well-defined limit metric g on Ω, with scalar curvature function R. The set Ω could a priori have an infinite number of connected components, for example if an infinite number of distinct 2-spheres simultaneously shrink to points at time T . For small ρ > 0, put Ωρ = {x ∈ Ω : R(x) ≤ ρ−2 }, a compact subset of M. The connected components of Ω can be divided into those that intersect Ωρ and those that do not. If a connected component does not intersect Ωρ then it is a “capped ǫ-horn” (consisting of a hornlike end capped off by a ball or a copy of RP 3 − B 3 ) or a “double ǫ-horn” (with two hornlike ends). If a connected component of Ω does intersect Ωρ then it has a finite number of ends, each being an ǫ-horn. Topologically, the surgery procedure of II.4 will amount to taking each connected component of Ω that intersects Ωρ , truncating each of its ǫ-horns and gluing a 3-ball onto each truncated horn. The connected components of Ω that do not intersect Ωρ are thrown away. Call the new manifold M ′ . At a time t slightly less than T , the region M − Ωρ consists of high-scalar-curvature regions. Using the characterization of such regions in I.12.1, one shows that M can be reconstructed from M ′ by taking the connected sum of its connected components, along possibly with a finite number of S 1 × S 2 and RP 3 factors. 57.2. II.4-II.5. Section II.4 defines the surgery procedure. A Ricci flow with surgery consists of a sequence of smooth 3-dimensional Ricci flows on adjacent time intervals with the property that for any two adjacent intervals, there is a a compact 3-dimensional submanifoldwith-boundary that is common to the final slice of the first time interval and the initial slice of the second time interval.


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There are two a priori assumptions on a Ricci flow with surgery, the pinching assumption and the canonical neighborhood assumption. The pinching assumption is a form of Hamilton-Ivey pinching. The canonical neighborhood assumption says that every spacetime point (x, t) with R(x, t) ≥ r(t)−2 has a neighborhood which, after rescaling, is ǫ-close to one of the neighborhoods that occur in a κ-solution or in a time slice of the standard solution. Here ǫ is a small but universal constant and r(·) is a decreasing function, which is to be specified. One wishes to define a Ricci flow with surgery starting from any compact oriented 3manifold, say with a normalized initial metric. There are various parameters that will enter into the definition : the above canonical neighborhood scale r(·), a nonincreasing function δ(·) that decays to zero, the truncation scale ρ(t) = δ(t)r(t) and the surgery scale h. In order to show that one can construct the Ricci flow with surgery, it turns out that one wants to perform the surgery only on necks with a radius that is very small compared to the canonical neighborhood scale; this is the role of the parameter δ(·). Suppose that the Ricci flow with surgery is defined at times less than T , with the a priori assumptions satisfied, and goes singular at time T . Define the open subset Ω ⊂ M as before and construct the compact subset Ωρ ⊂ M using ρ = ρ(T ). Any connected component N of Ω that intersects Ωρ has a finite number of ends, each of which is an ǫ-horn. This means that each point in the horn is in the 1center of2an ǫ-neck, i.e. has a neighborhood that, after 1 rescaling, is ǫ-close to a cylinder − ǫ , ǫ × S . In II.4.3 it is shown that as one goes down the end of the horn, there is a self-improvement phenomenon; for any δ > 0, one can find h < δρ so that if a point x in the horn has R(x) ≥ h−2 then it is actually in the center of a δ-neck. With δ = δ(T ), let h be the corresponding number. One then cuts off the ǫ-horn at a 2-sphere in the center of such a δ-neck and glues in a copy of a rescaled truncated standard solution. One does this for each ǫ-horn in N and each connected component N that intersects Ωρ , and throws away the connected components of Ω that do not intersect Ωρ . One lets the new manifold evolve under the Ricci flow. If one encounters another singularity then one again performs surgery. Based on an estimate on the volume change under a surgery, one concludes that a finite number of surgeries occur in any finite time interval. (However, one is not able to conclude from volume arguments that there is a finite number of surgeries altogether.) The preceding discussion was predicated on the condition that the a priori assumptions hold for all times. For the Ricci flow before the first surgery time, the pinching condition follows from the Hamilton-Ivey result. One shows that surgery can be performed so that it does not make the pinching any worse. Then the pinching condition will hold up to the second surgery time, etc. The main issue is to show that one can choose the parameters r(·) and δ(·) so that one knows a priori that the canonical neighborhood assumption, with parameter r(·), will hold for the Ricci flow with surgery. (For any singularity time T , one needs to know that the canonical neighborhood assumption holds for t ∈ [0, T ) in order to do the surgery at time T .)

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As a preliminary step, in Lemma II.4.5 it is shown that after one glues in a standard solution, the result will still look similar to a standard solution, for as long of a time interval as one could expect, unless the entire region gets removed by some exterior surgery. The result of II.5 is that the time-dependent parameters r(·) and δ(·) can be chosen so as to ensure that the a priori assumptions hold. The normalization of the initial metric implies that there is a time interval [0, C], for a universal constant C, on which the Ricci flow is smooth and has explicitly bounded curvature. On this time interval the canonical neighborhood assumption holds vacuously, if r(·) [0,C] is sufficiently small. To handle later times, the strategy is to divide [ǫ, ∞) into a countable sequence of finite time intervals and proceed by induction. In II.5 the intervals {[2j−1 ǫ, 2j ǫ]}∞ j=1 are used, although the precise choice of intervals is immaterial. We recall from I.12.1 that in the case of smooth flows, the proof of the canonical neighborhood assumption used the fact that one has κ-noncollapsing. It is not immediate that the method of proof of I.12.1 extends to a Ricci flow with surgery. (It is exactly for this reason that one takes δ(·) to be a time-dependent function which can be forced to be very small.) Hence one needs to prove κ-noncollapsing and the canonical neighborhoood assumption together. The main proposition of II.5 says that there are decreasing sequences rj , κj and δ j so that if δ(·) is a function with δ(·) [2j−1 ǫ,2j ǫ] ≤ δ j for each j > 0 then any Ricci flow with surgery, defined with the parameters r(·) and δ(·), is κj -noncollapsed on the time interval [2j−1 ǫ, 2j ǫ] at scales less than ǫ and satisfies the canonical neighborhood assumption there. Here we take r(·) [2j−1 ǫ,2j ǫ) = rj .

The proof of the proposition is by induction. Suppose that it is true for 1 ≤ j ≤ i. In the induction step, besides defining the parameters ri+1 , κi+1 and δ i+1 , one redefines δ i . As one only redefines δ in the previous interval, there is no circularity.

The first step of the proof, Lemma II.5.2, consists of showing that there is some κ > 0 so that for any r, one can find δ = δ(r) > 0 with the following property. Suppose that g(·) is a Ricci flow with surgery defined on [0, T ), with T ∈ [2i ǫ, 2i+1 ǫ], that satisfies the proposition on [0, 2iǫ]. Suppose that it also satisfies the canonical neighborhood assumption with parameter r on [2i ǫ, T ), and is constructed using a function δ(·) that satisfies δ(t) ≤ δ on [2i−1 ǫ, T ). Then it is κ-noncollapsed at all scales less than ǫ. The proof of this lemma is along the lines of the κ-noncollapsing result of I.7, with some important modifications. One again considers the L-length of curves γ(τ ) starting from the point at which one wishes to prove the noncollapsing. One wants to find a spacetime point (x, t), with t ∈ [2i−1 ǫ, 2i ǫ], at which one has an explicit upper bound on l. In I.7, the analogous statement came from a differential inequality for l. In order to use this differential equality in the present case, one needs to know that any curve γ(τ ) that is competitive to be a minimizer for L(x, t) will avoid the surgery regions. Choosing δ small enough, one can ensure that the surgeries in the time interval [2i−1 ǫ, T ) are done on very long thin necks. Using Lemma II.4.5, one shows that a curve γ(τ ) passing near such a surgery region obtains a large value of L, thereby making it noncompetitive as a minimizer for L(x, t). (This is the underlying reason that the surgery parameter δ(·) is chosen in a time-dependent way.) One


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also chooses the point (x, t) so that there is a small parabolic neighborhood around it with a bound on its geometry. One can then run the argument of I.7 to prove κ-noncollapsing in the time interval [2i ǫ, T ). The proof of the main proposition of II.5 is now by contradiction. Suppose that it is not α true. Then for some sequences r α → 0 and δ → 0, for each α there is a counterexample α to the proposition with ri+1 ≤ r α and δi , δi+1 ≤ δ . That is, there is some spacetime point in the interval [2i ǫ, 2i+1 ǫ] at which the canonical neighborhood assumption fails. Take a first such point (xα , tα ). By Lemma II.5.2, one has κ-noncollapsing up to the time of this first counterexample. Using this noncollapsing, one can consider taking rescaled limits. If there are no surgeries in an appropriate-sized backward spacetime region around (xα , tα ) then one can extract a convergent subsequence as α → ∞ and construct, as in the proof of Theorem I.12.1, a limit κ-solution, thereby giving a contradiction. If there are nearby interfering surgeries then one argues, using Lemma II.4.5, that the point (xα , tα ) is in fact in a canonical neighborhood, again giving a contradiction. Having constructed the Ricci flow with surgery, if the initial manifold is simply-connected then according to [24, 25, 53], there is a finite extinction time. One then concludes that the Poincaré Conjecture holds. 57.3. II.6-II.8. Sections II.6 and II.8 analyze the large-time behavior of a Ricci flow with surgery. Section II.6 establishes back-and-forth curvature estimates. Proposition II.6.3 is an analog of Theorem I.12.2 and Proposition II.6.4 is an analog of Theorem I.12.3. The proofs are along the lines of the proofs of Theorems I.12.2 and I.12.3, but are complicated by the possible presence of surgeries. The thick-thin decomposition for large-time slices is considered in Section II.7. Using monotonicity arguments of Hamilton, it is shown that as t → ∞ the metric on the w-thick part M + (w, t) becomes closer and closer to having constant negative sectional curvature. Using a hyperbolic rigidity argument of Hamilton, it is stated that the hyperbolic pieces stabilize in the sense that there is a finite collection {(Hi , xi )}ki=1 of pointed finite-volume 3-manifolds of constant sectional curvature − 14 so that for large t, the metric gb(t) = 1t g(t) S on the w-thick part M + (w, t) approaches the metric on the w-thick part of ki=1 Hi . It is stated that the cuspidal tori (if any) of the hyperbolic pieces are incompressible in M. To show this (following Hamilton), if there is a compressing 3-disk then one takes a minimal such 3-disk, say of area A(t), and shows from a differential inequality for A(·) that for large t the function A(t) is negative, which is a contradiction. Theorem II.7.4, a statement in Riemannian geometry, characterizes the thin part M − (w, t), for small w and large t, as a graph manifold. The main hypothesis of the theorem is that for each point x, there is a radius ρ = ρ(x) so that the ball B(x, ρ) has volume at most wρ3 and sectional curvatures bounded below by − ρ−2 . In this sense the manifold is locally volume collapsed with respect to a lower sectional curvature bound. Section II.8 contains an alternative proof of the incompressibility of cuspidal tori, using the functional λ1 (g) = λ1 (−4△+R). (At the beginning of Section 93, we give a simpler argument 2 using the functional Rmin (g) vol(M, g) 3 .) More generally, the functional λ1 (g) is used to

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define a topological invariant that determines the nature of the geometric decomposition. First, the manifold M admits a Riemannian metric g with λ1 (g) > 0 if and only if it admits a Riemannian metric with positive scalar curvature, which in turn is equivalent to saying that M is diffeomorphic to a connected sum of S 1 × S 2 ’s and round quotients of S 3 . If M 2 does not admit a Riemannian metric with λ1 > 0, let λ be the supremum of λ1 (g)·vol(M, g) 3 over all Riemannian metrics g on M. If λ = 0 then M is a graph manifold. If λ < 0 then the geometric decomposition of M contains a nonempty hyperbolic piece, with total volume 3 2 − 32 λ 2 . The proofs of these statements use the monotonicity of λ(g(t)) · vol(M, g(t)) 3 , when it is nonpositive, under a smooth Ricci flow. The main work is to show that in the case 2 of a Ricci flow with surgery, one can choose δ(·) so that λ(g(t)) · vol(M, g(t)) 3 is arbitrarily close to being nondecreasing in t.

58. II. Notation and terminology B(x, t, r) denotes the open metric ball of radius r, with respect to the metric at time t, centered at x. P (x, t, r, ∆t) denotes a parabolic neighborhood, that is the set of all points (x′ , t′ ) with x′ ∈ B(x, t, r) and t′ ∈ [t, t + ∆t] or t′ ∈ [t + ∆, t], depending on the sign of ∆t. Definition 58.1. We say that a Riemannian manifold (M1 , g1 ) has distance ≤ ǫ in the C N topologyPto another Riemannian manifold (M2 , g2 ) if there is a diffeomorphism φ : M2 → M1 1 so that |I| ≤ N |I|! k ∇I (φ∗ g1 − g2 ) k∞ ≤ ǫ. An open set U in a Riemannian 3-manifold M is an ǫ-neck if modulo rescaling, it has distance less than ǫ, in the C [1/ǫ]+1-topology, to the product of the round 2-sphere of scalar curvature 1 (and therefore Gaussian curvature 21 ) with an interval I of length greater than 2ǫ−1 . If a point x ∈ M and a neighborhood U of x are specified then we will understand that “distance” refers to the pointed topology, where the basepoint in S 2 × I projects to the center of I.

We make a similar definition of ǫ-closeness in the spacetime case, where ∇I now includes time derivatives. A subset of the form U ×[a, b] ⊂ M ×[a, b], where U ⊂ M is open, sitting in the spacetime of a Ricci flow is a strong ǫ-neck if after parabolic rescaling and time shifting, it has distance less than ǫ to the product Ricci flow defined on the time interval [−1, 0] which, at its final time, is isometric to the product of a round 2-sphere of scalar curvature 1 with an interval of length greater than 2ǫ−1 . (Evidently, the time-0 slice of the product has 3-dimensional scalar curvature equal to 1.) Our definition of an ǫ-neck differs in an insubstantial way from that on p. 1 of II. In the definition of [52], a ball B(x, t, ǫ−1 r) is called an ǫ-neck if, after rescaling the metric with a factor r −2 , it is ǫ-close, i.e. has distance less than ǫ, to the corresponding subset of the standard neck S 2 × I... (italicized words added by us). (The issue is that a large metric ball in the cylinder R × S 2 does not have a smooth boundary.) Clearly after a slight change of the constants, an ǫ-neck in our sense is contained in an ǫ-neck in the sense of [52], and vice versa. An important fact is that the notion of (x, t) being contained in an ǫ-neck is an open condition with respect to the pointed C [1/ǫ]+1-topology on Ricci flow solutions.


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With an ǫ-approximation f : S 2 × I → U being understood, a cross-sectional sphere in U will mean the image of S 2 × {λ} under f , for some λ ∈ (−ǫ−1 , ǫ−1 ). Any curve γ in U that intersects both f (S 2 ×{ǫ−1 }) and f (S 2 ×{−ǫ−1 }) must intersect each cross-sectional sphere. If γ is a minimizing geodesic and ǫ is small enough then γ will intersect each cross-sectional sphere exactly once. There is a typo in the definition of a strong ǫ-neck in [52] : the parabolic neighborhood should be P (x, t, ǫ−1 r, −r 2 ), i.e. it should go backward in time rather than forward. We note that the time interval involved in the definition of strong ǫ-neck, i.e. 1 after rescaling, is different than the rescaled time interval ǫ−1 in Theorem 52.7. In the next definition, I is an open interval and B 3 is an open ball. Definition 58.2. A metric on S 2 × I such that each point is contained in some ǫ-neck is called an ǫ-tube, or an ǫ-horn, or a double ǫ-horn, if the scalar curvature stays bounded on both ends, stays bounded on one end and tends to infinity on the other, or tends to infinity on both ends, respectively. A metric on B 3 or RP 3 − B 3 , such that each point outside some compact subset is contained in an ǫ-neck, is called an ǫ-cap or a capped ǫ-horn, if the scalar curvature stays bounded or tends to infinity on the end, respectively. An example of an ǫ-tube is S 2 × (−ǫ−1 , ǫ−1 ) with the product metric. For a relevant example of an ǫ-horn, consider the metric 1 r 2 dθ2 (58.3) g = dr 2 + 8 ln 1r on (0, R) × S 2 , where dθ2 is the metric on S 2 with R = 1. From [6], q the metric g models a rotationally symmetric neckpinch. Rescaling around r0 , we put s = 8 ln r10 rr0 − 1 and find   ! 1 8 ln r0 1 1 (58.4) 2+ s + O(s2) dθ2 . g = ds2 + 1 + q 1 2 1 r0 ln r0 8 ln r0

− 41 then the region with s ∈ (− ǫ−1 , ǫ−1 ) will be ǫFor small r0 , if we take ǫ ∼ ln r10 biLipschitz close to the standard cylinder. Note that as r0 → 0, the constant ǫ improves; this is related to Lemma 71.1.

An ǫ-cap is the result of capping off an ǫ-tube by a 3-ball or RP 3 − B 3 with an arbitrary metric. A capped ǫ-horn is the result of capping off an ǫ-horn by a 3-ball or RP 3 − B 3 with an arbitrary metric. Remark 58.5. Throughout the rest of these notes, ǫ denotes a small positive constant that is meant to be universal. The precise value of ǫ is unspecified. If the statement of a lemma or theorem invokes ǫ then the statement is meant to be true uniformly with respect to the other variables, provided ǫ is sufficiently small. When going through the proofs one is allowed to make ǫ small enough so that the arguments work, but one is only allowed to make a finite number of such reductions.

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Lemma 58.6. Let U be an ǫ-neck in an ǫ-tube (or horn) and let S be a cross-sectional sphere in U. Then S separates the two ends of the tube (or horn). Proof. Let W denote the tube (or horn). As any point m ∈ W lies in some ǫ-neck, there is a unique lowest eigenvalue of the Ricci operator Ric ∈ End(Tm W ) at m. Let ξm ⊂ Tm W be the corresponding eigenspace. As m varies, the ξm ’s form a smooth line field ξ on W , to which S is transverse. Suppose that S does not separate the two ends of W . Then S represents a trivial element of H2 (S 2 × I) ∼ = π2 (S 2 × I) and there is an embedded 3-disk D ⊂ W for which ∂D = S. This contradicts the fact that the line field ξ is transverse to S and extends over D.

59. II.1. Three-dimensional κ-solutions This section is concerned with properties of three-dimensional oriented κ-solutions. For brevity, in the rest of these notes we will generally omit the phrases “three-dimensional” and “oriented”. If (M, g(·)) is a κ-solution then its topology is easy to describe. By definition, (M, g(t)) has nonnegative sectional curvature. If it does not have strictly positive curvature then the universal cover splits off a line (see Theorem A.7), from which it follows (using Corollary 40.1 and the κ-noncollapsing) that (M, g(·)) is a standard shrinking cylinder R × S 2 or its Z2 quotient R ×Z2 S 2 . If (M, g(t)) has strictly positive curvature and M is compact then it is diffeomorphic to a spherical space form [35]. If (M, g(t)) has strictly positive curvature and M is noncompact then it is diffeomorphic to R3 [19]. The lemmas in this section give more precise geometric information. Recall that Mǫ consists of the points in a κ-solution which are not the center of an ǫ-neck. Lemma 59.1. If (M, g(t)) is a time slice of a noncompact κ-solution and Mǫ 6= ∅ then there is a compact submanifold-with-boundary X ⊂ M so that Mǫ ⊂ X, X is diffeomorphic to B 3 or RP 3 − B 3 , and M − int(X) is diffeomorphic to [0, ∞) × S 2 . Proof. If (M, g(t)) does not have positive sectional curvature and Mǫ 6= ∅ then M must be isometric to R×Z2 S 2 , in which case the lemma is easily seen to be true with X diffeomorphic to RP 3 − B 3 . Suppose that (M, g(t)) has positive sectional curvature. Choose x ∈ Mǫ . Let γ : [0, ∞) → M be a ray with γ(0) = x. As Mǫ is compact, there is some a > 0 so that if t > a then γ(t) ∈ / Mǫ . We can cover (a, ∞) by open intervals Vj so that γ is a geodesic Vj

segment in an ǫ-neck of rescaled length approximately 2ǫ−1 . Then we can find a cover of (a, ∞) by linearly ordered open intervals Ui , refining the previous cover, so that 1 −1 ǫ . 1. The rescaled length of γ is approximately 10 Ui 2. Choosing some xi ∈ Ui ∩ Ui+1 , the rescaled length (with rescaling at xi ) of γ is Ui ∩Ui+1 1 −1 approximately 40 ǫ and γ lies in an ǫ-neck Wi centered at xi . Ui ∩Ui+1


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Let φi be projection on the first factor in the assumed diffeomorphism Wi ∼ = (−ǫ−1 , ǫ−1 ) × −1 S . If ǫ is sufficiently small then the composition φi ◦ γ Ui : Ui → (−ǫ , ǫ−1 ) is a diffeo )) ⊂ Wi and pi = (φi ◦ γ )−1 ◦ φi : morphism onto its image. Put Ni = φ−1 (Im(φ ◦ γ i i Ui Ui Ni → Ui ⊂ (a, ∞). Then pi is ǫ-close to being a Riemannian submersion and on overlaps Ni ∩ Ni+1 , the maps pi and pi+1 are C K -close. Choosing an appropriate partition of unity P {bi } subordinate to the Ui ’s, if ǫ is small then the function f = i bi pi is a submersion from S 2 −1 (t, ∞). i Ni to (a, ∞). The fiber is seen to be S . Given t ∈ (a, ∞), put X = M − f −1 2 Then M − int(X) = f ([t, ∞)) is diffeomorphic to [0, ∞) × S . 2

Recall from Section 46 that we have an exhaustion of M by certain convex compact subsets. As M is one-ended, the subsets have connected boundary. As in Section 46, if the boundary of such a subset intersects an ǫ-neck then the intersection will be a nearly cross-sectional 2-sphere in the ǫ-neck. Hence with an appropriate choice of t, the set X will be isotopic to one of our convex subsets and so diffeomorphic to a closed 3-ball. Lemma 59.2. If (M, g(t)) is a time slice of a κ-solution with Mǫ = ∅ then the Ricci flow is the evolving round cylinder R × S 2 .

Proof. By assumption, each point (x, t) lies in an ǫ-neck. If ǫ is sufficiently small then piecing the necks together, we conclude that M must be diffeomorphic to S 1 × S 2 or R × S 2 ; see f is R × S 2 . the proof of Lemma 59.1 for a similar argument. Then the universal cover M As it has nonnegative sectional curvature and two ends, Toponogov’s theorem implies that f, e f (M g (t)) splits off an R-factor. Using the strong maximum principle, the Ricci flow on M f, e splits off an R-factor; see Theorem A.7. Using Corollary 40.1, it follows that (M g (t)) is 2 the evolving round cylinder R × S . From the κ-noncollapsing, the quotient M cannot be S 1 × S 2. A κ-solution has an asymptotic soliton (Section 39) that is either compact or noncompact. If the asymptotic soliton of a compact κ-solution (M, g(·)) is also compact then it must be a shrinking quotient of the round S 3 [35], so the same is true of M. Lemma 59.3. If a κ-solution (M, g(·)) is compact and has a noncompact asymptotic soliton then M is diffeomorphic to S 3 or RP 3 . Proof. We use Corollary 48.1 in Section 48. First, we claim that the time slices of the type-D κ-solutions of Corollary 48.1 have a universal upper bound on maxM R · diam(M)2 . To see this,√we can rescale at the point x ∈ Mǫ by R(x), after which the diameter is bounded above by 2 α. We then use Theorem 46.1 to get an upper bound on the rescaled scalar curvature, which proves the claim. Given an upper bound on maxM R · diam(M)2 , the asymptotic soliton cannot be noncompact. Thus we are in case C of Corollary 48.1. Take a sequence ti → −∞ and choose points xi , yi ∈ Mǫ (ti ) as in Corollary 48.1.C. Rescale by R(xi , ti ) and take a subsequence that converges to a pointed Ricci flow solution (M∞ , (x∞ , t∞ )). The limit M∞ cannot be compact, as otherwise we would have a uniform upper bound on R · diam2 for (M, g(ti )), which would contradict the existence of the noncompact asymptotic soliton. Thus M∞ is a noncompact 1 κ-solution. We can find compact sets Xi ⊂ M containing B(xi , αR(xi , ti )− 2 ) so that {Xi }

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converges to a set X∞ ⊂ M∞ as in Lemma 59.1. Taking a further subsequence, we find 1 similar compact sets Yi ⊂ M containing B(yi , αR(yi, ti )− 2 ) so that {Yi} converges to a set Y∞ ⊂ M∞ as in Lemma 59.1. In particular, for large i, Xi and Yi are each diffeomorphic to either B 3 or RP 3 − B 3 . Considering a minimizing geodesic segment xi yi as in the statement of Corollary 48.1.C, we can use an argument as in the proof of Lemma 59.1 to construct a submersion from M − (Xi ∪ Yi ) to an interval, with fiber S 2 . Hence M is diffeomorphic to the result of gluing Xi and Yi along a 2-sphere. As M has finite fundamental group, no more than one of Xi and Yi can be diffeomorphic to RP 3 − B 3 . Thus M is diffeomorphic to S 3 or RP 3 . From Lemma 50.1, every ancient solution which is a κ-solution for some κ is either a κ0 -solution or a metric quotient of the round S 3 . Lemma 59.4. There is a universal constant η such that at each point of every ancient solution that is a κ-solution for some κ, we have estimates (59.5)

3

|∇R| < ηR 2 ,

|Rt | < ηR2 .

Proof. This is obviously true for metric quotients of the round S 3 . For κ0 -solutions it follows from the compactness result in Theorem 46.1, after rescaling the scalar curvature at the given point to be 1. It is sometimes useful to rewrite (59.5) as a pair of estimates on the spacetime derivatives of the quantity R−1 at points where R 6= 0: 1 η |(R−1 )t | < η. (59.6) |∇(R− 2 )| < , 2 Lemma 59.7. For every sufficiently small ǫ > 0 one can find C1 = C1 (ǫ) and C2 = C2 (ǫ) such that for each point (x, t) in every κ-solution there is a radius r ∈ [R(x, t)−1/2 , C1 R(x, t)−1/2 ] and a neighborhood B, B(x, t, r) ⊂ B ⊂ B(x, t, 2r), which falls into one of the four categories: (a) B is a strong ǫ-neck (more precisely, B is the slice of a strong ǫ-neck at its maximal time, and an appropriate parabolic neighborhood of B satisfies the condition to be a strong ǫ-neck), or (b) B is an ǫ-cap, or (c) B is a closed manifold, diffeomorphic to S 3 or RP 3 , or (d) B is a closed manifold of constant positive sectional curvature. Furthermore: • The scalar curvature in B at time t is between C2−1 R(x, t) and C2 R(x, t). 3 • The volume of B in cases (a), (b) and (c) is greater than C2−1 R(x, t)− 2 . • In case (b), there is an ǫ-neck U ⊂ B with compact complement in B (i.e. the end of B is entirely contained in the ǫ-neck) such that the distance from x to U is at least 10000R(x, t)−1/2 . • In case (c) the sectional curvature in B at time t is greater than C2−1 R(x, t). Remark 59.8. The statement of the lemma is slightly stronger than the corresponding statement in II.1.5, in that we have r ≥ R(x, t)−1/2 as opposed to r > 0.


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Proof. We may assume that we are talking about a κ0 -solution, as if M is a metric quotient of a round sphere then it falls into category (d) for any r > π(R(xi , ti )/6)−1/2 (since then M = B(xi , ti , r) = B(xi , ti , 2r)). Fix a small ǫ and suppose that the claim is not true. Then there is a sequence of κ0 solutions Mi that together provide a counterexample. That is, there is a sequence Ci → ∞ and a sequence of points (xi , ti ) ∈ Mi ×(−∞, 0] so that for any r ∈ [R(xi , ti )−1/2 , Ci R(xi , ti )−1/2 ] one cannot find a B between B(xi , ti , r) and B(xi , ti , 2r) falling into one of the four categories and satisfying the subsidiary conditions with parameter C2 = Ci . Rescale the metric by R(xi , ti ) and take a convergent subsequence of (Mi , (xi , ti )) to obtain a limit κ0 -solution (M∞ , (x∞ , t∞ )). Then for any r > 1, one cannot find a B∞ between B(x∞ , t∞ , r) and B(x∞ , t∞ , 2r) falling into one of the four categories and satisfying the subsidiary conditions for any parameter C2 . If M∞ is compact then for any r greater than the diameter of the time-t∞ slice of M∞ , B(x∞ , t∞ , r) = M∞ = B(x∞ , t∞ , 2r) falls into category (c) or (d). For the subsidiary conditions, M∞ clearly has a lower volume bound, a positive lower scalar curvature bound and an upper scalar curvature bound. As a compact κ0 -solution has positive sectional curvature, M∞ also has a lower sectional curvature bound. This is a contradiction. If M∞ is noncompact then Lemma 59.1 (or more precisely its proof) and Lemma 59.2 imply that for some r > 1, there will be a B between B(x∞ , t∞ , r) and B(x∞ , t∞ , 2r) falling into category (a) or (b). In case (b), by choosing the parameter r sufficiently large, the existence of the ǫ-neck U with the desired properties follows from the proof of Lemma 59.1. For the other subsidiary conditions, B clearly has a lower volume bound, a positive lower scalar curvature bound and an upper scalar curvature bound. This contradiction completes the proof of the lemma. 60. II.2. Standard solutions The next few sections are concerned with the properties of special Ricci flow solutions on M = R3 . We fix a smooth rotationally symmetric metric g0 which is the result of gluing a hemispherical-type cap to a half-infinite cylinder of scalar curvature 1. Among other properties, g0 is complete and has nonnegative curvature operator. We also assume that g0 has scalar curvature bounded below by 1. Remark 60.1. In Section 72 we will further specialize the initial metric g0 of the standard solution, for technical convenience in doing surgeries. Definition 60.2. A Ricci flow (R3 , g(·)) defined on a time interval [0, a) is a standard solution if it has initial condition g0 , the curvature | Rm | is bounded on compact time intervals [0, a′ ] ⊂ [0, a), and it cannot be extended to a Ricci flow with the same properties on a strictly longer time interval. It will turn out that every standard solution is defined on the time interval [0, 1). To motivate the next few sections, let us mention that the surgery procedure will amount to gluing in a truncated copy of (R3 , g0 ). The metric on this added region will then evolve as part of the Ricci flow that takes up after the surgery is performed. We will need to

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understand the behavior of the Ricci flow after performing a surgery. Near the added region, this will be modeled by a standard solution. Hence one first needs to understand the Ricci flow of a standard solution. The main results of II.2 concerning the Ricci flow on a standard solution (Sections 61-64) are used in II.4.5 (Lemma 74.1) to show that, roughly speaking, the part of the manifold added by surgery acquires a large scalar curvature soon after the surgery time. This is used crucially in II.5 (Sections 79 and 80) to adapt the noncollapsing argument of I.7 to Ricci flows with surgery. Our order of presentation of the material in II.2 is somewhat different than that of [52]. In Sections 61-63, we cover Claims 2, 4 and 5 of II.2. These are what’s needed in the sequel. The other results of II.2, Claims 1 and 3, are concerned with proving the uniqueness of the standard solution. Although it may seem intuitively obvious that there should be a unique and rotationally-symmetric standard solution, the argument is not routine since the manifold is noncompact. In fact, the uniqueness is not really needed for the sequel. (For example, the method of proof of Lemma 74.1 produces a standard solution in a limiting argument and it is enough to know certain properties of this standard solution.) Because of this we will talk about a standard solution rather than the standard solution. Consequently, we present the material so that we do not logically need the uniqueness of the standard solution. Having uniqueness does not shorten the subsequent arguments any. Of course, one can ask independently whether the standard solution is unique. In Section 65 we show that a standard solution is rotationally symmetric. In Section 66 we sketch the argument for uniqueness. Papers concerning the uniqueness of the standard solution are [21, 41]. We end this section by collecting some basic facts about standard solutions. Lemma 60.3. Let (R3 , g(·)) be a standard solution. Then (1) The curvature operator of g is nonnegative. (2) All derivatives of curvature are bounded for small time, independent of the standard solution. (3) The scalar curvature satisfies limt→a− supx∈R3 R(x, t) = ∞.

(4) (R3 , g(·)) is κ-noncollapsed at scales below 1 on any time interval contained in [0, 2], where κ depends only on the choice of the initial condition g0 . (5) (R3 , g(·)) satisfies the conclusion of Theorem 52.7, in the sense that for any ∆t > 0, there is an r0 > 0 so that for any point (x0 , t0 ) with t0 ≥ ∆t and Q = R(x0 , t0 ) ≥ r0−2 , the solution in {(x, t) : dist2t0 (x, x0 ) < (ǫQ)−1 , t0 − (ǫQ)−1 ≤ t ≤ t0 } is, after scaling by the factor Q, ǫ-close to the corresponding subset of a κ-solution. Moreover, any Ricci flow which satisfies all of the conditions of Definition 60.2 except maximality of the time interval can be extended to a standard solution. In particular, using short-time existence [62, Theorem 1.1], there is at least one standard solution. Proof. (1) follows from [63, Theorem 4.14].


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(2) follows from Appendix D. (3) In view of (1), this is equivalent to saying that limt→a− supx∈R3 | Rm |(x, t) = ∞. The argument for this last assertion is in [22, Chapter 6.7.2]. The proof in [22, Chapter 6.7.2] is for the compact case but using the derivative estimates of Appendix D, the same argument works in the present case. (4) See Theorem 26.2. (5) See Theorem 52.7. The final assertion of the lemma follows from the method of proof of (3).

61. Claim 2 of II.2. The blow-up time for a standard solution is ≤ 1 Lemma 61.1. (cf. Claim 2 of II.2) Let [0, TS ) be the maximal time interval such that the curvature of all standard solutions is uniformly bounded for every compact subinterval [0, a] ⊂ [0, TS ). Then on the time interval [0, TS ), the family of standard solution converges uniformly at (spatial) infinity to the standard Ricci flow on the round infinite cylinder S 2 ×R of scalar curvature one. In particular, TS is at most 1. ∞ Proof. Let {Mi}∞ i=1 be a sequence of standard solutions, and let {xi }i=1 be a sequence tending to infinity in the time-zero slice M.

By (2) of Lemma 60.3, the gradient estimates in Appendix D, and Appendix E, every subsequence of {Mi, (xi , 0)}∞ i=1 has a subsequence which converges in the pointed smooth topology on the time interval [0, TS ). Therefore, it suffices to show that if {Mi, (xi , 0)}∞ i=1 converges to some pointed Ricci flow (M∞ , (x∞ , 0)) then M∞ is round cylindrical flow.

Since gi (0) = g0 for all i, the sequence of pointed time-zero slices {(M, xi , gi(0))}∞ i=1 converges in the pointed smooth topology to the round cylinder, i.e. (M∞ , g∞ (0)) is a round cylinder of scalar curvature 1. Each time slice (M∞ , g∞ (t)) is biLipschitz equivalent to (M∞ , g∞ (0)). In particular, it has two ends. As it also has nonnegative sectional curvature, Toponogov’s theorem implies that (M∞ , g∞ (t)) splits off an R-factor. Using the strong maximum principle, the Ricci flow M∞ splits off an R-factor; see Theorem A.7. Then using the uniqueness of the Ricci flow on the round S 2 , it follows that M∞ is a standard shrinking cylinder, which proves the lemma. In particular, TS ≤ 1.

62. Claim 4 of II.2. The blow-up time of a standard solution is 1 Lemma 62.1. (cf. Claim 4 of II.2) Let TS be as in Lemma 61.1. Then TS = 1. In particular, every standard solution survives until time 1. Proof. First, there is an α > 0 so that TS > α [62, Theorem 1.1]. In what follows we will apply Theorem 52.7. The hypothesis of Theorem 52.7 says that the flow should exist on a time interval of duration at least one, but by rescaling we can apply Theorem 52.7 just as well with the alternative hypothesis that the flow exists on a time interval of duration at least α.

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Suppose that TS < 1. Then there is a sequence of standard solutions {Mi }∞ i=1 , times ti → TS and points (xi , ti ) ∈ Mi so that limi→∞ R(xi , ti ) = ∞.

We first argue that no subsequence of the points xi can go to infinity (with respect to the time-zero slice). Suppose, after relabeling the subsequence, that {xi }∞ i=1 goes to infinity. From Lemma 61.1, for any fixed t′ < TS the pointed solutions (M, (xi , 0), gi(·)), defined for t ∈ [0, t′ ], approach that of the shrinking cylinder on the same time interval. Lemma 60.3 and the characterization of high-curvature regions from Theorem 52.7 implies a uniform bound on high-curvature regions of the time derivative of R, of the form (59.5). Then taking t′ sufficiently close to TS , we get a contradiction. We conclude that outside of a compact region the curvature stays uniformly bounded as t → TS ; compare with the proof of Lemma 52.11. (Alternatively, one could apply Theorem 30.1 to compact approximants, as is done in II.2.) Thus we may assume that the sequence {xi }∞ i=1 stays in a compact region of the timezero slice. By Theorem 52.7, there is a sequence ǫi → 0 so that after rescaling the pointed solution (M, (xi , ti )) by R(xi , ti ), the result is ǫi -close to the corresponding subset of an ancient solution. By Proposition 41.13, the ancient solutions have vanishing asymptotic volume ratio. Hence for every β > 0, there is some L < ∞ so that in the original unscaled 1

1

3

solution, for large i we have vol B(xi , ti , L R(xi , ti )− 2 ) ≤ β L R(xi , ti )− 2 . Applying the Bishop-Gromov inequality to the time-ti slices, we conclude that for any D > 0, limi→∞ D−3 vol(B(xi , ti , D)) = 0. However, this contradicts the previously-shown fact that the solution extends smoothly to time TS < 1 outside of a compact set. Thus TS = 1.

Lemma 62.2. The infimal scalar curvature on the time-t slice tends to infinity as t → 1− uniformly for all standard solutions. Proof. Suppose the lemma failed and let {(Mi, (xi , ti ))}∞ i=1 be a sequence of pointed standard solutions, with {R(xi , ti )}∞ uniformly bounded and lim i→∞ ti = 1. i=1

Suppose first that after passing to a subsequence, the points xi go to infinity in the timezero slice. From Lemma 61.1, for any t′ ∈ [0, 1) we have limi→∞ R−1 (xi , t′ ) = 1 − t′ . −1 Combining this with the derivative estimate ∂R∂t ≤ η at high curvature regions gives a contradiction; compare with the proof of Lemma 52.11. Thus the points xi stay in a compact region. We can now use the bounded-curvature-at-bounded-distance argument in Step 2 of the proof of Theorem 52.7 to extract a convergent subsequence of {(Mi, (xi , 0))}∞ i=1 with a limit Ricci flow solution (M∞ , (x∞ , 0)) that exists on the time interval [0, 1]. (In this case, the nonnegative curvature of the blowup region W comes from the fact that a standard solution has nonnegative curvature.) As in Step 3 of the proof of Theorem 52.7, M∞ will have bounded curvature for t ∈ [0, 1]. Note that M∞ is a standard solution. This contradicts Lemma 61.1. 63. Claim 5 of II.2. Canonical neighborhood property for standard solutions Let p be the center of the hemispherical region in the time-zero slice.


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Lemma 63.1. (cf. Claim 5 of II.2) Given ǫ > 0 sufficiently small, there are constants η = η(ǫ), C1 = C1 (ǫ) and C2 = C2 (ǫ) so that every standard solution M satisfies the conclusions of Lemmas 59.4 and 59.7, except that the ǫ-neck neighborhood need not be strong. (Here the constants do not depend on the standard solution.) More precisely, any point (x, t) is covered by one of the following cases : 1.The time t lies in ( 34 , 1) and (x, t) has an ǫ-cap neighborhood or a strong ǫ-neck neighborhood as in Lemma 59.7. 2. x ∈ B(p, 0, ǫ−1 ), t ∈ [0, 43 ] and (x, t) has an ǫ-cap neighborhood as in Lemma 59.7.

3. x ∈ / B(p, 0, ǫ−1 ), t ∈ [0, 34 ] and there is an ǫ-neck B(x, t, ǫ−1 r) such that the solution in P (x, t, ǫ−1 r, −t) is, after scaling with the factor r −2 , ǫ-close to the appropriate piece of the evolving round infinite cylinder. Moreover, we have an estimate Rmin (t) ≥ const. (1 − t)−1 , where the constant does not depend on the standard solution. Proof. We first show that the conclusion of Lemma 59.7 is satisfied. In view of Lemma 62.2, there is a δ > 0 so that if t ∈ (1 − δ, 1) then we can apply Theorem 52.7 and Lemma 59.7 to a point (x, t) to see that the conclusions of Lemma 59.7 are satisfied in this case. If t ∈ [0, 1 − δ] and x is sufficiently far from p (i.e. dist0 (x, p) ≥ D for an appropriate D) then Lemma 61.1 implies that (x, t) has a strong ǫ-neck neighborhood or there is an ǫ-neck B(x, t, ǫ−1 r) such that the solution in P (x, t, ǫ−1 r, −t) is, after scaling with the factor r −2 , ǫ-close to the appropriate piece of the evolving round infinite cylinder. (To elaborate a bit on the last possibility, the issue here is that there is no backward extension of the solution to t < 0. Because of this, if t > 0 is close to 0 then the backward neighborhood P (x, t, ǫ−1 r, −t) will not exist for rescaled time one, as required to have a strong ǫ-neck neighborhood. Since inf x∈M R(x, 0) = 1, we know from (B.2) that R(x, t) ≥ 1 . Then if t > 35 , the time from the initial slice to (x, t), after rescaled by the scalar 1− 2 t 3

curvature, is bounded below by t

1 1− 23 t

> 1. In particular, if t ≥

3 4

then r 2 t is at least one

and we are ensured that the backward neighborhood P (x, t, ǫ−1 r, −t) does contain a strong ǫ-neck neighborhood.) If t ∈ [0, 1 − δ] and dist0 (x, p) < D then, provided that D and ǫ are chosen appropriately, we can say that (x, t) has an ǫ-cap neighborhood. We now show that the conclusion of Lemma 59.4 is satisfied. If t ∈ [1 − δ, 1) then the conclusion follows from Theorem 52.7 and Lemma 59.4. If δ ′ > 0 is sufficiently small and t ∈ [0, δ ′ ] then the conclusion follows from Appendix D. If t ∈ [ 21 δ ′ , 1 − 12 δ] then we have an upper scalar curvature bound from Lemma 62.1. From Hamilton-Ivey pinching (see Appendix B), this implies a double-sided sectional curvature bound. The conclusion of Lemma 59.4, when t ∈ [ 12 δ ′ , 1 − 21 δ], now follows from the Shi estimates of Appendix D. −1 The last statement of the lemma follows from the estimate ∂R∂t ≤ const., which holds for t near 1 (see Lemmas 59.4, 60.3(5) and 62.2) and then can be extended to all t ∈ [0, 1) (see Lemma 61). From Lemma 62.2, limt→1 R−1 (x, t) = 0 for every x. Thus

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R−1 (x, t) ≤ const. (1 − t) for any (x, t). Equivalently,

R(x, t) ≥ const. (1 − t)−1 .

(63.2)

64. Compactness of the space of standard solutions Lemma 64.1. The family ST of pointed standard solutions {(M, (p, 0))} is compact with respect to pointed smooth convergence. Proof. This follows immediately from Appendix E and the fact that the constant TS from Lemma 61.1 is equal to 1, by Lemma 62.1. 65. Claim 1 of II.2. Rotational symmetry of standard solutions Consider a standard solution (M, g(·)). Since the time-zero metric g0 is rotationally symmetric, it is clear by separation of variables that there is a rotationally symmetric solution for some time interval [0, T ). In this section we show that every standard solution is rotationally symmetric for each t ∈ [0, 1). Of course this would follow from the uniqueness of the standard solution; see [21, 41]. But the direct argument given here is the first step toward a uniqueness proof as in [41]. Lemma 65.1. (cf. Claim 1 of II.2) Any Ricci flow solution in the space ST is rotationally symmetric for all t ∈ [0, 1). Proof. We first describe an evolution equation for vector fields which turns out Pto send m Killing vector fields to Killing vector fields. Suppose that a vector field u = m u ∂m evolves by m k m i um t = u ; k + R iu .

(65.2)

Then (65.3) m k ∂t (um;i ) = um t ;i + (∂t Γ ki ) u = (um; kk + Rmk uk ); i + (∂t Γmki ) uk = um; kki + Rmk;iuk + Rmk uk;i + (∂t Γmki ) uk = um; kik − Rmlki ul ; k − Rk lki um; l + Rmk;iuk + Rmk uk;i + (∂t Γmki) uk

= um; ki k − Rmlki ul ; k − Rli um; l + Rmk;i uk + Rmk uk;i + (∂t Γmki ) uk

= um; ik k − (Rmlkiul ); k − Rmlki ul ; k − Rki um; k + Rmk;iuk + Rmk uk;i + (∂t Γmki) uk

= um; ik k − Rmlki; k ul − 2Rmlki ul ; k − Rik um; k + Rmk;iuk + Rmk uk;i + (∂t Γmki) uk .

Contracting the second Bianchi identity gives (65.4)

Rmlki; k = Ril;m − Rim;l .


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Also, (65.5)

∂t Γmki = ∂t (g ml Γlki) = 2Rml Γlki − g ml (Rlk,i + Rli,k − Rik,l ) = −Rmk;i − Rmi;k + Rik;m .

Substituting (65.4) and (65.5) in (65.3) gives (65.6) Then (65.7)

∂t (um;i ) = um; ik k − 2Rmlki ul ; k − Rik um; k + Rmk uk;i . ∂t (uj;i) = ∂t (gjm um;i) = −2Rjm um;i + gjm ∂t (um;i ) = uj; ik k − 2Rjlki ul ; k − Rik uj ;k − Rjk uk;i

= uj; ik k + 2Rikjl ul ; k − Rik uj ;k − Rkj uk;i.

Equivalently, writing vij = uj;i gives

∂t vij = vij;kk + 2Ri kj l vkl − Ri k vkj − Rkj vik .

Then putting Lij = vij + vji gives

∂t Lij = Lij;kk + 2Ri kj l Lkl − Ri k Lkj − Rkj Lik . For any λ ∈ R, we have

(65.8) ∂t (e2λt Lij Lij ) = 2λ (e2λt Lij Lij ) + (e2λt Lij Lij );k k − 2 e2λt Lij;k Lij;k + Q(Rm, eλt L),

where Q(Rm, L) is an algebraic expression that is linear in the curvature tensor Rm and quadratic in L. Putting Mij = eλt Lij gives (65.9)

∂t (Mij M ij ) = 2λ Mij M ij + (Mij M ij );k k − 2 Mij;k M ij;k + Q(Rm, M).

Suppose that we have a Ricci flow solution g(t), t ∈ [0, T ], with g(0) = g0 . Let u(0) be a rotational Killing vector field for g0 . Let u∞ (0) be its restriction to (any) S 2 , which we will think of as the 2-sphere at spatial infinity. Solve (65.2) for t ∈ [0, T ] with u(t) bounded at spatial infinity for each t; due to the asymptotics coming from Lemma 61.1 (which is independent of the rotational symmetry question), there is no problem in doing so. Arguing as in the proof of Lemma 61.1, one can show that for any t ∈ [0, T ], at spatial infinity u(t) converges to u∞ (0). Construct Mij (t) from u(t). As u(0) is a Killing vector field, Mij (0) = 0. For any t ∈ [0, T ], at spatial infinity the tensor Mij (t) converges smoothly to zero. Suppose that λ is sufficiently negative, relative to the L∞ -norm of the sectional curvature on the time interval [0, T ]. We can apply the maximum principle to (65.9) to conclude that Mij (t) = 0 for all t ∈ [0, T ]. Thus u(t) is a Killing vector field for all t ∈ [0, T ]. To finish the argument, as Ben Chow pointed out, any Killing vector field u satisfies

(65.10)

um; kk + Rmi ui = 0.

To see this, we use the Killing field equation to write (65.11)

0 = um;k k + uk;m k = um;k k + uk;m k − uk; km = um;k k + uk;mk − uk;km = um; kk − Rkimk ui = um;kk + Rmi ui .

Then from (65.2), um = 0 and the Killing vector fields are not changing at all. This t implies that g(t) is rotationally symmetric for all t ∈ [0, T ].

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66. Claim 3 of II.2. Uniqueness of the standard solution In this section, which is not needed for the sequel, we outline an argument for the uniqueness of the standard solution. We do this for the convenience of the reader. Papers on the uniqueness issue are [21, 41]. Our argument is somewhat different than that of [52, Proof of Claim 3 of Section 2], which seems to have some unjustified statements. c ≡ In general, suppose that we have two Ricci flow solutions M ≡ (M, g(·)) and M (M, b g (·)) with bounded curvature on each compact time interval and the same initial condition. We want to show that they coincide. As the set of times for which g(t) = b g (t) is closed, it suffices to show that it is relatively open. Thus it is enough to show that g and b g agree on [0, T ) for some small T .

We will carry out the Deturck trick in this noncompact setting, using a time-dependent background metric as in [5, Section 2]. The idea is to define a 1-parameter family of metrics {h(t)}t∈[0,T ) by h(t) = φ−1 (t)∗ g(t), where {φ(t)}t∈[0,T ) is a 1-parameter family of diffeomorphisms of M whose generator is the negative of the time-dependent vector field (66.1)

W i(t) = hjk Γ(h)ijk − Γ(b g )ijk ,

with φ0 = Id. More geometrically, as in [33, Section 6], we consider the solution of the harmonic heat flow equation ∂F = △F for maps F : M → M between the manifolds ∂t (M, g(t)) and (M, b g (t)), with F (0) = Id.

We now specialize to the case when (M, g(·)) and (M, b g (·)) come from standard solutions. The technical issue, which we do not address here, is to show that a solution to the harmonic heat flow will exist for some time interval [0, T ) with uniformly bounded derivatives; see [21]. One is allowed to use the asymptotics of Section 61 here and from Section 65, one can also assume that all of the metrics are rotationally invariant. In the rest of this section we assume the existence of such a solution F . By further reducing the time interval if necessary, we may assume that F (t) is a diffeomorphism of M for each t ∈ [0, T ). Then h(t) = F −1 (t)∗ g(t). Clearly h(0) = g(0) = b g (0).

By Section 61, g and b g have the same spatial asymptotics, namely that of the shrinking cylinder. We claim that this is also true for h. That is, we claim that (M, h(·)) converges smoothly to the shrinking cylinder solution on [0, T ). It suffices to show that F converges smoothly to the identity on [0, T ). Suppose not. Let {xi }∞ i=1 be a sequence of points in the time-zero slice so that no subsequence of the pointed spacetime maps (F, (xi , 0)) converges to the identity. Using the derivative bounds, we can extract a subsequence that converges to some Fe : [0, T ) × R × S 2 → R × S 2 in the pointed smooth topology. However, Fe will satisfy the harmonic heat flow equation from the shrinking cylinder R × S 2 to itself, with Fe(0) being the identity, and will have bounded derivatives. The uniqueness of Fe follows by standard methods. Hence Fe(t) is the identity for all t ∈ [0, T ), which is a contradiction. By construction, the family of metrics {h(t)}t∈[0,T ) satisfies the equation

(66.2)

dhij = − 2 Rij (h) + ∇(h)i Wj + ∇(h)j Wi . dt


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In local coordinates, the right-hand side of (66.2) is a polynomial in hij , hij , hij,k and hij,kl . The leading term in (66.2) is dhij = hkl ∂k ∂l hij + . . . . dt A particular solution of (66.2) is h(t) = b g (t), since if we had g = gb then we would have W = 0 and φt = Id. (66.3)

Put w(t) = h(t) − b g (t). We claim that w satisfies an equation of the form

dw = − ∇(b g )∗ ∇(b g )w + P w + Qw, dt where P is a first-order operator and Q is a zeroth-order operator. To obtain the leading derivative terms in (66.4), using (66.3) we write

(66.4)

(66.5)

dwij = hkl ∂k ∂l hij − b g kl ∂k ∂l b gij + . . . dt = b g kl ∂k ∂l wij + hkl − b g kl ∂k ∂l hij + . . . = gbkl ∂k ∂l wij − gbka wab hbl ∂k ∂l hij + . . .

= gbkl ∂k ∂l wij − gbka hbl ∂k ∂l hij wab + . . .

A similar procedure can be carried out for the lower order terms, leading to (66.4). By construction the operators P and Q have smooth coefficients which, when expressed in terms of orthonormal frames, will be bounded on M. In fact, as h and gb have the same spatial asymptotics, it follows from [5, Proposition 4] that the operator on the right-hand side of (66.4) converges at spatial infinity to the Lichnerowicz Laplacian △L(b g ). By assumption, w(0) = 0. We now claim that w(t) = 0 for all t ∈ [0, T ). Let K ⊂ M be a codimension-zero compact submanifold-with-boundary. For any λ ∈ R, we have (66.6) Z 1 −2λt 2λt d 2 e e |w(t)| dvolbg(t) = dt 2 K Z dw R 2 |w| + hw, i dvolbg(t) = −λ − 2 dt K Z R 2 2 abcdi |w| − |∇(b g )w| + wab P ∇(b g )i wcd + hw, Qwi dvolbg(t) ± −λ − 2 K Z hw, ∇n wi dvol∂bg(t) = ∂K Z R 1 abcdi 1 abcdi 2 2 2 cd −λ − |w| − |∇(b g )i w − P wab | + |P wab | + hw, Qwi dvolbg(t) ± 2 2 4 K Z hw, ∇n wi dvol∂bg(t) . ∂K

Choose (66.7)

λ > sup v6=0

1 |P abcdi vab |2 4

+ hv, Qvi . hv, vi

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On any subinterval [0, T ′ ] ⊂ [0, T ), since w converges to zeroRat infinity and (M, b g (t)) is standard at infinity, by choosing K appropriately we can make ∂K hw, ∇n wi dvol∂bg(t) small. It follows that there is an exhaustion {Ki }∞ i=1 of M so that Z 1 −2λt 1 2 2λt d e |w(t)| dvolbg(t) ≤ (66.8) e dt 2 i Ki for t ∈ [0, T ′ ]. Then (66.9)

Z

2

Ki

|w(t)| dvolgb(t)

e2λt − 1 ≤ λi

for all t ∈ [0, T ′]. Taking i → ∞ gives w(t) = 0.

c ∈ ST then Thus h = b g . From (66.1), W = 0 and so h = g. This shows that if M, M c M = M. 67. II.3. Structure at the first singularity time

This section is concerned with the structure of the Ricci flow solution at the first singular time, in the case when the solution does go singular. Let M be a connected closed oriented 3-manifold. Let g(·) be a Ricci flow on M defined on a maximal time interval [0, T ) with T < ∞. One knows that limt→T − maxx∈M | Rm |(x, t) = ∞.

From Theorem 26.2 and Theorem 52.7, given ǫ > 0 there are numbers r = r(ǫ) > 0 and κ = κ(ǫ) > 0 so that for any point (x, t) with Q = R(x, t) ≥ r −2 , the solution 1 in P (x, t, (ǫQ)− 2 , (ǫQ)−1 ) is (after rescaling by the factor Q) ǫ-close to the corresponding subset of a κ-solution. By Lemma 59.4, the estimate (59.5) holds at (x, t), provided ǫ is sufficiently small. In addition, there is a neighborhood B of (x, t) as described in Lemma 59.7. In particular, B is a strong ǫ-neck, an ǫ-cap or a closed manifold with positive sectional curvature. If M has positive sectional curvature at some time t then it is diffeomorphic to a finite quotient of the round S 3 and shrinks to a point at time T [35]. The topology of M satisfies the conclusion of the geometrization conjecture and M goes extinct in a finite time. Therefore for the remainder of this section we will assume that the sectional curvature does not become everywhere positive. We now look at the behavior of the Ricci flow as one approaches the singular time T . Definition 67.1. Define a subset Ω of M by (67.2)

Ω = {x ∈ M : sup | Rm |(x, t) < ∞}. t∈[0,T )

Suppose that x ∈ M − Ω, so there is a sequence of times {ti } in [0, T ) with limi→∞ ti = T and limi→∞ | Rm |(x, ti ) = ∞. As minM R(·, t) in nondecreasing in t, the largest sectional curvature at (x, ti ) goes to infinity as i → ∞. Then by the Φ-almost nonnegative sectional curvature result of Appendix B, limi→∞ R(x, ti ) = ∞. From the time-derivative estimate of (59.5), limt→T − R(x, t) = ∞. Thus x ∈ M − Ω if and only if limt→T − R(x, t) = ∞.


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Lemma 67.3. Ω is open in M. Proof. Given x ∈ Ω, using the time-derivative estimate in (59.6) gives a bound of the form |R(x, t)| ≤ C for t ∈ [0, T ). Then using the spatial-derivative estimate in (59.6) gives a number rb > 0 so that so that |R(·, t)| ≤ 2C on B(x, t, rb), for each t ∈ [0, T ). The Φ-almost nonnegative sectional curvature implies a bound of the form | Rm(·, t)| ≤ C ′ on B(x, t, rb), for each t ∈ [0, T ). Then the length-distortion estimate of Section 27 implies that we can pick a neighborhood N of x so that | Rm | ≤ C ′ on N × [0, T ). Thus N ⊂ Ω. Lemma 67.4. Any connected component C of Ω is noncompact. Proof. Since M is connected, if C were compact then it would be all of M. This contradicts the assumption that there is a singularity at time T . We remark that a priori, the structure of M −Ω can be quite complicated. For example, it is not ruled out that an accumulating collection of 2-spheres in M can simultaneously shrink 2 2 to points. That is, M −Ω could have a subset of the form ({0} ∪{ 1i }∞ i=1 ) ×S ⊂ (−1, 1)×S , the picture being that Ω contains a sequence of smaller and smaller adjacent double horns. One could even imagine a Cantor set’s worth of 2-spheres simultaneously shrinking, although conceivably there may be additional arguments to rule out both of these cases. Lemma 67.5. If Ω = ∅ then M is diffeomorphic to S 3 , RP 3, S 1 × S 2 or RP 3 #RP 3 . Proof. The time-derivative estimate in (59.6) implies that for t slightly less than T , we have R(x, t) ≥ r −2 for all x ∈ M. Thus at that time, every x ∈ M has a neighborhood that is in an ǫ-neck or an ǫ-cap, as described in Lemma 59.7. (Recall that we have already excluded the positively-curved case of the lemma.) As in the proof of Lemma 59.1, by splicing together the projection maps associated with neck regions, one obtains an open subset U ⊂ M and a 2-sphere fibration U → N where the fibers are nearly totally geodesic, and the complement of U is contained in a union of ǫ-caps. It follows that U is connected. If there are any ǫ-caps then there must be exactly two of them U1 , U2 , and they may be chosen to intersect U in connected open sets Vi = Ui ∩ U which are isotopic to product regions in both U and in the Ui ’s. The caps being diffeomorphic to B 3 or RP 3 − B 3 , it follows that M is diffeomorphic to S 3 , RP 3 or RP 3 #RP 3 if U 6= M; otherwise M is diffeomorphic to an S 2 bundle over a circle, and the orientability assumption implies that this bundle is diffeomorphic to S 1 × S 2 . In the rest of this section we assume that Ω 6= ∅. From the local derivative estimates of Appendix D, there is a smooth Riemannian metric g = limt→T − g(t) on Ω. Let R denote Ω its scalar curvature. Thus the scalar curvature function extends to a continuous function on the subset (M × [0, T )) ∪ (Ω × {T }) ⊂ M × [0, T ]. Lemma 67.6. (Ω, g) has finite volume. Proof. From the lower scalar curvature bound of (B.2) and the formula dtd vol(M, g(t)) = R 3 − M R dvolM , we obtain an estimate of the form vol(M, g(t)) ≤ const. + const. t 2 , for t < T . The lemma follows.

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Lemma 67.7. There is a open neighborhood V of (M − Ω) × {T } in M × [0, T ] such that R−1 extends to a continuous function on V which vanishes on (M − Ω) × {T }. Proof. As observed above Lemma 67.3, x ∈ M − Ω if and only if limt→T − R−1 (x, t) = 0. The lemma follows by applying (59.6) to suitable spacetime paths. Definition 67.8. For ρ < 2r , put Ωρ = {x ∈ Ω : R(x) ≤ ρ−2 }. Lemma 67.9. The function R : Ω → R is proper; equivalently, if {xi } ⊂ Ω is a sequence which leaves every compact subset of Ω, then limi→∞ R(xi ) = ∞. In particular, Ωρ is a compact subset of M for every ρ < r. Proof. Suppose {xi } ⊂ Ω is a sequence such that {R(xi )} is bounded. After passing to a subsequence, we may assume that {xi } converges to some point x∞ ∈ M. But R−1 is well-defined and continuous on V , and vanishes on (M − Ω) × {T }, so we must have x∞ ∈ Ω. We now consider the connected components of Ω according to whether they intersect Ωρ or not. First, let C be a connected component of Ω that does not intersect Ωρ . Given x ∈ C, there is a neighborhood Bx of x which is ǫ-close to a region as described in Lemma 59.7. From Lemma 67.4, the neighborhood Bx cannot be of type (c) or (d) in the terminology of Lemma 59.7. We now introduce some terminology. If a manifold Z is diffeomorphic to R3 or RP 3 − B 3 then any embedded 2-sphere Σ ⊂ Z separates Z into two connected subsets, one of which has compact closure and the other contains the end of Z. We refer to the first component as the compact side and the other component as the noncompact side. An open subset R of a Riemannian manifold is a good cylinder if: • It is ǫ-close, modulo rescaling, to a segment of a round cylinder of scalar curvature 1. • The diameter of R is approximately 100 times its cross-section. • Every point in R, lies in an ǫ-neck in the ambient Riemannian manifold. From Lemma 59.7, every ǫ-cap neighborhood Bx contains a good cylinder lying in the ǫ-neck at the end of Bx . Lemma 67.10. Suppose that for all x ∈ C, the neighborhood Bx can be taken to be a strong ǫ-neck as in case (a) of Lemma 59.7. Then C is a double ǫ-horn. Proof. Each point x has an ǫ-neck neighborhood. We can glue these ǫ-necks together to form a submersion from C to a 1-manifold, with fiber S 2 ; cf. the proof of Lemma 59.1. (We can do the gluing by successively adding on good cylinders, where the intersections of successive cylinders have diameter approximately 10 times the diameter of the cross-sections.) In view of Lemma 67.7, it follows in this case that C is a double ǫ-horn.


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Lemma 67.11. Suppose that there is some x ∈ C whose neighborhood Bx is an ǫ-cap as in case (a) of Lemma 59.7. Then C is a capped ǫ-horn. Proof. Put p1 = x. Let R be a good cylinder in the ǫ-neck at the end of Bp1 . Now glue on successive good cylinders to R, as in the proof of the preceding lemma, going away from p1 . Case 1 : Suppose this gluing process can be continued indefinitely. Then taking the union of Bp1 with the good cylinders, we obtain an open subset W of C which is diffeomorphic to R3 or RP 3 − B 3 . We claim that W is a closed subset of Ω. If not then there is a sequence ∞ {xk }∞ k=1 ⊂ W converging to some x∞ ∈ Ω − W . This implies that {R(xk )}k=1 remains bounded. In view of the overlap condition between successive good cylinders, a subsequence of {xk }∞ k=1 lies in an infinite number of mutually disjoint good cylinders, whose volumes have a positive lower bound (because of the upper scalar curvature bound at the points xk ). This contradicts Lemma 67.6. Thus W is open and closed in Ω. Hence W = C and we are done. Case 2 : Now suppose that the gluing process cannot be continued beyond some good cylinder R1 . Then there must be a point p2 ∈ R1 such that Bp2 is an ǫ-cap. Also, note that the union W1 of Bp1 with the good cylinders is diffeomorphic to R3 or RP 3 − B 3 , and that R1 has compact complement in W1 . Let Σ ⊂ R1 be a cross-sectional 2-sphere passing through p2 . We first claim that if V is the compact side of Σ in W1 , then V coincides with the compact side V ′ of Σ in Bp2 . To see this, note that V and V ′ are both connected open sets disjoint from Σ, with topological frontiers ∂V = ∂V ′ = Σ. Then V − V ′ = V ∩ (C − (V ′ ∪ Σ)) and we obtain two open decompositions (67.12)

V = (V ∩ V ′ ) ⊔ (V − V ′ ),

V ′ = (V ∩ V ′ ) ⊔ (V ′ − V ).

If V ∩ V ′ = ∅, then V ∪ V ′ is a union of two compact manifolds with the same boundary Σ, and disjoint interiors. Hence it is an open and closed subset of the connected component C, which contradicts Lemma 67.4. Thus V ∩V ′ is nonempty. By (67.12) and the connectedness of V and V ′ , we get V ⊂ V ′ and V ′ ⊂ V , so V = V ′ as claimed.

Next, we claim that if R2 ⊂ Bp2 is a good cylinder with compact complement in Bp2 , then R2 is disjoint from W1 . To see this, note that R2 is disjoint from Σ because p2 ∈ Σ and the diameter of Σ is close to π(R(p2 )/6)−1/2 , whereas by Lemma 59.7 there is an ǫ-neck U ⊂ Bp2 with compact complement in Bp2 , at distance at least 9000R(p2 )−1/2 from p2 . Thus R2 must lie in the noncompact side of Σ in Bp2 , and hence is disjoint from V . As the good cylinder R1 ∋ p2 lies within B(p2 , 1000R(p2 )−1/2 ) ⊂ Ω, it follows that R2 is also disjoint from R1 , so R2 is disjoint from W1 = V ∪ R1 ⊂ Bp2 .

We continue adding good cylinders to R2 as long as we can. If we come to another cap point p3 then we jump to its cap Bp3 and continue the process. When so doing, we encounter successive cap points p1 , p2 , . . . with associated caps Bp1 ⊂ Bp2 ⊂ . . . and disjoint

good cylinders R1 , R2 , . . .. Since the ratio

supBp inf Bp

k

k

R

R

has an a priori bound by Lemma 59.7,

in view of the disjoint good cylinders in Bpk we get vol(Bpk ) ≥ const. k R(p1 )−3/2 . Then

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Lemma 67.6 gives an upper bound on k. Hence we encounter a finite number of cap points. Arguing as in Case 1, we conclude that C is a capped ǫ-horn. We note that there could be an infinite number of connected components of Ω that do not intersect Ωρ . Now suppose that C is a connected component of Ω that intersects Ωρ . As C is noncompact, there must be some point x ∈ C that is not in Ωρ . Again, any such x has a neighborhood B as in Lemma 59.7. If one of the boundary components of B intersects Ω2ρ then we terminate the process in that direction. For the directions of the boundary components of B that do not intersect Ω2ρ , we perform the above algorithm of looking for an adjacent ǫ-neck, etc. The only difference from before is that in at least one direction any such sequence of overlapping ǫ-necks will be finite, as it must eventually intersect Ω2ρ . (In the other direction it may terminate in Ω2ρ , in an ǫ-cap, or not terminate at all.) Once a cross-sectional 2-sphere intersects Ω2ρ , if ǫ is small then the entire 2-sphere lies in Ωρ . Thus any connected component of C − (C ∩ Ωρ ) is contained in an ǫ-tube with both boundary components in Ωρ , an ǫ-cap with boundary in Ωρ or an ǫ-horn with boundary in Ωρ . We note that Ωρ need not have a nice boundary. There is an a priori ρ-dependent lower bound for the volume of any such connected component of C − (C ∩ Ωρ ), in view of the fact that it contains ǫ-necks that adjoin Ωρ . From Lemma 67.6, there is a finite number of connected components of Ω that intersect Ωρ . Any such connected component has a finite number of ends, each being an ǫ-horn. Note that the ǫ-horns can be made disjoint, each with a quantitative lower volume bound. The surgery procedure, which will be described in detail in Section 73, is performed as follows. First, one throws away all connected components of Ω that do not intersect Ωρ . For each connected component Ωj of Ω that intersects Ωρ and for each ǫ-horn of Ωj , take a cross-sectional sphere that lies far in the ǫ-horn. Let X be what’s left after cutting the ǫ-horns at these 2-spheres and removing the tips. The (possibly-disconnected) postsurgery manifold M ′ is the result of capping off ∂X by 3-balls. We now discuss how to reconstruct the original manifold M from M ′ . Lemma 67.13. M is the result of taking connected sums of components of M ′ and possibly taking additional connected sums with a finite number of S 1 × S 2 ’s and RP 3 ’s. Proof. At a time shortly before T , each point of M − X has a neighborhood as in S Lemma ′ 59.7. The components of M − X are ǫ-tubes and ǫ-caps. Writing M = X ∪ B 3 and M = X ∪ (M − X), one builds M from M ′ as follows. If the boundary of an ǫ-tube of M − X lies in two disjoint components of X then it gives rise to a connected sum of two components of M ′ . If the boundary of an ǫ-tube lies in a single connected component of X then it gives rise to the connected sum of the corresponding component of M ′ with a new copy of S 1 × S 2 . If an ǫ-cap in M − X is a 3-ball it does not have any effect on M ′ . If an ǫ-cap is RP 3 − B 3 then it gives rise to the connected sum of the corresponding component of M ′ with a new copy of RP 3 . The lemma follows. Remark 67.14. We do not assume that the diameter of (M, g(t)) stays bounded as t → T ; it is an open question whether this is the case.


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68. Ricci flow with surgery: the general setting In this section we introduce some notation and terminology in order to treat Ricci flows with surgery. The principal purpose of sections II.4 and II.5 is to show that one can prescribe the surgery procedure in such a way that Ricci flow with surgery is well-defined for all time. This involves showing that • One can give a sufficiently precise description of the formation of singularities so that one can envisage defining a geometric surgery. In the case of the formation of the first singularity, such a description was given in Section 67. • The sequence of surgery times cannot accumulate.

The argument in Section 67 strongly uses both the κ-noncollapsing result of Theorem 26.2 and the characterization of the geometry in a spacetime region around a point (x0 , t0 ) with large scalar curvature, as given in Theorem 52.7. The proofs of both of these results use the smoothness of the solution at times before t0 . If surgeries occur before t0 then one must have strong control on the scales at which the surgeries occur, in order to extend the arguments of Theorems 26.2 and 52.7. This forces one to consider time-dependent scales. Section II.4 introduces Ricci flow with surgery, in varying degrees of generality. Our treatment of this material follows Perelman’s. We have added some terminology to help formalize the surgery process. There is some arbitrariness in this formalization, but the version given below seems adequate. For later use, we now summarize the relevant notation that we introduce. More precise definitions will be given below. We will avoid using new notation as much as possible. • M is a Ricci flow with surgery. • Mt is the time-t slice of M.

• Mreg is the set of regular points of M.

• If T is a singular time then MT− is the limit of time slices Mt as t → T − (called Ω in II.4.1) and MT+ is the outgoing time slice (for example, the result of performing surgery on + Ω). If T is a nonsingular time then M− T = MT = MT . The basic notion of a Ricci flow with surgery is simply a sequence of Ricci flows which “fit together” in the sense that the final (possibly singular) time slice of each flow is isometric, modulo surgery, to the initial time slice of the next one. Definition 68.1. A Ricci flow with surgery is given by + • A collection of Ricci flows {(Mk ×[t− k , tk ), gk (·))}1≤k≤N , where N ≤ ∞, Mk is a compact + − (possibly empty) manifold, tk = tk+1 for all 1 ≤ k < N, and the flow gk goes singular at t+ k for each k < N. We allow t+ to be ∞. N

• A collection of limits {(Ωk , g¯k )}1≤k≤N , in the sense of Section 67, at the respective final times t+ k that are singular if k < N. (Recall that Ωk is an open subset of Mk .)

− • A collection of isometric embeddings {ψk : Xk+ → Xk+1 }1≤k 0 is small then in going from Mt−s to Mt+s , the and Xk+ ∼ Mk+1 − Xk+1 = Xk+1 − . topological change is that we remove Mk − Xk+ from Mk and add Mk+1 − Xk+1

We let M(t,t′ ) = ∪t¯∈(t,t′ ) Mt¯ denote the time slab between t and t′ , i.e. the union of the time slices between t and t′ . The closed time slab M[t,t′ ] is defined to be the closure of − M(t,t′ ) in M, so M[t,t′ ] = M+ t ∪ M(t,t′ ) ∪ Mt′ . We (ab)use the notation (x, t) to denote a point x ∈ M lying in the time t slice Mt , even though M may no longer be a product.

The spacetime M has three types of points: + 1. The 4-manifold points, which include all points at nonsingular times in (t− 1 , tN ) and all + + − − points in π(Interior(Xk ) × {tk }) (or π(Interior(Xk ) × {tk })) for 1 ≤ k < N, + 2. The boundary points of M, which are the images in M of M1 × {t− 1 }, ΩN × {tN }, + + − − (Ωk − Xk ) × {tk } for 1 ≤ k < N, and (Mk − Xk ) × {tk } for 1 < k ≤ N, and 3. The “splitting” points, which are the images in M of ∂Xk+ × {t+ k } for 1 ≤ k < N.

Here the classification of points is according to the smooth structure, not the topology. In fact, M is a topological manifold-with-boundary. We say that (x, t) is regular if it is either a 4-manifold point, or it lies in the initial time slice Mt−1 or final time slice Mt+ . Let Mreg N denote the set of regular points. It has a natural smooth structure since the gluing maps ψk , being isometries between smooth Riemannian manifolds, are smooth maps.


129

Note that the Ricci flows on the Mk ’s define a Riemannian metric g on the “horizontal” subbundle of the tangent bundle of Mreg . It follows from the definition of the Ricci flow that g is actually smooth on Mreg .

We metrize each time slice Mt , and the forward and backward time slices M± t , by infimizing the path length of piecewise smooth paths. We allow our distance functions to be infinite, since the infimum will be infinite when points lie in different components. If (x, t) ∈ Mt and r > 0 then we let B(x, t, r) denote the corresponding metric ball. Similarly, ± B ± (x, t, r) denotes the ball in M± t centered at (x, t) ∈ Mt . A ball B(x, t, r) ⊂ Mt is proper if the distance function d(x,t) : B(x, t, r) → [0, r) is a proper function; a proper ball “avoids singularities”, except possibly at its frontier. Proper balls B ± (x, t, r) ⊂ M± t are defined likewise. An admissible curve in M is a path γ : [c, d] → M, with γ(t) ∈ Mt for all t ∈ [c, d], such + that for each k, the part of γ landing in M[t− t+ ] lifts to a smooth map into Mk × [t− k , tk ) ∪ k k Ωk × {t+ ˙ to denote the “horizontal” part of the velocity of an admissible k }. We will use γ curve γ. If t < t0 , a point (x, t) ∈ M is accessible from (x0 , t0 ) ∈ M if there is an admissible curve running from (x, t) to (x0 , t0 ). An admissible curve γ : [c, d] → M is static if its lifts to the product spaces have constant first component. That is, the points in the image of a static curve are “the same”, modulo the passage of time and identifications taking place at surgery times. A barely admissible curve is an admissible curve γ : [c, d] → M such that − the image is not contained in M+ c ∪ Mreg ∪ Md . If γ : [c, d] → M is barely admissible then − + there is a surgery time t = tk = tk+1 ∈ (c, d) such that γ(t) lies in (68.3)

− − π(∂Xk+ × {t+ k }) = π(∂Xk+1 × {tk+1 }).

If (x, t) ∈ M+ t , r > 0, and ∆t > 0 then we define the forward parabolic region P (x, t, r, ∆t) to be the union of (the images of) the static admissible curves γ : [t, t′ ] → M starting in B + (x, t, r), where t′ ≤ t + ∆t. That is, we take the union of all the maximal extensions of all static curves, up to time t + ∆t, starting from the initial time slice B + (x, t, r). When ∆t < 0, the parabolic region P (x, t, r, ∆t) is defined similarly using static admissible curves ending in B − (x, t, r). If Y ⊂ Mt , and t ∈ [c, d] then we say that Y is unscathed in [c, d] if every point (x, t) ∈ Y lies on a static curve defined on the time interval [c, d]. If, for instance, d = t then this will force Y ⊂ M− t . The term “unscathed” is intended to capture the idea that the set is unaffected by singularities and surgery. (Sometimes Perelman uses the phrase “the solution is defined in P (x, t, r, ∆t)” as synonymous with “the solution is unscathed in P (x, t, r, ∆t)”, for example in the definition of canonical neighborhood in II.4.1.) We may use the notation Y × [c, d] for the set of points lying on static curves γ : [c, d] → M which pass through Y , when Y is unscathed on [c, d]. Note that if Y is open and unscathed on [c, d] then we can think of the Ricci flow on Y × [c, d] as an ordinary (i.e. surgery-free) Ricci flow.

The definitions of ǫ-neck, ǫ-cap, ǫ-tube and (capped/double) ǫ-horn from Section 58 do not require modification for a Ricci flow with surgery, since they are just special types of Riemannian manifolds; they will turn up as subsets of forward or backward time slices of a Ricci flow with surgery. A strong ǫ-neck is a subset of the form U × [c, d] ⊂ M, where

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U ⊂ M− d is an open set that is unscathed on the interval [c, d], which is a strong ǫ-neck in the sense of Section 58. 69. II.4.1. A priori assumptions This section introduces the notion of canonical neighborhood. The following definition captures the geometric structure that emerges by combining Theorem 52.7 and its extension to Ricci flows with surgery (Section 77) with the geometric description of κ-solutions. The idea is that blowups either yield κ-solutions, whose structure is well understood from Section 59, or there are surgeries nearby in the recent past, in which case the local geometry resembles that of the standard solution. Both alternatives produce canonical neighborhoods. Definition 69.1. (Canonical neighborhoods, cf. Definition in II.4.1) Let ǫ > 0 be small enough so that Lemmas 59.7 and 63.1 hold. Let C1 be the maximum of 30ǫ−1 and the C1 (ǫ)’s of Lemmas 59.7 and 63.1. Let C2 be the maximum of the C2 (ǫ)’s of Lemmas 59.7 and 63.1. Let r : [a, b] → (0, ∞) be a positive nonincreasing function. A Ricci flow with surgery M defined on the time interval [a, b] satisfies the r-canonical neighborhood assumption if every −2 (x, t) ∈ M± has a canonical neighborhood in the t with scalar curvature R(x, t) ≥ r(t) corresponding (forward/backward) time slice, as in Lemma 59.7. More precisely, there is an 1 1 ± rˆ ∈ (R(x, t)− 2 , C1 R(x, t)− 2 ) and an open set U ⊂ M± ˆ) ⊂ U ⊂ B ± (x, t, 2ˆ r) t with B (x, t, r that falls into one of the following categories : (a) U × [t − ∆t, t] ⊂ M is a strong ǫ-neck for some ∆t > 0. (Note that after parabolic rescaling the scalar curvature at (x, t) becomes 1, so the scale factor must be ≈ R(x, t), which implies that ∆t ≈ R(x, t)−1 .) (b) U is an ǫ-cap which, after rescaling, is ǫ-close to the corresponding piece of a κ0 solution or a time slice of a standard solution (cf. Section 60). (c) U is a closed manifold diffeomorphic to S 3 or RP 3 . (d) U is ǫ-close to a closed manifold of constant positive sectional curvature. Moreover, the scalar curvature in U lies between C2−1 R(x, t) and C2 R(x, t). In cases (a), 3 (b), and (c), the volume of U is greater than C2−1 R(x, t)− 2 . In case (c), the infimal sectional curvature of U is greater than C2−1 R(x, t). Finally, we require that (69.2)

3 2

|∇R(x, t)| < ηR(x, t) ,

∂R < ηR(x, t)2 , (x, t) ∂t

where η is the constant from (59.5). Here the time dervative ∂R (x, t) should be interpreted ∂t as a one-sided derivative when the point (x, t) is added or removed during surgery at time t. Remark 69.3. Note that the smaller of the two balls in B ± (x, t, rˆ) ⊂ U ⊂ B ± (x, t, 2ˆ r) is closed, in order to make it easier to check the openness of the canonical neighborhood condition. The requirement that C1 be at least 30ǫ−1 will be used in the proof of Lemma 73.7; see Remark 73.8.


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Remark 69.4. For convenience, in case (b) we have added the extra condition that U is ǫ-close to the corresponding piece of a κ0 -solution or a time slice of a standard solution. One does not need this extra condition, but it is consistent to add it. We remark that when − surgery is performed according to the recipe of Section 73, if a point (p, t) lies in M+ t − Mt (i.e. it is “added” by surgery) then it will sit in an ǫ-cap, because M+ t will resemble a standard solution from Section 60 near (p, t). Points lying somewhat further out on the capped neck will belong to a strong ǫ-neck which extends backward in time prior to the surgery. The next condition, which will ultimately be guaranteed by the Hamilton-Ivey curvature pinching result and careful surgery, is also essential in blowup arguments à la Section 52. Definition 69.5. (Φ-pinching) Let Φ ∈ C ∞ (R) be a positive nondecreasing function such that for positive s, Φ(s) is a decreasing function which tends to zero as s → ∞. The Ricci s flow with surgery M satisfies the Φ-pinching assumption if for all (x, t) ∈ M, one has Rm(x, t) ≥ −Φ(R(x, t)). We remark that the notion of Φ-pinching here is somewhat different from Perelman’s φpinching. The purpose of this definition is to distill out the properties of the Hamilton-Ivey pinching condition which are needed in the rest of the proof. Definition 69.6. A Ricci flow with surgery satisfies the a priori assumptions if it satisfies the Φ-pinching and r-canonical neighborhood assumptions on the time interval of the flow. Note that the a priori assumptions depend on ǫ, the function r(t) of Definition 69.1 and the function Φ of Definition 69.5. 70. II.4.2. Curvature bounds from the a priori assumptions In this section we state some technical lemmas about Ricci flows with surgery that satisfy the a priori assumptions of the previous section. The first one is the surgery analog of Lemma 52.11. Lemma 70.1. (cf. Claim 1 of II.4.2) Given (x0 , t0 ) ∈ M put Q = |R(x0 , t0 )| + r(t0 )−2 . 1 Then R(x, t) ≤ 8Q for all (x, t) ∈ P (x0 , t0 , 12 η −1 Q− 2 , − 81 η −1 Q−1 ), where η is the constant from (69.2). Proof. The lemma follows from the estimates (69.2). One integrates these derivative bounds 1 along a subinterval of a path that goes in B(x0 , t0 , 12 η −1 Q− 2 ) and then backward in time along a static path. See the proof of Lemma 52.11. We also use the fact that if t′ ≤ t0 and R(x′ , t′ ) ≥ Q then the inequalities (69.2) are valid at (x′ , t′ ), since r(·) is nonincreasing. The next lemma expresses the main consequence of Claim 2 of II.4.2. Lemma 70.2. (cf. Claim 2 of II.4.2) If ǫ is small enough then the following holds. Suppose that M is a Ricci flow with surgery that satisfies the Φ-pinching assumption. Then for any A < ∞ and rb > 0 there exist ξ = ξ(A) > 0 and K = K(A, rb) < ∞ with the following property. Suppose that M also satisfies the r-canonical neighborhood assumption for some function r(·). Then for any time t0 , if (x0 , t0 ) is a point so that Q = R(x0 , t0 ) > 0 satisfies

132

BRUCE KLEINER AND JOHN LOTT 1

Φ(Q) Q

< ξ and (x, t0 ) is a point so that distt0 (x0 , x) ≤ AQ− 2 then R(x, t0 ) ≤ KQ, where K = K(A, r(t0 )). Proof. The proof is similar to the proof of Step 2 of Theorem 52.7. (The canonical neighborhood assumption replaces Step 1 of the proof of Theorem 52.7.) Assuming that the lemma fails, one obtains a piece of a nonflat metric cone as a blowup limit. Using the canonical neighborhood assumption, one concludes that the corresponding points in M have a neighborhood of type (a), i.e. a strong ǫ-neck, since the neighborhoods of type (b), (c) and (d) of Definition 69.1 are not close to a piece of metric cone. A strong ǫ-neck, has the time interval needed to apply the strong maximum principle as in Step 2 of the proof of Theorem 52.7, in order to get a contradiction. 71. II.4.3. δ-necks in ǫ-horns In this section we show that an ǫ-horn has a self-improving property as one goes down the horn. For any δ > 0, if the scalar curvature at a point is sufficiently large then the point actually lies in a δ-neck. In the statement of the next lemma we will write Ω synonymously with the M− T of Section 68. Lemma 71.1. (cf. Lemma II.4.3) Given the pinching function Φ, a number Tb ∈ (0, ∞), a positive nonincreasing function r : [0, Tb] → R and a number δ ∈ 0, 21 , there is a nonincreasing function h : [0, Tb] → R with 0 < h(t) < δ 2 r(t) so that the following property is satisfied. Let M be a Ricci flow with surgery defined on [0, T ), with T < Tb, which satisfies the a priori assumptions (Definition 69.6) and which goes singular at time T . Let (Ω, g) denote the time-T limit, in the sense of Section 67. Put ρ = δ r(T ) and (71.2)

Ωρ = {(x, t) ∈ Ω | R(x, T ) ≤ ρ−2 }.

Suppose that (x, T ) lies in an ǫ-horn H ⊂ Ω whose boundary is contained in Ωρ . Suppose 1 also that R(x, T ) ≥ h−2 (T ). Then the parabolic region P (x, T, δ −1 R(x, T )− 2 , −R(x, T )−1 ) is contained in a strong δ-neck. (As usual, ǫ is a fixed constant that is small enough so that the result holds uniformly with respect to the other variables.) Proof. Fix δ ∈ (0, 1). Suppose that the claim is not true. Then there is a sequence of Ricci flows with surgery Mα and points (xα , T α ) ∈ Mα with T α < Tb such that 1. Mα satisfies the Φ-pinching and r-canonical neighborhood assumptions, 2. Mα goes singular at time T α , 3. (xα , T α ) belongs to an ǫ-horn Hα ⊂ Ωα whose boundary is contained in Ωαρ , and 4. R(xα , T α ) → ∞, but 1 5. For each α, P (xα , T α , δ −1 R(xα , T α )− 2 , −R(xα , T α )−1 ) is not contained in a strong δ-neck.

Recall that when ǫ is small enough, any cross-sectional 2-sphere sitting in an ǫ-neck V ⊂ Hα separates the ends of Hα ; see Section 58. We may find a properly embedded minimizing geodesic γ α ⊂ Hα which joins the two ends of Hα . As γ α must intersect a


133 1

cross-sectional 2-sphere containing (xα , T α ), it must pass within distance ≤ 10R(xα , T α )− 2 of (xα , T α ), when ǫ is small. Let y α be the endpoint of γ α contained in Ωαρ and let ybα be the first point, moving along γ α from the noncompact end of Hα toward y α, where −1 R(b y α , T α ) = ρ−2 . As the gradient bound |∇R 2 | ≤ 21 η is valid along γ α starting from ybα and going out the noncompact end (since such points on γ α have scalar curvature greater 1 than r(T α )−2 ), we have distT α (xα , y α ) ≥ distT α (xα , ybα) ≥ η2 ρ − R(xα , T α )− 2 . Let Lα

denote the time-T α distance from xα to the other end of Hα . Since R goes to infinity as one 1 exits the end, Lemma 70.2 implies that limα→∞ R(xα , T α) 2 Lα = ∞. From the existence of 1 γ α , whose length in either direction from xα is large compared to R(xα , T α )− 2 , it is clear that for large α, the canonical neighborhood of (xα , T α) must be of type (a) or (b) in the terminology of Definition 69.1. By Lemmas 67.9 and 70.2, we also know that for any fixed 1 σ < ∞, for large α the ball B(xα , T α, σR(xα , T α )− 2 ) has compact closure in the time-T α slice of Mα .

By Lemma 70.2, after rescaling the metric on the time-T α slice by R(xα , T α ) we have uniform curvature bounds on distance balls. We also have a uniform lower bound on the injectivity radius at (xα , T α) of the rescaled solution, in view of its canonical neighborhood. Hence after passing to a subsequence, we may take a pointed smooth complete limit (M ∞ , x∞ , g∞ ) of the time-T α slices, where the derivative bounds needed to take a smooth limit come from the canonical neighborhood assumption. By the Φ-pinching assumption, M ∞ will have nonnegative curvature.

After passing to a subsequence, we can also assume that the γ α ’s converge to a minimizing geodesic γ in M ∞ that passes within distance 10 from x∞ . The rescaled length of γ α from 1 xα to y α is bounded below by η2 R(xα , T α) 2 ρ − 1 , which tends to infinity as α → ∞. We have shown that the rescaled length of γ α from xα to the other end of Hα also tends to infinity as α → ∞. It follows that γ is bi-infinite. Thus by Toponogov’s theorem, M ∞ splits off an R-factor. Then for large α, the canonical neighborhood of (xα , T α ) must be an ǫ-neck, and M ∞ = R × S 2 for some positively curved metric on S 2 . In particular, M ∞ has scalar curvature uniformly bounded above. Any point x b ∈ M ∞ is a limit of points (b xα , T α ) ∈ Mα . As R∞ (b x) > 0 and R(xα , T α ) → α α α α ∞, it follows that R(b x , T ) → ∞. Then for large α, (b x , T ) is in a canonical neighborhood ∞ which, in view of the R-factor in M , must be a strong ǫ-neck. From the upper bound on the scalar curvature of M ∞ , along with the time interval involved in the definition of a strong ǫ-neck, it follows that we can parabolically rescale the pointed flows (Mα , xα , T α ) by R(xα , T α ), shift time and extract a smooth pointed limiting Ricci flow (M∞ , x∞ , 0) which is defined on a time interval (ξ, 0], for some ξ < 0. In view of the strong ǫ-necks around the points (b xα , T α ), if we take ξ close to zero then we are ensured that the Ricci flow (M∞ , x∞ , 0) has positive scalar curvature R∞ . Given (b x, t) ∈ M∞ , as R∞ (b x, t) > 0 and R(xα , T α ) → ∞, the Φ-pinching implies that the time-t slice M∞ b. Thus M∞ has nonnegative curvature. The timet has nonnegative curvature at x ∞ 0 slice M0 splits off an R-factor, which means that the same will be true of all time slices; cf. the proof of Lemma 61.1. Hence M∞ is a product Ricci flow.

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Let ξ be the minimal negative number so that after parabolically rescaling the pointed flows (Mα , xα , T α ) by R(xα , T α ), we can extract a limit Ricci flow (M∞ , (x∞ , 0), g∞ (·)) which is the product of R with a positively curved Ricci flow on S 2 , and is defined on the time interval (ξ, 0]. We claim that ξ = −∞. Suppose not, i.e. ξ > −∞. Given (b x, t) ∈ M∞ , as R∞ (b x, t) > 0 and R(xα , T α ) → ∞, it follows that (x, t) is a limit of points α α α (b x , t ) ∈ M that lie in canonical neighborhoods. In view of the R-factor in M∞ , for large α these canonical neighborhoods must be strong ǫ-necks. This implies in particular that −2 ∂R∞ R∞ (b x , t) > 0, so ∂R∂t∞ (b x, t) > 0. Then there is a uniform upper bound Q for the ∂t 1 ∞ scalar curvature on M . Extending backward from a time-(ξ + 100Q ) slice, we can construct ′ ′ a limit Ricci flow that exists on some time interval (ξ , 0] with ξ < ξ. As before, using the strong ǫ-neck condition and the Φ-pinching, if ξ ′ is sufficiently close to ξ then we are ensured that the Ricci flow on (ξ ′ , 0] is the product of R with a positively curved Ricci flow on S 2 . This is a contradiction. Thus we obtain an ancient solution M∞ with the property that each point (x, t) lies in a strong ǫ-neck. Removing the R-factor gives an ancient solution on S 2 . In view of the fact that each time slice is ǫ-close to the round S 2 , up to rescaling, it follows that the ancient solution on S 2 must be the standard shrinking solution (see Sections 40 and 43). Then M∞ is the standard shrinking solution on R × S 2 . Hence for an infinite number 1 of α, P (xα , T α , δ −1 R(xα , T α )− 2 , −R(xα , T α )−1 ) is in fact in a strong δ-neck, which is a contradiction. Remark 71.3. If a given h makes Lemma 71.1 work for a given function r then one can check that logically, h also works for any r ′ with r ′ ≥ r. Because of this, we may assume that h only depends on min r = r(T ) and is monotonically nondecreasing as a function of r(T ). Similarly, if a given h makes Lemma 71.1 work for a given value of δ then h also works for any δ ′ with δ ′ ≥ δ. Thus we may assume that h is monotonically nondecreasing as a function of δ. 72. Surgery and the pinching condition This section describes how one can take a δ-neck satisfying the time-t Hamilton-Ivey pinching condition, and perform surgery so as to obtain a new manifold which also satisfies the time-t pinching condition, and which is δ ′ -close to the standard solution modulo rescaling. Here δ ′ is a nonexplicit function of δ but satisifes the important property that δ ′ (δ) → 0 as δ → 0.

The main geometric idea which handles the delicate part of the surgery procedure is contained in the following lemma. It says that one can “round off” the boundary of an approximate round half-cylinder so as to simultaneously increase the scalar curvature and the minimum of sectional curvature at each point. As the statement of the following lemma involves the curvature operator, we state our conventions. If M has constant sectional curvature k then the curvature operator acts on 2-forms as multiplication by 2k. This is consistent with the usual Ricci flow literature, e.g. [22]. Recall that ǫ is our global parameter, which is taken sufficiently small.


135

Lemma 72.1. Let gcyl denote the round cylindrical metric of scalar curvature 1 on R × S 2 . Let z denote the coordinate in the R-direction. Given A > 0, suppose that f : (−A, 0] → R is a smooth function such that • f (k) (0) = 0 for all k ≥ 0. • On (−A, 0),

(72.2)

f (z) < 0 ,

f ′ (z) > 0 ,

f ′′ (z) < 0.

• kf kC 2 < ǫ.

• For every z ∈ (−A, 0),

(72.3)

max (|f (z)|, |f ′ (z)|) ≤ ǫ |f ′′ (z)| .

Then if h0 is a smooth metric on (−A, 0] × S 2 with kh0 − gcyl kC 2 < ǫ and we set h1 = e h0 , it follows that for all p ∈ (−A, 0) × S 2 we have Rh1 (p) > Rh0 (p) − f ′′ (z(p)). Also, if λ1 (p) denotes the lowest eigenvalue of the curvature operator at p then λh1 1 (p) > λh1 0 (p) − f ′′ (z(p)). 2f (z)

Proof. We will use the variational characterization of λ1 (p) : (72.4)

ωij Rijkl ω kl ω6=0 ωij ω ij

λ1 (p) = inf

where ω ∈ Λ2 (Tp M). We will also use the following formulas about curvature quantities for conformally related metrics in dimension 3 : (72.5) Rh1 = e−2f Rh0 − 4 △f − 2 |∇f |2

and (72.6) ij ij j j j i j −2f i i j i 2 i i j e e e e R kl (h1 ) = e R kl (h0 ) − f k δ l + f l δ k + f k δ l − f l δ k − |∇f | δ k δ l − δ l δ k ,

where feij = f;ij − f;i f;j . That is, fe = Hess(f ) − df ⊗ df . The right-hand sides of these expressions are computed using the metric h0 .

To motivate the proof, let us first consider the linearization of these expressions around h0 . Keeping only the linear terms in f gives to leading order, (72.7) and (72.8)

Rh1 ∼ Rh0 −2f Rh0 − 4 △f Rijkl (h1 ) ∼ Rijkl (h0 ) − 2f Rijkl (h0 ) − f; ik δ jl + f; il δ jk + f; jk δ il − f; jl δ i k .

From the assumptions, Rh0 ∼ 1 and f < 0 on (−A, 0) × S 2 . As h0 is close to gcyl , we have △f ∼ f ′′ (z), so −2f Rh0 − 4 △f ≥ − f ′′ (z). Similarly, in the case of gcyl a minimizer ω in (72.4) is of the form ω = X ∧ ∂z , where X is a unit vector in the S 2 -direction. As h0 is close to gcyl , a minimizing ω for h0 will be close to something of the form X ∧ ∂z . Then (72.9)

λh1 1 ∼ λh1 0 − 2f λh1 0 − 2 f ′′ (z) ≥ λh1 0 + 2f (z) |λh1 0 | − 2 f ′′ (z). g

As h0 is close to gcyl , λh1 0 is close to λ1cyl = 0. Then we can use (72.3) to say that 2f (z)|λh1 0 | − 2 f ′′ (z) ≥ − f ′′ (z).

136


The remaining issue is to show that the increase in R and λ1 coming from the linear approximation is still approximately valid in the nonlinear case, provided that ǫ is sufficiently small. For this, we have to show that the increase from the linear approximation dominates the error terms that we have neglected. To deal with the scalar curvature first, from (72.5) we have (72.10) Rh1 = e−2f Rh0 − 4 △gcyl f + 4e−2f (△gcyl f − △f ) − 2 e−2f |∇f |2 ≥ Rh0 − 4 f ′′ (z) + 4e−2f (△gcyl f − △f ) − 2 e−2f |∇f |2 .

Next, there is an estimate of the form (72.11)

|△gcyl f − △f | ≤ const. k h0 − gcyl kC 2 (|f (z)| + |f ′ (z)| + |f ′′ (z)|) ≤ const. ǫ (|f (z)| + |f ′ (z)| + |f ′′(z)|) .

As e−2f ≤ e2ǫ , if ǫ is small then −2f e (72.12) (△gcyl f − △f ) ≤ const. ǫ (|f (z)| + |f ′(z)| + |f ′′ (z)|) Similarly,

e−2f |∇f |2 ≤ const. |f ′(z)|2 ≤ const. ǫ |f ′ (z)|.

(72.13)

When combined with (72.3), if ǫ is taken sufficiently small then (72.14)

− 4 f ′′ (z) + 4e−2f (△gcyl f − △f ) − 2 e−2f |∇f |2 ≥ − f ′′ (z).

This shows the desired estimate for Rh1 (p).

To estimate λh1 1 we use (72.4) and (72.6) to write (72.15)

λh1 1 (p) = e−2f (z)

Comparing with (72.16)

! ωij Rijkl (h0 ) ω kl − 4 ωij feik ω kj − 2 |∇f |2 (z) . inf ω6=0 ωij ω ij ωij Rijkl (h0 ) ω kl ω6=0 ωij ω ij

λh1 0 (p) = inf

gives (72.17)

λh1 0 (p) ≤ e2f (z) λh1 1 (p) +

where ω is a minimizer in (72.15), or (72.18)

4 ωij feik ω kj + 2 |∇f |2 (z), ωij ω ij

λh1 1 (p) ≥ e−2f (z) λh1 0 (p) − 4 e−2f (z)

ωij feik ω kj − 2 e−2f (z) |∇f |2(z). ωij ω ij

Using the variational formula (72.4), one can show that |λh1 0 (p)| ≤ const. ǫ. From eigenvalue perturbation theory [55, Chapter 12], ω will be of the form X ∧ ∂z + O(ǫ) for some unit vector X tangential to S 2 . Then we get an estimate (72.19)

λh1 1 (p) ≥ λh1 0 (p) − 2f ′′ (z) − const. ǫ (|f (z)| + |f ′ (z)| + |f ′′ (z)|) .

From (72.3), if ǫ is taken sufficiently small then λh1 1 (p) − λh1 0 (p) ≥ − f ′′ (z).


137

Recall that the initial condition S0 for the standard solution is an O(3)-symmetric metric g0 on R3 with nonnegative curvature operator, whose end is isometric to a round halfcylinder of scalar curvature 1. To facilitate the surgery procedure, we will assume that some metric ball around the O(3)-fixed point has constant positive curvature. Outside of this ball we use radial coordinates (z, θ) ∈ (−B, ∞) × S 2 , with g0 = e2F (z) gcyl . Here gcyl is the round cylindrical metric of scalar curvature one and F ∈ C ∞ (−B, ∞).

Lemma 72.20. Given A > 0, we can choose B > A and F ∈ C ∞ (−B, ∞) so that 1. F ≡ 0 on [0, ∞) × S 2 . 2. The restriction of F to (−A, 0] × S 2 satisfies the hypotheses of Lemma 72.1. 3. The metric e2F (z) gcyl on (−B, ∞) × S 2 has nonnegative sectional curvature and extends smoothly to a metric on R3 by adding a ball of constant positive curvature at {−B} × S 2 . Proof. For a metric of the forme2F (z) gcyl , one computes that the sectional curvatures are − e− 2F F ′′ and e− 2F 12 − (F ′)2 . In particular, the conditions for positive sectional curvature are F ′′ < 0 and |F ′ | < √12 . The 3-sphere of constant sectional curvature k 2 , with two points removed, has a metric given by √ ! √ 1 2 + √ z − log 1 + e 2z . (72.21) Fk (z) = log k 2

(Shifting z gives other metrics of constant curvature k 2 . We have normalized so that the z = 0 slice is the slice of maximal area.) Note that the derivative √

√ 1 e 2z √ D(z) = √ − 2 2 1 + e 2z

(72.22) is independent of k.

Given A > 0, we take F to be 0 on [0, ∞) and of the form c1 ec2 /z on (−A, 0]. We can take the constant c1 > 0 sufficiently small and the constant c2 < ∞ sufficiently large so that the hypotheses of Lemma 72.1 are satisfied. It remains to smoothly cap off ([−A, ∞) × S 2 , e2F (z) gcyl ) with something of positive sectional curvature. With our given choice of F , we have F ′ (−A) ∈ (0, ǫ). As limz→−∞ D(z) = √1 , (−A,0]

2

′ we can choose B > A so that D(−B) > F ′ (−A). As F ′′ (−A) < 0 and D (−B) < 0, we e ′ < 0 and which e : (−B, ∞) → 0, √1 which has D can extend F ′ to a smooth function D 2 coincides with D on a small interval (−B, −B + δ). Putting Z z e (72.23) F (z) = F (0) + D(w) dw, 0

∞

we obtain F ∈ C (B, ∞) which coincides with Fk on (−B, −B + δ), for some k > 0. Then we can glue on a round metric ball of constant curvature k 2 to {−B}×S 2 , in order to obtain the desired metric. In the statement of the next lemma we continue with the metric constructed in Lemma 72.20.

138


Lemma 72.24. There exists δ ′ = δ ′ (δ) with limδ→0 δ ′ (δ) = 0 and a constant δ0 > 0 such that the following holds. Suppose that δ < δ0 , x ∈ {0} × S 2 and h0 is a Riemannian metric on (−A, 1δ ) × S 2 with R(x) > 0 such that: • h0 satisfies the time-t Hamilton-Ivey pinching condition of Definition B.5. 1

• R(x)h0 is δ-close to gcyl in the C [ δ ]+1-topology.

Then there is a smooth metric h on R3 = D3 ∪ (−B, 1δ ) × S 2 such that • h satisfies the time-t pinching condition. • The restriction of h to [0, 1δ ) × S 2 is h0 .

• The restriction of R(x)h to (−B, −A)×S 2 is g0 , the initial metric of a standard solution.

• The restriction of R(x)h to D3 has constant curvature k 2 .

1 • R(x)h is δ ′ -close to e2F gcyl in the C [ δ′ ]+1 -topology on −B, 1δ × S 2 .

Proof. Put (72.25)

A U1 = (−B, − ) × S 2 , 2

1 U2 = (−A, ) × S 2 δ

and let {α1 , α2 } be a C ∞ partition of unity subordinate to the open cover {U1 , U2 } of (−B, 1δ ) × S 2 . We set (72.26)

h = α1 R(x)−1 g0 + α2 e2F h0

on −B, 1δ × S 2 and cap it off with a 3-ball of constant curvature k, as in Lemma 72.20.

Given δ ′ , we claim that if δ is sufficiently small then the conclusion of the lemma holds. The only of the lemma that is not obvious is the pinching condition. Note that on part A 1 2 − 2 , δ × S the metric h agrees with e2F h0 and hence, when δ is sufficiently small, the pinching condition will hold on − A2 , 1δ × S 2 by Lemmas 72.1 and B.6. On the other hand, when δ is sufficiently small, the restrictions of the metrics g0 = e2F gcyl and R(x)e2F h0 to (−A, − A2 ) × S 2 will be very close and will have strictly positive curvature. (The positive curvature for e2F h0 also follows from Lemma 72.1; if δ is small enough then λh1 0 will be close to zero, while −f ′′ (z) is strictly positive for z ∈ (−A, − A2 ).) Thus h will have positive curvature on (−B, − A2 ) × S 2 and the pinching condition will hold there. We have now fixed the initial condition g0 for a standard solution, along with the procedure to meld g0 to an approximate cylinder. 73. II.4.4. Performing surgery and continuing flows This section discusses the surgery procedure and shows how to prolong a Ricci flow with surgery, provided that the a priori assumptions hold.


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Definition 73.1. (Ricci flow with cutoff) Suppose that a ≥ 0 and let M be a Ricci flow with surgery defined on [a, b] that satisfies the a priori assumptions of Definition 69.6. Let δ : [a, b] → (0, δ0 ) be a nonincreasing function, where δ0 is the parameter of Lemma 72.24. Then M is a Ricci flow with (r, δ)-cutoff if at each singular time t, the forward time slice − M+ t is obtained from the backward time slice Ω = Mt by applying the following procedure: A. Discard each component of Ω that does not intersect

(73.2)

Ωρ = {(x, t) ∈ Ω | R(x, t) ≤ ρ−2 },

where ρ = δ(t)r(t). B. In each ǫ-horn Hij of each of the remaining components Ωi , find a point (xij , t) such that R(xij , t) = h−2 , where h = h(t) is as in Lemma 71.1. 1

C. Find a strong δ-neck Uij × [t − h2 , t] containing P (xij , t, δ −1 R(xij , t)− 2 , −R(xij , t)−1 ); this is guaranteed to exist by Lemma 71.1. S D. For each ij, let Sij ⊂ Uij be a cross-sectional 2-sphere containing (xij , t). Cut i Ωi along the Sij ’s and throw away the tips of the horns Hij , to obtain a compact manifoldwith-boundary X having a spherical boundary component for each ij. E. Glue caps onto X, using Lemma 72.24, to obtain the closed manifold M+ t . For concreteness, we take the parameter A of Lemma 72.24 to be 10. The neighborhood of a boundary component of X is parametrized as [−A, δ −1 ) × S 2 , with Sij = {−A} × S 2 . The metric on [0, δ −1 ) × S 2 is unaltered by the surgery procedure. The corresponding region in the new manifold M+ t , minus a metric ball of constant curvature, is parametrized by (−B, δ −1 ) × S 2 . Put Sij′ = {0} × S 2 ⊂ M− t . We will consider the part added by surgery + on Hij to be the 3-disk in Mt bounded by Sij′ . In terms of Definition 68.1, if t = t+ k then S ′ + − + − the subset Xk of Ωk = Mt has boundary ij Sij . The added part Mt − Xk+1 is a union of 3-balls. Remark 73.3. Our definition of surgery differs slightly from that in [52]. The paper [52] has two extra steps involving throwing away certain components of the postsurgery manifold. We omit these steps in order to simplify the definition of surgery, but there is no real loss either way. First, in the setup of [52, Section 4.4], any component of M+ t that is ǫ-close to a metric 3 quotient of the round S is thrown away. The motivation of [52] was to not have to include these in the list of canonical neighborhoods. Such components are topologically standard. We do include such manifolds in the list of canonical neighborhoods and do not throw them away in the surgery procedure. Second, when considering the long-time behavior of Ricci flow in [52, Section 7], any component of M+ t which admits a metric of nonnegative scalar curvature is thrown away. The motivation for this extra step is that any such component admits a metric that is either flat or has finite extinction time. In either case one concludes that the component is a graph manifold and, for the purposes of the geometrization conjecture, is standard. (Recall the definition of graph manifolds from Appendix I.) Again, we do not throw away such components.

140


Note that the definition of Ricci flow with (r, δ)-cutoff also depends on the function r(t) through the a priori assumption. We now state how the topology of the time slice changes when going backward through the singular time t. Recall that for t′ < t close to t, the time slices Mt′ are all diffeomorphic; we refer to this diffeomorphism type as the presurgery manifold, and the forward time slice M+ t as the postsurgery manifold. Lemma 73.4. The presurgery manifold may be obtained from the postsurgery manifold by applying the following operations finitely many times: • Replacing two connected components with their connected sum. • Taking connected sum of a connected component with S 1 × S 2 or RP 3 . • Taking the disjoint union with an additional S 1 × S 2 or an isometric quotient of the round S 3 . Proof. The proof is basically the same as that of Lemma 67.13. The only difference is that we must take into account the compact components of Ω that do not intersect Ωρ ; these are thrown away in Step A. (Such components did not occur in Lemma 67.13 because in Lemma 67.13 we were dealing with the first surgery for the Ricci flow on the initial connected manifold; see Lemma 67.4, which is valid for the first surgery time.) Any such component is diffeomorphic to S 1 × S 2 , RP 3 #RP 3 or a quotient of the round S 3 , in view of the canonical neighborhood assumption; see the proof of Lemma 67.5. − 3 Remark 73.5. When δ > 0 is sufficiently small, we will have vol(M+ t ) < vol(Mt ) − h(t) for each surgery time t ∈ (a, b). This is because each component that is discarded in step D contains at least “half” of the δ-neck Uij , which has volume at least const. δ −1 h(t)3 , while the cap added has volume at most const. h(t)3 .

Remark 73.6. For a Ricci flow with surgery whose original manifold is nonaspherical and irreducible, one wants to know that the Ricci flow goes extinct within a finite time [24, 25, 53]. Consider the effect of a first surgery, say at time t. Among the connected components of the postsurgery manifold M+ t , one will be diffeomorphic to the presurgery manifold and the others will be 3-spheres. Let Nt+ be a component of M+ t that is diffeomorphic to the presurgery manifold. By the nature of the surgery procedure, there is a function ξ defined on a small interval (t − α, t) so that limt′ →t ξ(t′ ) = 1 and for t′ ∈ (t − α, t), there is a homotopyequivalence from (Mt′ , g(t′)) to Nt+ that expands distances by at most ξ(t′ ). Following the subsequent evolution of Nt+ , there is a similar statement for the later singular times. This fact is needed in [24, 25, 53] in order to control the decay of a certain area functional as one goes through a surgery. We discuss how to continue Ricci flows after surgery. We recall that in Definition 68.1 of a Ricci flow with surgery defined on an interval [a, c], the final time slice Mc consists of a single manifold M− c = Ω that may or may not be singular. Lemma 73.7. (Prolongation of Ricci flows with cutoff ) Take the function Φ to be the timedependent pinching function associated to Definition B.5 in Appendix B. Suppose that r and δ are nonincreasing positive functions defined on [a, b]. Let M be a Ricci flow with (r, δ)-cutoff defined on an interval [a, c] ⊂ [a, b]. Provided sup δ is sufficiently small, either (1) M can be prolonged to a Ricci flow with (r, δ)-cutoff defined on [a, b], or


141

(2) There is an extension of M to a Ricci flow with surgery defined on an interval [a, T ] with T ∈ (c, b], where a. The restriction of the flow to any subinterval [a, T ′ ], T ′ < T , is a Ricci flow with (r, δ)-cutoff, but b. The r-canonical neighborhood assumption fails at some point (x, T ) ∈ M− T. In particular, the only obstacle to prolongation of Ricci flows with (r, δ)-cutoff is the potential breakdown of the r-canonical neighborhood assumption. Proof. Consider the time slice of Mc = M− c at time c. If it is singular then we perform steps + − A-E of Definition 73.1 to produce Mc ; otherwise we set M+ c = Mc . Since the surgery is done using Lemma 72.24, provided δ > 0 is sufficiently small, the forward time slice M+ c will satisfy the Φ-pinching assumption. We claim that the r-canonical neighborhood assumption holds in M+ c . More precisely, if + −1 − a point (x, c) ∈ Mc lies within a distance of 10ǫ h from the added part M+ c − Mc then −1 + it lies in an ǫ-cap, while if (x, c) lies at distance greater than 10ǫ h from Mc − M− c and −2 has scalar curvature greater than r(c) then it lies in a canonical neighborhood that was 1 present in the presurgery manifold M− c . (We are assuming that ǫ < 100 .) In view of Lemma 63.1, the only point to observe is that points at distance roughly 10ǫ−1 h lie in ǫ-necks, as they are unaltered by the surgery and they were in δ-necks before the surgery. This gives the ǫ-neck needed to define an ǫ-cap. We now prolong M by Ricci flow with initial condition M+ c . If the flow extends smoothly up to time b then we are done because either the canonical neighborhood assumption holds up to time b yielding (a), or it fails at some time in the interval (c, b], and we have (b). Otherwise, there is some time tsing ≤ b at which it goes singular. We add the singular limit Ω at time tsing to obtain a Ricci flow with surgery defined on [a, tsing ]. From Lemma 72.24, M satisfies the Hamilton-Ivey pinching condition of Definition B.5 on [a, tsing ]. As the function r is nonincreasing in t, it follows from Definition 69.1 that the set of times t ∈ [c, tsing ] for which the r-canonical neighborhood assumption holds is relatively open to the right (i.e. if the r-canonical neighborhood assumption holds at time t ∈ [c, tsing ) then it also holds within some interval [t, t′ )). Thus the set of times t ∈ [c, tsing ] for which the r-canonical neighborhood assumption holds is either an interval [c, T ), with T ≤ tsing , or [c, tsing ]. If the set of such times t is (c, T ) for some T ≤ tsing then the lemma holds. Otherwise, the r-canonical neighborhood assumption holds at tsing . In this case we repeat the construction with c replaced by tsing , and iterate if necessary. Either we will reach time b after a finite number of iterations, or we will reach a time T satisfying (2), or we will hit an infinite number of singular times before time b. However, the last possibility cannot occur. A singular time corresponds to a component going extinct or to a surgery. The number of components going extinct before time b can be bounded in terms of the number of surgeries before time b, so it suffices to show that the latter is finite. Each surgery removes a volume of at least h3 , but the lower bound on the scalar curvature during the flow, coming from the maximum principle, gives a finite upper bound on the total volume growth during the complement of the singular times. Remark 73.8. The condition C1 ≥ 30ǫ−1 in Definition 69.1 was in order to ensure that the ǫ-cap coming from a surgery satisfies the requirements to be a canonical neighborhood.

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74. II.4.5. Evolution of a surgery cap Let M be a Ricci flow with (r, δ)-cutoff. The next result says that provided δ is small, after a surgery at scale h there is a ball B of radius Ah ≫ h centered in the surgery cap whose evolution is close to that of a standard solution for an elapsed time close to h2 , unless another surgery occurs during which the entire ball is thrown away. Note that the elapsed time h2 corresponds, modulo parabolic rescaling, to the duration of the standard solution. Lemma 74.1. (cf. Lemma II.4.5) ˆ θ, rˆ) > 0 with the following For any A < ∞, θ ∈ (0, 1) and rˆ > 0, one can find δˆ = δ(A, property. Suppose that we have a Ricci flow with (r, δ)-cutoff defined on a time interval [a, b] ˆ with min r = r(b) ≥ rˆ. Suppose that there is a surgery time T0 ∈ (a, b), with δ(T0 ) ≤ δ. + Consider a given surgery at the surgery time and let (p, T0 ) ∈ MT0 be the center of the ˆ = h(δ(T0 ), ǫ, r(T0 ), Φ) be the surgery scale given by Lemma 71.1 and put surgery cap. Let h ˆ 2 ). Then one of the two following possibilities occurs : T1 = min(b, T0 + θh ˆ T1 − T0 ). The pointed solution there (with (1) The solution is unscathed on P (p, T0, Ah, respect to the basepoint (p, T0 )) is, modulo parabolic rescaling, A−1 -close to the pointed flow ˆ −2 ], where U0 is an open subset of the initial time slice S0 of a standard on U0 × [0, (T1 − T0 )h solution S and the basepoint is the center c of the cap in S0 . (2) Assertion (1) holds with T1 replaced by some t+ ∈ [T0 , T1 ), where t+ is a surgery time. ˆ becomes extinct at time t+ , i.e. P (p, T0, Ah, ˆ t+ − T0 ) ∩ Moreover, the entire ball B(p, T0 , Ah) + − Mt+ ⊂ Mt+ − Mt+ . Proof. We give a proof with the same ingredients as the proof in [52], but which is slightly rearranged. We first show the following result, which is almost the same as Lemma 74.1. ˆ θ, rˆ) > 0 with Lemma 74.2. For any A < ∞, θ ∈ (0, 1) and rˆ > 0, one can find δˆ = δ(A, the following property. Suppose that we have a Ricci flow with (r, δ)-cutoff defined on a time interval [a, b] with min r = r(b) ≥ rˆ. Suppose that there is a surgery time T0 ∈ (a, b), with ˆ Consider a given surgery at the surgery time and let (p, T0 ) ∈ M+ be the center δ(T0 ) ≤ δ. T0 ˆ = h(δ(T0 ), ǫ, r(T0 ), Φ) be the surgery scale given by Lemma 71.1 and of the surgery cap. Let h ˆ 2 ). Suppose that the solution is unscathed on P (p, T0 , Ah, ˆ T1 − T0 ). put T1 = min(b, T0 + θh Then the pointed solution there (with respect to the basepoint (p, T0 )) is, modulo parabolic ˆ −2 ], where U0 is an open subset rescaling, A−1 -close to the pointed flow on U0 × [0, (T1 − T0 )h of the initial time slice S0 of a standard solution S and the basepoint is the center c of the cap in S0 . Proof. Fix θ and rˆ. Suppose that the lemma is not true. Then for some A > 0, there is a α α sequence {Mα , (pα , T0α )}∞ α=1 of pointed Ricci flows with (r , δ )-cutoff that together provide a counterexample. In particular, 1. limα→∞ δ α (T0α ) = 0. ˆ α , T α − T α ). 2. Mα is unscathed on P (pα, T0α , Ah 1 0 α α c 3. If (M , (ˆ p , 0)) is the pointed Ricci flow arising from (Mα , (pα , T0α )) by a time shift of ˆ α then P (ˆ ˆ α )−2 ) is not A−1 -close to a T0α and a parabolic rescaling by h pα , 0, A, (T1α − T0α )(h pointed subset of a standard solution.


143

ˆ α )−2 . (We do not exclude that T2 = 0.) Clearly T2 ≤ θ. Put T2 = lim inf α→∞ (T1α − T0α )(h ˆ α )−2 . Let T3 be After passing to a subsequence, we can assume that T2 = limα→∞ (T1α −T0α )(h the supremum of the set of times τ ∈ [0, T2 ] with the property that we can apply Appendix cα , (ˆ E, if we want, to take a convergent subsequence of the pointed solutions (M pα , 0)) on the time interval [0, τ ] to get a limit solution with bounded curvature. (In applying Appendix E, we use the case l > 0 of Appendix D to get bounds on the curvature derivatives near time 0. In particular, if T2 > 0 then T3 > 0.) From the nature of the surgery gluing in Lemma 72.24, since limα→∞ δ α (T0α ) = 0 we know that we can at least take a limit of the cα , (ˆ pointed solutions (M pα , 0)) on the time interval [0, 0], so T3 is well-defined. Sublemma 74.3. T3 = T2 .

Proof. Suppose not. Consider the interval [0, T3 ) (where we define [0, 0) to be {0}). Given α σ ∈ (0, T2 −T3 ], for any subsequence of {Mα , (ˆ pα , 0)}∞ pα , 0)}∞ α=1 (which we relabel as {M , (ˆ α=1 ) either 1. There is some λ > 0 and an infinite number of α for which the set B(ˆ pα , 0, λ) becomes scathed on [0, T3 + σ], or 2. For each λ > 0 the set P (ˆ pα , 0, λ, T3 + σ) is unscathed for large α, but for each Λ > 0 there is some λΛ > 0 such that lim supα→∞ supP (ˆpα,0,λΛ ,T3 +σ) | Rm | ≥ Λ.

c∞ , (ˆ By Appendix E, after passing to a subsequence, there is a complete limit solution (M p∞ , 0)) defined on the time interval [0, T3 ) with bounded curvature on compact time intervals. Rec∞ , (ˆ label the subsequence by α. By Lemma 60.3, (M p∞ , 0)) must be the same as the c∞ restriction of some standard solution to [0, T3 ). From Lemma 62.1, the curvature of M is uniformly bounded on [0, T3 ); therefore by the canonical neighborhood assumption and equation (69.2), we can choose σ ∈ (0, T2 − T3 ] and Λ′ > 0 so that for any λ > 0, we have lim supα→∞ supP (ˆpα ,0,λ,T3 +σ) | Rm | ≤ Λ′ . However, limα→∞ δ α (T0α ) = 0 and surgeries only occur near the centers of δ-necks. From the curvature bound on the time interval [0, T3 + σ] and the length distortion estimates of Lemma 27.8, for a given λ the balls B(ˆ pα , 0, λ) will stay within a uniformly bounded distance from pˆα on the time interval [0, T3 + σ]. Hence they cannot be scathed on [0, T3 + σ] for an infinite number of α, as the collar length of the δ-neck around the supposed surgery locus would be large enough to prohibit the cap point pˆα from being within a bounded distance from the surgery locus. This, along with the fact that lim supα→∞ supP (ˆpα ,0,λ,T3 +σ) | Rm | ≤ Λ′ for all λ > 0, gives a contradiction.

cα pα , 0)}∞ , Returning to the original sequence {Mα , (pα , T0α )}∞ α=1 and its rescaling {M , (ˆ α=1 we can now take a subsequence that converges on the time interval [0, T2 ), again necessarcαβ , (ˆ ily to a standard solution. Then there will be an infinite subsequence {M pαβ , 0)}∞ β=1 αβ αβ ˆ αβ −2 αβ αβ cα , (ˆ of {M pα , 0)}∞ , with lim (T − T )( h ) = T , so that P (ˆ p , 0, A, (T β→∞ 1 2 α=1 0 1 − αβ ˆ αβ −2 −1 T0 )(h ) ) is A -close to a pointed subset of a standard solution (by the canonical neighborhood assumption, equation (69.2) and Appendix D). This is a contradiction. ˆ T1 −T0 ) We now finish the proof of Lemma 74.1. If the solution is unscathed on P (p, T0 , Ah, then we can apply Lemma 74.2 to see that we are in case (1) of the conclusion of Lemma ˆ T1 − T0 ). 74.1. Suppose, on the other hand, that the solution is scathed on P (p, T0 , Ah,

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ˆ t − T0 ). We can Let t+ be the largest t so that the solution is unscathed on P (p, T0 , Ah, apply Lemma 74.2 to see that conclusion (1) of Lemma 74.1 holds with T1 replaced by t+ . As surgery is always performed near the middle of a δ-neck, if δˆ 0, we can find A = A(l, rˆ) < ∞ and θ = θ(l, rˆ) with the following property. Suppose that we are in the situation ˆ θ, rˆ). As usual, h ˆ will be the surgery scale coming from of Lemma 74.1, with δ(T0 ) < δ(A, Lemma 71.1. Let γ : [T0 , Tγ ] → M be an admissible curve, with Tγ ∈ (T0 , T1 ]. Suppose that ˆ ˆ Tγ − T0 ), and either γ(T0 ) ∈ B(p, T0 , A2h ), γ([T0 , Tγ )) ⊂ P (p, T0 , Ah, ˆ 2, a. Tγ = T1 = T0 + θ(h) or ˆ × [T0 , Tγ ]. b. γ(Tγ ) ∈ ∂B(p, T0 , Ah) Then

(75.2)

Z

Tγ

T0

2 R(γ(t), t) + |γ(t)| ˙ dt > l.

ˆ θ, rˆ) so as to satisfy Proof. For the moment, fix A < ∞ and θ ∈ (0, 1). Choose δˆ = δ(A, Lemma 74.1. Let M, (p, T0 ), etc., be as in the hypotheses of Lemma 74.1. Let γ : [T0 , Tγ ] → M be a curve as in the hypotheses of the Corollary. From Lemma 74.1, we know that ˆ Tγ − T0 ) ⊂ M, there is a standard solution S such that the parabolic region P (p, T0 , Ah, ˆ −2 ) A−1 -close to a pointed flow with basepoint (p, T0 ), is (after parabolic rescaling by h ˆ −2 . U0 × [0, Tˆγ ] ⊂ S, the latter having basepoint (c, 0). Here U0 ⊂ S0 and Tˆγ = (Tγ − T0 )h −1 Then the image of γ, under the diffeomorphism implicit in the definition of A -closeness, gives rise to a smooth curve γ0 : [0, Tˆγ ] → U0 × [0, Tˆγ ] so that (if A is sufficiently large) :


145

3 γ0 (0) ∈ B(c, 0, A), 5 Z Tγ Z Tˆγ 1 |γ| ˙ 2 dt ≥ |γ˙ 0 |2 dt, 2 T0 0 Z Tˆγ Z Tγ 1 R(γ0 (t), t)dt, R(γ(t), t)dt ≥ 2 0 T0

(75.3) (75.4) (75.5) and (a) Tˆγ = θ, or

(b) γ0 (Tˆγ ) 6∈ P (c, 0, 45 A, Tˆγ ).

In case (a) we have, by Lemma 63.1, Z Z Z Tγ 1 θ 1 θ R(γ0 (t), t)dt ≥ const.(1 − t)−1 dt = const. log(1 − θ). (75.6) R(γ(t), t)dt ≥ 2 0 2 0 T0

If we choose θ sufficiently close to 1 then in this case, we can ensure that Z Tγ Z Tγ 2 (75.7) R(γ(t), t) + |γ(t)| ˙ dt ≥ R(γ(t), t) dt > l. T0

T0

In case (b), we may use the fact that the Ricci curvature of the standard solution is everywhere nonnegative, and hence the metric tensor is nonincreasing with time. So if π : S = S0 × [0, 1) → Sθ is projection to the time-θ slice and we put η = π ◦ γ0 then Z ˆ Z ˆ Z Tγ 2 1 Tγ 1 1 Tγ 2 2 2 ˆ (75.8) d(η(0), η(Tγ )) |γ˙0 (t)| dt ≥ |η(t)| ˙ dt ≥ |γ(t)| ˙ dt ≥ 2 0 2 0 2Tˆγ T0 2 1 ≥ d(η(0), η(Tˆγ )) . 2 With our given value of θ, in view of (b), if we take A large enough then we can ensure that 2 1 d(η(0), η(Tˆγ )) > l. This proves the lemma. 2

76. II.4.7. A technical estimate The next result is a technical result that will not be used in the sequel. Corollary 76.1. (cf. Corollary II.4.7) For any Q < ∞ and rˆ > 0, there is a θ = θ(Q, rˆ) ∈ (0, 1) with the following property. Suppose that we are in the situation of Lemma 74.1, with ˆ θ, rˆ) and A > ǫ−1 . If γ : [T0 , Tx ] → M is a static curve starting in B(p, T0 , Ah), ˆ δ(T0 ) < δ(A, and (76.2)

Q−1 R(γ(t)) ≤ R(γ(Tx )) ≤ Q(Tx − T0 )−1

ˆ 2. for all t ∈ [T0 , Tx ], then Tx ≤ T0 + θh

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Remark 76.3. The hypothesis (76.2) in the corollary means that in the scale of the scalar curvature R(γ(Tx )) at the endpoint γ(Tx ), the scalar curvature on γ is bounded and the elapsed time of γ is bounded. The conclusion says that given these bounds, the elapsed time is strictly less than that of the corresponding rescaled standard solution. ˆ 2 then by Lemma 63.1 and 74.1, Proof. If Tx > T0 + θh ˆ 2 )) ≥ const.(1 − θ)−1 h ˆ −2 . (76.4) R(γ(T0 + θh Thus by (76.2) we get (76.5) or

ˆ −2 ≤ R(γ(Tx )) ≤ Q(Tx − T0 )−1 , Q−1 const.(1 − θ)−1 h

ˆ 2, Tx − T0 ≤ const. Q2 (1 − θ)h ˆ 2 is less than θh ˆ 2 , which gives a If we choose θ close enough to 1 then const. Q2 (1 − θ)h contradiction. (76.6)

77. II.5. Statement of the the existence theorem for Ricci flow with surgery Our presentation of this material follows Perelman’s, except for some shuffling of the material. We will be using some terminology introduced in Section 68, as well as results about the L-function and noncollapsing from Sections 78 and 79. Definition 77.1. A compact Riemannian 3-manifold is normalized if | Rm | ≤ 1 everywhere, and the volume of every unit ball is at least half the volume of the Euclidean unit ball. We will use the fact that a smooth normalized Ricci flow, with bounded curvature on compact time intervals, satisfies the Hamilton-Ivey pinching condition of Definition B.5. The main result of the surgery procedure is Proposition 77.2 (cf. II.5.1), which implies that one can choose positive nonincreasing functions r : R+ → (0, ∞), δ : R+ → (0, ∞) such that the Ricci flow with (r, δ)-surgery flow starting with any normalized initial condition will be defined for all time. The actual statement is structured to facilitate a proof by induction: Proposition 77.2. (cf. Proposition II.5.1) There exist decreasing sequences 0 < rj < ǫ2 , κj > 0, 0 < δ¯j < ǫ2 for 1 ≤ j < ∞, such that for any normalized initial data and any nonincreasing function δ : [0, ∞) → (0, ∞) such that δ < δ¯j on [2j−1 ǫ, 2j ǫ], the Ricci flow with (r, δ)-cutoff is defined for all time and is κ-noncollapsed at scales below ǫ. Here, and in the rest of this section, r and κ will always denote functions defined on an interval [0, T ] ⊆ [0, ∞) with the property that r(t) = rj and κ(t) = κj for all t ∈ [0, T ] ∩ [2j−1 ǫ, 2j ǫ). By “κ-noncollapsed at scales below ǫ”, we mean that for each ρ < ǫ and all (x, t) ∈ M with t ≥ ρ2 , whenever P (x, t, ρ, −ρ2 ) is unscathed and | Rm | ≤ ρ−2 on P (x, t, ρ, −ρ2 ), then we also have vol(B(x, t, ρ)) ≥ κ(t)ρ3 .

Recall that ǫ is a “global” parameter which is assumed to be small, i.e. all statements involving ǫ (explicitly or otherwise) are true provided ǫ is sufficiently small. Proposition 77.2


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does not impose any serious new constraints on ǫ. For example, instead of using the time intervals {[2j−1ǫ, 2j ǫ]}∞ j=1 , we could have taken any collection of adjoining time intervals starting at a small positive time. Also, we just need some fixed upper bound on rj and δ j . We will follow [52] and write these somewhat arbitrary constants in terms of the single global parameter ǫ. Note also that having normalized initial data sets a length scale for the Ricci flow. The phrase “the Ricci flow with (r, δ)-cutoff is defined for all time” allows for the possibility that the entire manifold goes extinct, i.e. that after some time we are talking about the flow on the empty set. In the rest of this section we give a sketch of the proof. The details are in the subsequent sections. Given positive nonincreasing functions r and δ, if one has a normalized initial condition (M, g(0)) then there will be a maximal time interval on which the Ricci flow with (r, δ)-cutoff is defined. This interval can be finite only if it is of the form [0, T ) for some T < ∞, and the Ricci flow with (r, δ)-cutoff on [0, T ) extends to a Ricci flow with surgery on [0, T ] for which the r-canonical neighborhood assumption fails at time T ; see Lemma 73.7. The main point here is that the r-canonical neighborhood assumption allows one to run the flow forward up to the singular time, and then perform surgery, while volume considerations rule out an accumulation of surgery times. Thus the crux of the proof is showing that the functions r and δ can be chosen so that the r-canonical neighborhood assumption will continue to hold, and the Ricci flow with surgery satisfies a noncollapsing condition. The strategy is to argue by induction on i that ri , δ¯i , and κi can be chosen (and δ¯i−1 can be adjusted) so that the statement of the proposition holds on the the finite time interval [0, 2i ǫ]. In the induction step, one establishes the canonical neighborhood assumption using an argument by contradiction similar to the proof of Theorem 52.7. (We recommend that the reader review this before proceeding). The main difference between the proof of Theorem 52.7 and that of Proposition 77.2 is that the non-collapsing assumption, the key ingredient that allows one to implement the blowup argument, is no longer available as a direct consequence of Theorem 26.2, due to the presence of surgeries. We now discuss the augmentations to the non-collapsing argument of Theorem 26.2 necessitated by surgery; this is treated in detail in sections 78 and 79. We first recall Theorem 26.2 and its proof: if a parabolic ball P (x0 , t0 , r0 , −r02 ) in Ricci flow (without surgery) is sufficiently collapsed then one uses the L-function with basepoint (x0 , t0 ), and the L-exponential map based at (x0 , t0 ), to get a contradiction. One considers the reduced volume of a suitably chosen time slice Mt . There is a positive lower bound on the reduced volume coming from the selection of a point where the reduced distance is at most 23 , which in turn comes from an application of the maximum principle to the L-function. On the other hand, there is an upper bound on the reduced volume, which the collapsing forces to be small, thereby giving the contradiction. The upper bound comes from the monotonicity of the weighted Jacobian of the L-exponential map. In fact, this upper bound works without significant modification in the presence of surgery, provided one considers only the reduced volume contributed by those points in the time t slice which may be joined to (x0 , t0 ) by minimizing L-geodesics lying in the regular part of spacetime (see Lemma 78.11).

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To salvage the lower bound on the reduced volume, the basic idea is that by making the surgery parameter δ small, one can force the L-length of any curve passing close to the surgery locus to be large (Lemma 79.3). This implies that if (x, t) is a point where L isn’t too large, then there will necessarily be an L-geodesic from (x0 , t0 ) to (x, t). To construct the minimizer, one takes a sequence of admissible curves from (x, t) to (x0 , t0 ) with L-length tending to the infimum, and argues that they must stay away from the surgeries; hence they remain in a compact part of spacetime, and subconverge to a minimizer. Therefore the calculations from Sections 15-26 will be valid near such a point (x, t). The maximum principle can then be applied as before to show that the minimum of the reduced length is ≤ 32 on each time slice (see Lemma 78.6). To be more precise, if one makes the surgery parameter δ(t′ ) small for a surgery at a given time t′ then one can force the L-length of any curve passing close to the time-t′ surgery locus to be large, provided that the endtime t0 of the curve is not too large compared to t′ . (If t0 is much larger than t′ then the curve may spend a long time in regions of negative scalar curvature after time t′ . The ensuing negative effect on L could overcome the positive effect of the small surgery parameter.) In the proof of Theorem 26.2, in order to show noncollapsing at time t0 , one went all the way back to a time slice near the initial time and found a point there where l was at most 32 . There would be a problem in using this method for Ricci flows with surgery - we would have to constantly redefine δ(t′ ) to handle the case of larger and larger t0 . The resolution is to not go back to a time slice near the initial time slice. Instead, in order to show κ-noncollapsing in the time slice [2i ǫ, 2i+1 ǫ], we will want to get a lower bound on the reduced volume for a time t-slice with t lying in the preceding time interval [2i−1 ǫ, 2i ǫ]. As we inductively have control over the geometry in the time slice [2i−1 ǫ, 2i ǫ], the argument works equally well.

Finally, as mentioned, after obtaining the a priori κ-noncollapsing estimate on the interval [2 ǫ, 2i+1 ǫ], one proves that the r-canonical neighborhood assumption holds at time T ∈ [2i ǫ, 2i+1 ǫ]. One difference here is that because of possible nearby surgeries, there are two ways to obtain the canonical neighborhood : either from closeness to a κ-solution, as in the proof of Theorem 52.7, or from closeness to a standard solution. i

78. The L-function of I.7 and Ricci flows with surgery In this section we examine several points which arise when one adapts the noncollapsing argument of Theorem 26.2 to Ricci flows with surgery. This material is implicit background for Lemma 79.12 and Proposition 84.1. We will use notation and terminology introduced in Section 68. Let M be a Ricci flow with surgery, and fix a point (x0 , t0 ) ∈ M. One may define the L-length of an admissible curve γ from (x0 , t0 ) to some (x, t), for t < t0 , using the formula Z t0 p t0 − t¯ R + |γ| ˙ 2 dt¯, (78.1) L(γ) = t

where γ˙ denotes the spatial part of the velocity of γ. One defines the L-function on M(−∞,t0 ) by setting L(x, t) to be the infimal L-length of the admissible curves from (x0 , t0 ) to (x, t) if such an admissible curve exists, and infinity otherwise. We note that if (x, t) is in a surgery


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time slice M− t and is actually removed by the surgery then there will not be an admissible curve from (x0 , t0 ) to (x, t). If γ is an admissible curve lying in Mreg then the first variation formula applies. Hence an admissible curve in Mreg from (x0 , t0 ) to (x, t) whose L-length equals L(x, t) will satisfy the L-geodesic equation. If γ is a stable L-geodesic in Mreg then the proof of the monotonicity 3 along γ of the weighted Jacobian τ − 2 exp(−l(τ ))J(τ ) remains valid. Similarly, if U ⊂ M(−∞,t0 ) is an open set such that every (x, t) ∈ U is accessible from (x0 , t0 ) by a minimizing L-geodesic (i.e. an L-geodesic of L-length L(x, t)) contained in Mreg , then the arguments of Section 24 imply that the differential inequality (78.2)

¯ τ + ∆L ¯≤6 L

¯ = 2√τ L and l = holds in U, in the barrier sense, where τ = t0 − t, L

¯ L . 4τ

Lemma 78.3 (Existence of L-minimizers). Let M be a Ricci flow with surgery defined on [a, b]. Suppose that (x0 , t0 ) ∈ M lies in the backward time slice M− t0 .

(1) For each (x, t) ∈ M[a,t0 ) with L(x, t) < ∞, there exists an L-minimizing admissible path γ : [t, t0 ] → M from (x, t) to (x0 , t0 ) which satisfies the L-geodesic equation at every time t ∈ (t, t0 ) for which γ(t) ∈ Mreg .

(2) L is lower semicontinuous on M[a,t0 ) and continuous on Mreg ∩ M[a,t0 ) . (Note that M+ a ⊂ Mreg .)

(3) Every sequence (xj , tj ) ∈ M[a,t0 ) with lim supj L(xj , tj ) < ∞ has a convergent subsequence. Proof. (1) Let {γj : [t, t0 ] → M}∞ j=1 be a sequence of admissible curves from (x, t) to (x0 , t0 ) such that limj→∞ L(γj ) = L(x, t) < ∞. By restricting the sequence, we may assume that supj L(γj ) < 2L(x, t). We claim that there is a subsequence of the γj ’s that (a) converges uniformly to some γ∞ : [t, t0 ] → M,

and

(b) converges weakly to γ∞ in W 1,2 on any subinterval [t′ , t′′ ] ⊂ [t, t0 ) such that [t′ , t′′ ] is free of singular times. To see this, note that on any time interval [c, d] ⊂ [t, t0 ) which is free of singular times, one may apply the Schwarz inequality to the L-length, along with the fact that the metrics on the time slices Mt , t ∈ [c, d], are uniformly biLipschitz to each other, to conclude that the + γj ’s are uniformly Hölder-continuous on [c, d]. We know that γj (t′ ) lies in M− t′ ∩ Mt′ for each surgery time t′ ∈ (t, t0 ), and so one can use similar reasoning to get Hölder control on a short time interval of the form [t′′ , t′ ]. Using a change of variable as in (17.6), one obtains uniform Hölder control near t0 after reparametrizing with s. It follows that the γj ’s are equicontinuous and map into a compact part of spacetime, so Arzela-Ascoli applies; therefore, by passing to a subsequence we may assume that (a) holds. To show (b), we apply weak compactness to the sequence (78.4)

{γj |[t′ ,t′′ ] };

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this is justified by the fact that the paths γj |[t′ ,t′′ ] ’s remain in a part of M with bounded geometry. Thus we may assume that our sequence {γj } converges uniformly on [t, t0 ] and weakly on every subinterval [t′ , t′′ ] as in (b). By weak lower semicontinuity of L-length, it follows that the W 1,2 -path γ∞ has L-length ≤ L(x, t). Since any W 1,2 path may be approximated in W 1,2 by admissible curves with the same endpoints, it follows that γ∞ minimizes L-length among W 1,2 paths, and therefore it restricts to a smooth solution of the L-geodesic equation on each time interval [t′ , t′′ ] ⊂ [t, t0 ] such that (t′ , t′′ ) is free of singular times. Hence γ∞ is an L-minimizing admissible curve. (2) Pick (x, t) ∈ M[a,t0 ) . To verify lower semicontinuity at (x, t) we suppose the sequence {(xj , tj )} ⊂ M[a,t0 ) converges to (x, t) and lim inf j→∞ L(xj , tj ) < ∞. By (1) there is a sequence {γj } of L-minimizing admissible curves, where γj runs from (xj , tj ) to (x0 , t0 ). By the reasoning above, a subsequence of {γj } converges uniformly and weakly in W 1,2 to a W 1,2 curve γ∞ : [t, t0 ] → M going from (x, t) to (x0 , t0 ), with (78.5)

L(γ∞ ) ≤ lim inf L(γj ). j→∞

Therefore L(x, t) ≤ lim inf j→∞ L(xj , tj ), and we have established semicontinuity. If (x, t) ∈ Mreg , the opposite inequality obviously holds, so in this case (x, t) is a point of continuity.

(3) Because {L(xj , tj )} is uniformly bounded, any sequence {γj } of L-minimizing paths with γj (tj ) = (xj , tj ) will be equicontinuous, and hence by Arzela-Ascoli a subsequence converges uniformly. Therefore a subsequence of {(xj , tj )} converges. The fact that (78.2) can hold locally allows one to appeal – under appropriate conditions – to the maximum principle as in Section 24 to prove that min l ≤ 32 on every time slice. L L = 4τ . Recall that l = 2√ τ

Lemma 78.6. Suppose that M is a Ricci flow with surgery defined on [a, b]. Take t0 ∈ (a, b] and (x0 , t0 ) ∈ M− t0 . Suppose that for every t ∈ [a, t0 ), every admissible curve [t, t0 ] → M ending at (x0 , t0 ) which does not lie in Mreg ∪ M− t0 has reduced length strictly greater than 3 + . Then there is a point (x, a) ∈ M where l(x, a) ≤ 23 . a 2 Remark 78.7. In the lemma we consider the Ricci flow with surgery to begin at time a. − + Hence Mreg ∪ M− t0 = Ma ∪ Mreg ∪ Mt0 and so the hypothesis of the lemma is a statement about the reduced lengths of barely admissible curves, in the sense of Section 68. Proof. As in the case when there are no surgeries, the proof relies on the maximum principle and a continuity argument. Let β : M[a,t0 ) → R ∪ {∞} be the function (78.8)

¯ − 6τ = 4τ β = L

3 l− , 2

where as usual, τ (x, t) = t0 − t. Note that for each τ ∈ (0, t0 − a], the function β attains a minimum βmin (τ ) < ∞ on the slice Mt0 −τ , because by (2) of Lemma 78.3, it is continuous on the compact manifold M+ t0 −τ (as seen by changing the parameter a of Lemma 78.3 to t0 − τ ), and β ≡ ∞ on Mt0 −τ − M+ t0 −τ . Thus it suffices to show that βmin (t0 − a) ≤ 0.


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From Lemma 24.3, βmin (τ ) < 0 for τ > 0 small. Let τ1 ∈ (0, t0 − a] be the supremum of the τ¯ ∈ (0, t0 − a] such that βmin < 0 on the interval (0, τ¯). We claim that

(a) βmin is continuous on (0, τ1 ) and (b) The upper right τ -derivative of βmin is nonpositive on (0, τ1 ). To see (a), pick τ ∈ (0, τ1 ), suppose that {τj } ⊂ (0, τ1 ) is a sequence converging to τ and choose (xj , t0 − τj ) ∈ M+ t0 −τj such that β(xj , t0 − τj ) = βmin (τj ) < 0. By Lemma 78.3 part (3), the sequence {(xj , t0 − τj )} subconverges to some (x, t0 − τ ) ∈ M+ t0 −τ for which β(x, t0 − τ ) ≤ lim inf j→∞ β(xj , t0 − τj ). Thus βmin is lower semicontinuous at τ . On the other hand, since βmin (τ ) < 0, the minimum of β on Mt0 −τ will be attained at a point − + (x, t0 − τ ) ∈ M+ t0 −τ lying in the interior of Mt0 −τ ∩ Mt0 −τ , as β > 0 elsewhere on Mt0 −τ (by Lemma 78.3 and the hypothesis on admissible curves). Therefore β is continuous at (x, t0 − τ ), which implies that βmin is upper semicontinuous at τ . This gives (a).

Part (b) of the claim follows from the fact that if τ ∈ (0, τ1 ) and the minimum of β on Mt0 −τ is attained at (x, t0 − τ ) then l(x, τ ) < 23 , so there is a neighborhood U of (x, t0 − τ ) such that the inequality ∂β (78.9) + ∆β ≤ 0 ∂τ holds in the barrier sense on U (by Lemma 78.3 and the hypothesis on admissible curves). d Hence the upper right derivative ds β(x, τ + s) is nonpositive, so the upper right τ s=0

derivative of βmin (τ ) is also nonpositive.

The claim implies that βmin is nonincreasing on (0, τ1 ), and so lim supτ →τ1− β(τ ) < 0. By parts (2) and (3) of Lemma 78.3, we have βmin(τ1 ) < 0, and the minimum is attained at some (x, t0 − τ1 ) ∈ Mreg . (Recall that Ma ⊂ Mreg .) This implies that τ1 = t0 − a, for otherwise βmin (τ ) would be strictly negative for τ ≥ τ1 close to τ1 , contradicting the definition of τ1 . The notion of local collapsing can be adapted to Ricci flows with surgery, as follows. Definition 78.10. Let M be a Ricci flow with surgery defined on [a, b]. Suppose that (x0 , t0 ) ∈ M and r > 0 are such that t0 − r 2 ≥ a, B(x0 , t0 , r) ⊂ M− t0 is a proper ball and the parabolic ball P (x0 , t0 , r, −r 2 ) is unscathed. Then M is κ-collapsed at (x0 , t0 ) at scale r if | Rm | ≤ r −2 on P (x0 , t0 , r, −r 2 ) and vol(B(x0 , t0 , r)) < κr3 ; otherwise it is κ-noncollapsed. We make use of the following variant of the noncollapsing argument from Section 26.

Lemma 78.11. (Local version of reduced volume comparison) There is a function κ′ : R+ → R+ , satisfying limκ→0 κ′ (κ) = 0, with the following property. Let M be a Ricci flow with surgery defined on √ [a, b]. Suppose that we are given t0 ∈ (a, b], (x0 , t0 ) ∈ Mt0 ∩ Mreg , t ∈ [a, t0 ) and r ∈ (0, t0 − t). Let Y be the set of points (x, t) ∈ Mt that are accessible from (x0 , t0 ) by means of minimizing L-geodesics which remain in Mreg . Assume in addition that M is κ-collapsed at (x0 , t0 ) at scale r, i.e. P (x0 , t0 , r, −r 2 )∩M[t0−r2 ,t0 ) ⊂ Mreg , | Rm | ≤ r −2

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on P (x0 , t0 , r, −r 2 ), and vol(B(x0 , t0 , r)) < κr3 . Then the reduced volume of Y is at most κ′ (κ). Proof. Let Yˆ ⊂ Tx0 Mt0 be the set of vectors v ∈ Tx0 Mt0 such that there is a minimizing L-geodesic γ : [t, t0 ] → Mreg running from (x0 , t0 ) to some point in Y , with p ˙ t¯) = −v. (78.12) lim t0 − t¯ γ( t¯→t0

The calculations from Sections 17-23 apply to L-geodesics sitting in Mreg . In particular, n the monotonicity of the weighted Jacobian τ − 2 exp(−l(τ ))J(τ ) holds. Now one repeats the proof of Theorem 26.2, working with the set Yˆ instead of the set of initial velocities of all minimizing L-geodesics. 79. Establishing noncollapsing in the presence of surgery The key result of this section, Lemma 79.12, gives conditions under which one can deduce noncollapsing on a time interval I2 , given a noncollapsing bound on a preceding interval I1 and lower bounds on r on I1 ∪ I2 . Definition 79.1. The L+ -length of an admissible curve γ is Z t0 √ 2 (79.2) L+ (γ, τ ) = t0 − t R+ (γ(t), t) + |γ(t)| ˙ dt, t0 −τ

where R+ (x, t) = max(R(x, t), 0).

Lemma 79.3. (Forcing L+ to be large, cf. Lemma II.5.3)

For all Λ < ∞, r¯ > 0 and rˆ > 0, there is a constant F0 = F0 (Λ, r¯, rˆ) with the following property. Suppose that • M is a Ricci flow with (r, δ)-cutoff defined on an interval containing [t, t0 ], where r([t, t0 ]) ⊂ [ˆ r , ǫ],

• r0 ≥ r¯, B(x0 , t0 , r0 ) is a proper ball which is unscathed on [t0 − r02 , t0 ], and | Rm | ≤ r0−2 on P (x0 , t0 , r0 , −r02 ),

• γ : [t, t0 ] → M is an admissible curve ending at (x0 , t0 ) whose image is not contained in Mreg ∪ Mt0 , and • δ < F0 (Λ, r¯, rˆ) on [t, t0 ].

Then L+ (γ) > Λ.

Proof. The idea is that the hypotheses on γ imply that it must touch the part of the manifold added during surgery at some time t¯ ∈ [t, t0 ]. Then either γ has to move very fast at times close to t¯ or t0 , or it will stay in the surgery region while it develops large scalar curvature. In the first case L+ (γ) will be large because of the |γ| ˙ 2 term in the formula for L+ , and in the second case it will be large because of the R(γ) term.


153

q r¯ . Then since First, we can assume that F0 is small enough so that F0 < 100ǫ 2 (79.4) max h(t) ≤ max δ max r(t) ≤ F02 ǫ, [t,t0 ]

r¯ 100 −2

we have max[t,t0 ] h(t)
102 r0−2 , while | Rm | ≤ r0−2 on P (x0 , t0 , r0 , −r02 ). Therefore when going backward in time from (x0 , t0 ), γ must leave the parabolic region P (x0 , t0 , r0 , −r02 ) before it arrives at (x, t). If it exits at a time t˜ > t0 − ∆t then applying the Schwarz inequality we get (79.8) Z t0 2 Z t0 −1 Z t0 √ 1 2 2 −1/2 t0 − s |γ(s)| ˙ ds ≥ |γ(s)| ˙ ds (t0 − s) ds ≥ r0 (∆t)−1/2 > Λ, 100 ˜ ˜ ˜ t t t

1 where the factor of 100 comes from the length distortion estimate of Section 27, using the −2 fact that | Rm | ≤ r0 on P (x0 , t0 , r0 , −r02 ). So we can restrict to the case when γ exits P (x0 , t0 , r0 , −∆t) through the initial time slice at time t0 − ∆t. In particular, by (79.7), γ must exit the parabolic region P (x, t, Ah(t), θh2 (t)) by time t0 − ∆t.

By Lemma 74.1, the parabolic region P (x, t, Ah, θh2 ) is either unscathed, or it coincides (as a set) with the parabolic region P (x, t, Ah, s) for some s ∈ (0, θh2 ) and the entire final time slice P (x, t, Ah, s) ∩ Mt+s of P (x, t, Ah, s) is thrown away by a surgery at time t + s.

One possibility is that γ exits P (x, t, Ah, θh2 ) through the final time slice. If this is the case then P (x, t, Ah, θh2 ) must be unscathed (as otherwise the final face is removed by surgery at time t + s < t + θh2 and γ would have nowhere to go after this time), so γ lies in P (x, t, Ah, θh2 ) for the entire time interval [t, t + θh2 ]. The other possibility is that γ leaves P (x, t, Ah, θh2 ) before the final time slice of P (x, t, Ah, θh2 ), in which case it exits the ball B(x, t, Ah) by time t + θh2 .

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Corollary 75.1 applies to either of these two possibilities. Putting Tγ = sup{t¯ ∈ [t, t + θh2 ] | γ([t, t¯]) ⊂ P (x, t, Ah, θh2 )}

(79.9)

and using the fact that Tγ ≤ t0 − ∆t, we have (79.10) Z t0

Z

Tγ √ 2 t0 − s R+ (γ(s), s) + |γ(s)| ˙ ds ≥ t0 − s R+ (γ(s), s) + |γ(s)| ˙ ds ≥ t t Z Tγ 1/2 2 (∆t) R+ (γ(s), s) + |γ(s)| ˙ ds ≥ (∆t)1/2 l = Λ,

√

2

t

where the last inequality comes from Corollary 75.1 and the choice of A, θ, and δ in (79.5) and (79.6). This completes the proof. Lemma 79.11. If M is a Ricci flow with surgery, with normalized initial condition at time zero, then for all t ≥ 0, R(x, t) ≥ − 23 t+1 1 . 4

Proof. From the initial conditions, Rmin (0) ≥ −6. If the Ricci flow is smooth then (B.2) implies that Rmin (t) ≥ − 32 t+1 1 . If there is a surgery at time t0 then Rmin on M+ t0 equals 4

Rmin on M− t0 , as surgery is done in regions of high scalar curvature. The lemma follows by applying (B.2) on the time intervals between the singular times.

In the statement of the next lemma, one has successive time intervals [a, b) and [b, c). As a mnemonic we use the subscript − for quantities attached to the earlier interval [a, b), and + for those associated with [b, c). We will also assume that the global parameter ǫ is small enough that the Φ-pinching condition implies that whenever | Rm(x, t)| ≥ ǫ−2 , then R(x, t) > | Rm(x,t)| . (We remind the reader of the role of the parameter ǫ; see Remark 58.5.) 100 Lemma 79.12. (Noncollapsing estimate) (cf. Lemma II.5.2) Suppose ǫ ≥ r− ≥ r+ > 0, κ− > 0, E− > 0 and E < ∞. Then there are constants δ = δ(r− , r+ , κ− , E− , E) and κ+ = κ+ (r− , κ− , E− , E) with the following property. Suppose that • a < b < c,

b − a ≥ E− ,

c − a ≤ E,

• M is a Ricci flow with (r, δ)-cutoff with normalized initial condition defined on a time interval containing [a, c), • r ≥ r− on [a, b) and r ≥ r+ on [b, c), •r≤ǫ,

• M is κ− -noncollapsed at scales below ǫ on [a, b) and • δ ≤ δ¯ on [a, c),

Then M is κ+ -noncollapsed at scales below ǫ on [b, c).

Remark 79.13. The important point to notice here is that δ¯ is allowed to depend on the lower bound r+ on [b, c), but the noncollapsing constant κ+ does not depend on r+ .


155

p r+ Proof. In the proof, we can assume that 100 ≤ E− /3. If this were not the case then we p could prove the lemma with r+ replaced by 100 E− /3. Then the lemma would also hold for the original value of r+ . First, from Lemma 79.11, R ≥ −6 on M[a,c).

Suppose that r0 ∈ (0, ǫ), (x0 , t0 ) ∈ M[b,c), B(x0 , t0 , r0 ) is a proper ball unscathed on the interval [t0 − r02 , t0 ], and | Rm | ≤ r0−2 on P (x0 , t0 , r0 , −r02 ). p r+ We first assume that r0 ≤ E− /3 and r0 ≥ 100 . We will consider L-length, L+ -length, etc in M[a,t0 ) with basepoint at (x0 , t0 ). Suppose that b t ∈ [a, t0 ). Then for any admissible curve γ : [b t, t0 ] → M[a,t0 ] ending at (x0 , t0 ), we have

(79.14)

L(γ) ≤ L+ (γ) ≤ L(γ) +

Z

c

a

√ 3 6 c − t dt ≤ L(γ) + 4E 2 3

and l(x, t) ≥

L+ − 4E 2 1

.

2E 2

1 3 r+ Assume that δ¯ ≤ F0 (4E 2 +4E 2 , 100 , r+ ) where F0 is the function from Lemma 79.3. Then by (79.14) and Lemma 79.3, we conclude that any admissible curve [b t, t0 ] → M[a,t0 ] ending at (x0 , t0 ) which does not lie in Mreg ∪ Mt0 has reduced length bounded below by 2 = 32 + 12 . By Lemma 78.6 there is an admissible curve γ : [a, t0 ] → M ending at (x0 , t0 ) such that √ √ (79.15) L(γ) = L(γ(a)) = 2 t0 − a l(γ(a)) ≤ 3 t0 − a,

so by (79.14) it follows that (79.16)

√ √ 3 3 L+ (γ) ≤ 3 t0 − a + 4E 2 ≤ 3 E + 4E 2 .

Set (79.17)

t1 = a +

b−a , 3

t2 = a +

2(b − a) 3

and (79.18)

√ 1 − 32 3 E− . ρ = 3 E + 4E 2 3

By construction, t2 ≤ t0 − r02 . Note that there is a t¯ ∈ [t1 , t2 ] such that R(γ(t¯)) ≤ ρ. Otherwise we would get (79.19) r r Z t2 Z t2 √ √ 3 1 1 1 L+ (γ) > E− ρ dt ≥ E− E− ρ = 3 E + 4E 2 , t0 − t R+ (γ(t)) dt ≥ 3 3 3 t1 t1 contradicting (79.16). Put x = γ(t). By Lemma 70.1, there is an estimate of the form (79.20)

R ≤ const. s−2

156


−2 ). Appealing to Hamiltonon the parabolic ball Pˆ = P (x, t¯, s, −s2 ) with s−2 = const.(ρ+r− −2 Ivey curvature pinching as usual, we get that | Rm | ≤ const. s in Pˆ . If t − s2 < a then we shrink s (as little as possible) to ensure that Pˆ ⊂ M[a,b) . Provided that δ¯ is less than a small constant c1 = c1 (r− , E− , E), we can guarantee that Pˆ is unscathed, by forcing the curvature in a surgery cap to exceed our bound (79.20) on R. Put U = Mt¯− 1 s2 ∩ Pˆ . If s < ǫ 2 then the κ− -noncollapsing assumption on [a, b) gives a lower bound on vol(B(¯ x, t¯, s))s−3 . If s ≥ ǫ then the κ− -noncollapsing assumption gives a lower bound on vol(B(¯ x, t¯, ǫ/2))(ǫ/2)−3 . In either, case, we get a lower bound on vol(B(¯ x, t¯, s)) and hence a lower bound vol(U) ≥ v = v(r− , κ− , E− , E). Now every point in U can be joined to (x0 , t0 ) by a curve of L+ length at most Λ+ = Λ+ (r− , E− , E), by concatenating an admissible curve [t¯− 12 s2 , t¯] → M (of controlled L+ -length) with γ | . Shrinking δ¯ again, we can apply Lemmas 78.3 and [t¯,t0 ]

79.3 with (79.14) to ensure that every point in U can be joined to (x0 , t0 ) by a minimizing L-geodesic lying in Mreg ∪ Mt0 . Lemma 78.11 then implies that vol(B(x0 , t0 , r0 ))r0−3 ≥ κ1 = κ1 (r− , κ− , E− , E).

(79.21)

(We briefly recall the argument. We have a parabolic ball around (¯ x, t¯), of small but controlled size, on which we have uniform curvature bounds. The lower volume bound coming from the κ− -noncollapsing assumption on [a, b) means that we have bounded geometry on the parabolic ball. As we have a fixed upper bound on l(¯ x, t¯), we can estimate from below + the reduced volume of the accessible points Y ⊂ Mt¯− 1 s2 . Then we obtain a lower bound on vol(B(x0 , t0 , r0 ))r0−3 as in Theorem 26.2.)

2

p r+ . This completes the proof of the lemma when r0 ≤ E− /3 and r0 ≥ 100 p Now suppose that r0 > E− /3. Applying our noncollapsing estimate (79.21) to the ball p of radius E− /3 gives (79.22) 3 (E /3) 23 p (E− /3) 2 − −3 − 23 ≥ κ1 = κ2 , vol(B(x0 , t0 , r0 ))r0 ≥ vol(B(x0 , t0 , E− /3)(E− /3) r03 ǫ3 where κ2 = κ2 (r− , κ− , E− , E).

The next sublemma deals with the case when r0 < Sublemma 79.23. If r0
0 such that a standard solution is κ-noncollapsed as scales < 1.) The distance distortion estimate ensures that B(¯ x, t, 10−9 s) ⊂ B(z, t0 , 10−6 s) ⊂ B(x0 , t0 , s). Then the standard volume distortion estimate implies that vol(B(x0 , t0 , s)) ≥ const. vol(B(¯ x, t, 10−9s)) ≥ const. s3 , again for some universal constant. Finally we use Bishop-Gromov volume comparison to get vol(B(x0 , t0 , r0 ))r0−3 ≥ const. vol(B(x0 , t0 , s))s−3 .

In case (c) we apply (79.21), replacing the r0 parameter there by s, and Bishop-Gromov volume comparison as in case (b). 80. Construction of the Ricci flow with surgery The proof is by induction on i. To start the induction process, we observe that the initial normalization | Rm | ≤ 1 at t = 0 implies that a smooth solution exists for some definite time [22, Corollary 7.7]. The curvature bound on this time interval, along with the volume assumption on the initial time balls, implies that the solution is κ-noncollapsed below scale 1 and satisfies the ρ-canonical neighborhood assumption vacuously for small ρ > 0. Now assume inductively that rj , κj , and δ¯j have been selected for 1 ≤ j ≤ i, thereby defining the functions r, κ, and δ¯ on [0, 2iǫ], such that for any nonincreasing function δ on ¯ [0, 2i ǫ] satisfying 0 < δ(t) ≤ δ(t), if one has normalized initial data then the Ricci flow with i (r, δ)-cutoff is defined on [0, 2 ǫ] and is κ(t)-noncollapsed at scales < ǫ.

158


We will determine κi+1 using Lemma 79.12. So suppose it is not possible to choose ri+1 and δ¯i+1 (and, if necessary, make δ¯i smaller) so that if we put κi+1 = κ+ (ri , κi , 2i−1 ǫ, 3(2i−1 )ǫ) (where κ+ denotes the function from Lemma 79.12) then the inductive statement above holds with i replaced by i + 1. Then given sequences r α → 0 and δ¯α → 0, for each α there must be a counterexample, say (M α , g α (0)), to the statement with ri+1 = r α and δ¯i = δ¯i+1 = δ¯α . We assume that ˆ 1 − 1 , rα) (80.1) δ¯α < δ(α, α where δˆ is the quantity from Lemma 74.1; this will guarantee that for any A < ∞ and θ ∈ (0, 1) we may apply Lemma 74.1 with parameters A and θ for sufficiently large α. We also assume that ¯ i , r α , κi , 2i−1 ǫ, 3(2i−1 )ǫ) (80.2) δ¯α < δ(r ¯ i , r α , κi , 2i−1 ǫ, 3(2i−1 )ǫ) is from Lemma 79.12. By Lemma 73.7 each initial condition where δ(r (M α , g α (0)) will prolong to a Ricci flow with surgery Mα defined on a time interval [0, T α ] with T α ∈ (2iǫ, ∞], which restricts to a Ricci flow with (r, δ)-cutoff on any proper subinterval [0, τ ] of [0, T α ], but for which the r α -canonical neighborhood assumption fails at some point (¯ xα , T α ) lying in the backward time slice Mα− T α . (It is implicit in this statement that 1 α α α α R(¯ x , T ) ≥ (rα )2 .) Since (M , g (0)) violates the theorem, we must have T α ∈ (2i ǫ, 2i+1 ǫ]. By (80.2) and Lemma 79.12, it follows that Mα is κi+1 -noncollapsed at scales below ǫ on the interval [2i ǫ, T α ), where κi+1 = κ+ (ri , κi , 2i−1ǫ, 3(2i−1 )ǫ) and κ+ denotes the function from Lemma 79.12. cα , (¯ Let (M xα , 0)) be the pointed Ricci flow with surgery obtained from (Mα , (¯ xα , T α )) α α α by shifting time by T and parabolically rescaling by R(¯ x , T ). We also remove the part α α c cα to be diffeomorphic to of (M , (¯ x , 0)) after time zero and we take the time-zero slice M 0 α− MT α . In brief, the rest of the proof goes as follows. If surgeries occur further and further away from (¯ xα , 0) in spacetime as α → ∞, then the reasoning of Theorem 52.7 applies and we obtain a κ-solution as a limit. This would contradict the fact that (¯ xα , T α) does not have a canonical neighborhood. Thus there must be surgeries in a parabolic ball of a fixed size centered at (¯ xα , 0), for arbitrarily large α. Then one argues using Lemma 74.1 that the solution will be close to the (suitably rescaled and time-shifted) standard solution, which again leads to a canonical neighborhood and a contradiction. We now return to the proof. Recall that a metric ball B is proper if the distance function from the center is a proper function on B. If T is a surgery time for a Ricci flow with surgery then a metric ball in M− T need not be proper.

cα whose scalar curvature is strictly greater than Note that by continuity, every point in M α that of (¯ x , 0) has a neighborhood as in Definition 69.1, except that the error estimate is 2ǫ instead of ǫ. cα is proper for sufficiently large Sublemma 80.3. For all λ < ∞, the ball B(¯ xα , 0, λ) ⊂ M 0 α.

cα0 is Proof. As in Lemma 70.2, for each ρ < ∞ the scalar curvature on B(¯ xα , 0, ρ) ⊂ M uniformly bounded in terms of α. (In carrrying out the proof of Lemma 70.2, we now use the


159

aforementioned property of having canonical neighborhoods of quality 2ǫ.) Thus B(¯ xα , 0, ρ) cα (see Lemma 67.9), from which the sublemma follows. has compact closure in M 0

Let T1 ∈ [−∞, 0] be the infimum of the set of numbers τ ′ ∈ (−∞, 0] such that for all cα is proper, and unscathed on [τ ′ , 0] for sufficiently large λ < ∞, the ball B(¯ xα , 0, λ) ⊂ M 0 α.

cα , (¯ Lemma 80.4. After passing to a subsequence if necessary, the pointed flows (M xα , 0)) ∞ ∞ converge on the time interval (T1 , 0] to a Ricci flow (without surgery) (M , (¯ x , 0)) with a smooth complete nonnegatively-curved Riemannian metric on each time slice, and scalar curvature globally bounded above by some number Q < ∞. (We interpret (0, 0] to mean {0} rather than the empty set.) Proof. Suppose first that T1 < 0. Then the arguments of Theorem 52.7 apply in the time interval (T1 , 0], to give the Ricci flow (without surgery) (M∞ , (¯ x∞ , 0)). Since r α → 0, ∞ Hamilton-Ivey pinching implies that M will have nonnegative curvature. The fact that cα allows the canonical neighborhood assumption, with ǫ replaced by 2ǫ, holds for each M us to deduce that the scalar curvature of M∞ is globally bounded above by some number Q < ∞; compare with Section 46 and Step 4 of the proof of Theorem 52.7.

Now suppose that T1 = 0. The argument is similar to Steps 2 and 3 of the proof of Theorem 52.7. As in Step 2, or more precisely as in Lemma 70.2, for each ρ < ∞ the cα is uniformly bounded in terms of α. Given ρ < ∞ scalar curvature on B(¯ xα , 0, ρ) ⊂ M 0 cα , if the parabolic region of Lemma 70.1 (centered around and (xα , 0) ∈ B(¯ xα , 0, ρ) ⊂ M cα ) is unscathed then we can apply the 2ǫ-canonical neighborhood assumption (xα , 0) ∈ M cα , Lemma 70.1 and Appendix D to derive bounds on the curvature derivatives at on M cα that depend on ρ but are independent of α. If the parabolic region is scathed (xα , 0) ∈ M 0 then we can apply Lemma 74.1, along with our scalar curvature bound at (xα , 0), to again obtain uniform bounds on the curvature derivatives at (xα , 0). Hence there is a subsequence cα , x¯α )}∞ that converges to a smooth complete of the pointed Riemannian manifolds {(M 0 α=1 ∞ ∞ pointed Riemannian manifold (M0 , x ¯ ). As in the previous case, it will have bounded nonnegative sectional curvature. This proves the lemma. Alternatively, in the case T1 = 0 one can argue directly that if the parabolic region of Lemma 70.1 is scathed then (xα , T α ) has a canonical neighborhood; see the rest of the proof of Proposition 77.2.

If we can show that T1 = −∞ then (M∞ , (¯ x∞ , 0)) will be a κ-solution, which will α α contradict the assumption that (¯ x , T ) does not admit a canonical neighborhood. Suppose that T1 > −∞. We know that for all τ ′ ∈ (T1 , 0] and λ < ∞, the scalar curvature in P (¯ xα , 0, λ, τ ′ ) is bounded by Q + 1 when α is sufficiently large. By Lemma 70.1, there cα is unscathed on exists σ < T1 such that for all λ < ∞, if (for large α) the solution M P (¯ xα , 0, λ, tα ) for some tα > σ then (80.5)

R(x, t) < 8(Q + 2) for all (x, t) ∈ P (¯ xα , 0, λ, tα).

(In applying Lemma 70.1, we use the fact that in the unscaled variables, r(T α )−2 ≤ R(¯ xα , T α ) by assumption, along with the fact that r(·) is a nonincreasing function.)

160


By the definition of T1 , and after passing to a subsequence if necessary, there exist λ < ∞ cα of static curves so that and a sequence γ α : [σ α , 0] → M α α 1. γ (0) ∈ B(¯ x , 0, λ) and 2. The point γ α (σ α ) is inserted during surgery at time σ α > σ. For each α, we may assume that σ α is the largest number having this property. Put ξ α = σ α + (hα (σ α ))2 . (In the notation of [52], (hα (σ α ))2 would be written as R(¯ x, t¯) h2 (T0 ). Note that we have no a priori control on hα (σ α ).) Then ξ α is the blowup time of the cα , γ α (σ α )). We rescaled and shifted standard solution that Lemma 74.1 compares with (M α claim that lim inf α→∞ ξ > 0. Otherwise, Lemma 74.1 would imply that after passing to cα , starting from time σ α , that are better and better a subsequence, there are regions of M approximated by rescaled and shifted standard solutions whose blowup times ξ α have a limit that is nonpositive, thereby contradicting (80.5). Lemma 74.1, along with the fact that R(¯ xα , 0) = 1, also gives a uniform upper bound on ξ α . cα to the time interval Now Lemma 74.1 implies that for large α, the restriction of M α [σ , 0] is well approximated by the restriction to [σ , 0] of a rescaled and shifted standard solution. Then Lemma 63.1 implies that (¯ xα , T α) has a canonical neighborhood. The canonical neighborhood may be either a strong ǫ-neck or an ǫ-cap. (Note a strong ǫ-neck may arise when an ǫ-neck around (¯ xα , T α ) extends smoothly backward in time to form a strong ǫ-neck that incorporates part of the Ricci flow solution that existed before the surgery time σ α .) α

This is a contradiction.

81. II.6. Double sided curvature bound in the thick part Having shown that for a suitable choice of the functions r and δ, the Ricci flow with (r, δ)cutoff exists for all time and for every normalized initial condition, one wants to understand its implications. The main results in II.6 are noncollapsing and curvature estimates which form the basis of the analysis of the large-time behavior given in II.7. Lemma 81.1. If M is a Ricci flow with (r, δ)-cutoff on a compact manifold and g(0) has positive scalar curvature then the solution goes extinct after a finite time, i.e. MT = ∅ for some T > 0. Proof. We apply (B.2). This formula is initially derived for smooth flows but because surgeries are performed in regions of high scalar curvature, it is also valid for a Ricci flow with surgery; cf. the proof of Lemma 79.11. It follows that the flow goes extinct by time 3 . 2Rmin (0) Lemma 81.2. If M is a Ricci flow with surgery that goes extinct after a finite time, then the initial (compact connected orientable) 3-manifold is diffeomorphic to a connected sum of S 1 × S 2 ’s and quotients of the round S 3 . Proof. This follows from Lemma 73.4.


161

According to [24, 25] and [53], if none of the prime factors in the Kneser-Milnor decomposition of the initial manifold are aspherical then the Ricci flow with surgery again goes extinct after a finite time. Along with Lemma 81.2, this proves the Poincaré Conjecture. Passing to Ricci flow solutions that may not go extinct after a finite time, the main result of II.6 is the following : Corollary 81.3. (cf. Corollary II.6.8) For any w > 0 one can find τ = τ (w) > 0, K = K(w) > 0, r = r(w) > 0 and θ = θ(w) > 0 with the following property. Suppose we have a solution to the Ricci flow with (r, δ)-cutoff on the time interval [0, t0 ], with normalized initial data. Let hmax (t0 ) be the maximal surgery radius on [t0 /2, t0 ]. (If there are no surgeries on [t0 /2, t0 ] then√hmax (t0 ) = 0.) Let r0 satisfy 1. θ−1 (w)hmax (t0 ) ≤ r0 ≤ r t0 . 2. The ball B(x0 , t0 , r0 ) has sectional curvatures at least − r0−2 at each point. 3. vol(B(x0 , t0 , r0 )) ≥ wr03. Then the solution is unscathed in P (x0 , t0 , r0 /4, −τ r02 ) and satisfies R < Kr0−2 there. Corollary 81.3 is an analog of Corollary 55.1, but there are some differences. One minor difference is that Corollary 55.1 is stated as the contrapositive of Corollary 81.3. Namely, Corollary 55.1 assumes that −r0−2 is achieved as a sectional curvature in B(x0 , t0 , r0 ), and its conclusion is that vol(B(x0 , t0 , r0 )) ≤ wr03. The relation with Corollary 81.3 is the following. Suppose that assumptions 1 and 2 of Corollary 81.3 hold. If − r0−2 is achieved somewhere as a sectional curvature in B(x0 , t0 , r0 ) then Hamilton-Ivey pinching implies that the scalar curvature is very large at that point, which contradicts the conclusion of Corollary 81.3. Hence assumption 3 of Corollary 81.3 must not be satisfied. A more substantial difference is that the smoothness of the flow in Corollary 55.1 is guaranteed by the setup, whereas in Corollary 81.3 we must prove that the solution is unscathed in P (x0 , t0 , r0 /4, −τ r02 ).

The role of the parameter r in Corollary 81.3 is essentially to guarantee that we can use Hamilton-Ivey pinching effectively. 82. II.6.5. Earlier scalar curvature bounds on smaller balls from lower curvature bounds and a later volume bound A simpler analog of Corollary 81.3 is the following. Lemma 82.1. (cf. Lemma II.6.5(a)) Given w > 0, there exist τ0 = τ0 (w) > 0 and K0 = K0 (w) < ∞ with the following property. Suppose that we have a Ricci flow with (r, δ)-cutoff such that 1. The parabolic neighborhood P (x0 , 0, r0, −τ r02 ) is unscathed, where τ ≤ τ0 . 2. The sectional curvatures are bounded below by − r0−2 on P (x0 , 0, r0 , −τ r02 ). 3. vol(B(x0 , 0, r0 )) ≥ wr03. Then R ≤ K0 τ −1 r0−2 on P (x0 , 0, r0 /4, −τ r02 /2).

162


Proof. Let us first note a consequence of Corollary 45.13. Sublemma 82.2. Given w > 0, there exist τ0 = τ0 (w) > 0 and K = K(w) < ∞ with the following property. Suppose that we have S a Ricci flow with (r, δ)-cutoff such that 1. There are no surgeries on a family t∈[−τ r2 ,0] B(x0 , t, r0 ) of time-dependent balls, where 0 τ ≤ τ0 . 2. The sectional curvatures are bounded below by − r0−2 on the above family of balls. 3. vol(B(x0 , 0, r0 )) ≥ wr03. Then R ≤ K τ −1 r0−2 on

S

t∈[− 43 τ r02 ,0]

B(x0 , t, r0 /2).

Here weShave changed the conclusion of Corollary 45.13 to obtain an upper curvature S bound on t∈[− 3 τ r2 ,0] B(x0 , t, r0 /2) instead of t∈[− 3 τ r2 ,0] B(x0 , t, r0 /4), but this clearly fol0 0 4 4 lows from the arguments of the proof of Corollary 45.13. We now return to the situation of Lemma 82.1.

If τ0 is sufficiently small, then for t ∈ [−τ r02 , 0] and (x, t) ∈ B(x0 , t, 9r0 /10), the lower curvature bound Rm ≥ − r0−2 on P (x0 , 0, r0, −τ r02 ) implies that (x, 0) ∈ B(x0 , 0, r0) (more precisely, that (x, t) lies on a static curve S with one endpoint in B(x0 , 0, r0 ), or equivalently, 2 that (x, t) ∈ P (x0 , 0, r0, −τ r0 )). Thus t∈[−τ r2 ,0] B(x0 , t, 9r0 /10) ⊂ P (x0 , 0, r0 , −τ r02 ) and so 0 S Rm ≥ −r0−2 ≥ −(9r0 /10)−2 on t∈[−τ (9r0 /10)2 ,0] B(x0 , t, 9r0 /10).

Applying Sublemma 82.2 with S r0 replaced by 9r0 /10, and slightly redefining w, gives that −1 −2 R ≤ K τ (9r0 /10) on t∈[− 3 τ (9r0 /10)2 ,0] B(x0 , t, 9r0 /20). Then the length distortion 4 estimate of Lemma 27.8 implies that for sufficiently small τ0 , if (x, 0) ∈ B(x0 , 0, r0/4) then (x, t) ∈ B(x0 , t, 9r0 /20) for t ∈ [−τ r02 /2, 0]. That is,

(82.3)

P (x0 , 0, r0 /4, −τ r02 /2) ⊂

[

B(x0 , t, 9r0/20).

t∈[−τ r02 /2,0]

In applying the length distortion estimate we use the fact that the change in distance is p estimated by ∆d ≤ const. K τ −1 (9r0 /10)−2 · τ r02 /2 which, for small τ0 , is a small fraction of r0 .

Thus we have shown that R ≤ K τ −1 (9r0 /10)−2 on P (x0 , 0, r0 /4, −τ r02 /2). This proves the lemma.

The formulation of [52, Lemma II.6.5] specializes Lemma 82.1 to the case w = 1 − ǫ. It includes the statement [52, Lemma II.6.5(b)] saying that vol(B(x0 , r0 /4, −τ r02 )) is at least 1 of the volume of the Euclidean ball of the same radius. This follows from the proof of 10 Corollary 45.1(b), provided that τ0 is sufficiently small. There is an evident analogy between Lemma 82.1 and Corollary 81.3. However, there is the important difference that Corollary 81.3 (along with Corollary 55.1) only assumes a lower sectional curvature bound at the final time slice.


163

83. II.6.6. Locating small balls whose subballs have almost Euclidean volume The result of this section is a technical lemma about volumes of subballs. Lemma 83.1. (cf. Lemma II.6.6) For any b ǫ, w > 0 there exists θ0 = θ0 (b ǫ, w) such that if B(x, 1) is a metric ball of volume at least w, compactly contained in a manifold without boundary with sectional curvatures at least −1, then there exists a subball B(y, θ0) ⊂ B(x, 1) such that every subball B(z, r) ⊂ B(y, θ0 ) of any radius has volume at least (1 − b ǫ) times the volume of the Euclidean ball of the same radius. The proof is similar to that of Lemma 54.1. Suppose that the claim is not true. Then there is a sequence of Riemannian manifolds {Mi }∞ i=1 and balls B(xi , 1) ⊂ Mi with compact closure so that Rm ≥ − 1 and vol(B(xi , 1)) ≥ w, along with a sequence ri′ → 0 so that each B(xi ,1)

subball B(x′i , ri′ ) ⊂ B(xi , 1) has a subball B(x′′i , ri′′ ) ⊂ B(x′i , ri′ ) with vol(B(x′′i , ri′′)) < (1 − b ǫ) ω3 (r ′′ )n . After taking a subsequence, we can assume that limi→∞ (B(xi , 1), xi ) = (X, x∞ ) in the pointed Gromov-Hausdorff topology, where (X, x∞ ) is a pointed Alexandrov space with curvature bounded below by −1. From [12, Theorem 10.8], the Riemannian volume forms dvolMi converge weakly to the three-dimensional Hausdorff measure µ of X. If x′∞ is a regular point of X then there is some δ > 0 so that B(x′∞ , δ) has compact closure in X and b ǫ for all r < δ, µ(B(x′∞ , r)) ≥ (1 − 10 ) ω3 r 3 . Fixing such an r for the moment, for large i there are balls B(x′i , r) ⊂ B(xi , 1) with vol(B(x′i , r)) ≥ (1 − 5bǫ ) ω3 r 3 . Recalling the sequence {ri′ }, by hypothesis there is a subball B(x′′i , ri′′ ) ⊂ B(x′i , ri′ ) with vol(B(x′′i , ri′′)) < (1 − b ǫ) ω3 (ri′′ )3 . ′ ′′ ′ Clearly B(xi , r) ⊂ B(xi , r + ri ). From the Bishop-Gromov inequality, R r+ri′ sinh2 (s) ds vol(B(x′′i , r + ri′ )) 0 ≤ . R ri′′ vol(B(x′′i , ri′′ )) sinh2 (s) ds

(83.2)

0

Then (83.3)

vol(B(x′i , r))

≤

vol(B(x′′i , r

For large i we obtain (83.4)

vol(B(x′i , r))

+

ri′ ))

≤ (1 − b ǫ) ω3 R r′′ i 0

b ǫ ≤ (1 − ) ω3 · 3 2

Z

r

(ri′′ )3 2

sinh (s) ds

Z

r+ri′

sinh2 (s) ds.

0

sinh2 (s) ds.

0

Then if we choose r to be sufficiently small, we contradict the fact that vol(B(x′i , r)) ≥ (1 − 5bǫ ) ω3 r 3 for all i. Remark 83.5. By similar reasoning, for every L > 1 one may find θ1 = θ1 (b ǫ, L) such that under the hypotheses of Lemma 83.1, there is a subball B(y, θ1 ) ⊂ B(x, 1) which is LbiLipschitz to the Euclidean unit ball.

164


84. II.6.8. Proof of the double sided curvature bound in the thick part, modulo two propositions In this section we explain how Corollary 81.3 follows from Lemma 83.1 and two other propositions, which will be proved in subsequent sections. We first state the other propositions, which are Propositions 84.1 and 84.2. Proposition 84.1. (cf. Proposition II.6.3) For any A < ∞ one can find positive constants κ(A), K1 (A), K2 (A), r(A), such that for any t0 < ∞ there exists δ A (t0 ) > 0, decreasing in t0 , with the following property. Suppose that we have a Ricci flow with (r, δ)-cutoff on a time interval [0, T ], where δ(t) < δ A (t) on [t0 /2, t0 ], with normalized initial data. Assume that 1. The solution is unscathed on a parabolic ball P (x0 , t0 , r0 , −r02 ), with 2r02 < t0 . 2. | Rm | ≤ 3r12 on P (x0 , t0 , r0 , −r02 ). 0 3. vol(B(x0 , t0 , r0 )) ≥ A−1 r03 . Then (a) The solution is κ-noncollapsed on scales less than r0 in B(x0 , t0 , Ar0 ). (b) Every point x ∈ B(x0 , t0 , Ar0 ) with R(x, t0 ) ≥ K1 r0−2 has a canonical neighborhood in the sense of Definition 69.1. √ (c) If r0 ≤ r t0 then R ≤ K2 r0−2 in B(x0 , t0 , Ar0 ). Proposition 84.1(a) is an analog of Theorem 28.2. (The reason for the “3” in the hypothesis | Rm | ≤ 3r12 comes from Remark 28.3.) Propo0 sition 84.1(c) is an analog of Theorem 53.1, but the hypotheses are slightly different. In Proposition 84.1 one assumes a lower bound on the volume of the time-t0 ball B(x0 , t0 , r0 ), while in Theorem 53.1 one assumes assumes a lower bound on the volume of the time(t0 − r02 ) ball B(x0 , t0 − r02 , r0 ). In view of the curvature assumption on P (x0 , t0 , r0 , −r02 ), the hypotheses are essentially equivalent. Conclusions (a), (b) and (c) of Proposition 84.1 are similar to the conclusions of Theorem 28.2, Lemma 53.3 and Theorem 53.1, respectively. Conclusions (a) and (b) of Proposition 84.1 are also related to what was proved in Proposition 77.2 to construct the Ricci flow with surgery. The difference is that the noncollapsing and canonical neighborhood results of Proposition 77.2 are statements at or below the scale √ r(t), whereas Proposition 84.1 is a statement about much larger scales, comparable to t0 . We note that the parameter δ A in Proposition 84.1 is independent of the function δ used to define the Ricci flow with (r, δ)-cutoff. In the proof of the next proposition we will apply Lemma 83.1 with b ǫ equal to the global parameter ǫ, so we will write θ(w) instead of θ(ǫ, w).

Proposition 84.2. (cf. Proposition II.6.4) There exist τ, r, C1 > 0 and K < ∞ with the following property. Suppose that we have a Ricci flow with (r, δ)-cutoff on √ the time interval [0, t0 ], with normalized initial data. Let r0 satisfy 2C1 hmax (t0 ) ≤ r0 ≤ r t0 , where hmax (t0 ) is the maximal cutoff radius for surgeries in [t0 /2, t0 ]. (If there are no surgeries on [t0 /2, t0 ] then hmax (t0 ) = 0.)


165

Assume 1. The ball B(x0 , t0 , r0 ) has sectional curvatures at least −r0−2 at each point. 2. The volume of any subball B(x, t0 , r) ⊂ B(x0 , t0 , r0 ) with any radius r > 0 is at least (1 − ǫ) times the volume of the Euclidean ball of the same radius. Then the solution is unscathed on P (x0 , t0 , r0 /4, −τ r02 ) and satisfies R < Kr0−2 there. Proposition 84.2 is an analog of Theorem 54.2. However, there is the important difference that in Proposition 84.2 we have to prove that no surgeries occur within P (x0 , t0 , r0 /4, −τ r02 ).

Assuming the validity of Propositions 84.1 and 84.2, suppose that the hypotheses of Corollary 81.3 are satisfied. We will allow ourselves to shrink the parameter r in order to apply Hamilton-Ivey pinching when needed. Put r0′ = θ0 (w) r0 , where θ0 (w) is from Lemma 83.1. By Lemma 83.1, there is a subball B(x′0 , t0 , r0′ ) ⊂ B(x0 , t0 , r0 ) such that every subball of B(x′0 , t0 , r0′ ) has volume at least (1 − ǫ) times the volume of the Euclidean ball of the same radius. As the sectional curvatures are bounded below by −r0−2 on B(x0 , t0 , r0 ), they are bounded below by −(r0′ )−2 on B(x′0 , t0 , r0′ ). By an appropriate 0 (w) , we choice of the parameters θ(w) and r of Corollary 81.3, in particular taking θ(w) ≤ θ2C 1 ′ ′ can ensure that Proposition 84.2 applies to B(x0 , t0 , r0 ). Then the solution is unscathed on P (x′0 , t0 , r0′ /4, −τ (r0′ )2 ) and satisfies | Rm | ≤ K (r0′ )−2 there, where the lower bound on Rm comes from Hamilton-Ivey pinching. With τ being the parameter of Proposition 84.2 and putting r0′′ = min(K −1/2 , τ 1/2 , 14 ) r0′ , for all t′′0 ∈ [t0 − (r0′′ )2 , t0 ] the solution is unscathed on P (x′0 , t′′0 , r0′′, −(r0′′ )2 ) and satisfies | Rm | ≤ (r0′′ )−2 there. From the curvature bound Rm ≥ − (r0′ )−2 on P (x′0 , t0 , r0′ /4, −τ (r0′ )2 ) (coming from pinching) and the fact that B(x′0 , t0 , r0′′ ) has almost Euclidean volume, we obtain a bound vol(B(x′0 , t′′0 , r0′′ )) ≥ const. (r0′′ )3 . Applying 0 Proposition 84.1 with A = 100r gives R ≤ K2 (r0′′ )−2 on B(x0 , t′′0 , 10r0 ) ⊂ B(x′0 , t′′0 , 100r0), r0′′ for all t′′0 ∈ [t0 − (r0′′ )2 , t0 ]. Writing this as R ≤ const. r0−2 , if we further restrict θ(w) to be sufficiently small then we can ensure that R ≤ const. θ2 (w) h−2 ≤ .01 h−2 . As surgeries only occur at spacetime points (x, t) where R(x, t) ∼ h(t)−2 , there are no surgeries S on t′′ ∈[t0 −(r′′ )2 ,t0 ] B(x0 , t′′0 , 10r0). Using length distortion estimates, we can find a parabolic 0 0 S neighborhood P (x0 , t0 , r0 /4, −τ r02 ) ⊂ t′′ ∈[t0 −(r′′ )2 ,t0 ] B(x0 , t′′0 , 10r0 ) for some fixed τ . This 0 0 proves Corollary 81.3. 85. II.6.3. Canonical neighborhoods and later curvature bounds on bigger balls from curvature and volume bounds We now prove Proposition 84.1. We first recall its statement. Proposition 85.1. (cf. Proposition II.6.3) For any A > 0 one can find positive constants κ(A), K1 (A), K2 (A), r(A), such that for any t0 < ∞ there exists δ A (t0 ) > 0, decreasing in t0 , with the following property. Suppose that we have a Ricci flow with (r, δ)-cutoff on a time interval [0, T ], where δ(t) < δ A (t) on [t0 /2, t0 ], with normalized initial data. Assume that 1. The solution is unscathed on a parabolic neighborhood P (x0 , t0 , r0 , −r02 ), with 2r02 < t0 . 2. | Rm | ≤ r0−2 on P (x0 , t0 , r0 , −r02 ).

166


3. vol(B(x0 , t0 , r0 )) ≥ A−1 r03 . Then (a) The solution is κ-noncollapsed on scales less than r0 in B(x0 , t0 , Ar0 ). (b) Every point x ∈ B(x0 , t0 , Ar0 ) with R(x, t0 ) ≥ K1 r0−2 has a canonical neighborhood in the sense of Definition 69.1. √ (c) If r0 ≤ r t0 then R ≤ K2 r0−2 in B(x0 , t0 , Ar0 ). Proof. The proof of part (a) is analogous to the proof of Theorem 28.2. The proof of part (b) is analogous to the proof of Lemma 53.3. The proof of part (c) is analogous to the proof of Theorem 53.1. We will be brief on the parts of the proof of Proposition 84.1 that are along the same lines as was done before, and will concentrate on the differences. 0) 0) For part (a) : we can reduce the case r0 < r(t to the case r0 ≥ r(t as in Sublemma 100 100 79.23; a canonical neighborhood of type (d) with small volume cannot occur, in view of condition 3 of Proposition 84.1. (We remark that the κ-noncollapsing that we want does not follow from the noncollapsing estimate used in the proof of Proposition 77.2, which p r(t0 ) would give a time-dependent κ.) So we may assume that 100 ≤ r0 ≤ t0 /2. For fixed t0 , this sets a lower bound on r0 .

Now suppose that (x, t0 ) ∈ B(x0 , t0 , Ar0 ), ρ < r0 , the parabolic neighborhood P (x, t0 , ρ, −ρ2 ) is unscathed and | Rm | ≤ ρ−2 there. We want to get a lower bound on ρ−3 vol(B(x, t0 , ρ)). We recall the idea of the proof of Theorem 28.2. With the notation of Theorem 28.2, after rescaling so that r0 = t0 = 1, we had a point x ∈ B(x0 , 1, A) around which we wanted to prove noncollapsing. Defining l using curves starting at (x, 1), we wanted to find a point (y, 1/2) ∈ B(x0 , 1/2, 1/2) so that l(y, 1/2) was bounded above by a universal constant. Given such a point, we concatenated a minimizing L-geodesic (from (x, 1) to (y, 1/2)) with curves emanating backward in time from (y, 1/2). Then the bounded geometry near (y, 1/2) allowed us to estimate from below the reduced volume at a time slightly less than 1/2. We knew that there was some point y ∈ M so that l(y, 1/2) ≤ 23 , but the issue in Theorem 28.2 was to find a point (y, 1/2) ∈ B(x0 , 1/2, 1/2) with l(y, 1/2) bounded above by a universal constant. The idea was to take the proof that some point y ∈ M has l(y, 1/2) ≤ 32 and localize it near x0 . The proof of Theorem 28.2 used the function h(y, t) = nondecreasing function that is one φ(d(y, t)−A(2t−1))(L(y, 1−t)+7). Here φ was a certain√ on (−∞, 1/20) and infinite on [1/10, ∞), and L(q, τ ) = 2 τ L(q, τ ). Clearly min h(·, 1) ≤ 7 and min h(·, 1/2) is achieved in B(x0 , 1/2, 1/10). The equation h ≥ −(6 + C(A))h implied that dtd min h ≥ −(6 + C(A)) min h, and so (min h)(t) ≤ 7 e(6+C(A))(1−t) .

In the present case, if one knew that the possible contribution of a barely admissible curve to h(y, t) was greater than 7 e(6+C(A))(1−t) + ǫ then one could still apply the maximum principle to find a point (y, 1/2) with h(y, 1/2) ≤ 7 e(6+C(A))/2 . For this, it suffices to know that the possible contribution of a barely admissible curve to L(q, τ ) can be bounded below by a sufficiently large number. However, Lemma 79.3 only says that we can make the contribution of a barely admissible curve to L large √ (using the lower scalar curvature bound to pass from L+ to L). Because of the factor 2 τ in the definition of L(q, τ ), we cannot necessarily say that its contribution to L(q, τ ) is large. To salvage the argument, the idea


167

is to redefine h and redo the proof of Theorem 28.2 in order to get an extra factor of min h.

√

τ in

(The use of Lemma 79.3 is similar to what was done in the proof of Proposition 77.2. However, there is a difference in scales. In Proposition 77.2 one was working at a microscopic scale in order to construct the Ricci flow with surgery. The function δ(t) in Proposition 77.2 was relevant to this scale. In the present case we are working at the macroscopic scale √ r0 ∼ t0 in order to analyze the long-time behavior of the Ricci flow with surgery. The function δA (t) of Proposition 84.1 is relevant to this scale. Thus we will end up further reducing the surgery function δ(t) of Proposition 77.2 in order to be able to apply Proposition 84.1.) By assumption, | Rm | ≤ 1 at t = 0. From Lemma 79.11, R ≥ − t ∈ [t0 − r02 /2, t0 ], (85.2)

R

r02

1 3 . 2 t+1/4

Then for

r02 3 r02 3 ≥ − = −1. ≥ − 2 t0 − r02 /2 2 2r02 − r02 /2

After rescaling so that r0 = 1, the time interval [t0 − r02 , t0 ] is shifted to [0, 1]. Then for t ∈ [ 21 , 1], we certainly have R ≥ −3. √ Rτ √ ˆ τ) ≡ From this, if 0 < τ ≤ 21 then L(y, τ ) ≥ −6 τ 0 v dv = −4τ 2 , so L(y, √ L(y, τ ) + 2 τ > 0. Putting ˆ τ) h(y, τ ) = φ(dt (x0 , y) − A(2t − 1)) L(y, √ √ and using the fact that dtd τ = − dτd τ = − 2√1 τ , the computations of the proof of Theorem 28.2 give

(85.3)

√ 1 h ≥ − L + 2 τ C(A) φ − 6 φ − √ φ τ 1 φ. = − C(A)h − 6 + √ τ

(85.4)

Then if h0 (τ ) = min h(·, τ ), we have d h0 (τ ) 1 1 φ 1 −1 dh0 (85.5) log √ = h0 − ≤ C(A) + 6 + √ − dτ τ dτ 2τ τ h0 2τ 1 1 1 = C(A) + 6 + √ √ − 2τ τ L+2 τ √ 6 τ + 1 1 . = C(A) + √ − 2τ τ L + 2τ As L ≥ − 4τ 2 , (85.6)

d dτ

√ h0 (τ ) 6 τ + 1 1 50 √ − log √ ≤ C(A) + ≤ C(A) + √ . 2 τ 2τ − 4τ τ 2τ τ

168


As τ → 0, the Euclidean space computation gives L(q, τ ) ∼ |q|2 , so limτ →0 Then √ √ (85.7) h0 (τ ) ≤ 2 τ exp(C(A)τ + 100 τ ). √ This estimate has the desired extra factor of τ .

h0 (τ ) √ τ

= 2.

It now suffices to show that for a barely admissible curve γ that hits a surgery region at time 1 − τ , Z τ √ √ 2 v R(γ(1 − v), v) + |γ(v)| ˙ dv ≥ exp(C(A)τ + 100 τ ) + ǫ, (85.8) 0

where 0 < τ ≤ 12 . Choosing δ A (t0 ) small enough, this follows from Lemma 79.3 along with the lower scalar curvature bound. Then we can apply the maximum principle and follow the proof of Theorem 28.2. In our case the bounded geometry near (y, 1/2) ∈ B(x0 , 1/2, 1/2) comes from assumptions 2 and 3 of the Proposition. The function δ A is now determined. After reintroducing the scale r0 , this proves part (a) of the proposition.

The proof of part (b) is similar to the proofs of Lemma 53.3 and Proposition 77.2. Suppose that for some A > 0 the claim is not true. Then there is a sequence of Ricci flows Mα which together provide a counterexample. In particular, some point (xα , tα ) ∈ B(xα0 , tα0 , Ar0α ) has R(xα , tα ) ≥ K1α (r0α )−2 but does not have a canonical neighborhood, where K1α → ∞ as α → ∞. Because of the canonical neighborhood assumption, we must have K1α (r0α )−2 ≤ r(tα0 )−2 . Then 2K1α ≤ K1α tα0 (r0α )−2 ≤ tα0 r(tα0 )−2 . Since K1α → ∞ and the function t → tr(t)−2 is bounded on any finite t-interval, it follows that tα0 → ∞. Applying point selection to each Mα and removing the superscripts, there are points x ∈ B(x0 , t, 2Ar0 ) with t ∈ [t0 −r02 /2, t0 ] such that Q ≡ R(x, t) ≥ K1 r0−2 and (x, t) does not have a canonical neighborhood, but each point (x, t) ∈ P with R(x, t) ≥ 4Q does have a canonical neighborhood, where −1 1/2 −1/2 P = {(x, t) : dt (x0 , x) ≤ dt (x0 , x) + K1 Q , t ∈ [t − 14 K1 Q , t]}. From (a), we have −1 noncollapsing in P . Rescaling by Q , we have bounded curvature at bounded distances from x; see Lemma 70.2. Then we can extract a pointed limit X∞ , which we think of as a time zero slice, that will have nonnegative sectional curvature. (The required pinching for the last statement comes from the assumption that 2r02 < t0 , along with the fact that K1α → ∞.) The fact that points (x, t) ∈ P with R(x, t) ≥ 4Q have a canonical neighborhood implies that regions of large scalar curvature in X∞ have canonical neighborhoods, from which one can deduce as in Section 46 that the sectional curvatures of X∞ are globally bounded above by some Q0 > 0. Then for each A, Lemmas 27.8 and 70.1 imply that for large α, the −1/2 −1 , −ǫη −1 Q−1 parabolic neighborhood P (x, t, A Q 0 Q ) is contained in P . (Here ǫ is a small parameter, which we absorb in the global parameter ǫ.) In applying Lemma 27.8 we use the curvature bound near x0 coming from the hypothesis of the proposition along with the curvature bound near x just derived; cf. the proof of Lemma 53.3. In addition, we claim −1/2 −1 that P (x, t, A Q , −ǫη −1 Q−1 0 Q ) is unscathed. This is proved as in Section 80. Recall −1/2 −1 that the idea is to show that a surgery in P (x, t, A Q , −ǫη −1 Q−1 0 Q ) implies that (x, t) lies in a canonical neighborhood, which contradicts our assumption. In the argument we use the fact that tα0 → ∞ implies δ(tα0 ) → 0 in order to rule out surgeries; this is the replacement α for the condition δ → 0 that was used in Section 80.


169

We extend X∞ to the maximal backward-time limit and obtain an ancient κ-solution, which contradicts the assumption that the points (x, t) did not have canonical neighborhoods. This proves part (b) of the proposition. To prove part (c), we can rescale t0 to 1 and then apply Lemma 70.2; see the end of the proof of Theorem 53.1. The Φ-pinching that we use comes from the Hamilton-Ivey estimate of (B.4). We recall that in the proof of Theorem 53.1 we need to get nonnegative curvature in the region W near the blowup point; this comes from the fact that r α → 0 in the contradiction argument, along with the Hamilton-Ivey pinching. This proves the proposition. In what follows, we will want to apply it freely for arbitrary A, provided that t0 is large enough. To do so, we reduce the function δ used to define the Ricci flow with (r, δ)-cutoff, if necessary, in order to ensure that δ(t) ≤ δ 2t (2t). Here δ 2t (2t) is the quantity δ A (2t) from Proposition 84.1 evaluated at A = 2t. 86. II.6.4. Earlier scalar curvature bounds on smaller balls from lower curvature bounds and volume bounds, in the presence of possible surgeries In this section we prove Proposition 84.2. We first recall its statement. Proposition 86.1. (cf. Proposition II.6.4) There exist τ, r, C1 > 0 and K < ∞ with the following property. Suppose that we have a Ricci flow with (r, δ)-cutoff on √ the time interval [0, t0 ], with normalized initial data. Let r0 satisfy 2C1 hmax (t0 ) ≤ r0 ≤ r t0 , where hmax (t0 ) is the maximal cutoff radius for surgeries in [t0 /2, t0 ]. (If there are no surgeries on [t0 /2, t0 ] then we put hmax (t0 ) = 0.) Assume 1. The ball B(x0 , t0 , r0 ) has sectional curvatures at least −r0−2 at each point. 2. The volume of any subball B(x, t0 , r) ⊂ B(x0 , t0 , r0 ) with any radius r > 0 is at least (1 − ǫ) times the volume of the Euclidean ball of the same radius. Then the solution is unscathed on P (x0 , t0 , r0 /4, −τ r02 ) and satisfies R < Kr0−2 there. Proposition 84.2 is an analog of Theorem 54.2. However, the proof of Proposition 84.2 is more complicated, due to the need to deal with possible surgeries. The idea of the proof is to put oneself in a setting in which one can apply Lemma 82.1. To do this, one needs to first show that the solution is unscathed in a parabolic region P (x0 , t0 , r0 , −τ0 r02 ) and that one has Rm ≥ −r0−2 there. Proof. The constants C1 , K and τ are fixed numbers, but the requirements on them will be specified during the proof. The number r will emerge from the proof, via a contradiction argument. We first dispose of the case when r0 ≤ r(t0 ). Suppose that r0 ≤ r(t0 ). We claim that R ≤ r0−2 on B(x0 , t0 , r20 ). If not then R(x, t0 ) > r0−2 for some x ∈ B(x0 , t0 , r20 ), and so R(x, t0 ) > r(t0 )−2 . This implies that R(x, t0 ) is in a canonical neighborhood, which contradicts the almost-Euclidean-volume assumption on subballs of B(x0 , t0 , r0 ).

170


1 −1 2 Thus R ≤ r0−2 on B(x0 , t0 , r20 ). Lemma 70.1 implies that R ≤ 16r0−2 on P (x0 , t0 , 12 r0 , − 16 η r0 ). Furthermore, if (*.1) C1 ≥ 100 1 −1 2 then R ≤ 2h12 on P (x0 , t0 , r0 /4, − 16 η r0 ). As surgeries only occur when R ≥ h−2 , there cannot be any surgeries in the region. Hence if we have (*.2) K ≥ 200 and 1 −1 (*.3) τ ≤ 16 η then there are no counterexamples to the proposition with r0 ≤ r(t0 ).

Continuing with the proof of the proposition, suppose that we have a sequence of Ricci flows with (r, δ)-cutoff Mα satisfying the assumptions of the proposition, with r α → 0, so that the conclusion of the proposition is violated for each Mα . Let tα0 be the first time when the conclusion is violated for Mα and let B(xα0 , tα0 , r0α) be a time-tα0 ball of smallest radius which provides a counterexample. (Such a ball exists since we have already shown that the proposition holds if r0α ≤ r(tα0 ).) That is, either there is a surgery in P (xα0 , tα0 , r0α/4, −τ (r0α )2 ) or R ≥ K(r0α )−2 somewhere on P (xα0 , tα0 , r0α /4, −τ (r0α )2 ). From the previous paragraph, r0α > r(tα0 ).

Let τb be the supremum of the numbers τe with the property that for large α, P (xα0 , tα0 , r0α, −e τ (r0α )2 ) is unscathed and Rm ≥ − (r0α )−2 there.

Lemma 86.2. τb is bounded below by the parameter τ0 of Lemma 82.1, where we take w = 1 − ǫ in Lemma 82.1.

Proof. Suppose that τb < τ0 . Put b tα = tα0 − (1 − ǫ′ ) τb (r0α )2 , where ǫ′ will eventually be taken to be a small positive number. Applying Lemma 82.1 to the solution on P (xα0 , tα0 , r0α , − (1 − 1 of the ǫ′ ) τb (r0α )2 ) (see the end of Section 82), the volume of B(xα0 , b tα , r0α /4) is at least 10 volume of the Euclidean ball of the same radius. From Lemma 83.1, there is a subball B(xα1 , b tα , r α ) ⊂ B(xα0 , b tα , r0α ) of radius r α = θ0 (1/10) r0α /4 with the property that all of its subballs have volume at least (1 − ǫ) times the volume of the Euclidean ball of the same radius. The sectional curvature on B(xα1 , b tα , r α ) is bounded below by − (r α )−2 . As we can apply the conclusion of the proposition to this subball (in view of its earlier time or smaller radius than B(xα0 , tα0 , r0α )), it follows that the solution on P (xα1 , b tα , r α /4, −τ (r α )2 ) is unscathed and has R < K(r α )−2 . As

(r α )2 (r α )2 (r α )2 (r α )2 (r0α )2 /tα0 ≥ α 2 α 0 α 2 = α 2 , b (r0 ) t0 − τ0 (r0 ) (r0 ) 1 − τ0 (r0α )2 /tα0 tα the fact that rα → 0 as α → ∞ implies that the Hamilton-Ivey pinching improves with α. In particular, for large α, | Rm | < K(r α )−2 on P (xα1 , b tα , r α /4, −τ (r α )2 ). Putting re0α = K −1/2 r α , if 1 (*.4) K −1 ≤ 10 τ α −2 then | Rm | ≤ (e r0 ) on the parabolic ball P (xα1 , e tα , e r0α, −(e r0α )2 ) for any e tα ∈ [b tα − 12 τ (r α )2 , b tα ]. α α Taking A = 100r0 /e r0 , Proposition 84.1(c) now implies that for large α, we have R ≤ α −2 α eα K2 (A) (e r0 ) on B(x1 , t , 100r0α). Provided that 1 C12 (*.5) K2 (A) K (θ0 (1/10))−2 ≤ 1000 1 −2 α −2 we will have K2 (A) (e r0 ) < 2 h and so there will not be any surgeries on such balls. Then the length distortion estimates of Lemma 27.8 imply that there is some c > 0 so that (86.3)


171

R ≤ K2 (A) (e r0α )−2 on P (xα0 , r0α , b tα , −c(r0α )2 ). Hamilton-Ivey pinching now implies that for large α, Rm ≥ − (r0α )−2 on P (xα0 , r0α , b tα , −c(r0α )2 ). As c can be taken independent of the ′ ′ small number ǫ , taking ǫ → 0 we contradict the maximality of τb. We can now apply Lemma 82.1 to obtain R ≤ K0 τ0−1 (r0α )−2 on P (xα0 , tα0 , r0α /4, −τ0 (r0α )2 /2). This will give a contradiction provided that (*.6) K0 τ0−1 < K/2, (*.7) τ < τ0 /2 and (*.8) K0 τ0−1 ≤ C12 , where the last condition rules out surgeries in P (xα0 , tα0 , r0α /4, −τ (r0α )2 ).

We choose τ to satisfy (∗.3) and (∗.7). We choose K to satisfy (∗.2), (∗.4) and (∗.6). Finally, we choose C1 to satisfy (∗.1), (∗.5) and (∗.8). This proves the proposition.

Remark 86.4. In subsequent sections we will want to know that for any w > 0, with the notation of Corollary 81.3, we have θ−1 (w) hmax (t0 ) ≤ r(t0 ) if t0 is sufficiently large (as a function of w). We can always achieve this by lowering the function δ(·) used to define the (t0 ) Ricci flow with (r, δ)-cutoff so that limt0 →∞ hmax = 0. We will assume hereafter that this r(t0 ) is the case. 87. II.7.1. Noncollapsed pointed limits are hyperbolic In this section we start the analysis of the long-time decomposition into hyperbolic and graph manifold pieces. In the section, M will denote a Ricci flow with (r, δ)-cutoff whose initial time slice (M0 , g(0)) is compact and has normalized metric. From Lemma 81.1, if g(0) has positive scalar curvature then the solution goes extinct in a finite time. From Lemma 81.2, these manifolds are understood topologically. If g(0) has nonnegative scalar curvature then either it acquires positive scalar curvature or it is flat, so again the topological type is understood. Hereafter we assume that the flow does not become extinct, and that Rmin < 0 for all t. − 3 Lemma 87.1. V (t) t + 14 2 is nonincreasing in t.

Proof. Suppose first that the flow is nonsingular. In the case the lemma follows from Lemma 79.11 and the equation Z dV (87.2) = − R dV ≤ − Rmin V. dt M

If there are surgeries then it only has the effect of causing further decrease in V . − 3 2 ˆ Definition 87.3. Put V = limt→∞ V (t) t + 41 2 and R(t) = Rmin (t) V (t) 3 .

Lemma 87.4. On any time interval which is free of singular times, and on which Rmin (t) ≤ 0 for all t (which we are assuming), we have Z ˆ dR 2 ˆ −1 (87.5) ≥ RV (Rmin − R) dV. dt 3 M

172


Proof. From (B.1), (87.6)

dRmin dt

ˆ dRmin dR = V dt dt

≥ 2 3

2 3

2 Rmin . Then

1 dV 2 2 2 ≥ Rmin V + Rmin V − 3 3 dt 3

2 3

1 2 − Rmin V − 3 3

from which the lemma follows.

Z

R dV,

M

ˆ is nondecreasCorollary 87.7. If Rmin (t) ≤ 0 for all t (which we are assuming) then R(t) ing. Proof. If M is a nonsingular flow then the corollary follows from Lemma 87.4. If there are ˆ (since surgeries then it only has the effect of decreasing V (t), and so possibly increasing R(t) Rmin (t) ≤ 0). ˆ Put R = limt→∞ R(t). Lemma 87.8. If V > 0 then R V

−2/3

= − 23 .

Proof. Suppose that V > 0. Using Lemma 79.11, (87.9) − 32 !− 23 2 3 1 1 −2/3 Rmin (t) ≥ − . RV = lim Rmin (t)V (t) 3 V (t) t + = lim t + t→∞ t→∞ 4 4 2 In particular, there is a limit as t → ∞ of t + 14 Rmin (t). Suppose that 3 1 −2/3 Rmin (t) = c > − . (87.10) RV = lim t + t→∞ 4 2

Combining this with (87.2) gives that for any µ > 0, V (t) ≤ const. tµ−c whenever t is sufficiently large. Then V (t)t−3/2 ≤ const. t−(c+3/2−µ) . Taking µ = 21 (c + 23 ), we contradict the assumption that V > 0. From the proof of Lemma 87.8, if V > 0 then Rmin (t) ∼ −

3 . 2t

The next proposition shows that a long-time limit will necessarily be hyperbolic.

Proposition 87.11.√Given the flow M, suppose that we have a sequence of parabolic neighborhoods P (xα , tα , r tα , −r 2 tα ), for tα → ∞ and some fixed r ∈ (0, 1), such that the scalings of the parabolic neighborhoods with factor tα smoothly converge to some limit solution (M∞ , (x, 1), g∞ (·)) defined in a parabolic neighborhood P (x, 1, r, −r 2 ). Then g∞ (t) has constant sectional curvature − 4t1 . Proof. Suppose first that the flow is surgery-free. Because of the assumed existence of the limit (M∞ , (x, 1), g∞(·)), the original solution M has V > 0. We claim that the scalar curvature on P (x, 1, r, −r 2) is spatially constant. If not then there are numbers c < 0 and s0 , µ > 0 so that Z ′ (87.12) (Rmin (s) − R(x, s)) dV ≤ c B(x,s,r)


173

′ whenever s ∈ (s0 −µ, s0 +µ) ⊂ [1−r 2 , 1], where Rmin (s) is the minimum of R over B(x, s, r). Then for large α, Z c√α ′ α α (87.13) (R (st ) − R(x, st )) dV < t , min √ 2 B(xα ,stα ,r tα ) √ ′ where Rmin (stα ) is now the minimum of R over B(xα , stα , r tα ). Thus Z c√α t . (Rmin (stα ) − R(x, stα )) dV < (87.14) √ 2 B(xα ,stα ,r tα )

After passing to a subsequence, we can assume that

tα+1 tα

>

s0 +µ s0 −µ

for all α. From (87.5),

(87.15)

Z Z 2 ∞ ˆ −1 ˆ R(t) V (t) (Rmin (t) − R(x, t)) dV (x) dt R − R(0) ≥ 3 0 M Z Z 2 X α s0 +µ ˆ α α −1 ≥ t R(st ) V (st ) (Rmin (stα ) − R(x, stα )) dV (x) ds 3 α s0 −µ M Z Z s +µ 0 2 X α ˆ α ) V (stα )−1 (Rmin (stα ) − R(x, stα )) dV (x) ds. t R(st ≥ √ 3 α s0 −µ B(xα ,stα ,r tα ) 2

Using the definitions of V and R = − 23 V 3 , along with (87.14), it follows that the right-hand side of (87.15) is infinite. This contradicts the fact that R < ∞.

Thus R is spatially constant on P (x, 1, r, −r 2). As Rmin (t) ∼ − 2t3 on M, we know that the scalar curvature R at (x, t) ∈ P (x, 1, r, −r 2 ), which only depends on t, satisfies R(t) ≥ − 2t3 . It does not immediately follow that the scalar curvature on P (x, 1, r, −r 2 ) equals − 2t3 , as Rmin (t) is the minimum of the scalar curvature on all of M. However, if the scalar curvature is not identically − 2t3 on P (x, 1, r, −r 2 ) then again we can find c < 0 and s0 , µ > 0 so that for large α, (87.14) holds for s ∈ (s0 − µ, s0 + µ) ⊂ [1 − r 2 , 1]. Again we get a contradiction using (87.5). Thus R(t) = − 2t3 on P (x, 1, r, −r 2 ). Then from (B.1), each time-slice of P (x, 1, r, −r 2) has an Einstein metric. Thus the sectional curvature on P (x, 1, r, −r 2 ) is − 4t1 . The argument goes through if one allows surgeries. The main ingredient was the monotonicity formulas, which √ still hold if there are surgeries. Note that for large α there are no surgeries in P (xα , tα , r tα , −r 2 tα ) by assumption. 88. II.7.2. Noncollapsed regions with a lower curvature bound are almost hyperbolic on a large scale √ In this section it is shown that for fixed A, r, w > 0 and large time t0 , if B(x0 , t0 , r t0 ) ⊂ 3

3 2 M+ curvatures at least −r −2 t−1 t0 has volume at least wr t0 and sectional√ 0 then the Ricci flow 2 on the parabolic neighborhood P (x0 , t0 , Ar t0 , Ar t0 ) is close to the flow on a hyperbolic manifold.

We retain the assumptions of the previous section.

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Lemma 88.1. (cf. Lemma II.7.2) √ (a) Given w, r, ξ > 0 one can find T = T (w, r, ξ) > ∞ such that if the ball B(x0 , t0 , r t0 ) ⊂ 3

3 2 −2 −1 M+ t0 at some time t0 ≥ T has volume at least wr t0 and sectional curvatures at least −r t0 then the curvature at (x0 , t0 ) satisfies

(88.2)

|2tRij (x0 , t0 ) + gij |2 = (2tRij (x0 , t0 ) + gij )(2tRij (x0 , t0 ) + g ij ) < ξ 2.

(b) Given in addition√A < ∞ and allowing T to depend on A, we can ensure (88.2) for all points in B(x0 , t0 , Ar t0 ). √ (c) The same is true for P (x0 , t0 , Ar t0 , Ar 2 t0 ). Note that the time T will depend on the initial metric. Proof. To prove (a), suppose that there is a sequence of points (xα0 , tα0 ) with tα0 → ∞ that provide a counterexample. We wish to apply Corollary 81.3 with the parameter r0 of the √α corollary equal to r t0 . Putting ! Rx 2 sinh (s) ds 0 (88.3) w b = min w, R1 x∈[0,1] x3 sinh2 (s) ds 0 b where r is from Corollary 81.3, then the hypotheses of the lemma will still be if r > r(w), b Thus after redefining w, if necessary, we satisfied upon replacing w by w b and r by r(w). may assume that r ≤ r(w). As√the function hmax (t) is nonincreasing, if tα0 is sufficiently large then θ−1 (w)hmax (tα0 ) ≤ r tα0 . Using Corollary 81.3 with a redefinition of w, we can take a convergent pointed subsequence as α → ∞ of the tα0 -rescalings, whose limit is defined in an abstract parabolic neighborhood. From Proposition 87.11 the limit will be hyperbolic, which is a contradiction.

−2 For part (b), Corollary 81.3 gives a bound R ≤ Kr√ 0 in the unscathed parabolic neighbor′ 2 hood P (x0 , t0 , r0 /4, −τ r0 ), where r0 = min(r, r(w )) t0 . We apply Proposition 84.1 to the −2 ′ ′ 2 ′ −2 parabolic neighborhood P (x Proposition 84.1(b), √0 , t0 , r0 , − (r0 ) ) where K r0 = (r0 ) . By ′ each point y ∈ B(x0 , t0 , Ar t0 ) with scalar curvature at least Q = K (A)r0−2 has a canonical neighborhood. Suppose that there is such a point. From part (a) we have √ R(x0 , t0 ) < 0, ′ so along a geodesic from x0 to y there will be some point x0 ∈ B(x0 , t0 , Ar t0 ) with scalar curvature Q. It also has a canonical neighborhood, necessarily of type (a) or (b). We can apply part (a) to a ball around x′0 with a radius on the order of (K ′ (A))−1/2 r0 , and with a value of w coming from the canonical neighborhood condition, to get a contradiction for √ ′ −1/2 large t0 . (Note r0 is proportionate to t0 .) Thus R ≤ K ′ (A)r0−2 on √ that (K (A)) B(x0 , t0 , Ar t0 ). If T is large enough then the Φ-almost nonnegative curvature implies that | Rm | ≤ K ′ (A)r0−2 . Then the noncollapsing in Proposition 84.1(a) gives a lower local volume bound. √ Hence we can apply part (a) of the lemma to appropriate-sized balls in A B(x0 , t0 , 2 r t0 ). As A is arbitrary, this proves (b) of the lemma.

For part (c), without loss of generality we can take ξ small. Suppose that the claim is not true. Then there is a point (x0 , t0 ) that√satisfies the hypotheses of the lemma but for which ξ. Without there is a point (x1 , t1 ) ∈ P (x0 , t0 , Ar t0 , Ar 2 t0 ) with |2t1 Rij (x1 , t1 ) + gij | ≥ √ loss of generality, we can take (x1 , t1 ) to be a first such point√in P (x0 , t0 , Ar t0 , Ar 2 t0 ). By part (b), t1 > t0 . Then |2tRij + gij | ≤ ξ on P (x0 , t0 , Ar t0 , t1 − t0 ). If ξ is small then this region has negative sectional curvature and there are no surgeries in the region.


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′ Using the length distortion estimates of Section 27, we can find r ′ = r√ (r, A) > 0 so that ′ −2 −1 ′ the sectional curvature is bounded below by − (r ) t1 on B(x0 , t1 , r t1 ). Also, by the evolution of volume under Ricci flow, there will be a w ′ = w ′ (r, w, ξ, A) so that the volume √ 3 of B(x0 , t1 , r ′√ t1 ) is bounded below by w ′ (r ′)3 (t1 ) 2 . Thus for large t0 we can apply (b) to B(x0 , t1 , A′ r ′ t1 ) with an appropriate choice of A′ to obtain a contradiction.

89. II.7.3. Thick-thin decomposition This section is concerned with the large-time decomposition of the manifold in “thick” and “thin” parts. Definition 89.1. For x ∈ M+ t , let ρ(x, t) be the unique number ρ ∈ (0, ∞) such that inf B+ (x,t,ρ) Rm = − ρ−2 , if such a ρ exists, and put ρ(x, t) = ∞ otherwise. The function ρ(x, t) is well-defined because M+ t is a compact smooth Riemannian manifold, so for fixed (x, t) ∈ M+ the quantity inf B + (x,t,ρ) Rm is a continuous nonincreasing t function of ρ which is negative for sufficiently large ρ if and only if (x, t) lies in a connected component with negative sectional curvature somewhere; on the other hand the function −ρ−2 is continuous and strictly increasing. We note that when it is finite, the quantity ρ(x, t) may be larger than the diameter of the component of Mt containing (x, t).

As an example, if M is the flow √ on a manifold M with spatially constant negative curvature then for large t, ρ(x, t) ∼ 2 t uniformly on M. The “thin” part of Mt , in the sense of hyperbolic geometry, can then be characterized as the points x so that vol(B(x, t, ρ(x, t)) < w ρ3 (x, t), for an appropriate constant w. Lemma 89.2.√For any w > 0 we can find ρ = ρ(w) > 0 and T = T (w) such that if t ≥ T and ρ(x, t) < ρ t then (89.3)

vol(B(x, t, ρ(x, t))) < wρ3 (x, t).

Proof. If the lemma is not true then there is a sequence (xα , tα ) with tα → ∞, ρ(xα , tα )(tα )−1/2 → 0 and vol(B(xα , tα , ρ(xα , tα ))) ≥ wρ3 (xα , tα ). The first step is to apply Corollary 81.3, but we need to know that for large α we have ρ(xα , tα ) ≥ θ−1 (w) hmax (tα ), where θ(w) and hmax are from Corollary 81.3. Suppose that this is not the case. Then after passing to a subsequence we have ρ(xα , tα ) < θ−1 (w) hmax (tα ) ≤ r(tα ) for all α, where we used Remark 86.4 in the last inequality. There are points xα,′ ∈ B(xα , tα , ρ(xα , tα )) with a sectional curvature equal to − ρ−2 (xα , tα ). Applying the Hamilton-Ivey pinching estimate of (B.4) with X α = ρ−2 (xα , tα ), and using the fact that limα→∞ tα X α = ∞, gives (89.4)

lim R(xα,′ , tα ) ρ2 (xα , tα ) = ∞.

α→∞

We claim that the curvatures at the centers of the balls satisfy (89.5)

lim R(xα , tα ) ρ2 (xα , tα ) = ∞.

α→∞

Suppose not. Then there is some number C ∈ (0, ∞) so that after passing to a subsequence, R(xα , tα ) ρ2 (xα , tα ) ≤ C for all α. By continuity and (89.4), after passing to another subsequence we can assume that there is a point xα′′ on a time-tα geodesic segment between

176


xα and xα′ so that R(xα′′ , tα ) ρ2 (xα , tα ) = 2C, for all α. We now apply Lemma 70.2 around (xα′′ , tα ) to get a contradiction to (89.4). (More precisely, we apply a version of Lemma 70.2 that applies along geodesics, as in Claim 2 of II.4.2.) In applying Lemma 70.2, we use the −2 (xα ,tα )) fact that limα→∞ tα ρ−2 (xα , tα ) = ∞ in order to say that limα→∞ Φ(2Cρ = 0, with 2Cρ−2 (xα ,tα ) the notation of Lemma 70.2. Making a similar argument centered at other points in B(xα , tα , ρ(xα , tα )), we deduce that (89.6) lim inf R B(xα ,tα ,ρ(xα ,tα )) ρ2 (xα , tα ) = ∞. α→∞ α α

In particular, since ρ(x , t ) ≤ r(tα ), if α is large then each point y α ∈ B(xα , tα , ρ(xα , tα )) is the center of a canonical neighborhood. As α → ∞, the intrinsic scales R(y α , tα )−1/2 become arbitrarily small compared to ρ(xα , tα ). However, by Lemma 83.1 there is a subball B α,′ of B(xα , tα , ρ(xα , tα )) with radius θ0 (w)ρ(xα , tα ) so that every subball of B α,′ has almost Euclidean volume. This contradicts the existence of a small canonical neighborhood around each point of B(xα , tα , ρ(xα , tα )).

We now know that for large α, ρ(xα , tα ) ≥ θ−1 (w) hmax (tα ). Thus we can apply Corollary 81.3 to get an unscathed solution on the parabolic neighborhood P (xα , tα , ρ(xα , tα )/4, −τ ρ2 (xα , tα )), with R < K0 ρ(xα , tα )−2 there. Applying Proposition 84.1(c), along with the fact that limα→∞ tα ρ−2 (xα , tα ) = ∞, gives an estimate R ≤ K2 ρ(xα , tα )−2 on B(xα , tα , ρ(xα , tα )). But then for large α, the Hamilton-Ivey pinching gives Rm > − 21 ρ(xα , tα )−2 on B(xα , tα , ρ(xα , tα )), which is a contradiction. Remark 89.7. Another approach to the above proof would be to use the canonical neighborhood at the center (xα , tα ) of the ball, along with the Bishop-Gromov inequality, to contradict the fact that vol(B(xα , tα , ρ(xα , tα ))) ≥ wρ3 (xα , tα ). For this to work we would 3 1 have to know that the relative volume (ǫ2 R(xα , tα )) 2 vol(B(xα , tα , ǫ−1 R(xα , tα )− 2 ) of the canonical neighborhood (of type (a) or (b)) around (xα , tα ) is small compared to w. This will be the case if we take the constant ǫ to be small enough, but in a w-dependent way. Although ǫ is supposed to be a universal constant, this approach will work because when characterizing the graph manifold part in Section 92, w can be taken to be a small but fixed constant. Definition 89.8. The w-thin part M − (w, t) ⊂ M+ t is the set of points x ∈ M so that either ρ(x, t) = ∞ or vol(B(x, t, ρ(x, t))) < w (ρ(x, t))3 .

(89.9)

− The w-thick part is M + (w, t) = M+ t − M (w, t).

Lemma 89.10. Given w > 0, there are w ′ = w ′ (w) > 0 and T ′ = T ′ (w) < ∞ so that taking √ r = ρ(w) (with reference to Lemma 89.2), if x0 ∈ M + (w, t) and t0 ≥ T ′ then B(x0 , t0 , r t0 ) 3

has volume at least w ′ r 3 t02 and sectional curvature at least − r −2 t−1 0 .

+ Proof. Suppose that x0 ∈ M √ (w, t0 ). From Lemma 89.2,−2if t0 is big enough (as a function of w) then ρ(x0 , t0 ) ≥ r t0 . As √ Rm ≥ − ρ(x0 , t0 ) on B(x0 , t0 , ρ(x0 , t0 )), we3 have Rm ≥ − r −2 t−1 on B(x , t , r t0 ). As vol(B(x0 , t0 , ρ(x0 , t0 ))) ≥ w (ρ(x0 , t0 )) , the 0 0 0 √ −3 √ Bishop-Gromov inequality gives a lower bound on r t0 vol(B(x0 , t0 , r t0 )) in terms of w.


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90. Hyperbolic rigidity and stabilization of the thick part Lemma 89.10 implies that Lemma 88.1 applies to M + (w, t) if t is sufficiently large (as a function of w). That is, if one takes a sequence of points in the w-thick parts at a sequence of times tending to infinity then the pointed time slices subconverge, modulo rescaling the metrics by t−1 , to complete finite volume hyperbolic manifolds with sectional curvatures equal to − 14 . (The − 41 comes from the Ricci flow equation along with the equation g(t) = t g(1) for the rescaled limit, which implies that g(1) has Einstein constant − 12 .) In what follows we will take the word “hyperbolic” for a 3-manifold to mean “constant sectional curvature − 14 ”. The next step, following Hamilton [34], is to show that for large time the picture stabilizes, i.e. the limits are unique in a strong sense. Proposition 90.1. There exist a number T0 < ∞, a nonincreasing function α : [T0 , ∞) → (0, ∞) with limt→∞ α(t) = 0, a (possibly empty) collection {(H1 , x1 ), . . . , (Hk , xk )} of complete connected pointed finite-volume hyperbolic 3-manifolds and a family of smooth maps k [ 1 (90.2) f (t) : Bt = B xi , −→ Mt , α(t) i=1 defined for t ∈ [T0 , ∞), such that

1. f (t) is close to an isometry:

(90.3)

kt−1 f (t)∗ gMt − gBt k

1

< α(t),

C α(t)

2. f (t) defines a smooth family of maps which changes slowly with time: (90.4)

1 |f˙(p, t)| < α(t)t− 2

for all p ∈ Bt , where f˙ refers to the time derivative (as defined with admissible curves), and

3. f (t) parametrizes more and more of the thick part: M + (α(t), t) ⊂ im(f (t)) for all t ≥ T0 . Remark 90.5. The analogous statement in [52, Section 7.3] is in terms of a fixed w. That is, −1 ∞ for a given w one considers pointed limits of {(M+ tj , (xj , tj ), tj g(tj ))}j=1 with limj→∞ tj = ∞ and the basepoint satisfying xj ∈ M + (w, tj ) ⊂ M+ tj for all j. Considering the possible limit spaces in a certain order, as described below, one extracts complete pointed finite-volume hyperbolic manifolds {(Hi, xi )}ki=1 with xi in the w-thick part of Hi . There is a number w0 > 0 so that as long as w ≤ w0 , the hyperbolic manifolds Hi are independent of w. For Sk ′ ′ any w > 0, as time goes on the w -thick part of i=1 Hi better approximates M + (w ′ , t). Hence the formulation of Proposition 90.1 is equivalent to that of [52, Section 7.3]. Rather than proving Proposition 90.1 using harmonic maps as in [34], we give a simple proof using smooth compactness and a smoothing argument. Roughly speaking, the idea is to exploit a variant of Mostow rigidity to show that for large t, the components of the wthick part change slowly with time, and are close to hyperbolic manifolds which are isolated (due to a refinement of Mostow-Prasad rigidity). This forces them to eventually stabilize.

178


Definition 90.6. If (X, x) and (Y, y) are pointed smooth Riemannian manifolds and ǫ > 0 then (X, x) is ǫ-close to (Y, y) if there is a pointed map f : (X, x) → (Y, y) such that (90.7)

f |B(x,ǫ−1 ) : B(x, ǫ−1 ) → Y

is a diffeomorphism onto its image and (90.8)

kf ∗ gY − gX kC ǫ−1 < ǫ,

where the norm is taken on B(x, ǫ−1 ). Note that nothing is required of f on the complement of B(x, ǫ−1 ). Such a map f is called an ǫ-approximation. We will sometimes refer to a partially defined map f : (X, x) ⊃ (W, x) → (Y, y) as an ǫ-approximation provided that W contains B(x, ǫ−1 ) and the conditions above are satisfied. By convention we will permit X and Y to be disconnnected, in which case ǫ-closeness only says something about the components containing the basepoints. We say that two maps f1 , f2 : (X, x) → Y (not necessarily basepoint-preserving) are ǫ-close if (90.9)

sup

dY (f1 (p), f2 (p)) < ǫ.

p∈B(x,ǫ−1 )

We recall some facts about hyperbolic manifolds. There is a constant µ0 > 0, the Margulis constant, such that if X is a complete connected finite-volume hyperbolic 3-manifold (orientable, as usual), µ ≤ µ0 , and (90.10)

Xµ = {x ∈ X | InjRad(X, x) ≥ µ}

is the µ-thick part of X, then Xµ is a nonempty compact manifold-with-boundary whose complement U is a finite union of components U1 , . . . , Uk , where each Ui is isometric either to a geodesic tube around a closed geodesic or to a cusp. In particular, Xµ is connected and there is a one-to-one correspondence between the boundary components of Xµ and the “thin” components Ui . For each i, let ρi : Ui → R denote either the distance function from the core geodesic or a Busemann function, in the tube and cusp cases respectively. In the latter case we normalize ρi so that ρ−1 i (0) = ∂Ui . (The Busemann function goes to − ∞ as one goes down the cusp.) The radial direction is the direction field on Ui − core(Ui ) defined by ∇ρi , where core(Ui ) is the core geodesic when Ui is a geodesic tube and the empty set otherwise. Lemma 90.11. Let (X, x) be a pointed complete connected finite-volume hyperbolic 3manifold. Then for each ζ > 0 there exists ξ > 0 such that if X ′ is a complete finitevolume hyperbolic manifold with at least as many cusps as X, and f : (X, x) → X ′ is a ξ-approximation, then there is an isometry fˆ : (X, x) → X ′ which is ζ-close to f . This was stated as Theorem 8.1 in [34] as going back to the work of Mostow. We give a proof here. The hypothesis about cusps is essential because every pointed noncompact finite-volume hyperbolic 3-manifold (X, x) is a pointed limit of a sequence {(Xi , xi )}∞ i=1 of compact hyperbolic manifolds. Hence for every ξ > 0, if i is sufficiently large then there is a ξ-approximation f : (X, x) → (Xi , xi ), but there is no isometry from X to Xi .


179

Proof. The main step is to show that for the fixed (X, x), if ξ is sufficiently small then for any ξ-approximation f : (X, x) → X ′ satisfying the hypotheses of the lemma, the manifolds X and X ′ are diffeomorphic. The proof of this will use the Margulis thick-thin decomposition. The rest of the assertion then follows readily from Mostow-Prasad rigidity [47, 54]. Pick µ1 ∈ (0, µ0 ) so that X − Xµ1 consists only of cusps U1 , . . . , Uk . The thick part Xµ1 is compact and connected. Given ξ > 0, let f : (B(x, ξ −1 ), x) → (X ′ , x′ ) be a ξ-approximation as in (90.8). The intuitive idea is that because of the compactness of Xµ1 , if ξ is sufficiently small then f Xµ is close to being an isometry from Xµ1 to its image. Then f (Xµ1 ) is close 1 to a connected component of the thick part Xµ′ 1 of X ′ . As f (Xµ1 ) and Xµ′ 1 are connected, this means that f (Xµ1 ) is close to Xµ′ 1 . We will show that in fact Xµ1 is diffeomorphic to Xµ′ 1 . The boundary components of Xµ1 correspond to the cusps of X and the boundary components of Xµ′ 1 correspond to the connected components of X ′ − Xµ′ 1 . As X ′ has at least as many cusps as X by assumption, it follows that the connected components of X ′ − Xµ′ 1 are all cusps. Hence X and X ′ are diffeomorphic. In order to show that Xµ1 is diffeomorphic to Xµ′ 1 , we will take a larger region W ⊃ Xµ1 that is diffeomorphic to Xµ1 and show that f (W ) can be isotoped to Xµ′ 1 by sliding it inward along the radial direction. More precisely, in each cusp Ui put Vi = ρ−1 i ([−3L, −L]), where −1 L ≫ 1 is large enough that every cuspidal torus ρi (s) with s ∈ [−3L, −L] has diameter much less than one. Let W ⊂ X be the complement of the open horoballs at height −2L, i.e. (90.12)

W =X−

k [

i=1

ρ−1 i (−∞, −2L).

When ξ is sufficiently small, f will preserve injectivity radius to within a factor close to 1 for points p ∈ B(x, ξ −1 ) with d(p, ∂B(x, ξ −1 )) > 2 InjRad(X, p). Therefore when ξ is small, f will map each Vi into X ′ − Xµ′ 1 , and hence into one of the connected components Uk′ i of ′ ′ X ′ − Xµ′ 1 . Let Zi′ be the image of Zi = ρ−1 i (−2L) under f . Note that d(core(Uki ), Zi ) & L (if core(Uk′ i ) 6= ∅), for otherwise f −1 (core(Uk′ i )) would be a closed curve with small diameter and curvature lying in Ui , which contradicts the fact that the horospheres have principal curvatures − 21 (because of our normalization that the sectional curvatures are − 41 ). Thus for each point z ′ ∈ Zi′ there is a minimizing radial geodesic segment γ ′ passing through z ′ with d(z ′ , ∂γ ′ ) & L. The preimage of γ ′ under f is a curve γ ⊂ X with small curvature and length & L passing through Zi . This forces the direction of γ to be nearly radial and so transverse to Zi . Hence Zi′ is transverse to the radial direction in Uk′ i . Combining this with the fact that Zi′ is embedded implies that Zi′ is isotopic in Uk′ i to ∂Uk′ i . It follows that f (W ) is isotopic to Xµ′ 1 . Then by the preceding argument involving counting the number of cusps, X and X ′ are diffeomorphic. We apply Mostow-Prasad rigidity [47, 54] to deduce that X is isometric to X ′ . We now claim that for any ζ > 0, if ξ is sufficiently small then the map f is ζ-close to an isometry from X to X ′ . Suppose not. Then there are a number ζ > 0 and a sequence of 1i -approximations fi : (X, x) → (Xi′ , x′i ) so that none of the fi ’s are ζ-close to any isometry from (X, x) to (Xi′ , x′i ). Taking a convergent subsequence of the maps fi gives a

180


′ limit isometry f∞ : (X, x) → (X∞ , x′∞ ). From what has already been proven, for large i ′ ′ we know that Xi is isometric to X, and so isometric to X∞ . This is a contradiction.

Recall the statement of Lemma 88.1. Definition 90.13. Given w > 0, let Λw be the space of complete pointed finite-volume −1 ∞ hyperbolic 3-manifolds that arise as pointed limits of sequences {(M+ ti , (xi , ti ), ti g(ti ))}i=1 + with limi→∞ ti = ∞ and the basepoint (xi , ti ) satisfying (xi , ti ) ∈ M + (w, ti ) ⊂ Mti for all i. The space Λw is compact in the smooth pointed topology. Any element of Λw has volume at most V , the latter being defined in Definition 87.3. The next lemma summarizes the content of Lemma 88.1. Lemma 90.14. Given w > 0, there is a decreasing function β : [0, ∞) → (0, ∞] with lims→∞ β(s) = 0 such that if (x, t) ∈ M + (w, t) ⊂ M+ t , and Zt denotes the forward time −1 slice M+ rescaled by t , then t 1. Some (X, x) ∈ Λw is β(t)-close to (Zt , (x, t)). √ 2. B(x, t, β(t)−1 t) ⊂ M+ t is unscathed on the interval [t, 2t] and if γ : [t, 2t] → M is a static curve starting at (x, t), t¯ ∈ [t, 2t], then the map √ √ (90.15) B(x, t, β(t)−1 t) → P (x, t, β(t)−1 t, t) ∩ Mt¯ defined by following static curves induces a map (90.16) satisfying (90.17)

it,t¯ : (Zt , (x, t)) ⊃ (B(x, t, β(t)−1 ), (x, t)) → (Zt¯, γ(t¯)) k (it,t¯)∗ gZt¯ − gZt kC β(t)−1 < β(t).

Proof. This follows immediately from Lemma 88.1.

Proof of Proposition 90.1. If for some w > 0 we have Λw = ∅ then M + (w, t) = ∅ for large t. Thus if Λw = ∅ for all w > 0, we can take the empty collection of pointed hyperbolic manifolds and then 1 and 2 will be satisfied vacuously, and α(t) may be chosen so that 3 holds. So we assume that Λw 6= ∅ for some w > 0.

Since every complete finite-volume hyperbolic 3-manifold has a point with injectivity radius ≥ µ0 , there is a w0 > 0 such that the collections {Λw }w≤w0 contain the same sets of underlying hyperbolic manifolds (although the basepoints have more freedom when w is small). We let H1 be a hyperbolic manifold from this collection with the fewest cusps and we choose a basepoint x1 ∈ H1 so that (H1 , x1 ) ∈ Λw0 . Put w1 = w20 . Note that x1 lies in the w1 -thick part of H1 . In what follows we will use the fact that if f is a ǫ-approximation from H1 , for sufficiently small ǫ, then f (x1 ) will lie in the .9w0 -thick part of the image. The idea of the first step of the proof is to define a family {f0 (t)} of δ-approximations (H1 , x1 ) → Zt , for all t sufficiently large, by taking a δ-approximation (H1 , x1 ) → Zt , pushing it along static curves, and arguing using Lemma 90.11 that one can make small


181

adjustments from time to time to keep it a δ-approximation. The family {f0 (t)} will not vary continuously with time, but it will have controlled “jumps”. More precisely, pick T0 < ∞ and let ξ1 , . . . , ξ4 > 0 be parameters to be specified later. We assume that T0 is large enough so that 2β(T0 ) < ξ1 , where β is from Lemma 90.14. By the definition of Λw1 , we may pick T0 so that there is a point (x0 , T0 ) ∈ M + (w1 , T0 ) ⊂ M+ T0 and a ξ1 -approximation f0 (T0 ) : (H1 , x1 ) → (ZT0 , x0 ). To do the induction step, for a given j ≥ 0 suppose that at time 2j T0 there is a j point (xj , 2j T0 ) ∈ M + (w1 , 2j T0 ) ⊂ M+ 2j T0 and a ξ1 -approximation f0 (2 T0 ) : (H1 , x1 ) → (Z2j T0 , xj ). As mentioned above, if ξ1 is small then in fact xj ∈ M + (.9w0 , 2j T0 ).

By part 2 of Lemma 90.14, provided that T0 is sufficiently large we may define, for all t ∈ [2j T0 , 2j+1T0 ], a 2ξ1-approximation f0 (t) : (H1 , x1 ) → Zt by moving f0 (2j T0 ) along static curves. Provided that ξ1 is sufficiently small we will have f0 (2j+1T0 )(x1 ) ∈ M + (w1 , 2j+1T0 ) and then part 1 of Lemma 90.14 says there is some (H ′ , x′ ) ∈ Λw1 with a β(2j+1T0 )approximation φ : (H ′, x′ ) → (Z2j+1 T0 , f0 (2j+1 T0 )(x1 )). Provided that β(2j+1T0 ) and ξ1 are sufficiently small, the partially defined map φ−1 ◦ f0 (2j+1 T0 ) will define a ξ2 -approximation from (H1 , x1 ) to (H ′ , x′ ). Hence provided that ξ2 is sufficiently small, by Lemma 90.11 the map will be ξ3 -close to an isometry ψ : (H1 , x1 ) → H ′. (In applying Lemma 90.11 we use the fact that H1 is also a manifold with the fewest number of cusps in Λw1 .) Put φ1 = φ◦ψ. Provided that ξ3 is sufficiently small, f0 (2j+1T0 ) and φ1 will be ξ4 -close as maps from (H1 , x1 ) to Z2j+1 T0 . Since φ1 is a β(2j+1T0 )-approximation precomposed with an isometry which shifts basepoints a distance at most ξ3 , it will be a 2β(2j+1T0 )-approximation provided that ξ3 < 1 and β(2j+1T0 ) < 21 . We now redefine f0 (2j+1T0 ) to be φ1 and let xj+1 be the image of x1 under φ1 . This completes the induction step. In this way we define a family of partially defined maps {f0 (t) : (H1 , x1 ) → Zt }t∈[T0 ,∞) . From the construction, f0 (2j T0 ) is a 2β(2j T0 )-approximation for all j ≥ 0. Lemma 90.14 then implies that there is a function α1 : [T0 , ∞) → (0, ∞) decreasing to zero at infinity such that for all t ∈ [T0 , ∞), f0 (t) is an α1 (t)-approximation, and for every t¯ ∈ [t, 2t] we may slide f0 (t) along static curves to define an α1 (t)-approximation h(t¯) : (H1 , x1 ) → Zt which is α1 (t)-close to f0 (t¯). One may now employ a standard smoothing argument to convert the family {f0 (t)}t∈[T0 ,∞) into a family {f1 (t)}t∈[T0 ,∞) which satisfies the first two conditions of the proposition. If condition 3 fails to hold then we redefine the Λw ’s by considering limits of only those −1 + ∞ + {(M+ ti , (xi , ti ), ti g(ti ))}i=1 with ti → ∞ and xi ∈ M (w, ti ) ⊂ Mti not in the image of f1 (ti ). Repeating the construction we obtain a pointed hyperbolic manifold (H2 , x2 ) and a family {f2 (t)} defined for large t satisfying conditions 1 and 2, where im(f2 (t)) is disjoint from im(f1 (t)) for large t. Iteration of this procedure must stop after k steps for some finite number k, in view of the fact that V < ∞ and the fact that there is a positive lower bound on the volumes of complete hyperbolic 3-manifolds. We get the desired family {f (t)} by taking the union of the maps f1 (t), . . . , fk (t).

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91. Incompressibility of cuspidal tori By Proposition 90.1, we know that for large times the thick part of the manifold can be parametrized by a collection of (truncated) finite volume hyperbolic manifolds. In this section we show that each cuspidal torus maps to an embedded incompressible torus in Mt . (An alternative argument is given in Section 93.) The strategy, due to Hamilton, is to argue by contradiction. If such a torus were compressible then there would be an embedded compressing disk of least area at each time. By estimating the rate of change of the area of such disks one concludes that the area must go to zero in finite time, which is absurd. Let T0 , α, {(H1 , x1 ), . . . , (Hk , xk )}, Bt , and f (t) be as in Proposition 90.1. We will consider a fixed Hi , with 1 ≤ i ≤ k, which is noncompact. Choose a number a > 0 much smaller than the Margulis constant and let {V1 , . . . , Vl } ⊂ Hi be the cusp regions bounded by tori of diameter a. Each Vj is an embedded 3-dimensional submanifold (with boundary) of Hi and is isometric to the quotient of a horoball in hyperbolic 3-space H3 by the action of a copy of Z2 sitting in the stabilizer of the horoball. The boundary ∂Vj is a totally umbilic torus whose principal curvatures are equal to 21 everywhere. We let Y ⊂ Hi be the closure S of the complement of lj=1 Vj in Hi . Let Ta < ∞ be large enough that BTa (defined as in Proposition 90.1) contains Y . In order to focus on a given cusp, we now fix an integer 1 ≤ j ≤ l and put

(91.1)

Z = ∂Vj ,

Zˆt = f (t)(Z) Yˆt = f (t)(Y ),

for every t ≥ Ta . The objective of this section is:

ˆ t = M+ ˆ W t − int(Yt )

Proposition 91.2. The homomorphism (91.3)

π1 (f (t)) : π1 (Z, ⋆) → π1 (M+ t , f (t)(⋆))

is a monomorphism for all t ≥ Ta . Proof. The proof will occupy the remainder of this section. The first step is: Lemma 91.4. The kernels of the homomorphisms (91.5)

π1 (f (t)) : π1 (Z, ⋆) → π1 (M+ t , f (t)(⋆)),

ˆ t , f (t)(⋆)) π1 (f (t)) : π1 (Z, ⋆) → π1 (W

are independent of t, for all t ≥ Ta . Proof. We prove the assertion for the first homomorphism. The argument for the second one is similar. The kernel obviously remains constant on any time interval which is free of singular times. − Suppose that t0 ≥ Ta is a singular time. Then the intersection M+ t0 ∩ Mt0 includes into + Mt0 and, by using static curves, into Mt for t 6= t0 close to t0 . By Van Kampen’s theorem, these inclusions induce monomorphisms of the fundamental groups. Therefore for t close to t0 , the kernel of (91.5) is the same as the kernel of (91.6)

− π1 (f (t)) : π1 (Z, ⋆) → π1 (M+ t0 ∩ Mt0 ),

which is independent of t for times t close to t0 .


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We now assume that the kernel of (91.7)

π1 (f (t)) : π1 (Z, ⋆) → π1 (M+ t , f (t)(⋆))

is nontrivial for some, and hence every, t ≥ Ta . By Van Kampen’s theorem and the fact that the cuspidal torus Z ⊂ Y is incompressible in Y , it follows that the kernel K of ˆ t , f (t)(⋆)) (91.8) π1 (f (t)) : π1 (Z, ⋆) → π1 (W ˆ t ; R) → H1 (∂ W ˆ t ; R) is a Lain nontrivial for all t ≥ Ta . By Poincaré duality, Im H1 (W ˆ t ; R). In particular, Im H1 (W ˆ t ; R) → H1 (Z; R) has rank one. grangian subspace of H1 (∂ W c Dually, Ker H1 (Z; R) → H1 (Wt ; R) has rank one and so K, a subgroup of a rank-two free abelian group, has rank one. We note that for all large t, Zˆt is a convex boundary compoˆ t . The main theorem of [43] implies that for every such t, there is is a least-area nent of W compressing disk ˆ t , Zˆt ). (91.9) (N 2 , ∂N 2 ) ⊂ (W t

t

We recall that a compressing disk is an embedded disk whose boundary curve is essential ˆ t is a compact manifold even when t is a singular time. in Zˆt . We note that by definition, W −1 The embedded curve f (t) (∂Nt ) ⊂ Z represents a primitive element of π1 (Z) which, since K has rank one, must therefore generate K. It follows that modulo taking inverses, the homotopy class of f (t)−1 (∂Nt ) ⊂ Z is independent of t. We define a function A : [Ta , ∞) → (0, ∞) by letting A(t) be the infimum of the areas of such embedded compressing disks. We now show that the least-area compressing disks avoid the surgery regions.

Lemma 91.10. Let δ(t) be the surgery parameter from Section 73. There is a T = T (a) < ˆ t is ∞ so that whenever t ≥ T , no point in any area-minimizing compressing disk Nt ⊂ W in the center of a 10δ(t)-neck. Proof. If the lemma were not true then there would be a sequence of times tk → ∞ and ˆ t , Zˆt ), along with a for each k an area-minimizing compressing disk (Ntk , ∂Ntk ) ⊂ (W k k point xk ∈ Ntk that is in the center of a 10δ(tk )-neck. Note that the scalar curvature near ∂Ntk is comparable to − 2t3k . We now rescale by R(xk , tk ), and consider the map of ˆ t , Zˆt , xk ) where the domain is equipped with pointed manifolds fk : (Ntk , ∂Ntk , xk ) ֒→ (W k k the pullback Riemannian metric. By [57] and standard elliptic regularity, for all ρ < ∞ and every integer j, the j th covariant derivative of the second fundamental form of fk is uniformly bounded on the ball B(xk , ρ) ⊂ Ntk , for sufficiently large k. Therefore the pointed Riemannian manifolds (Ntk , ∂Ntk , xk ) subconverge in the smooth topology to a pointed, complete, connected, smooth manifold (N∞ , x∞ ). Using the same bounds on the derivatives of the second fundamental form, we may extract a limit mapping φ∞ : N∞ → R × S 2 which is a 2-sided isometric stable minimal immersion. By [58, Theorem 2], φ∞ is a totally geodesic immersion whose normal vector field in M has vanishing Ricci curvature. It follows that φ∞ is a cover of a fiber {pt}×S 2 . This contradicts the fact that N∞ is noncompact. We redefine Ta if necessary so that Ta is greater than the T of Lemma 91.10.

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We can isotope the surface Z by moving it down the cusp Vj . In doing so we do not change the group K but we can make the diameter of Z as small as desired. The next lemma refers to this isotopy freedom. Lemma 91.11. Given D > 0, there is a number a0 > 0 so that for any a ∈ (0, a0 ),√ if R diam(Z) = a and t is sufficiently large then ∂Nt κ∂Nt ds ≤ D2 and length(∂Nt ) ≤ D2 t, where κ∂Nt is the geodesic curvature of ∂Nt ⊂ Nt . Proof. This is proved in [34, Sections 11 and 12]. We just state the main idea. For the ˆ t the metric t−1 g(t). First, ∂Nt is the intersection of Nt purposes of this proof, we give W with Zˆt . Because Nt is minimal with respect to free boundary conditions (i.e. the only constraint is that ∂Nt is in the right homotopy class in Zˆt ), it follows that Nt meets Zˆt ˆt orthogonally. Then κ∂Nt = Π(v, v), where Π is the second fundamental form of Zˆt in W and v is the unit tangent vector of ∂Nt . Given a > 0, let Z be the horospherical torus in Vj of diameter a. By Proposition 90.1, for large t the map f (t) is close to being an isometry of pairs (Y, Z) → (Yˆt , Zˆt ). As Z has principal curvatures 21 , we may assume that Π(v, v) is close to 21 . This reduces the problem to showing that with an appropriate choice of a0 , if a ∈ (0, a0 ) then for large values of t the length of ∂Nt is guaranteed to be small. The ˆ t is close to being the standard cusp Vj , a large piece of the minimal intuition is that since W disk Nt should be like a minimal surface N∞ in Vj that intersects Z in the given homotopy class. Such a minimal surface in Vj essentially consists of a geodesic curve in Z going all the way down the cusp. The length of the intersection of N∞ with the horospherical torus of diameter a is proportionate to a. Hence if a0 is small enough, one would expect that if a < a0 and if t is large then the length of ∂Nt is small. In particular, the length of ∂Nt is uniformly bounded with respect to a. A detailed proof appears in [34, Section 12]. R Rescaling from the metric t−1 g(t) to the original metric g(t), ∂Nt κ∂Nt ds is unchanged √ and length(∂Nt ) is multiplied by t. Lemma 91.12. For every D > 0 there is a number a0 > 0 with the following property. Given a ∈ (0, a0 ), suppose that we take Z to be the torus cross-section in Vj of diameter a. Then there is a number Ta′ < ∞ so that as long as t0 ≥ Ta′ , there is a smooth function A¯ ¯ 0 ) = A(t0 ), A¯ ≥ A everywhere, and defined on a neighborhood of t0 such that A(t 3 1 ′ (91.13) A¯ (t0 ) < A(t0 ) − 2π + D. 4 t0 + 41 Proof. Take a0 as in Lemma 91.11. For t0 > Ta , we begin with the minimizing compressing disk Nt0 ⊂ M+ t0 . If t0 is a surgery + − time and Nt0 intersected the surgery region M+ − (M ∩ M ) then N t0 would have to pass t0 t0 t0 through a 10δ(t0 )-neck, which is impossible by Lemma 91.10. Thus Nt0 avoids any parts added by surgery. For t close to t0 we define an embedded compressing disk St ⊂ M+ t as follows. We take Nt0 + ′ and extend it slightly to a smooth surface Nt0 ⊂ Mt0 which contains Nt0 in its interior. The surface Nt′0 will be unscathed on some open time interval containing t0 . If we let St′ ⊂ M+ t be the surface obtained by moving Nt′0 along static curves then for some b > 0, the surface


185

ˆ t transversely for all t ∈ (t0 − b, t0 + b). Putting St′ will intersect ∂ W ˆt St = St′ ∩ W

(91.14)

ˆ t. defines a compressing disk for Zˆt ⊂ W Define A¯ : (t0 − b, t0 + b) → R by

¯ = area(St ). A(t)

(91.15)

¯ 0 ) = A(t0 ) and A¯ ≥ A. Clearly A(t

For the rest of the calculation, we will view St as a surface sitting in a fixed manifold M ˆ t ) with a varying metric g(·), and put S = St0 = Nt0 . (a fattening of W

By the first variation formula for area, Z Z d ′ ¯ (91.16) A (t0 ) = hX, ν∂S i ds + dvolS , ∂S S dt t=t0 where X denotes the variation vector field for Zˆt , viewed as a surface moving in M, and ν∂S is the outward normal vector along ∂S. By Proposition 90.1, there is an estimate −1 |X| ≤ α(t0 )t0 2 , where α(t0 ) → 0 as t0 → ∞. Therefore Z −1 hX, ν∂St0 i ds ≤ α(t0 ) t0 2 length(∂Nt0 ). (91.17) ∂St0 By Lemma 91.11, the right-hand side of (91.17) is bounded above by

D 2

if t0 is large.

We turn to the second term in (91.16). Pick p ∈ S and let e1 , e2 , e3 be an orthonormal basis for Tp M with e1 and e2 tangent to S. Then 2 d 1 X dg 1 dvolS = (91.18) (ei , ei ) = − Ric(e1 , e1 ) − Ric(e2 , e2 ). dvolS dt t=t0 2 i=1 dt t=t0

Now

− Ric(e1 , e1 ) − Ric(e2 , e2 )) = −R + Ric(e3 , e3 ) = −R + K(e3 , e1 ) + K(e3 , e2 ) R R = − − K(e1 , e2 ) = − − KS + GKS , 2 2 where KS denotes the Gauss curvature of S and GKS denotes the product of the principal curvatures. Applying the Gauss-Bonnet formula Z Z (91.20) κ∂S ds = 2π − KS volS , (91.19)

∂S

S

the fact that GKS ≤ 0 (since S is time-t0 minimal) and the inequality 1 3 (91.21) Rmin (t) ≥ − 2 t + 41

from Lemma 79.11, we obtain Z Z Z d 3 1 (91.22) dvolS ≤ dvolS + κ∂S ds − 2π. t0 + 41 S dt t=t0 S 4 ∂S

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By Lemma 91.11, if a ∈ (0, a0 ), diam(Z) = a and t0 is sufficiently large then Using (91.16), (91.17) and (91.22), if t0 is large then 3 1 ′ ¯ (91.23) A (t0 ) < A(t0 ) − 2π + D. 4 t0 + 41

R

∂S

κ∂S ds ≤

This proves the lemma.

D . 2

Proof of Proposition 91.2. Pick D < 2π. Let a < a0 and Ta′ be as in Lemma 91.12. By Lemma 91.12, A is bounded on compact subsets of [Ta′ , ∞). By Lemma 91.10, for ′ ′ any t ∈ [Ta′ , ∞) we can find a compact set Kt ⊂ M+ t so that for all t ∈ [Ta , ∞) sufficiently ′ close to t, the compressing disk Nt′ lies in Kt . If (Kt , g(t)) and (Kt , g(t )) are eσ -biLipschitz equivalent then A(t) ≤ e2σ area (Nt′ ) = e2σ A(t′ ) and A(t′ ) ≤ e2σ area (Nt ) = e2σ A(t). It follows that A is continuous on [Ta′ , ∞). For t ≥ Ta′ , put

(91.24)

F (t) =

1 t+ 4

− 43

1 1 4 A(t) + 4(2π − D) t + . 4

We claim that F (t) ≤ F (Ta′ ) for all t ≥ Ta′ . Suppose not. Put t0 = inf{t ≥ Ta′ : F (t) > F (Ta′ )}. By continuity, F (t0 ) = F (Ta′ ). Consider the function A¯ of Lemma 91.12. Put − 3 1 1 4 1 4 ¯ ¯ A(t) + 4(2π − D) t + . (91.25) F (t) = t + 4 4 Then F¯ (t0 ) = F (t0 ) and in a small interval around t0 , we have F¯ ≥ F . However, (91.13) implies that F¯ ′ (t0 ) < 0. There is some σ > 0 so that for t ∈ (t0 , t0 + σ), we have 1 (91.26) F (t) ≤ F¯ (t) ≤ F¯ (t0 ) + F¯ ′ (t0 ) (t − t0 ) < F¯ (t0 ) = F (t0 ) = F (Ta′ ), 2 which contradicts the definition of t0 . Thus if t ≥ Ta′ then F (t) ≤ F (Ta′ ). This implies that A(t) is negative for large t, which contradicts the fact that an area is nonnegative. We have shown that the homomorphism (91.3) is injective if t is sufficiently large. In view of Lemma 91.4, the same statement holds for all t ≥ Ta . 92. II.7.4. The thin part is a graph manifold This section is concerned with showing that the thin part M − (w, t) is a graph manifold. We refer to Appendix I for the definition of a graph manifold. We remind the reader that this completes the proof the geometrization conjecture. The next two theorems are purely Riemannian. They say that if a 3-manifold is locally volume-collapsed, with sectional curvature bounded below, then it is a graph manifold. They differ slightly in their hypotheses. Theorem 92.1. (cf. Theorem II.7.4) Suppose that (M α , g α ) is a sequence of compact oriented Riemannian 3-manifolds, closed or with convex boundary, and w α → 0. Assume that


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(1) for each point x ∈ M α there exists a radius ρ = ρα (x) not exceeding the diameter of M α such that the ball B(x, ρ) in the metric g α has volume at most w αρ3 and sectional curvatures at least − ρ−2 . (2) each component of the boundary of M α has diameter at most w α , and has a (topologically trivial) collar of length one, where the sectional curvatures are between − 14 − ǫ and − 14 − ǫ. Then for large α, M α is diffeomorphic to a graph manifold.

Remark 92.2. A proof of Theorem 92.1 appears in [64, Section 8]. The proof in [64] is for closed manifolds, but in view of condition (2) the method of proof clearly goes through to manifolds-with-boundary as considered in Theorem 92.1. The statement of the theorem in [52, Theorem II.7.4] also has a condition ρ < 1, which seems to be unnecessary. Theorem 92.3. (cf. Theorem II.7.4) Suppose that (M α , g α ) is a sequence of compact oriented Riemannian 3-manifolds, closed or with convex boundary, and w α → 0. Assume that (1) for each point x ∈ M α there exists a radius ρ = ρα (x) such that the ball B(x, ρ) in the metric g α has volume at most w αρ3 and sectional curvatures at least − ρ−2 . (2) each component of the boundary of M α has diameter at most w α , and has a (topologically trivial) collar of length one, where the sectional curvatures are between − 41 − ǫ and − 14 − ǫ. (3) for every w ′ > 0 and m ∈ {0, 1, . . . , [ǫ−1 ]}, there exist r = r(w ′ ) > 0 and Km = Km (w ′ ) < ∞ such that for sufficiently large α, if r ∈ (0, r] and a ball B(x, r) in the metric g α has volume at least w ′ r 3 and sectional curvatures at least − r −2 then |∇m Rm |(x) ≤ Km r −m−2 . Then for large α, M α is diffeomorphic to a graph manifold.

Remark 92.4. The statement of this theorem in [52, Theorem II.7.4] has the stronger assumption that (3) holds for all m ≥ 0. In the application to the locally collapsing part of the Ricci flow, it is not clear that this stronger condition holds. However, one does get a bound on a large number of derivatives, which is good enough. Remark 92.5. As pointed out in [52, Section 7.4], adding condition (3) simplifies the proof and allows one to avoid both Alexandrov spaces and Perelman’s stability theorem. (A proof of Perelman’s stability theorem appears in [38]). Proofs of Theorem 92.3 are in [9], [40] and [46]. Remark 92.6. Comparing Theorems 92.1 and 92.3, Theorem 92.1 has the extra assumption that ρα (x) does not exceed the diameter of M α . Without this extra assumption, the Alexandrov space arguments could give that for large α, M α is homeomorphic to a nonnegatively curved Alexandrov space [64, Theorem 1.1(2)]. This does not immediately imply that M α is a graph manifold. Remark 92.7. We give some simple examples where the collapsing theorems apply. Let (Σ, ghyp ) be a closed surface with the hyperbolic metric. Let S 1 (µ) be a circle of length µ and consider the Ricci flow on S 1 × Σ with the initial metric S 1 (µ) × (Σ, c0 ghyp ). The Ricci flow solution at time t is S 1 (µ) × (Σ, (c0 + 2t)ghyp ). In the rest of this example we consider the rescaled metric t−1 g(t). Its diameter goes like O(t0 ) and its sectional curvatures go like O(t0 ). If we take ρ to be a small constant then for large t, B(x, t, ρ) is approximately a circle bundle over a ball in a hyperbolic surface of constant sectional curvature − 21 , with

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circle lengths that go like t−1/2 . The sectional curvature on B(x, t, ρ) is bounded below by − ρ−2 , and ρ−3 vol(B(x, t, ρ)) ∼ t−1/2 . Theorems 92.1 and 92.3 both apply.

Next, consider a compact 3-dimensional nilmanifold that evolves under the Ricci flow. Let θ1 , θ2 , θ3 be affine-parallel 1-forms on M which lift to Maurer-Cartan forms on the Heisenberg group, with dθ1 = dθ2 = 0 and dθ3 = θ1 ∧ θ2 . Consider the metric (92.8)

g(t) = α2 (t) θ12 + β 2 (t) θ22 + γ 2 (t) θ32 . 2

Its sectional curvatures are R1212 = − 34 αγ2 β 2 and R1313 = R2323 = 41 is 1 γ2 (92.9) Ric = − α2 θ12 − β 2 θ22 + γ 2 θ32 . 2 2 2 α β The general solution to the Ricci flow equation is of the form (92.10)

γ2 . α2 β 2

The Ricci tensor

α2 (t) = A0 (t + t0 )1/3 ,

β 2 (t) = B0 (t + t0 )1/3 , A0 B0 γ 2 (t) = (t + t0 )−1/3 . 3 In the rest of this example we consider the rescaled metric t−1 g(t). Its diameter goes like t−1/3 , its volume goes like t−4/3 and its sectional curvatures go like t0 . If we take ρ(x) = diam then ρ−3 vol(B(x, ρ)) ∼ t−1/3 , so both Theorem 92.1 and Theorem 92.3 apply. We could also take ρ(x) to be a small constant c > 0, in which case Theorem 92.3 applies. In general, among the eight maximal homogeneous geometries, the rescaled solution for a ^ compact 3-manifold with geometry H 2 × R or SL 2 (R) will collapse to a hyperbolic surface 1 of constant sectional curvature − 2 . The rescaled solution for a Sol geometry will collapse to a circle. The rescaled solution for an R3 or Nil geometry will collapse to a point. We remark that although these homogeneous solutions are collapsing in the sense of Theorem 92.1, there is no contradiction with the no local collapsing result of Theorem 26.2, which only rules out local collapsing on a finite time interval. Returning to our Ricci flow with surgery, recall the statement of Proposition 90.1. If the b i (t) be the collection {H1 , . . . , Hk } of Proposition 90.1 is nonempty then for large t, let H result of removing from Hi the horoballs whose boundaries are at distance approximately 1 b i (t) = Hi .) Put α(t) from the basepoint xi . (If there are no such horoballs then H 2 (92.11)

b b Mthin (t) = M+ t − f (t)(H1 (t) ∪ . . . Hk (t)).

Proposition 92.12. For large t, Mthin (t) is a graph manifold.

Proof. We give two closely related proofs, one using Theorem 92.3 and one using Theorem 92.1. If the proposition is not true then there is a sequence tα → ∞ so that for each α, Mthin (tα ) is not a graph manifold. Let M α be the manifold obtained from Mthin (tα ) by throwing away connected components which are closed and admit metrics of nonnegative sectional curvature, and put g α = (tα )−1 g(tα ). Since any closed manifold of nonnegative


189

sectional curvature is a graph manifold by [35], for each α the manifold M α is not a graph manifold. We first show that the assumptions of Theorem 92.3 are verified. Lemma 92.13. Condition (3) in Theorem 92.3 holds for the M α ’s. ′ Proof. With w ′ being a parameter as in Condition (3) in Theorem 92.3, let r = r(w√ ) be the parameter of Corollary 81.3. It is enough to show that for large α, if r ∈ (0, r tα ], α α ′ 3 xα ∈ M+ tα , and B(x , t , r) has volume at least w r and sectional curvatures bounded below by − r −2 , then |∇m Rm |(xα , tα ) ≤ Km r −m−2 for an appropriate choice of constants Km .

To prove this by contradiction, we assume that after passing to a subsequence if necessary, √ α α α ′ α 3 α α α there are r ∈ (0, r t ] and x ∈ M+ tα such that B(x , t , r ) has volume at least w (r ) α −2 α m+2 m α α and sectional curvature at least −(r ) , but limα→∞ (r ) |∇ Rm |(x , t ) = ∞ for −1 some m ≤ [ǫ ].

In the notation of Corollary 81.3, if r α ≥ θ−1 (w ′ )hmax (tα ) for infinitely many α then for these α, Corollary 81.3 gives a curvature bound on an unscathed parabolic neighborhood P (xα , tα , r α /4, −τ (r α )2 ) and hence, by Appendix D, derivative bounds at (xα , tα ). This is a contradiction. Therefore we may assume that r α < θ−1 (w ′ )hmax (tα ) ≤ r(tα ) for all α, where we used Remark 86.4 for the last inequality.

Suppose first that R(xα , tα ) ≤ (r(tα ))−2 . By Lemma 70.1, there is an estimate R ≤ 1 −1 α −2 α α 1 −1 α α 2 16 (r(t )) on the parabolic neighborhood P x , t , 4 η r(t ), − 16 η (r(t )) . A surgery in this neighborhood could only occur where R ≥ hmax (tα )−2 . For large α, hmax (tα )−2 >> r(tα )−2 by Remark 86.4. Hence this neighborhood is unscathed. Appendix D now gives bounds of the form |∇m Rm |(xα , tα ) ≤ const. (r(tα ))−m−2 ≤ const. (r α )−m−2 , which is a contradiction.. Suppose now that R(xα , tα ) > (r(tα ))−2 . Then (xα , tα ) is in the center of a canonical m+2 neighborhood and there are universal estimates |∇m Rm |(xα , tα ) ≤ const.(m) R(xα , tα ) 2 for all m ≤ [ǫ−1 ]. Hence in this case, it suffices to show that R(xα , tα ) is bounded above by a constant times (r α )−2 , i.e. it suffices to get a contradiction just to the assumption that limα→∞ (r α )2 R(xα , tα ) = ∞. So suppose that limα→∞ (r α )2 R(xα , tα ) = ∞. We claim that limα→∞ (r α )2 inf R = B(xα ,tα ,r α )

∞. Suppose not. Then there is some C ∈ (0, ∞) so that after passing to a subsequence, there are points xα′ ∈ B (xα , tα , r α ) with (r α )2 R(xα′ , tα ) < C. Considering points along the time-tα geodesic segment from xα′ to xα , for large α we can find points xα′′ ∈ B (xα , tα , r α ) with (r α )2 R(xα′′ , tα ) = 2C. Applying Lemma 70.2 at (xα′′ , tα ), or more precisely a version that applies along geodesics as in Claim 2 of II.4.2, we obtain a contradiction to the assumption that limα→∞ (r α )2 R(xα , tα ) = ∞. In applying Lemma 70.2 we use that r α < r(tα ) and Φ(2C(r α )−2 ) limα→∞ r(tα ) = 0, giving limα→∞ (r α )−2 tα = ∞, in order to say that limα→∞ 2C(rα )−2 = 0. Hence for large α, every point x ∈ B (xα , tα , r α ) is in the center of a canonical neighbor1 hood of size comparable to R(x, tα )− 2 , which is small compared to r α . On the other hand,

190


from Lemma 83.1, there is a ball B ′ of radius θ0 (w ′) r α in B (xα , tα , r α ) so that every subball of B ′ has almost-Euclidean volume. This is a contradiction. This proves the lemma.

We continue with the proof of Proposition 92.12. By construction there is a sequence 1 w α → 0 so that conditions (1) and (2) of Theorem 92.3 hold with ρα (xα ) = (tα )− 2 ρ(xα , tα ). Hence for large α, M α is diffeomorphic to a graph manifold. This is a contradiction to the choice of the M α ’s and proves the theorem. We now give a proof that instead uses Theorem 92.1. Let dα denote the diameter of 1 M α . If we take ρα (xα ) = (tα )− 2 ρ(xα , tα ) then we can apply Theorem 92.1 as long as that the diameter statement in condition (1) of Theorem 92.1 is satisfied. If it is not satisfied then there is some point xα ∈ M α with ρα (xα ) > dα . The sectional curvatures of M α are bounded below by − ρα (x1α )2 , and so are bounded below by − (dα1 )2 . If there is a subsequence with

ρα (xα ) dα

(92.14)

≤ C < ∞ then

vol(M α ) = vol(B(xα , ρα (xα ))) ≤ w α ρα (xα )3 ≤ w α C 3 (dα )3

and we can apply Theorem 92.1 with ρα = dα , after redefining w α . Thus we may assume α) α α → 0 then we can apply that limα→∞ ρ d(xα ) = ∞. If there is a subsequence with vol(M α 3 (d ) α

) Theorem 92.1 with ρα = dα . Thus we may assume that vol(M is bounded away from zero. (dα )3 After rescaling the metric to make the diameter one, we are in a noncollapsing situation with the lower sectional curvature bound going to zero. By the argument in the proof of Lemma 92.13 (which used Corollary 81.3) there are uniform L∞ -bounds on Rm(M α ) and its covariant derivatives. After passing to a subsequence, there is a limit (M∞ , g∞ ) in the smooth topology which is diffeomorphic to M α for large α, and carries a metric of nonnegative sectional curvature. As any boundary component of M∞ would have to have a neighborhood of negative sectional curvature (see the definition of Mthin and condition (2) of Theorem 92.1), M∞ is closed. However, by construction M α has no connected components which are closed and admit metrics of nonnegative sectional curvature. This contradiction shows that the diameter statement in condition (1) of Theorem 92.1 is satisfied. The other conditions of Theorem 92.1 are satisfied as before.

b b Thus for large t, M+ t has a decomposition into a piece f (t)(H1 (t) ∪ . . . Hk (t)), whose interior admits a complete finite-volume hyperbolic metric, and the complement, which is a graph manifold. In addition, by Section 91 the cuspidal tori are incompressible in M+ t . By Lemma 73.4, the initial (connected) manifold M0 is diffeomorphic to a connected sum of the connected components of Mt , along with some possible additional connected sums with a finite number of S 1 × S 2 ’s and quotients of the round S 3 . This proves the geometrization conjecture of Appendix I.


191

93. II.8. Alternative proof of cusp incompressibility The goal of this section is to prove Perelman’s Proposition II.8.2, which gives a numerical characterization of the geometric type of a compact 3-manifold. It also contains an independent proof of the incompressibility of the cuspidal ends of the hyperbolic piece in the geometric decomposition. We recall from Section 7 that λ(g) is the first eigenvalue of −4△ + R, and can also be expressed as R (4|∇Φ|2 + R Φ2 ) dV M R (93.1) λ(g) = inf . Φ∈C ∞ (M ) : Φ6=0 Φ2 dV M From Lemma 7.11, if g(·) is a Ricci flow and λ(t) = λ(g(t)) then d 2 λ(t) ≥ λ2 (t). dt 3

(93.2) 2

From Lemma 8.1, λ(t) V (t) 3 is nondecreasing when it is nonpositive. For any metric g, there are inequalities (93.3)

min R ≤ λ(g) ≤

R

R dV , vol(M, g) M

where the first inequality follows directly from (93.1) and the second inequality comes from using 1 as a test function in (93.1). 2

Perelman’s proof of his Proposition II.8.2 uses the functional λ(g)V (g) 3 . The functional 2 2 b = Rmin (t)V (t) 3 Rmin V (g) 3 plays a similar role. For example, from Corollary 87.7, R(t) is nondecreasing when it is nonpositive. We first give a proof of an analog of Proposition 2 2 II.8.2 that uses Rmin V (g) 3 instead of λ(g)V (g) 3 . The technical simplification is that when 2 Rmin V (g) 3 is nonpositive, it is nondecreasing under a surgery, as surgeries are only done in regions of large positive scalar curvature, so Rmin doesn’t change, and a surgery reduces 2 volume. (A possible extinction of a component clearly doesn’t change Rmin V (g) 3 .) We show that a minimal-volume hyperbolic submanifold of M has incompressible tori, which gives a different approach to Section 91. 2

Perelman’s alternative approach to Section 91 uses the functional λ(g)V (g) 3 instead of 2 2 2 Rmin V (g) 3 . Our use of Rmin V (g) 3 and the sigma-invariant σ(M), instead of λ(g)V (g) 3 and λ, is inspired by [4]. 2

We then give the arguments using λ(g)V (g) 3 , thereby proving Perelman’s Proposition 2 II.8.2. The main technical difficulty is to control how λ(g)V (g) 3 changes under a surgery. 93.1. The approach using the σ-invariant. We first give some well-known results about the sigma-invariant. We recall that the sigma-invariant of a closed connected manifold M of dimension n ≥ 3 is given by R R(g) dvol(g) (93.4) σ(M) = sup inf M , n−2 C g∈C vol(M, g) n

192


where C runs over the conformal classes of Riemannian metrics on M. From the solution to the Yamabe problem, the infimum in (93.4) is realized by a metric of constant scalar curvature in the given conformal class. It follows that if σ(M) > 0 then M admits a metric with positive scalar curvature. Conversely, suppose that M admits a metric g0 with positive scalar curvature. Let C be the conformal class containing g0 . Then R 4(n−1) R 2 2 |∇u| + R(g0 ) u dvolM (g0 ) R(g) dvol(g) M n−2 = inf (93.5) inf M n−2 n−2 R u>0 g∈C 2n n vol(M, g) n n−2 dvolM (g0 ) u M is positive, in view of the Sobolev embedding theorem, and so σ(M) > 0. We claim that if σ(M) ≤ 0 then (93.6)

2

σ(M) = sup Rmin (g) V (g) n . g

To see this, as the infimum in (93.4) is realized by a metric of constant scalar curvature 2 in the given conformal class, it follows that σ(M) ≤ supg Rmin (g) V (g) n . Now given a Riemannian metric g, the infimum in (93.4) within the corresponding conformal class C 4 2 e V (e e Then equals R g ) n for a metric e g = u n−2 g with constant scalar curvature R. R 4(n−1) |∇u|2 + R u2 dvolM n−2 M 2 e n (93.7) R V (e g ) = inf . R n−2 u>0 2n n u n−2 dvolM M As

(93.8)

R 2 2 |∇u| + R u dvolM u2 dvolM n−2 M ≥ R (g) min R R n−2 n−2 2n 2n n n n−2 dvol n−2 dvol u u M M M M

R 4(n−1) M

2 2 e V (e and Rmin (g) ≤ 0, Holder’s inequality implies that R g ) n ≥ Rmin (g) V (g) n . It follows 2 that σ(M) ≥ supg Rmin (g) V (g) n .

The next proposition answers conjectures of Anderson [2].

Proposition 93.9. Let M be a closed connected oriented 3-manifold. (a) If σ(M) > 0 then M is diffeomorphic to a connected sum of a finite number of S 1 × S 2 ’s and metric quotients of the round S 3 . Conversely, each such manifold has σ(M) > 0. (b) M is a graph manifold if and only if σ(M) ≥ 0. 3 (c) If σ(M) < 0 then − 32 σ(M) 2 is the minimum of the numbers V with the following property : M can be decomposed as a connected sum of a finite collection of S 1 ×S 2 ’s, metric quotients of the round S 3 and some other components, the union of which is denoted by M ′ , and there exists a (possibly disconnected) complete finite-volume manifold N with constant sectional curvature − 41 and volume V which can be embedded in M ′ so that the complement M ′ − N (if nonempty) is a graph manifold. 3 Moreover, if vol(N) = − 23 σ(M) 2 then the cusps of N (if any) are incompressible in M ′.


193

Proof. If σ(M) > 0 then M has a metric g of positive scalar curvature. From Lemmas 81.1 and 81.2, M is a connected sum of S 1 × S 2 ’s and metric quotients of the round S 3 . Conversely, if M is a connected sum of S 1 × S 2 ’s and metric quotients of the round S 3 then M admits a metric g of positive scalar curvature and so σ(M) > 0. Now suppose that σ(M) ≤ 0. If M is a graph manifold then M volume-collapses with bounded curvature, so (93.6) implies that σ(M) = 0. Suppose that M is not a graph manifold. Suppose that we have a given decomposition of M as a connected sum of a finite collection of S 1 ×S 2 ’s, metric quotients of the round S 3 and some other components, the union of which is denoted by M ′ , and there exists a (possibly disconnected) finite-volume complete manifold N with constant sectional curvature − 14 which can be embedded in M ′ so that the complement (if nonempty) is a graph manifold. Let Vhyp denote the hyperbolic volume of N. We do not assume that the cusps of N are incompressible in M ′ . For any ǫ > 0, we claim that there is a metric gǫ on M with R ≥ − 6 · 41 − ǫ and volume V (gǫ ) ≤ Vhyp + ǫ. This comes from collapsing the graph manifold pieces, along with the fact that the connected sum operation can be performed while decreasing the scalar curvature arbitrarily little and increasing the volume arbitrarily 2 2/3 2/3 little. Then Rmin (gǫ ) V (gǫ ) 3 ≥ − 23 Vhyp − const. ǫ. Thus σ(M) ≥ − 32 Vhyp . Let Vb denote the minimum of Vhyp over all such decompositions of M. (As the set of volumes of complete finite-volume 3-manifolds with constant curvature − 41 is well-ordered, there is a minimum.) Then σ(M) ≥ − 23 Vb 2/3 .

Next, take an arbitrary metric g0 on M and consider the Ricci flow g(t) with initial metric g0 . From Sections 90 and 92, there is a nonempty manifold N with a complete finite+ volume metric of constant curvature − 14 so that for large t, there is a decomposition M t = 1 M1 (t)∪M2 (t) of the time-t manifold, where M1 (t) is a graph manifold and (M2 (t), t g(t) M2 (t) ) is close to a large piece of N. In terms of condition (c) of Proposition 93.9, we will think of 3 M ′ as being M+ t . Because of the presence of N, we know that t Rmin (t) ≤ − 2 + ǫ(t) and V (t) ≥ t2/3 Vhyp (N) − ǫ(t) for a function ǫ(t) with limt→∞ ǫ(t) = 0. The monotonicity of Rmin (t) V 2/3 (t), even through surgeries, implies that 3 3 (93.10) Rmin (g0 ) V 2/3 (g0 ) ≤ − Vhyp (N)2/3 ≤ − Vb 2/3 . 2 2 Thus σ(M) ≤ − 3 Vb 2/3 . 2

This shows that σ(M) = − 23 Vb 2/3 . Now take a decomposition of M as in condition (c) of Proposition 93.9, with Vhyp (N) = Vb . We claim that the cuspidal 2-tori of N are incompressible in M ′ . If not then there would be a metric g on M with R(g) ≥ − 32 and vol(g) < Vhyp (N) [3, Pf. of Theorem 2.9]. This would contradict the fact that σ(M) = − 23 Vhyp (N)2/3 . 93.2. The approach using the λ-invariant.

Proposition 93.11. (cf. II.8.2) Let M be a closed connected oriented 3-manifold. (a) If M admits a metric g with λ(g) > 0 then it is diffeomorphic to a connected sum of a finite number of S 1 × S 2 ’s and metric quotients of the round S 3 . Conversely, each such

194


manifold admits a metric g with λ(g) > 0. (b) Suppose that M does not admit any metric g with λ(g) > 0. Let λ denote the supremum 2 of λ(g)V (g) 3 over all metrics g on M. Then M is a graph manifold if and only if λ = 0. 3 (c) Suppose that M does not admit any metric g with λ(g) > 0, and λ < 0. Then − 32 λ 2 is the minimum of the numbers V with the following property : M can be decomposed as a connected sum of a finite collection of S 1 × S 2 ’s, metric quotients of the round S 3 and some other components, the union of which is denoted by M ′ , and there exists a (possibly disconnected) complete manifold N with constant sectional curvature − 14 and volume V which can be embedded in M ′ so that the complement M ′ − N (if nonempty) is a graph manifold. 3 Moreover, if vol(N) = − 32 λ 2 then the cusps of N (if any) are incompressible in M ′ . Proof. We first give the argument for Proposition 93.11 under the pretense that all Ricci flows are smooth, except for possible extinction of components. (Of course this is not the case, but it will allow us to present the main idea of the proof.) If λ(g) > 0 for some metric g then from (93.2), the Ricci flow starting from g will become 3 extinct within time 2λ(g) . Hence Lemma 81.2 applies. Conversely, if M is a connected sum 1 2 of S × S ’s and metric quotients of the round S 3 then M admits a metric g of positive scalar curvature. From (93.3), λ(g) > 0. Now suppose that M does not admit any metric g with λ(g) > 0. If M is a graph manifold then M volume-collapses with bounded curvature, so (93.3) implies that λ = 0. Suppose that M is not a graph manifold. Suppose that we have a given decomposition of M as a connected sum of a finite collection of S 1 × S 2 ’s, metric quotients of the round S 3 and some other components, the union of which is denoted by M ′ , and there exists a (possibly disconnected) complete manifold N with constant sectional curvature − 41 which can be embedded in M ′ so that the complement (if nonempty) is a graph manifold. Let Vhyp denote the hyperbolic volume of N. We do not assume that the cusps of N are incompressible in M ′ . For any ǫ > 0, we claim that there is a metric gǫ on M with R ≥ − 6 · 14 − ǫ and volume V (gǫ ) ≤ Vhyp + ǫ. This comes from collapsing the graph manifold pieces, along with the fact that the connected sum operation can be performed while decreasing the scalar curvature arbitrarily little and increasing the volume arbitrarily little. Then (93.3) implies 2 2/3 2/3 that λ(gǫ ) V (gǫ ) 3 ≥ − 23 Vhyp − const. ǫ. Thus λ ≥ − 23 Vhyp .

Let Vb denote the minimum of Vhyp over all such decompositions of M. (As the set of volumes of complete finite-volume 3-manifolds with constant curvature − 14 is well-ordered, there is a minimum.) Then λ ≥ − 32 Vb 2/3 .

Next, take an arbitrary metric g0 on M and consider the Ricci flow g(t) with initial metric g0 . From Sections 90 and 92, there is a nonempty manifold N with a finite-volume complete metric of constant curvature − 14 so that for large t, there is a decomposition of the time-t + 1 manifold Mt = M1 (t) ∪ M2 (t) where M1 (t) is a graph manifold and (M2 (t), t g(t) M2 (t) ) is


195

close to a large piece of N. As N has finite volume, a constant function on N is squareintegrable and so inf spec(−△N ) = 0. Equivalently, R |∇f |2 dvolN N R (93.12) inf = 0. f ∈Cc∞ (N ),f 6=0 f 2 dvolN N

Taking an appropriate test function Φ on M with compact support in M2 (t) gives t λ(t) ≤ − 32 +ǫ1 (t), with limt→∞ ǫ1 (t) = 0. In terms of condition (c) of Proposition 93.11, we will think 2/3 of M ′ as being M+ Vhyp (N) − ǫ2 (t), with t . From the presence of N, we know that V (t) ≥ t limt→∞ ǫ2 (t) = 0. As we are assuming that the Ricci flow is nonsingular, the monotonicity of λ(t) V 2/3 (t) implies that (93.13)

λ(g0 ) V 2/3 (g0 ) ≤ −

Thus λ ≤ − 32 Vb 2/3 .

3 3 Vhyp (N)2/3 ≤ − Vb 2/3 . 2 2

This shows that λ = − 23 Vb 2/3 . Now take a decomposition of M as in condition (c) of Proposition 93.11, with Vhyp (N) = Vb . We claim that the cuspidal 2-tori of N are incompressible in M ′ . If not then there would be a metric g on M with R(g) ≥ − 23 and vol(g) < Vhyp (N) [3, Pf. of Theorem 2.9]. Using (93.3), one would obtain a contradiction to the fact that λ = − 32 Vhyp (N)2/3 . 2

To handle the behaviour of λ(t) V 3 (t) under Ricci flows with surgery, we first state a couple of general facts about Schrödinger operators. Lemma 93.14. Given a closed Riemannian manifold M, let X be a codimension-0 submanifoldwith-boundary of M. Given R ∈ C ∞ (M), let λM be the lowest eigenvalue of −4△ + R on M, with corresponding eigenfunction ψ. Let λX be the lowest eigenvalue of the corresponding operator on X, with Dirichlet boundary conditions, and similarly for λM −int(X) . Then for all η ∈ Cc∞ (int(X)), we have (93.15)

λM ≤ min(λX , λM −int(X) )

and (93.16)

λX ≤ λM + 4

R

|∇η|2 ψ 2 dV . η 2 ψ 2 dV M

M R

Proof. Equation (93.15) follows from Dirichlet-Neumann bracketing [55, Chapter XIII.15]. To prove (93.16), ηψ is supported in int(X) and so R (4|∇(ηψ)|2 + R η 2 ψ 2 ) dV R λX ≤ M (93.17) η 2 ψ 2 dV M R (4|∇η|2 ψ 2 + 8h∇η, ∇ψiηψ + 4η 2 |∇ψ|2 + R η 2 ψ 2 ) dV R . = M η 2 ψ 2 dV M As −4△ψ + Rψ = λM ψ, we have

196

(93.18)


λM

Z

2

Z

2

Z

2

η ψ dV = − 4 η ψ△ψ dV + Rη 2 ψ 2 dV M M Z M Z 2 = 4 h∇(η ψ), ∇ψi dV + Rη 2 ψ 2 dV ZM Z M Z 2 2 = 4 η |∇ψ| dV + 8 h∇η, ∇ψiηψ dV + Rη 2 ψ 2 dV. M

M

M

Equation (93.16) follows.

The next result is an Agmon-type estimate. Lemma 93.19. With the notation of Lemma 93.14, given a nonnegative function φ ∈ C ∞ (M), suppose that f ∈ C ∞ (M) satisfies 4|∇f |2 ≤ R − λM − c

(93.20)

on supp(φ), for some c > 0. Then

kef φψk2 ≤ 4 c−1 kef △φk∞ + kef ∇φk∞ (λM − min R)1/2 kψk2 .

(93.21)

Proof. Put H = − 4△ + R. By assumption,

φ(R − 4|∇f |2 − λM ) φ ≥ c φ2

(93.22)

and so there is an inequality of operators on L2 (M) : (93.23)

φ(H − 4|∇f |2 − λM ) φ = 4φd∗ dφ + φ(R − 4|∇f |2 − λM ) φ ≥ c φ2 .

In particular,

Z

(93.24)

f

2

f

e ψφ (H − 4|∇f | − λM ) φe ψdV ≥ c

M

Z

φ2 e2f ψ 2 dV.

M

For ρ ∈ C ∞ (M), (93.25) and so

ef H(e−f ρ) = Hρ + 4 ∇ · ((∇f )ρ) + 4 h∇f, ∇ρi − 4 |∇f |2ρ Z

(93.26)

f

−f

ρe H(e ρ) dV =

M

f

Z

M

ρ(H − 4 |∇f |2 )ρ dV.

Taking ρ = e φψ gives Z Z 2f (93.27) e φψH(φψ) dV = ef φψ(H − 4 |∇f |2)ef φψ dV. M

M

From (93.24) and (93.27), (93.28) Z Z f 2 2f cke φψk2 ≤ e φψ(H − λM )(φψ) dV = e2f φψ[H, φ]ψ dV = hef φψ, ef [H, φ]ψi2 . M

M

Thus

(93.29)

ckef φψk2 ≤ kef [H, φ]ψk2 .


197

Now (93.30) Then (93.31) Finally,

ef [H, φ]ψ = − 4 ef (△φ)ψ − 8 ef h∇φ, ∇ψi. kef [H, φ]ψk2 ≤ 4 kef △φk∞ kψk2 + 8 kef ∇φk∞ k∇ψk2 .

(93.32) and so (93.33) This proves the lemma.

4

k∇ψk22

=

Z

M

(λM − R) ψ 2 dV

2k∇ψk2 ≤ (λM − min R)1/2 kψk2 .

Clearly Lemma 93.19 is also true if f is just assumed to be Lipschitz-regular. We now apply Lemmas 93.14 and 93.19 to a Ricci flow with surgery. A singularity caused by extinction of a component will not be a problem, so let T0 be a surgery time + and let M+ = M+ T0 be the postsurgery manifold. We will write λ instead of λM+ . Let + − + − Mcap = M+ T0 − (MT0 ∩ MT0 ) be the added caps and put X = M+ − Mcap = MT0 ∩ MT0 . For simplicity, let us assume that Mcap has a single component; the argument in the general case is similar. From the nature of the surgery procedure, the surgery is done in an ǫ-horn extending from Ωρ , where ρ = δ(T0 )r(T0 ). In fact, because of the canonical neighborhood assumption, we can extend the ǫ-horn inward until R ∼ r(T0 )−2 . Appplying (93.1) with a test function supported in an ǫ-tube near this inner boundary, it follows that λ+ ≤ c′ r(T0 )−2 for some universal constant c′ >> 1. In what follows we take δ(T0 ) to be small. As R is much greater than r(T0 )−2 on Mcap , it follows that λM −int(X) is much greater than r(T0 )−2 . Then from (93.15), λ+ ≤ λX . We can apply (93.16) to get an inequality the other way. We take the function η to interpolate from being 1 outside of the h(T0 )-neighborhood Nh Mcap of Mcap , to being 0 on Mcap . In terms of the normalized eigenfunction ψ on M+ , this gives a bound of the form R ψ 2 dV Nh Mcap R . (93.34) λX ≤ λ+ + const. h(T0 )−2 1 − Nh Mcap ψ 2 dV R We now wish to show that Nh Mcap ψ 2 dV is small. For this we apply Lemma 93.19 with c = c′ r(T0 )−2 . Take an ǫ-tube U, in the ǫ-horn, whose center has scalar curvature roughly 200 c′ r(T0 )−2 and which is the closest tube to the cap with this property. Let x : U → (− ǫ−1 , ǫ−1 ) be the longitudinal parametrization of the tube, which we take to be increasing in the direction of the surgery cap. Let Φ : (−1, 1) → [0, 1] be a fixed nondecreasing smooth function which is zero on (−1, 1/4) and one on (1/2, 1). Put φ = Φ◦x on U. Extend φ to M+ by making it zero to the left of U and one to the right of U, where “right of U” means the connected component of M+ − U containing the surgery cap. Dimensionally, |∇φ|∞ ≤ const. r(T0 )−1 and |△φ|∞ ≤ const. r(T0 )−2 . Define a function f to the right of x−1 (0) by setting it to be the distance from x−1 (0) with respect to the metric 41 (R − λ+ − c) gM+ . (Note that to the right of x−1 (0), we have R ≥ 200c′r(T0 )−2 ≥

198


λ+ + c.) Then equation (93.21) gives |ef φψ|2 ≤ const.(T0 ). The point is that const.(T0 ) is independent of the (small) surgery parameter δ(T0 ). Hence (93.35)

Z

Nh Mcap

ψ 2 dV ≤

sup e−2f Nh Mcap

!Z

Nh Mcap

e2f ψ 2 dV ≤ const. sup e−2f . Nh Mcap

To estimate supNh Mcap e−2f , we use the fact that the ǫ-horn consists of a sequence of ǫ-tubes stacked together. In the region of M+ from x−1 (0) to the surgery cap, the scalar curvature ranges from roughly 200 c′ r(T0 )−2 to h(T0 )−2 . On a given ǫ-tube, if ǫ is sufficiently small then the ratio of the scalar curvatures between the two ends is bounded by e. Hence in going from x−1 (0) to the surgery cap, one must cross at least N disjoint ǫ-tubes, with 1 2 −2 eN = 200c . Traversing a given ǫ-tube (say of radius r ′ ) in going towards the ′ r(T0 ) h(T0 ) R ǫ−1 r′ surgery cap, f increases by roughly const. −ǫ−1 r′ (r ′ )−1 ds, which is const. ǫ−1 . Hence near the surgery cap, we have (93.36)

sup e−2f ≤ const. e− const. N ǫ

Nh Mcap

−1

−1

= const. (r(T0 )2 h(T0 )−2 )− const. ǫ .

Combining this with (93.34) and (93.35), we obtain (93.37)

−1

λX ≤ λ+ + const. h(T0 )−2 (r(T0 )2 h(T0 )−2 )− const. ǫ .

By making a single redefinition of ǫ, we can ensure that λX ≤ λ+ + const. h(T0 )4 . The last constant will depend on r(T0 ) but is independent of δ(T0 ). Thus if δ(T0 ) is small enough, we can ensure that |λX − λ+ | is small in comparison to the volume change V − (T0 ) − V + (T0 ), which is comparable to h(T0 )3 . If λ− is the smallest eigenvalue of − 4 △ + R on the presurgery manifold Mt , for t slightly less than T0 , then we can estimate |λX − λ− | in a similar way. Hence for an arbitrary positive continuous function ξ(t), we can make the parameters δ j of Proposition 77.2 small enough to ensure that (93.38)

|λ+ (T0 ) − λ− (T0 )| ≤ ξ(T0 ) (V − (T0 ) − V + (T0 ))

for a surgery at time T0 . We now redo the argument for the proposition, as given above in the surgery-free case, in the presence of surgeries. Suppose first that λ(g0 ) > 0 for some metric g0 on M. After possible rescaling, we can assume that g0 is the initial condition for a Ricci flow with surgery (M, g(·)), with normalized initial condition. Using the lower scalar curvature bound of 3 Lemma 79.11 and the Ricci flow equation, the volume on the time interval [0, λ(0) ] has an a priori upper bound of the form V (0). As a surgery at time T0 removes a volume P const. 3 3 comparable to h(T0 ) , we have h(T0 ) ≤ const. V (0), where the sum is over the surgeries and T0 denotes the surgery time. FromPthe above discussion, the change in λ due to the surgeries is bounded below by − const. T0 h(T0 )4 . Then the decrease in λ due to surgeries

NOTES ON PERELMAN’S PAPERS 3 on the time interval [0, λ(0) ] is bounded above by

(93.39)

const.

X

3 ] T0 ∈[0, λ(0)



199



h(T0 )4 ≤ const.  sup h(t) V (0). 3 t∈[0, λ(0) ]

By choosing the function δ(t) to be sufficiently small, the decrease in λ due to surgeries is 3 ] coming from the increase not enough to prevent the blowup of λ on the time interval [0, λ(0) of λ between the surgeries. Hence the solution goes extinct. Now suppose that M does not admit a metric g with λ(g) > 0. Again, if M is a graph manifold then λ = 0. Suppose that M is not a graph manifold. As before, λ ≥ − 32 Vb 2/3 . Given an initial metric g0 , we wish to show that by choosing the function δ(t) small enough we can make the function λ(t)V 2/3 (t) arbitrarily close to being nondecreasing. To see this, we consider the effect of a surgery on λ(t)V 2/3 (t). Upon performing a surgery the volume decreases, which in itself cannot decrease λ(t)V 2/3 (t). (We are using the fact that λ(t) is nonpositive.) Then from the above discussion, the change in λ(t)V 2/3 (t) due from a surgery at time T0 , is bounded below by −ξ(T0 ) (V − (T0 ) −V + (T0 )) V − (T0 )2/3 . With normalized initial conditions, we have an a priori upper bound on V (t) in terms of V (0) and t. Over any time interval [T1 , T2 ], we must have X V − (T0 ) − V + (T0 ) ≤ sup V (t), (93.40) T0 ∈[T1 ,T2 ]

t∈[T1 ,T2 ]

where the sum is over the surgeries in the interval [T1 , T2 ]. Then along with the monotonicity of λ(t)V 2/3 (t) in between the surgery times, by choosing the function δ(t) appropriately we can ensure that for any σ > 0 there is a Ricci flow with (r, δ)-cutoff starting from g0 so that λ(g0 ) V 2/3 (g0 ) ≤ λ(t) V 2/3 (t) + σ for all t. It follows that λ ≤ − 32 Vb 2/3 .

This shows that λ = − 23 Vb 2/3 . The same argument as before shows that if we have a decomposition with the hyperbolic volume of N equal to Vb then the cusps of N (if any) are incompressible in M ′ .

Remark 93.41. It follows that if the three-manifold M does not admit a metric of positive scalar curvature then σ(M) = λ. In fact, this is true in any dimension n ≥ 3 [1]. Appendix A. Maximum principles

In this appendix we list some maximum principles and their consequences. Our main source is [23], where references to the original literature can be found. The first type of maximum principle is a weak maximum principle which says that under certain conditions, a spatial inequality on the initial condition implies a time-dependent inequality at later times. Theorem A.1. Let M be a closed manifold. Let {g(t)}t∈[0,T ] be a smooth one-parameter family of Riemannian metrics on M and let {X(t)}t∈[0,T ] be a smooth one-parameter family

200


of vector fields on M. Let F : R × [0, T ] → R be a Lipschitz function. Suppose that u = u(x, t) is C 2 -regular in x, C 1 -regular in t and ∂u ≤ △g(t) u + X(t)u + F (u, t). ∂t

(A.2)

Let φ : [0, T ] → R be the solution of u(·, t) ≤ φ(t) for all t ∈ [0, T ].

dφ dt

= F (φ(t), t) with φ(0) = α. If u(·, 0) ≤ α then

There are various noncompact versions of the weak maximum principle. We state one here. Theorem A.3. Let (M, g(·)) be a complete Ricci flow solution on the interval [0, T ] with 1,2 uniformly bounded curvature. If u = u(x, t) is a Wloc function that weakly satisfies ∂u ≤ ∂t △g(t) u, with u(·, 0) ≤ 0 and Z TZ 2 (A.4) e− c dt (x,x0 ) u2 (x, s) dV (x) ds < ∞ 0

M

for some c > 0, then u(·, t) ≤ 0 for all t ∈ [0, T ].

A strong maximum principle says that under certain conditions, a strict inequality at a given time implies strict inequality at later times and also slightly earlier times. It does not require complete metrics. Theorem A.5. Let M be a connected manifold. Let {g(t)}t∈[0,T ] be a smooth one-parameter family of Riemannian metrics on M and let {X(t)}t∈[0,T ] be a smooth one-parameter family of vector fields on M. Let F : R × [0, T ] → R be a Lipschitz function. Suppose that u = u(x, t) is C 2 -regular in x, C 1 -regular in t and (A.6)

∂u ≤ △g(t) u + X(t)u + F (u, t). ∂t

= F (φ(t), t). If u(·, t) ≤ φ(t) for all t ∈ [0, T ] Let φ : [0, T ] → R be a solution of dφ dt and u(x0 , t0 ) < φ(t) for some x0 ∈ M and t0 ∈ (0, T ] then there is some ǫ > 0 so that u(·, t) < φ(t) for t ∈ (t0 − ǫ, T ]. A consequence of the strong maximum principle is a statement about restricted holonomy for Ricci flow solutions with nonnegative curvature operator Rm. Theorem A.7. Let M be a connected manifold. Let {g(t)}t∈[0,T ] be a smooth one-parameter family of Riemannian metrics on M with nonnegative curvature operator that satisfy the Ricci flow equation. Then for each t ∈ (0, T ], the image Im(Rmg(t) ) of the curvature operator is a smooth subbundle of Λ2 (T ∗ M) which is invariant under spatial parallel translation. There is a sequence of times 0 = t0 < t1 < . . . < tk = T such that for each 1 ≤ i ≤ k, Im(Rmg(t) ) is a Lie subalgebra of Λ2 (Tm∗ M) ∼ = o(n) that is independent of t for t ∈ (ti−1 , ti ]. Furthermore, Im(Rmg(ti ) ) ⊂ Im(Rmg(ti+1 ) ). In particular, under the hypotheses of Theorem A.7, a local isometric splitting at a given time implies a local isometric splitting at earlier times.


201

Appendix B. φ-almost nonnegative curvature In three dimensions, the Ricci flow equation implies that dR 2 R = △R + R2 + 2 |Rij − gij |2 . dt 3 3

(B.1)

The maximum principle of Appendix A implies that if (M, g(·)) is a Ricci flow solution defined for t ∈ [0, T ), with complete time slices and bounded curvature on compact time intervals, then (inf R)(t) ≥

(B.2) In particular, tR(·, t) > −

3 2

(inf R)(0) . 1 − 32 t(inf R)(0)

for all t ≥ 0 (compare [34, Section 2]).

Recall that the curvature operator is an operator on 2-forms. We follow the usual Ricci flow convention that if a manifold has constant sectional curvature k then its curvature operator is multiplication by 2k. In general, the trace of the curvature operator equals the scalar curvature. In three dimensions, having nonnegative curvature operator is equivalent to having nonnegative sectional curvature. Each eigenvalue of the curvature operator is twice a sectional curvature. Hamilton-Ivey pinching, as given in [34, Theorem 4.1], says the following. Assume that at t = 0 the eigenvalues λ1 ≤ λ2 ≤ λ3 of the curvature operator at each point satisfy λ1 ≥ −1. (One can always achieve this by rescaling. Note that it implies R(·, t) ≥ − 23 t+1 1 .) Given a point (x, t), put X = −λ1 . If X > 0 then 4

(B.3)

R(x, t) ≥ X (ln X + ln(1 + t) − 3) ,

or equivalently, (B.4)

1+t )−3 . tR(x, t) ≥ tX ln(tX) + ln( t

Definition B.5. Given t ≥ 0, a Riemannian 3-manifold (M, g) satisfies the time-t HamiltonIvey pinching condition if for every x ∈ M, if λ1 ≤ λ2 ≤ λ3 are the eigenvalues of the curvature operator at x, then either • λ1 ≥ 0, i.e. the curvature is nonnegative at x, or

• If λ1 < 0 and X = −λ1 then tR(x) ≥ tX ln(tX) + ln This condition has the following monotonicity property:

1+t t

−3 .

Lemma B.6. Suppose that Rm and Rm′ are 3-dimensional curvature operators whose scalar curvatures and first eigenvalues satisfy R′ ≥ R ≥ 0 and λ′1 ≥ λ1 . If Rm satisfies the time-t Hamilton-Ivey pinching condition then so does Rm′ .

202


− 3 > 0, since otherwise the Proof. We may assume that λ′1 < 0 and log(tX ′ ) + ln 1+t t ′ condition will be satisfied (because R > 0 by hypothesis). The function 1+t (B.7) Y 7→ tY ln(tY ) + ln − 3 t 1+t − 3 is nonnegative, is monotone increasing on the interval on which ln(tY ) + ln t 1+t 1+t ′ ′ so tX ln(tX) + ln t − 3 ≥ tX ln(tX ) + ln t − 3 . Hence Rm′ satisfies the pinching condition too. The content of the pinching equation is that for any s ∈ R, if tR(·, t) ≤ s then there is a lower bound t Rm(·, t) ≥ const.(s, t). Of course, this is a vacuous statement if s ≤ − 23 .

Using equation (B.4), we can find a positive function Φ ∈ C ∞ (R) such that 1. Φ is nondecreasing. is decreasing. 2. For s > 0, Φ(s) s 3. For large s, Φ(s) ∼ lns s . 4. For all t, (B.8)

Rm(·, t) ≥ − Φ(R(·, t)).

This bound has the most consequence when s is large. We note that for the original unscaled Ricci flow solution, the precise bound that we obtain depends on t0 and the time-zero metric, through its lower curvature bound. Appendix C. Ricci solitons Let {V (t)} be a time-dependent family of vector fields on a manifold M. The solution to the equation dg = LV (t) g dt

(C.1) is (C.2)

g(t) = φ−1 (t)∗ g(t0 )

where {φ(t)} is the 1-parameter group of diffeomorphisms generated by −V , normalized by φ(t0 ) = Id. (If M is noncompact then we assume that V can be integrated. The reason for the funny signs is that if a 1-parameter family η(t) is generated by of diffeomorphisms ∗ dη−1 (t+ǫ)∗ −1 ∗ dη(t+ǫ) = − vector fields W (t) then LW (t) = η (t) η(t)∗ .) dǫ dǫ ǫ=0

ǫ=0

The equation for a steady soliton is

(C.3)

2 Ric + LV g = 0,

where V is a time-independent vector field. The corresponding Ricci flow is given by (C.4)

g(t) = φ−1 (t)∗ g(t0 ),

where {φ(t)} is the 1-parameter group of diffeomorphisms generated by −V . (Of course, in this case {φ−1 (t)} is the 1-parameter group of diffeomorphisms generated by V .)


203

A gradient steady soliton satisfies the equations ∂gij = − 2Rij = 2∇i ∇j f, ∂t ∂f = |∇f |2 . ∂t

(C.5)

It follows from (C.5) that (C.6)

∂ ij g ∂j f = 2 Rij ∇j f + g ij ∇j |∇f |2 = −2 (∇i ∇j f ) ∇j f + ∇i |∇f |2 = 0, ∂t

showing that V = ∇f is indeed constant in t. The solution to (C.5) is (C.7)

g(t) = φ−1 (t)∗ g(t0 ), f (t) = φ−1 (t)∗ f (t0 ).

Conversely, given a metric b g and a function fb satisfying

(C.8)

bij + ∇ b i∇ b j fb = 0, R

b fb. If we define g(t) and f (t) by put V = ∇

(C.9)

then they satisfy (C.5).

g(t) = φ−1 (t)∗ b g, f (t) = φ−1 (t)∗ fb

A solution to (C.5) satisfies (C.10)

∂f = |∇f |2 − △f − R, ∂t

or (C.11)

∂ −f e = − △e−f + R e−f . ∂t

This perhaps motivates Perelman’s use of the backward heat equation (5.23). A shrinking soliton lives on a time interval (−∞, T ). For convenience, we take T = 0. Then the equation is g (C.12) 2 Ric + LV g + = 0. t The vector field V = V (t) satisfies V (t) = − given by (C.13)

1 t

V (−1). The corresponding Ricci flow is

g(t) = − t φ−1 (t)∗ g(−1),

where {φ(t)} is the 1-parameter group of diffeomorphisms generated by −V , normalized by φ(−1) = Id.

204


A gradient shrinking soliton satisfies the equations (C.14)

gij ∂gij = − 2Rij = 2∇i ∇j f + , ∂t t ∂f = |∇f |2 . ∂t

It follows from (C.14) that V = ∇f satisfies V (t) = − 1t V (−1). The solution to (C.14) is (C.15)

g(t) = − t φ−1 (t)∗ g(−1),

f (t) = φ−1 (t)∗ f (−1).


(C.16)

bij + ∇ b i∇ b j fb − 1 gb = 0, R 2

b fb. If we define g(t) and f (t) by put V (t) = − 1t ∇ (C.17)


g(t) = − t φ−1 (t)∗ b g, f (t) = φ−1 (t)∗ fb

An expanding soliton lives on a time interval (T, ∞). For convenience, we take T = 0. Then the equation is g (C.18) 2 Ric + LV g + = 0. t The vector field V = V (t) satisfies V (t) = (C.19)

1 t

V (1). The corresponding Ricci flow is given by

g(t) = t φ−1 (t)∗ g(1),

where {φ(t)} is the 1-parameter group of diffeomorphisms generated by −V , normalized by φ(1) = Id. A gradient expanding soliton satisfies the equations (C.20)

gij ∂gij = − 2Rij = 2∇i ∇j f + , ∂t t ∂f = |∇f |2 . ∂t

It follows from (C.20) that V = ∇f satisfies V (t) = (C.21)

1 t

V (1). The solution to (C.20) is

g(t) = t φ−1 (t)∗ g(1), f (t) = φ−1 (t)∗ f (1).


(C.22)

bij + ∇ b i∇ b j fb + 1 gb = 0, R 2


put V (t) = (C.23)

1 t

205

b fb. If we define g(t) and f (t) by ∇

g(t) = t φ−1 (t)∗ b g, f (t) = φ−1 (t)∗ fb


Obvious examples of solitons are given by Einstein metrics, with V = 0. Any steady or expanding soliton on a closed manifold comes from an Einstein metric. Other examples of solitons (see [22, Chapter 2]) are : 1. (Gradient steady soliton) The cigar soliton on R2 and the Bryant soliton on R3 . 2 . 2. (Gradient shrinking soliton) Flat Rn with f = − |x| 4t 2 3. (Gradient shrinking soliton) The shrinking cylinder R × S n−1 with f = − x4t , where x is the coordinate on R. 4. (Gradient shrinking soliton) The Koiso soliton on CP 2 #CP 2 . Appendix D. Local derivative estimates Theorem D.1. For any α, K, K ′, l ≥ 0 and m, n ∈ Z+ , there is some C = C(α, K, K ′, l, m, n) with the following property. Given r > 0, suppose that g(t) is a Ricci flow solution for 2 t ∈ [0, t], where 0 < t ≤ αr , defined on an open neighborhood U of a point p ∈ M n . K Suppose that B(p, r, 0) is a compact subset of U, that K (D.2) | Rm(x, t)| ≤ 2 r for all x ∈ U and t ∈ [0, t], and that |∇β Rm(x, 0)| ≤

(D.3) for all x ∈ U and |β| ≤ l. Then (D.4)

|∇β Rm(x, t)| ≤

r |β|+2 for all x ∈ B p, 2r , 0 , t ∈ (0, t] and |β| ≤ m. In particular, |∇β Rm(x, t)| ≤

C r |β|+2

K′

r |β|+2 C max(m−l,0) t 2

r2

whenever |β| ≤ l.

The main case l = 0 of Theorem D.1 is due to Shi [62]. The extension to l ≥ 0 appears in [41, Appendix B]. Appendix E. Convergent subsequences of Ricci flow solutions Theorem E.1. [32] Let {gi (t)}∞ i=1 be a sequence of Ricci flow solutions on connected pointed manifolds (Mi , mi ), defined for t ∈ (A, B) and complete for each i and t, with −∞ ≤ A < 0 < B ≤ ∞. Suppose that the following two conditions are satisfied : 1. For each r > 0 and each compact interval I ⊂ (A, B), there is an Nr,I < ∞ so that for

206


all t ∈ I and all i, supBt (mi ,r) | Rm(gi (t))| ≤ Nr,I , and 2. The time-0 injectivity radii {inj(gi (0))(mi )}∞ i=1 are uniformly bounded below by a positive number. Then after passing to a subsequence, the solutions converge smoothly to a complete Ricci flow solution g∞ (t) on a connected pointed manifold (M∞ , m∞ ), defined for t ∈ (A, B). That is, for any compact interval I ⊂ (A, B) and any compact set K ⊂ M∞ containing m∞ , there are pointed time-independent diffeomorphisms φK,i : K → Ki (with Ki ⊂ Mi ) so that {φ∗K,igi}∞ i=1 converges smoothly to g∞ on I × K. Given the sectional curvature bounds, the lower bound on the injectivity radii is equivalent to a lower bound on the volumes of balls around m0 [20, Theorem 4.7]. There are many variants of the theorem with alternative hypotheses. One can replace the interval (A, B) with an interval (A, B], −∞ ≤ A < 0 ≤ B < ∞. One can also replace the interval (A, B) with an interval [A, B), −∞ < A < 0 < B ≤ ∞, if in addition one has uniform time-A bounds supBA (mi ,r) |∇j Rm(gi(A))| ≤ Cr,j . Then using Appendix D, one gets smooth convergence to a limit solution g∞ on the time interval [A, B). (Without the time-A bounds one would only get C 0 -convergence on [A, B) and C ∞ -convergence on (A, B).) There is a similar statement if one replaces the interval (A, B) with an interval [A, B], −∞ < A < 0 ≤ B < ∞.

In the incomplete setting, if one has curvature bounds on spacetime product regions I × B0 (mi , r), with I ⊂ (A, B) and r < r0 < ∞, and an injectivity radius bound at (mi , 0), then one can still extract a subsequence which converges on compact subsets of (A, B) times compact subsets of B0 (m∞ , r0 ). There are analogous statements when (A, B) is replaced by (A, B], [A, B) or [A, B], and the ball is replaced by a metric annulus.

Appendix F. Harnack inequalities for Ricci flow We first recall the statement of the matrix Harnack inequality. Put (F.1)

Pabc = ∇a Rbc − ∇b Rac , Rab 1 . Mab = △Rab − ∇a ∇b R + 2 Racbd Rcd − Rac Rbc + 2 2t

Given a 2-form U and a 1-form W , put (F.2)

Z(U, W ) = Mab Wa Wb + 2 Pabc Uab Wc + Rabcd Uab Ucd .

Suppose that we have a Ricci flow for t > 0 on a complete manifold with bounded curvature on each compact time interval and nonnegative curvature operator. Hamilton’s matrix Harnack inequality says that for all t > 0 and all U and W , Z(U, W ) ≥ 0 [33, Theorem 14.1]. Taking Wa = Ya and Uab = (Xa Yb − Ya Xb )/2 and using the fact that (F.3)

Rict (Y, Y ) = (△Rab )Y a Y b + 2 Racbd Rcd Y a Y b − 2 Rac Rbc Y a Y b ,


207

we can write 2Z(U, W ) = H(X, Y ), where (F.4) H(X, Y ) = − HessR (Y, Y ) − 2hR(Y, X)Y, Xi + 4 (∇X Ric(Y, Y ) − ∇Y Ric(Y, X)) 2 1 + 2 Rict (Y, Y ) + 2 Ric(Y, ·) + Ric(Y, Y ). t

Substituting the elements of an orthonormal basis {ei }ni=1 for Y and summing over i gives (F.5) X i

H(X, ei ) = − △R + 2 Ric(X, X) + 4(h∇R, Xi −

X i

∇ei Ric(ei , X)) + 2

X i

Rict (ei , ei ) + 2| Ric |2 +

1 R. t

Tracing the second Bianchi identity gives X 1 (F.6) ∇ei Ric(ei , X) = h∇R, Xi. 2 i From (F.3), (F.7)

X

Rict (ei , ei ) = △R.

X

H(X, ei ) = H(X),

i

Putting this together gives (F.8)

i

where (F.9)

H(X) = Rt +

1 R + 2h∇R, Xi + 2 Ric(X, X). t

We obtain Hamilton’s trace Harnack inequality, saying that H(X) ≥ 0 for all X.

In the rest of this section we assume that the solution is defined for all t ∈ (−∞, 0). Changing the origin point of time, we have (F.10)

Rt +

1 R + 2h∇R, Xi + 2 Ric(X, X) ≥ 0 t − t0

whenever t0 ≤ t. Taking t0 → −∞ gives (F.11)

Rt + 2h∇R, Xi + 2 Ric(X, X) ≥ 0

In particular, taking X = 0 shows that the scalar curvature is nondecreasing in t for any ancient solution with nonnegative curvature operator, assuming again that the metric is complete on each time slice with bounded curvature on each compact time interval. More generally, (F.12)

0 ≤ Rt + 2h∇R, Xi + 2 Ric(X, X) ≤ Rt + 2h∇R, Xi + 2RhX, Xi.

208


gives If γ : [t1 , t2 ] → M is a curve parametrized by s then taking X = 21 dγ ds dR(γ(s), s) dγ 1 dγ dγ (F.13) = Rt (γ(s), s) + , ∇R ≥ − R , . ds ds 2 ds ds Integrating gives (F.14)

d ln R(γ(s),s) ds

with respect to s and using the fact that g(t) is nonincreasing in t

d2t1 (x1 , x2 ) R(x2 , t2 ) ≥ exp − R(x1 , t1 ). 2(t2 − t1 )

d2 (x1 ,x2 )

t1 whenever t1 < t2 and x1 , x2 ∈ M. (If n = 2 then one can replace 2(t by 2 −t1 ) particular, if R(x2 , t2 ) = 0 for some (x2 , t2 ) then g(t) must be flat for all t.

d2t (x1 ,x2 ) 1 .) 4(t2 −t1 )

In

Appendix G. Alexandrov spaces We recall some facts about Alexandrov spaces (see [11, Chapter 10], [12]). Given points e p (x, y) denote the comparison p, x, y in a nonnegatively curved Alexandrov space, we let ∠ angle at p, i.e. the angle of the Euclidean comparison triangle at the vertex corresponding to p. The Toponogov splitting theorem says that if X is a proper nonnegatively curved Alexandrov space which contains a line, then X splits isometrically as a product X = R × Y , where Y is a proper, nonnegatively curved Alexandrov space [11, Theorem 10.5.1]. Let M be an n-dimensional Alexandrov space with nonnegative curvature, p ∈ M, and λk → 0. Then the sequence (λk M, p) of pointed spaces Gromov-Hausdorff converges to the Tits cone CT M (the Euclidean cone over the Tits boundary ∂T M) which is a nonnegatively curved Alexandrov space of dimension ≤ n [7, p. 58-59]. If the Tits cone splits isometrically as a product Rk × Y , then M itself splits off a factor of Rk ; using triangle comparison, one finds k orthogonal lines passing through a basepoint, and applies the Toponogov splitting theorem. Now suppose that xk ∈ M is a sequence with d(xk , p) → ∞ and rk ∈ R+ is a sequence with d(xrkk,p) → 0. Then the sequence ( r1k M, xk ) subconverges to a pointed Alexandrov space (N∞ , x∞ ) which splits off a line. To see this, observe that since ( d(x1k ,p) M, p) converges to a e x (p, yk ) → π. Observe cone, we can find a sequence yk ∈ M such that d(yk ,xk ) → 1, and ∠ d(xk ,p)

k

that for any ρ < ∞, we can find sequences pk ∈ pxk , zk ∈ xk yk such that d(xk ,zk ) rk

d(xk ,pk ) rk

→ ρ,

→ ρ, and by monotonicity of comparison angles [11, Chapter 4.3] we will have e ∠xk (pk , zk ) → π. Passing to the Gromov-Hausdorff limit, we find p∞ , z∞ ∈ N∞ such that e x∞ (p∞ , z∞ ) = π. Since this construction applies for all ρ, d(p∞ , x∞ ) = d(z∞ , x∞ ) = ρ and ∠ it follows that N∞ contains a line passing through x∞ . Hence, by the Toponogov splitting theorem, it is isometric to a metric product R × N ′ for some Alexandrov space N ′ .

If M is a complete Riemannian manifold of nonnegative sectional curvature and C ⊂ M is a compact connected domain with weakly convex boundary then the subsets Ct = {x ∈ C | d(x, ∂C) ≥ t} are convex in C [18, Chapter 8]. If the second fundamental form of ∂C is


209

≥ 1r at each point of ∂C, then for all x ∈ C we have d(x, ∂C) ≤ r, since the first focal point of ∂C along any inward pointing normal geodesic occurs at distance ≤ r.

Finally, we recall the statement of the Bishop-Gromov volume comparison inequality. Suppose that M is an n-dimensional Riemannian manifold. Given p ∈ M and r2 ≥ r1 > 0, suppose that B(p, r2 ) has compact closure in M and that the sectional curvatures of B(p, r2 ) are bounded below by K ∈ R. Then  R r2 n−1 sin (kr) dr  R0r1 if K = k 2 ,  n−1 (kr) dr  sin 0      n vol(B(p, r2 )) r if K = 0, (G.1) ≤ r2n 1  vol(B(p, r1 ))     Rr  n−1  (kr) dr  R0r12 sinhn−1 if K = −k 2 . sinh (kr) dr 0

(If K = k with k > 0 then we restrict to r2 ≤ πk .) The same inequality holds if we just assume that B(p, r2 ) has Ricci curvature bounded below by (n − 1) K. Equation (G.1) also holds if M is an Alexandrov space and B(p, r2 ) has Alexandrov curvature bounded below by K. 2

Appendix H. Finding balls with controlled curvature Lemma H.1. Let X be a Riemannian manifold with R ≥ 0 and suppose B(x, 5r) is a compact subset of X. Then there is a ball B(y, r¯) ⊂ B(x, 5r), r ≤ r, such that R(z) ≤ 2R(y) for all z ∈ B(y, r¯) and R(y)¯ r 2 ≥ R(x)r 2 . Proof. Define sequences xi ∈ B(x, 5r), ri > 0 inductively as follows. Let x1 = x, r1 = r. For i > 1, let xi+1 = xi , ri+1 = ri if R(z) ≤ 2R(xi ) for all z ∈ B(xi , ri ); otherwise let ri+1 = √ri2 , and let xi+1 ∈ B(xi , ri ) be a point such that R(xi+1 ) > 2R(xi ). The sequence of balls B(xi , ri ) is contained in B(x, 5r), so the sequences xi , ri are eventually constant, and we can take y = xi , r¯ = ri for large i. There is an evident spacetime version of the lemma. Appendix I. Statement of the geometrization conjecture Let M be a connected orientable closed (= compact boundaryless) 3-manifold. One formulation of the geometrization conjecture says that M is the connected sum of closed 3-manifolds {Mi }ni=1 , each of which admits a codimension-0 compact submanifold-withboundary Gi so that • Gi is a graph manifold • Mi − Gi is hyperbolic, i.e. admits a complete Riemannian metric with constant negative sectional curvature and finite volume • Each component T of ∂Gi is an incompressible torus in Mi , i.e. with respect to a basepoint t ∈ T , the induced map π1 (T, t) → π1 (Mi , t) is injective.

210


We allow Gi = ∅ or Gi = Mi .

A reference for graph manifolds is [42, Chapter 2.4]. The definition is as follows. One takes a collection {Pi}N i=1 of pairs of pants (i.e. closed 2-disks with two balls removed) and ′ 1 N 1 2 N′ a collection of closed 2-disks {Dj2 }N j=1 . The 3-manifolds {S × Pi }i=1 ∪ {S × Dj }j=1 have toral boundary components. One takes an even number of these tori, matches them in pairs 1 2 N′ by homeomorphisms, and glues {S 1 × Pi }N i=1 ∪ {S × Dj }j=1 by these homeomorphisms. The resulting 3-manifold G is a graph manifold, and all graph manifolds arise in this way. We will assume that the gluing homeomorphisms are such that G is orientable. Clearly the boundary of G, if nonempty, is a disjoint union of tori. It is also clear that the result of gluing two graph manifolds along some collection of boundary tori is a graph manifold. The connected sum of two 3-manifolds is a graph manifold if and only if each factor is a graph manifold [42, Proposition 2.4.3]. The reason to require incompressibility of the tori T in the statement of the geometrization conjecture is to exclude phony decompositions, such as writing S 3 as the union of a solid torus and a hyperbolic knot complement. A more standard version of the geometrization conjecture uses some facts from 3-manifold theory [59]. First M has a Kneser-Milnor decomposition as a connected sum of uniquely defined prime factors. Each prime factor is S 1 × S 2 or is irreducible, i.e. any embedded S 2 bounds a 3-ball. If M is irreducible then it has a JSJ decomposition, i.e. there is a minimal collection of disjoint incompressible embedded tori {Tk }K to isotopy, k=1 in M, unique up SK ′ with the property that if M is the metric completion of a component of M − k=1 Tk (with respect to an induced Riemannian metric from M) then

• M ′ is a Seifert 3-manifold or • M ′ is non-Seifert and any embedded incompressible torus in M ′ can be isotoped into ∂M ′ .

The second version of the geometrization conjecture reduces to the conjecture that in the latter case, the interior of M ′ is hyperbolic. Thurston proved that this is true when ∂M ′ 6= ∅. The reason for the word “geometrization” is explained in [59, 65]. An orientable Seifert 3-manifold is a graph manifold [42, Proposition 2.4.2]. It follows that the second version of the geometrization conjecture implies the first version. One can show directly that any graph manifold is a connected sum of prime graph manifolds, each of which can be split along incompressible tori to obtain a union of Seifert manifolds [42, Proposition 2.4.7], thereby showing the equivalence of the two versions. References [1] K. Akutagawa, M. Ishida and C. Lebrun, “Perelman’s Invariant, Ricci Flow, and the Yamabe Invariants of Smooth Manifolds”, Arch. Math. 88, p. 71-76 (2007) [2] M. Anderson, “Scalar Curvature and Geometrization Conjectures for 3-Manifolds”, in Comparison Geometry, MSRI Publ. 30, Cambridge University Press, Cambridge, p. 49-82 (1997) [3] M. Anderson, “Scalar Curvature and the Existence of Geometric Structures on 3-Manifolds. I”, J. Reine Angew. Math. 553, p. 125-182 (2002) [4] M. Anderson, “Canonical Metrics on 3-Manifolds and 4-Manifolds”, Asian Journal of Math. 10, p. 127-164 (2006)


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[5] G. Anderson and B. Chow, “A Pointwise Bound for Solutions of the Linearized Ricci Flow Equation on 3-Manifolds”, Calculus of Variations 23, p. 1-12 (2005) [6] S. Angenent and D. Knopf, “Precise Asymptotics of the Ricci Flow Neckpinch”, Comm. Anal. Geom. 15, p. 773-844 (2007) [7] W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, Progress in Mathematics 61, Birkh¨ auser, Boston (1985) [8] W. Beckner, “Geometric Asymptotics and the Logarithmic Sobolev Inequality”, Forum Math. 11, p. 105-137 (1999) [9] L. Bessières, G. Besson, M. Boileau, S. Maillot and J. Porti, “Weak Collapsing and Geometrisation of Aspherical 3-Manifolds”, preprint, http://arxiv.org/abs/0706.2065 (2007) [10] J.-P. Bourguignon, “Une Stratification de l’Espace des Structures Riemanniennes”, Compositio Math. 30, p. 1-41 (1975) [11] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics 33, American Mathematical Society, Providence (2001) [12] Y. Burago, M. Gromov and G. Perelman, “A. D. Aleksandrov Spaces with Curvatures Bounded Below”, Russian Math. Surveys 47, p. 1-58 (1992) [13] E. Calabi, “An Extension of E. Hopf’s Maximum Principle with an Application to Riemannian Geometry”, Duke Math. J. 25, p. 45-56 (1957) [14] H-D. Cao and B. Chow, “Recent Developments on the Ricci Flow”, Bull. of the AMS 36, p. 59-74 (1999) [15] H.-D. Cao and X.-P. Zhu, “A Complete Proof of the Poincaré and Geometrization Conjectures - Application of the Hamilton-Perelman theory of the Ricci Flow”, Asian Journal of Math. 10, p. 165-492 (2006) Erratum to “A Complete Proof of the Poincaré and Geometrization Conjectures - Application of the Hamilton-Perelman theory of the Ricci Flow”, Asian Journal of Math. 10, p. 663-664 (2006) [16] A. Chau, L.-F. Tam and C. Yu, “Pseudolocality for the Ricci Flow and Applications”, preprint, http://arxiv.org/abs/math/0701153 (2007) [17] I. Chavel, Riemannian Geometry - a Modern Introduction, Cambridge Tracts in Mathematics 108, Cambridge University Press, Cambridge (1993) [18] J. Cheeger and D. Ebin, Comparison Theorems in Riemannian Geometry. North-Holland Publishing Co., Amsterdam-Oxford (1975) [19] J. Cheeger and D. Gromoll, “On the Structure of Complete Manifolds of Nonnegative Curvature”, Ann. of Math. 96, p. 413-443 (1972) [20] J. Cheeger, M. Gromov and M. Taylor, “Finite Propagation Speed, Kernel Estimates for Functions of the Laplace Operator, and the Geometry of Complete Riemannian Manifolds”, J. Diff. Geom. 17, p. 15-53 (1982) [21] B. Chen and X. Zhu, “Uniqueness of the Ricci Flow on Complete Noncompact Manifolds”, J. of Diff. Geom. 74, p. 119-154 (2006) [22] B. Chow and D. Knopf, The Ricci Flow : An Introduction, AMS Mathematical Surveys and Monographs, Amer. Math. Soc., Providence (2004) [23] B. Chow, P. Lu and L. Ni, Hamilton’s Ricci Flow, Graduate Studies in Mathematics 77, AMS, Providence (2006) [24] T. Colding and W. Minicozzi, “Estimates for the Extinction Time for the Ricci Flow on Certain 3Manifolds and a Question of Perelman”, J. Amer. Math. Soc. 18, p. 561-569 (2005) [25] T. Colding and W. Minicozzi, “Width and Finite Extinction Time of Ricci Flow”, preprint, http://arxiv.org/abs/0707.0108 (2007) [26] J. Dodziuk, “Maximum Principle for Parabolic Inequalities and the Heat Flow on Open Manifolds”, Indiana Univ. Math. J. 32, p. 703-716 (1983) [27] J. Eells and J. Sampson, “Harmonic Mappings of Riemannian Manifolds”, Amer. J. Math. 86, p. 109-160 (1964) [28] J.-H. Eschenburg, “Local Convexity and Nonnegative Curvature—Gromov’s Proof of the Sphere Theorem”, Inv. Math. 84, p. 507-522 (1986) [29] R. Hamilton, “Four-Manifolds with Positive Curvature Operator”, J. Diff. Geom. 24, p. 153-179 (1986)

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[30] R. Hamilton, “The Ricci Flow on Surfaces”, in Mathematics and General Relativity, Contemp. Math. 71, AMS, Providence, RI, p. 237-262 (1988). [31] R. Hamilton, “The Harnack Estimate for the Ricci Flow”, J. Diff. Geom. 37, p. 225-243 (1993) [32] R. Hamilton, “A Compactness Property for Solutions of the Ricci Flow” Amer. J. Math. 117, p. 545-572 (1995) [33] R. Hamilton, “The Formation of Singularities in the Ricci Flow”, in Surveys in Differential Geometry, Vol. II, Internat. Press, Cambridge, p. 7-136 (1995) [34] R. Hamilton, “Nonsingular Solutions of the Ricci Flow on Three-Manifolds”, Comm. Anal. Geom. 7, p. 695-729 (1999) [35] R. Hamilton, “Three-Manifolds with Positive Ricci Curvature”, J. Diff. Geom. 17, p. 255-306 (1982) [36] R. Hamilton, “Four-Manifolds with Positive Isotropic Curvature”, Comm. Anal. Geom. 5, p. 1-92 (1997) [37] H. Ishii, “On the Equivalence of Two Notions of Weak Solutions, Viscosity Solutions and Distribution Solutions”, Funkcial. Ekvac. 38, p. 101-120 (1995) [38] V. Kapovitch, “Perelman’s Stability Theorem,” in Surveys in Differential Geometry, vol. XI, Metric and Comparison Geometry, eds. J. Cheeger and K. Grove, International Press, Somerville, MA, p. 103-136 [39] B. Kleiner and J. Lott, webpage at http://www.math.lsa.umich.edu/˜lott/ricciflow/perelman.html [40] B. Kleiner and J. Lott, “Locally Collapsed 3-Manifolds”, to appear [41] P. Lu and G. Tian, “Uniqueness of Standard Solutions in the Work of Perelman”, preprint, http://www.math.lsa.umich.edu/˜lott/ricciflow/perelman.html (2005) [42] S. Matveev, Algorithmic Topology and Classification of 3-Manifolds, Springer-Verlag, Berlin (2003) [43] W. Meeks III and S.-T. Yau, “Topology of Three-Dimensional Manifolds and the Embedding Problems in Minimal Surface Theory”, Ann. of Math. 112, p. 441-484 (1980) [44] J. Morgan, “Recent Progress on the Poincaré Conjecture and the Classification of 3-Manifolds”, Bull. Amer. Math. Soc. 42, p. 57-78 (2005) [45] J. Morgan and G. Tian, Ricci Flow and the Poincaré Conjecture, Clay Mathematics Monographs 3, Amer. Math. Soc., Providence (2007) [46] J. Morgan and G. Tian, “Completion of Perelman’s Proof of the Geometrization Conjecture”, to appear [47] G. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Studies No. 78, Princeton University Press, Princeton (1973) [48] L. Ni, “The Entropy Formula for Linear Heat Equation”, J. Geom. Anal. 14, p. 87-100 (2004) [49] L. Ni, “Ricci Flow and Nonnegativity of Sectional Curvature”, Math. Res. Let. 11, p. 883-904 (2004) [50] L. Ni, “A Note on Perelman’s LYH Inequality”, Comm. Anal. Geom. 14, p. 883-905 (2006) [51] G. Perelman, “The Entropy Formula for the Ricci Flow and its Geometric Applications”, http://arXiv.org/abs/math.DG/0211159 [52] G. Perelman, “Ricci Flow with Surgery on Three-Manifolds”, http://arxiv.org/abs/math.DG/0303109 [53] G. Perelman, “Finite Extinction Time for the Solutions to the Ricci Flow on Certain Three-Manifolds”, http://arxiv.org/abs/math.DG/0307245 [54] G. Prasad, “Strong Rigidity of Q-Rank 1 Lattices”, Invent. Math. 21, p. 255-286 (1973) [55] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV, Analysis of Operators, Academic Press, New York (1978) [56] O. Rothaus, “Logarithmic Sobolev Inequalities and the Spectrum of Schrödinger Operators”, J. Funct. Anal. 42, p. 110-120 (1981) [57] R. Schoen, “Estimates for Stable Minimal Surfaces in Three-Dimensional Manifolds”, in Seminar on Minimal Submanifolds, Ann. of Math. Stud. 103, Princeton Univ. Press, Princeton, N.J., p. 111-126 (1983). [58] R. Schoen and S.-T. Yau, “Complete Three-Dimensional Manifolds with Positive Ricci Curvature and Scalar Curvature”, in Seminar on Differential Geometry, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, N.J., p. 209-228 (1982) [59] P. Scott, “The Geometries of 3-Manifolds”, Bull. London Math. Soc. 15, p. 401-487 (1983) [60] V. Sharafutdinov, “The Pogorelov-Klingenberg Theorem for Manifolds that are Homeomorphic to Rn ”, Siberian Math. J. 18, p. 649-657 (1977)


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[61] W.-X. Shi, “Complete Noncompact Three-Manifolds with Nonnegative Ricci Curvature”, J. Diff. Geom. 29, p. 353-360 (1989) [62] W.-X. Shi, “Deforming the Metric on Complete Riemannian Manifolds”, J. Diff. Geom. 30, p. 223-301 (1989) [63] W.-X. Shi, “ Ricci Deformation of the Metric on Complete Noncompact Riemannian Manifolds”, J. Diff Geom. 30, p. 303-394 (1989) [64] T. Shioya and T. Yamaguchi, “Volume-Collapsed Three-Manifolds with a Lower Curvature Bound”, Math. Ann. 333, p. 131-155 (2005) [65] W. Thurston, “Three-Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry”, Bull. Amer. Math. Soc. 6, p. 357-381 (1982) [66] P. Topping, Lectures on the Ricci Flow, LMS Lecture Notes 325, London Mathematical Society and Cambridge University Press, http://www.maths.warwick.ac.uk/ topping/RFnotes.html [67] R. Ye, “On the l-Function and the Reduced Volume of Perelman I,II”, Trans. Amer. Math. Soc. 360, p. 507-531, 533-544 (2008) [68] R. Ye, “On the Uniqueness of 2-Dimensional κ-Solutions”, http://www.math.ucsb.edu/˜yer/ricciflow.html Department of Mathematics, Yale University, New Haven, CT 06520-8283, USA E-mail address: [email protected] Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA E-mail address: [email protected]

arXiv:math/0602337v1 [math.DG] 15 Feb 2006

A NOTE ON PERELMAN’S LYH INEQUALITY Lei Ni Abstract We give a proof to the Li-Yau-Hamilton type inequality claimed by Perelman on the fundamental solution to the conjugate heat equation. The rest of the paper is devoted to improving the known differential inequalities of Li-Yau-Hamilton type via monotonicity formulae.

1. Introduction In [P], Perelman proved a Li-Yau-Hamilton type (also called differential Harnack) inequality for the fundamental solution of the conjugate heat equation, in the presence of the Ricci flow. More precisely, let (M, gij (t)) be a solution to Ricci flow: (1.1)

∂ gij = −2Rij ∂t

on M ×[0, T ] and let H(x, y, t) =

e−f n (4πτ ) 2

(where τ = T −t) be the funda-

mental solution to the conjugate heat equation uτ −∆u+Ru = 0. (More precisely we should write the fundamental solution as H(y, t; x, T ), which ∂ + ∆y + R(y, t) H = 0 for any (x, t) with t < T and satisfiesR − ∂t limt→T M H(y, t; x, T )f (y, t) dµt (y) = f (x, T ).) Define vH = τ 2∆f − |∇f |2 + R + f − n H.

Here all the differentiations are taken with respect to y, and n = dimR (M ). Then vH ≤ 0 on M × [0, T ]. This result is a differential inequality of Li-Yau type [LY], which has important consequences in the later part of [P]. For example it is essential in proving the pseudolocality theorems. It is also crucial in localizing the entropy formula [N3].

Mathematics Subject Classification. Primary 58J35. Key words and phrases. Heat equation, differential Harnack inequality, entropy formula, local monotonicity formulae, mean value theorem. The author was supported in part by NSF Grants and an Alfred P. Sloan Fellowship, USA. Received September 2005. PROOF COPY

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NOT FOR DISTRIBUTION

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In Section 9 of [P], the following important differential equation 1 ∂ − ∆ + R vu = −2τ |Rij + ∇i ∇j f − gij |2 u (1.2) ∂τ 2τ

is stated for any positive solution u to the conjugate heat equation, whose integration on M gives the celebrated entropy formula for the Ricci flow. One can consult various resources (e.g. [N1]) for the detailed computations of this equation, which can also be done through a straightforward calculation, after knowing the result. [P] then proceeds the proof of the claim R vH ≤ 0 in a clever way by checking that for any τ∗ with T ≥ τ∗ > 0, M vH (y)h(y) dµτ∗ (y) ≤ 0, for any smooth function h(y) ≥ 0 with compact support. In order to achieve this, in [P] the heat ∂ equation ∂t − ∆ h(y, t) = 0 with the ‘initial data’ h(y, T − τ∗ ) = h(y) (more precisely t = T − τ∗ ), the given compactly supported nonnegative function, is solved. Applying (1.2) to u(y, τ ) = H(x, y, τ ), one can easily derive as in [P], via integration by parts, that Z Z 1 d vH h dµτ = −2 τ |Rij + ∇i ∇j f − gij |2 Hh dµτ ≤ 0. (1.3) dτ M 2τ M The Li-Yau type inequality vH ≤ 0 then follows from the above monotonicity, provided the claim that Z (1.4) lim vH h dµτ ≤ 0. τ →0 M

The main purpose of this note is to prove (1.4), hence provide a complete proof of the claim vH ≤ 0. This will be done in Section 3 after some preparations in R Section 2. It was written in [P] that ‘it is easy to see’ that limτ →0 M vH h dµτ = 0. It turns out that the proof found here need to use some gradient estimates for positive solutions, a quite precise estimate on the ‘reduced distance’ and the monotonicity formula (1.3). (We shall focus on the proof of (1.4) for the case when M is compact and leave the more technical details of generalizing it to the noncompact setting to the later refinements.) Indeed the claim that R limτ →0 M vH h dµτ = 0 follows from a blow-up argument of [P], after we have established (1.4). Since our argument is a bit involved, this may not be the proof. In Section 4 we derive several monotonicity formulae, which improve various Li-Yau-Hamilton inequalities for linear heat equation (systems) as well as for Ricci flow, including the original Li-Yau’s inequality. In Section 5 we illustrate the localization of them by applying a general scheme of [EKNT]. 2. Estimates and results needed We shall collect some known results and derive some estimates needed for proving (1.4) in this section. We need the asymptotic behavior of PROOF COPY


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the fundamental solution to the conjugate heat equation for small τ . Let dτ (x, y) be the distance function with respect to the metric g(τ ). Let Bτ (x, r) (V olτ ) be the ball of radius r centered at x (the volume) with respect to the metric g(τ ). Theorem 2.1. Let H(x, y, τ ) be the fundamental solution to the (backward in t) conjugate heat equation. Then as τ → 0 we have that 2 d (x,y) ∞ exp − 0 4τ X τ j uj (x, y, τ ). (2.1) H(x, y, τ ) ∼ n (4πτ ) 2 j=0 By (2.1) we mean that exists T > 0 and sequence uj ∈ C ∞ (M × M × [0, T ]) such that 2 d (x,y) k exp − 0 4τ X τ j uj (x, y, τ ) = wk (x, y, τ ) H(x, y, τ ) − n 2 (4πτ ) j=0

with

n wk (x, y, τ ) = O τ k+1− 2

as τ → 0, uniformly for all x, y ∈ M . The function u0 (x, y, τ ) can be chosen so that u0 (x, x, 0) = 1. This result was proved in details, for example in [GL], when there is ∂ no zero order term R(y, τ )u(y, τ ) in the equation ∂τ u − ∆u + Ru = 0 and replacing d0 (x, y) by dτ (x, y). However, one can check that the argument carries over to this case if one assumes that the metric g(τ ) is C ∞ near τ = 0. One can consult [SY, CLN] for intrinsic presentations. Let Z Wh (g, H, τ ) = vH h dµτ M

where h is the previously described solution to the heat equation. It is clear that for any τ with T ≥ τ > 0, Wh (g, H, τ ) is a well-defined quantity. A priori it may blow up as τ → 0. It turns out that in our course of proving that limτ →0 Wh (g, H, τ ) ≤ 0 we need to show first that exists C > 0, which may depends on the geometry of the Ricci flow solution (M, g(τ )) defined on M × [0, T ], but independent of τ (as τ → 0) so that Wh (g, H, τ ) ≤ C for all T ≥ τ > 0. The following lemma supplies the key estimates for this purpose. Lemma 2.2. Let (M, g(t)) be a smooth solution to the Ricci flow on M × [0, T ]. Assume that there exist k1 ≥ 0 and k2 ≥ 0, such that the Ricci curvature Rij (g(τ )) ≥ −k1 gij (τ ) and max(R(y, τ ), |∇R|2 (y, τ )) ≤ k2 , on M × [0, t]. (i) If u ≤ A is a positive solution to the conjugate heat equation on M × [0, T ], then there exists C1 and C2 depending on k1 , k2 and n such PROOF COPY


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that for 0 < τ ≤ min(1, T ), (2.2)

τ

A |∇u|2 ≤ (1 + C τ ) log + C τ 1 2 u2 u

(ii) If u is a positive solution to the conjugate heat equation on M × [0, T ], then there exists B, depending on (M, g(τ )) so that for 0 ≤ τ ≤ min(T, 1), Z |∇u|2 B (2.3) τ u dµτ + C2 τ . ≤ (2 + C1 τ ) log n u2 uτ 2 M Remark 2.3. Here and thereafter we use the same Ci (B) at the differentR lines if they just differ only by a constant depending on n. Notice that M u dµτ is independent od τ and equals to 1 if u is the fundamental solution. The proof to the lemma given below is a modification of some arguments in [H]. Proof. Direct computation, under a unitary frame, gives ∂ ui uj 2 |∇u|2 |∇u|2 2 −∆ = − uij − R + ∂τ u u u u −4Rij ui uj − 2h∇(Ru), ∇ui + u 2 |∇u| ≤ (4 + n)k1 + 2|∇R||∇u| u |∇u|2 ≤ [(4 + n)k1 + 1] + k2 u u and ∂ A |∇u|2 A −∆ u log = + Ru − Ru log ∂τ u u u 2 |∇u| A ≥ − nk1 u − k2 u log . u u Combining the above two equations together we have that ∂ −∆ Φ≤0 ∂τ where A |∇u|2 k2 τ − e u log − (k2 + nk1 ek2 )τ u Φ=ϕ u u τ with ϕ = 1+[(4+n)k , which satisfies 1 +1]τ d ϕ + [(4 + n)k1 + 1] ϕ < 1. dτ By the maximum principle we have that A |∇u|2 k2 τ ≤ e u log + (k2 + nk1 ek2 )τ u. ϕ u u PROOF COPY



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From this one can derive (2.2) easily. To prove the second part, we claim that for u, a positive solution to the conjugate heat equation, there exists a C depending on (M, g(τ )) such that Z C u(z, τ ) dµτ (z). (2.4) u(y, τ ) ≤ n τ2 M

This is a mean-value type inequality, which can be proved via, for example the Moser iteration. Here we follow [H]. We may assume n that supy∈M,0≤τ ≤1 τ 2 u(y, τ ) is finite. Otherwise we may replacing τ by τǫ = τ − ǫ and let ǫ → 0 after establishing the claim for τǫ . Now let n (x0 , τ0 ) ∈ M × [0, 1] be such a space-time point that max τ 2 u(y, τ ) = n τ02 u(y0 , τ0 ). Then we have that n n n 2 2 2 sup u(y, t) ≤ τ0 u(y0 , τ0 ) = 2 2 u(y0 , t0 ). τ0 τ0 M ×[ ,τ0 ] 2

Noticing this upper bound, we apply (2.2) to u on M × [ τ20 , τ0 ], and conclude that ! ! n τ0 |∇u|2 2 2 u(y0 , τ0 ) (y, τ0 ) ≤ (1 + C1 τ0 ) log + C2 τ0 . 2 u2 u(y, τ0 ) n 2 2 u(y0 ,τ0 ) + C2 τ0 . The above can be written as Let g = log u(y,τ0 ) √ |∇ g| ≤

r

1 + C1 τ0 2τ0

which implies that

sup q

Bτ0 y0 ,

τ0 1+C1 τ0

1 √ √ g(y, τ0 ) ≤ g(y0 , τ0 ) + √ . 2

Rewriting the above in terms of u we have that √n n − 1 + √2 log 2+C2 u(y, τ0 ) ≥ 2 2 u(y0 , τ0 )e 2 2 2 = C3 u(y0 , τ0 ) q τ0 . Here we have also used τ0 ≤ 1. Noticing for all y ∈ Bτ0 y0 , 1+C 1 τ0 that r n τ0 V olτ0 Bτ0 y0 , ≥ C4 τ02 1 + C1 τ0

for some C4 depending on the geometry of (M, g(τ0 )). Therefore we have that Z C5 u(y, τ0 ) dµτ0 (y) ≥ u(y0 , τ0 ) n τ02 M

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for some C5 depending on C3 and C4 . By the way we choose (y0 , τ0 ) we have that Z Z n n 2 2 τ u(y, τ ) ≤ τ0 u(y0 , τ0 ) ≤ C5 u(y, τ0 ) dµτ0 (y) = C5 u(y, τ ) dµτ (y). M

M

This proves the claim (2.4). Now the estimate (2.3) follows from (2.2), applying to u on M × [ τ2 , τ ], and the just proved (2.4), which ensures the needed upper bound for applying the estimate (2.2). q.e.d. e−f n (4πτ ) 2

is the fundamental solution to the conjugate heat equaR tion we have that M u dµτ = 1. Therefore, by (2.3), we have that Z Z 2 (2.5) τ |∇f | uh dµτ ≤ (2 + C1 τ ) (log B + f + C2 τ ) uh dµτ . If u =

M

M

On the other hand, integration by parts can rewrite Z Z Wh (g, u, τ ) = τ |∇f |2 uh dµτ − 2τ h∇f, ∇hiu dµτ M M Z Z +τ Ruh dµτ + (f − n)uh dµτ M

=

M

I + II + III + IV.

The I term can be estimated by (2.5), whose right hand side contains R only one ‘bad’ term M f uh dµτ in the sense that it could possibly blow up. The second term Z Z II = 2τ h∇u, ∇hi dµ = −2τ u∆h dµτ M

M

is clearly bounded as τ → 0. In fact II → 0 as τ → 0. The same conclusion obviously holds for III. Summarizing above, we reduce the question of bounding from above the quantity Wh (u, g, τ ) to bounding one single term Z V = f uh dµτ M

from above (as τ → 0). We shall show later that limτ →0 V ≤ 0. To do this we need to use the ‘reduced distance’, introduced by Perelman in [P] for the Ricci flow geometry. Let x be a fixed point in M . Let ℓ(y, τ ) be the reduced distance in [P], with respect to (x, 0) (more precisely τ = 0). We collect the relevant properties of ℓ(y, τ ) in the following lemma (Cf. [Ye, CLN]). ¯ τ ) = 4τ ℓ(y, τ ). Lemma 2.4. Let L(y, (i) Assume that there exists a constant k1 such that Rij (g(τ )) ≥ ¯ τ ) is a local Lipschitz function on M × [0, T ]; −k1 gij (τ ), L(y, PROOF COPY



7

(ii) Assume that there exist constant k1 and k2 so that −k1 gij (τ ) ≤ Rij (g(τ )) ≤ k2 gij (τ ). Then ¯ τ ) ≤ e2k2 τ d20 (x, y) + 4k2 n τ 2 L(y, 3

(2.6) and (2.7)

d20 (x, y)

(iii) (2.8)

2k1 τ

≤e

4k1 n 2 ¯ τ ; L(y, τ ) + 3

¯  L(y,τ ) exp − 4τ ∂  ≤ 0. −∆+R  n ∂τ (4πτ ) 2



Proof. The first two claims follow from the definition by straight forward checking. For (iii), it was proved in Section 7 of [P]. By now there are various resources where the detailed proof can be found. See for example [Ye] and [CLN]. q.e.d. The consequence of (2.6) and (2.7) is that ¯ ) exp − L(y,τ 4τ = δx (y), lim n τ →0 (4πτ ) 2 which together with (2.8) implies that H, the fundamental solution to the conjugate heat equation, is bounded from below as ¯ ) exp − L(y,τ 4τ . H(x, y, τ ) ≥ n (4πτ ) 2 Hence ¯ τ) L(y, . (2.9) f (y, τ ) ≤ 4τ This was proved in [P] out of the claim vH ≤ 0. Since we are in the middle of proving vH ≤ 0, we have to show the above alternative of obtaining (2.9). 3. Synthesis Now we assembly the results in the previous section to prove (1.4). As the first step we show that Wh (g, H, τ ) is bounded (thanks to the monotonicity (1.3), it is sufficient to bound it from above) as τ → 0, where H(x, y, τ ) is the fundamental solution to the conjugate heat equation with H(x, y, 0) = δx (y). By the reduction done in the previous section we only need to show that Z V = f uh dµτ M

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is bounded from above as τ → 0. By (2.9) we have that Z Z ¯ L(y, τ ) H(x, y, τ )h(y, τ ) dµτ (y) lim sup f Hh dµτ ≤ lim sup 4τ τ →0 τ →0 M M Z d20 (x, y) H(x, y, τ )h(y, τ ) dµτ (y) ≤ lim sup 4τ τ →0 M Z k2 τ k2 n e −1 2 d0 (x, y) + τ H(x, y, τ )h(y, τ ) dµτ (y). + lim τ →0 M 4τ 3 Here we have used (2.6) in the last inequality. By Theorem 2.1, some elementary computations give that Z d20 (x, y) n lim H(x, y, τ )h(y, τ ) dµτ (y) = h(x, 0). τ →0 M 4τ 2 k τ

Since e 24τ−1 d20 (x, y) + k23n τ is a bounded continuous function even at τ = 0, we have that Z k2 τ k2 n e −1 2 d0 (x, y) + τ H(x, y, τ )h(y, τ ) dµτ (y) = 0. lim τ →0 M 4τ 3 R This completes our proof on finiteness of lim supτ →0 M f Hh dµτ . In fact we have proved that Z n (3.1) lim sup (f − )Hh dµτ ≤ 0. 2 τ →0 M By the just proved finiteness of Wh (g, H, τ ) as τ → 0, and the (entropy) monotonicity (1.3), we know that the limit limτ →0 Wh (g, H, τ ) exists. Let Z lim Wh (g, H, τ ) = lim vH h dµτ = α τ →0

τ →0 M

for some finite α. Hence limτ →0 Wh (g, H, τ ) − Wh (g, H, τ2 ) = 0. By (1.3) and the mean-value theorem we can find τk → 0 such that Z 1 2 lim τk |Rij + ∇i ∇j f − gij |2 Hh dµτk = 0. τk →0 2τk M

By the Cauchy-Schwartz inequality and the Hölder inequality we have that Z n lim τk R + ∆f − Hh dµτk = 0. τk →0 2τk M This implies that lim Wh (g, H, τ ) = lim

τ →0

PROOF COPY

Z

τk →0 M

τk (∆f − |∇f |2 ) + f −

n Hh dµτk . 2



9

Again the integration by parts shows that Z Z 2 τk (∆f − |∇f | )Hh dµτk = τk h∇H, ∇hi dµτk M M Z = −τk H∆h dµτk → 0. M

Hence by (3.1) lim Wh (g, H, τ ) = lim

τ →0

Z

τk →0 M

(f −

n )Hh dµτk ≤ 0. 2

This proves α ≤ 0, namely (1.4). The claim that α = limτ →0 Wh (g, H, τ ) = 0 can now be proved by the blow-up argument as in Section 4 of [P]. Assume that α < 0. One can easily check that this would imply that limτ →0 µ(g, τ ) < 0. Here µ(g, τ ) is the invariant defined in Section 4 of [P]. In fact, noticing that h(y, τ ) > 0 for all τ ≤ τ∗ (where the τ∗ is the one we 1 fixed in the introduction). Therefore by multiple h(x,0) (more pre1 cisely h(x,·) at τ = 0) to the original h(y, τ ), we may assume that R h(x, 0) = M H(x, y, τ )h(y, τ ) dµτ = 1. Let u ˜(y, τ ) = H(x, y, τ )h(y, τ ) n ˜ and f = − log u ˜ − 2 log(4π). Now direct computation yields that Z |∇h|2 − h log h H dµτ . Wh (g, H, τ ) = W(g, u˜, τ ) + τ h M Noticing that the second integration goes to 0 as τ → 0, we can deduce that W(g,R u˜, τ ) < 0 for sufficient small τ if α < 0. This, together with the fact M u ˜ dµ = 1, implies that µ(g, τ ) < 0 for sufficiently small τ . Now Perelman’s blow-up argument in the Section 4 of [P] gives a contradiction with the sharp logarithmic Sobolev inequality on the Euclidean space [G]. (One can consult, for example [N1, STW], for more details of this part.) Remark 3.1. The method of proof here follows a similar idea used in [N1], where the asymptotic limit of the entropy as τ → ∞ was computed. Note that we have to use properties of the reduced distance, introduced in Section 7 of [P], in our proof, while the similar, but slightly easier, claim that limτ →0 W(g, H, τ ) = 0 appears much earlier in Section 4 of [P]. The proof can be easily modified to give the asymptotic behavior of the entropy defined in [N1] for the fundamental solution to the linear heat equation, with respect to a fixed Riemannian metric. Indeed if we restrict to the class of complete Riemannian manifolds with nonnegative Ricci curvature we have the following estimates. PROOF COPY


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Proposition 3.2. For any δ > 0, there exists C(δ) such that H(x, y, τ ) d2 (x, y) |∇H|2 (x, y, τ ) ≤ 2 C(δ) + (3.2) H τ (4 − δ)τ and (3.3) |∇H|2 H(x, y, τ ) ∆H(x, y, τ ) + (x, y, τ ) ≤ 2 H τ

d2 (x, y) C(δ) + 4 (4 − δ)τ

.

The previous argument for the Ricci flow case can be transplanted to show that τ (2∆f − |∇f |2 ) + f − n ≤ 0 where H(y, τ ; x, 0) =

1 e−f (4πτ )n/2

is the fundamental solution to the heat

∂ ∂τ

operator − ∆. This gives a rigorous argument for the inequality (1.5) (Theorem 1.2) of [N1], for both the compact manifolds and complete manifolds with non-negative Ricci (or Ricci curvature bounded from below). For the full detailed account please see [CLN]. 4. Improving Li-Yau-Hamilton estimates via monotonicity formulae The proof of (1.4) indicates a close relation between the monotonicity formulae and the differential inequalities of Li-Yau type. The hinge is simply Green’s second identity. This was discussed very generally in [EKNT]. Moreover if we chose h in the introduction to be the ∂ fundamental solution to the time dependent heat equation ( ∂t − ∆) centered at (x0 , t0 ) we can have a better upper bound on vH (x0 , t0 ) in terms of the a weighted integral which is non-positive. In fact, this follows from the representation formula for the solutions to the nonhomogenous conjugate heat equation. More precisely, since h(y, t; x0 , t0 ) is the fundamental solution to the heat equation (to make it very clear, vH is defined with respect to H = H(y, t; x, T ), the fundamental solution to the conjugate heat equation centered at (x, T ) with T > t0 ), we have that Z lim

t→t0

h(y, t; x0 , t0 )vH (y, t) dµt (y) = vH (x0 , t0 ).

M

On the other hand from (1.2) we have that (by Green’s second identity) Z Z 1 d hvH dµt = 2τ |Rij + fij − |2 Hh dµt . dt M 2τ M

Therefore Z Z lim hvH dµt − vH (x0 , t0 ) = t→T

M

PROOF COPY

T

2τ t0

Z

M

|Rij + fij −

1 gij |2 Hh dµt dt. 2τ



11

Using the fact that limt→T vH = 0 we have that 2 Z T Z 1 Hh dµt dt ≤ 0, vH (x0 , t0 ) = −2 (T − t) R + f − ij ij 2(T − t) t0

M

which sharpens the estimate vH ≤ 0 by providing a non-positive upper bound. Noticing also the duality h(y, t; x0 , t0 ) = H(x0 , t0 ; y, t) for any t > t0 we can express everything in terms of the fundamental solution to the (backward) conjugate heat equation. Below we show a few new monotonicity formulae, which expand the list of examples shown in the introduction of [EKNT] on the monotonicity formulae, and more importantly improve the earlier established Li-Yau-Hamilton estimates in a similar way as the above. For the simplicity let us just consider the Kähler-Ricci flow case even though often the discussions are also valid for the Riemannian (Ricci flow) case, after replacing the assumption on the nonnegativity of the bisectional curvature by the nonnegativity of the curvature operator whenever necessary. We first let (M, gαβ¯ (x, t)) (m = dimC M ) be a solution to the KählerRicci flow: ∂ g ¯ = −Rαβ¯. ∂t αβ Let Υαβ¯(x, t) be a Hermitian symmetric tensor defined on M × [0, T ], which is deformed by the complex Lichnerowicz-Laplacian heat equation (or L-heat equation in short): ∂ 1 − ∆ Υγ δ¯ = Rβ αγ Rγ p¯kpδ¯ + Rpδ¯Υγ p¯ . ¯ δ¯Υαβ¯ − ∂t 2 ¯

¯

Let div(Υ)α = gγ δ ∇γ Υαδ¯ and div(Υ)β¯ = gγ δ ∇δ¯Υγ β¯. Consider the quantity αβ¯ γ δ¯ 1 Z = g g ∇β¯∇γ + ∇γ ∇β¯ Υαδ¯ + Rαδ¯Υγ β¯ 2 K + ∇γ Υαδ¯Vβ¯ + ∇β¯Υαδ¯Vγ + Υαδ¯Vβ¯Vγ + t 1 αβ¯ ¯ [g ∇β¯div(Υ)α + gγ δ ∇γ div(Υ)δ¯] = 2 K ¯ ¯ +gαβ gγ δ [Rαδ¯Υγ β¯ + ∇γ Υαδ¯Vβ¯ + ∇β¯Υαδ¯Vγ + Υαδ¯Vβ¯Vγ ] + t where K is the trace of Υαβ¯ with respect to gαβ¯ (x, t). In [NT] the following result, which is the Kähler analogue of an earlier result in [CH], was showed by the maximum principle.

Theorem 4.1. Let Υαβ¯ be a Hermitian symmetric tensor satisfying the L-heat equation on M × [0, T ]. Suppose Υαβ¯(x, 0) ≥ 0 (and satisfies some growth assumptions in the case M is noncompact). Then Z ≥ 0 on M × (0, T ] for any smooth vector field V of type (1, 0). PROOF COPY


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The use of the maximum principle in the proof can be replaced by the integration argument as in the proof of (1.4). For any T ≥ t0 > 0, to prove that Z ≥ 0 at t0 it suffices to show that when t = t0 , Rin order 2 Zh dµ ≥ 0 for any compact-supported nonnegative function h. t t M ∂ − ∆ + R h(y, τ ) = 0 Now we solve the conjugate heat equation ∂τ with τ = t0 − t and h(y, τ = 0) = h(y), the given compact-supported function at t0 . By the perturbation argument we may as well as assume that Υ > 0. Then let Zm (y, t) = inf V Z(y, t). It was shown in [NT] that Zm ∂ − ∆ Zm = Y1 + Y2 − 2 ∂t t where

and

Y1 = Υp¯q ∆Rp¯q + Rp¯qαβ¯Rαβ q Vα ¯ + ∇α ¯ Rp¯ q Vα ¯ + ∇α Rp¯ Rp¯q + Rp¯qαβ¯ Vα¯ Vβ + t Y2

1 1 = Υγ α¯ ∇p Vγ¯ − Rp¯γ − gp¯γ ∇p¯Vα − Rα¯p − gp¯α t t +Υγ α¯ ∇p¯Vγ¯ ∇p Vα ≥ 0.

Notice that in the above expressions, at every point (y, t) the V is the minimizing vector. This implies the monotonicity Z Z d 2 2 t Zm h dµt = t (Y1 + Y2 ) h dµt ≥ 0. dt M M

Since limt→0 t2 Zm = 0, which is certainly the case if Υ is smooth at t = 0 and Rcan be assumed so in general by shifting t with a ǫ > 0, we have that M t2 Zm h dµ|t=t0 ≥ 0. This proof via the integration by parts implies the following monotonicity formula. Proposition 4.2. Let (M, g(t)), Υ and Z be as in Theorem 4.1. For any space-time point (x0 , t0 ) with 0 < t0 ≤ T , let ℓ(y, τ ) be the reduced distance function with respect to (x0 , t0 ). Then (4.1) Z Z exp(−ℓ) exp(−ℓ) d 2 2 t Zm dµt ≥ t (Y1 + Y2 ) dµt ≥ 0. dt M (πτ )m (πτ )m M In particular, Z t0 Z exp(−ℓ) 2 2 (4.2) t0 Z(x0 , t0 ) ≥ t (Y1 + Y2 ) dµt dt ≥ 0. (πτ )m 0 M Notice that (4.2) sharpens the original Li-Yau-Hamilton estimate of [NT], by encoding the rigidity (such as Hamilton’s result on eternal solutions), out of the equality case in the Li-Yau-Hamilton estimate PROOF COPY



13

Z ≥ 0, into the integral of the right hand side. The result holds for the Riemannian case if one uses computation from [CH]. In [N3], the author discovered a new matrix Li-Yau-Hamilton inequality for the K¨ ahler-Ricci flow. (We also showed a family of equations which connects this matrix inequality to Perelman’s entropy formula.) More precisely we showed that for any positive solution u to the forward ∂ conjugate heat equation ∂t − ∆ − R u = 0, we have that 1 (4.3) Υαβ¯ := u ∇α ∇β¯ log u + Rαβ¯ + gαβ¯ ≥ 0 t under the assumption that (M, g(t)) has bounded nonnegative bisectional curvature. Using the above argument we can also obtain a new monotonicity related to (4.3). Indeed, tracing (1.21) of [N3] gives that 2 1 ∂ − ∆ Q = RQ − Rαβ¯ Υβ α¯ − Q + |Υαβ¯|2 + u |∇α ∇β log u|2 + Y3 ∂t t u ¯

where Q = gαβ Υαβ¯ and Y3 = u ∆R + |Rαβ¯ |2 + ∇α R∇α¯ log u + ∇α log u∇α¯ R 1 + Rαβ¯ ∇α¯ log u∇β log u + R t ≥ 0.

Hence we have the following monotonicity formula, noticing that Y4 := RQ − Rαβ¯ Υβ α¯ ≥ 0. Proposition 4.3. Let (M, g(t)) and (x0 , t0 ) be as in Proposition 4.2. Then Z d exp(−ℓ) 2 (4.4) t Q dµt dt M (πτ )m Z exp(−ℓ) 1 2 2 2 |Υ ¯| + u |∇α ∇β log u| + Y3 + Y4 dµt ≥ t u αβ (πτ )m M ≥ 0. In particular,

(4.5) t20 Q(x0 , t0 ) Z t0 Z 1 exp(−ℓ) 2 2 2 ≥ t |Υ ¯| + u |∇α ∇β log u| + Y3 + Y4 . u αβ (πτ )m 0 M Again the advantage of the above monotonicity formula is that it encodes the consequence on equality case (which is that (M, g(t)) is an gradient expanding soliton) into the the right hand side integral. Without Ricci flow, we can apply the similar argument to prove LiYau’s inequality and obtain a monotonicity formula. More precisely, let (M, g) (n = dimR M ) be a complete Riemannian manifold with PROOF COPY


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nonnegative Ricci curvature. Let u(x, t) be a positive solution to the heat equation on M × [0, T ]. Li and Yau proved that n ≥ 0. ∆ log u + 2t Another way of proving the above Li-Yau’s inequality is through the above integration by parts argument and the differential equation 2 2 2 ∂ − ∆ Q = |Υij |2 − Q + Rij ∇i u∇j u ∂t u t u where ui uj u Υij = ∇i ∇j u + gij − 2t u n ). This together with Cheeger-Yau’s and Q = gij Υij = u(∆ log u + 2t theorem [CY] on lower bound of the heat kernel, gives the following monotonicity formula, which also give characterization on the manifold if the equality holds somewhere for some positive u. Proposition 4.4. Let (M, g) be a complete Riemannian manifold with non-negative Ricci curvature. Let (x0 , t0 ) be a space-time point with t0 > 0. Let τ = t0 − t. Then Z d 2 ˆ (4.6) t Q(y, t)H(x0 , y, τ ) dµ(x) dt M ! 2 Z 1 2 ∇i ∇j log u + gij + Rij ∇i log u∇j log u uH ˆ dµ ≥ 0, ≥ 2t 2t M 2 d (x0 ,y) 1 ˆ 0 , y, τ ) = with d(x0 , y) being the diswhere H(x exp − n 4τ (4πτ ) 2

tance function between x0 and y. In particular, we have that n u∆ log u + u (x0 , t0 ) (4.7) 2t ! 2 Z t0 Z 2 1 2 ˆ ∇i ∇j log u + gij + Rij ∇i log u∇j log u uH. ≥ 2 t 2t t0 0 M

It is clear that (4.7) improves the estimate of Li-Yau slightly by providing the lower estimate, from which one can see easily that the equality (for Li-Yau’s estimate) holding somewhere implies that M = Rn (this was first observed in [N1], with the help of an entropy formula). The expression in the right hand side of (4.4) also appears in the linear entropy formula of [N1]. One can write down similar improving results for the Li-Yau type estimate proved in [N1], which is a linear analogue of Perelman’s estimate vH ≤ 0, and the one in [N2], which is a linear version of Theorem 4 above. For example, when M is a complete Riemannian manifold with −f the nonnegative Ricci curvature, if u = H(x, y, t) = e n , the fun(4πt) 2

damental solution to the heat equation centered at x at t = 0, letting PROOF COPY



15

ˆ is the ‘pseudo W = t(2∆f − |∇f |) + f − n, we have that W ≤ 0. If H backward heat kernel’ defined as in Proposition 4.4 we have that Z d ˆ 0 , y, τ ) dµ(y) (−W )uH(x dt M ! 2 Z 1 ∇i ∇j f − gij + Rij ∇i f ∇j f uH(x ˆ 0 , y, τ ) dµ(y) ≥ 0 = 2t 2t M and

(−W u) (x0 , t0 ) ≥ 2

Z

t0

t 0

Z

M

! 2 1 ˆ ∇i ∇j f − gij + Rij ∇i f ∇j f uH. 2t

If we assume further that M is a complete Kähler manifold with nonnegative bisectional curvature and u(y, t) is a strictly plurisubharmonic solution to the heat equation with w = ut , then Z Z d w ˆ ˆ 0 , y, t) dµ(y) ≥ 0, t2 Zm H(x0 , y, t) dµ(y) = t2 Y5 H(x dt M M where w w Zm (y, t) = inf wt + ∇α wVα¯ + ∇α¯ wVα + uαβ¯Vα¯ Vβ + t V ∈T 1,0 M and 1 1 Y5 = uγ α¯ ∇p Vγ¯ − gp¯γ ∇p¯Vα − gp¯α t t +uγ α¯ ∇p¯Vγ¯ ∇p Vα + Rαβs u V ¯ t¯ s¯t β Vα ¯ ≥ 0 w . In particwith V being the minimizing vector in the definition of Zm ular, Z t0 Z ∂2 ˆ 0 , y, t) dµ(y) dt. u(y, t) (x0 , t0 ) ≥ t2 Y5 H(x ∂(log t)2 0 M

This sharpens the logarithmic-convexity of [N2]. Finally we should remark that in all the discussions above one can ˆ replace the ‘pseudo backward heat kernel’ H(y, t; x0 , t0 ) = (or

exp(−ℓ(y,τ )) , n (4πτ ) 2

r 2 (x0 ,y) ) 4(t0 −t) n 2 (4π(t0 −t))

exp(−

centered at (x0 , t0 ) in the case of Ricci flow), which we

ˆ wrote before as H(y, x0 , τ ) by abusing the notation, by the fundamental solution to the backward heat equation (even by constant 1 in the case of compact manifolds). Also it still remains interesting on how to make effective uses of these improved estimates, besides the rigidity results out of the inequality being equality somewhere. There is also a small point that should not be glossed over. When the manifold is complete noncompact, one has to justify the validity of the Green’s second identity (for example in Proposition 4.4 we need to justify that PROOF COPY


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ˆ ˆ dµ = 0). This can be done when t0 is sufficiently H∆Q − Q∆ H M small together with integral estimates on the Li-Yau-Hamilton quantity (cf. [CLN]). The local monotonicity formula that shall be discussed in the next section provides another way to avoid possible technical complications caused by the non-compactness.

R

5. Local monotonicity formulae In [EKNT], a very general scheme on localizing the monotonicity formulae is developed. It is for any family of metrics evolved by the equa∂ gij = −2κij . The localization is through the so-called ‘heat ball’. tion ∂t More precisely for a smooth positive space-time function v, which often is the fundamental solution to the backward conjugate heat equation or ˆ 0 , y, τ ) = the ‘pseudo backward heat kernel’ H(x

r 2 (x0 ,y)

e− 4τ n (4πτ ) 2

(or

e−ℓ(y,τ ) n (4πτ ) 2

in

the case of Ricci flow), with τ = t0 − t, one defines the ‘heat ball’ by Er = {(y, t)| v ≥ r −n ; t < t0 }. For all interesting cases we can check that Er is compact for small r (cf. [EKNT]). Let ψr = log v + n log r. For any ‘Li-Yau-Hamilton’ quantity Q we define the local quantity: Z |∇ψr |2 + ψr (trg κ) Q dµt dt. P (r) := Er

The finiteness of the integral can be verified via the localization of Lemma 2.2, a local gradient estimate. The general form of the theorem, which is proved in Theorem 1 of [EKNT], reads as the following. Theorem 5.1. Let I(r) = (5.1)I(r2 ) − I(r1 ) = −

Z

r2

r1

+ψr

P (r) rn .

n

Then Z

Q ∂ + ∆ − tr g κ v ∂t v

r n+1 Er ∂ − ∆ Q dµt dt dr. ∂t

It gives the monotonicity of I(r) in the cases that Q ≥ 0, which is ensured by the Li-Yau-Hamilton estimates in the case we shall consider, ∂ ∂ and both ∂t + ∆ − trg κ v and ∂t − ∆ Q are nonnegative. The non ∂ + ∆ − tr g κ v comes for free if we chose v to be the negativity of ∂t ∂ ‘pseudo backward heat kernel’. The nonnegativity of ∂t − ∆ Q follows from the computation, which we may call as in [N3] the pre-LiYau-Hamilton equation, during the proof of the corresponding Li-YauHamilton estimate. Below we illustrate examples corresponding to the monotonicity formulae derived in the previous section. These new ones expand the list of examples given in Section 4 of [EKNT]. PROOF COPY



17

For the case of Ricci/Kähler-Ricci flow, for a fixed (x0 , t0 ), let v = the ‘pseudo backward heat kernel’, where ℓ is the reduced dis-

e−ℓ(y,τ ) n , (4πτ ) 2

tance centered at (x0 , t0 ). Example 5.2. Let Zm , Y1 and Y2 be as in Proposition 4.2. Let Q = t2 Zm . Then Z 2 n d I(r) ≤ − n+1 t ψr (Y1 + Y2 ) dµt dt ≤ 0 dr r Er

and

Q(x0 , t0 ) ≥ I(¯ r) +

Z

r¯ 0

n r n+1

Z

Er

t2 Q,

t2 ψr (Y1 + Y2 ) dµt dt dr.

Example 5.3. Let u, Q = Υαβ¯, Y3 and Y4 be as in Proposition 4.3. Then Z n 1 d 2 2 2 I(r) ≤ − n+1 |Υ ¯| + u |∇α ∇β log u| + Y3 + Y4 ≤ 0 t ψr dr r u αβ Er and

Q(x0 , t0 ) ≥ I(¯ r )+

Z

r¯

n r n+1

0

Z

t2 ψr Er

1 |Υαβ¯|2 + u |∇α ∇β log u|2 + Y3 + Y4 . u

For the fixed metric case, we may choose either v = H(x0 , y, τ ), ˆ 0 , y, τ ) = the backward heat kernel or v = H(x

d2 (x0 ,y)

e− 4τ n (4πτ ) 2

, the ‘pseudo

backward heat kernel’. Example 5.4. Let u and Q be as in Proposition 4.4. Let Q = t2 Q and f = log u. Then ! 2 Z 1 d 2n 2 t uψr ∇i ∇j f + gij + Rij ∇i f ∇j f dµ dt ≤ 0 I(r) ≤ − n+1 dr r 2t Er and

Q(x0 , t0 ) ≥ I(¯ r )+

Z

r¯

0

2n r n+1

Z

Example 5.5. Let u =

2

t uψr Er e−f n (4πt) 2

! 2 1 ∇i ∇j f + gij + Rij ∇i f ∇j f . 2t

be the fundamental solution to the

(regular) heat equation. Let W = t(2∆f −|∇f |2 )+f −n and Q = −uW . Then ! 2 Z 2n 1 d I(r) ≤ − n+1 tuψr ∇i ∇j f − gij + Rij ∇i f ∇j f dµ dt ≤ 0 dr r 2t Er

and

Q(x0 , t0 ) ≥ I(¯ r )+ PROOF COPY

Z

r¯

0

2n r n+1

Z

tuψr Er

! 2 1 ∇i ∇j f − gij + Rij ∇i f ∇j f . 2t NOT FOR DISTRIBUTION

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Note that this provides another localization of entropy other than the one in [N3] (see also [CLN]). Example 5.6. Let M be a complete Kähler manifold with nonnegw and Y be as in the last case ative bisectional curvature. Let u, Zm 5 2 w considered in Section 4. Let Q = t Zm . Then Z n d I(r) ≤ − n+1 t2 Y5 ψr dµ dt dr r Er and

Z r¯ Z ∂2 n u(x, t) (x , t ) ≥ I(¯ r ) + t2 Y5 ψr dµ dt dr. 0 0 n+1 ∂(log t)2 r 0 Er

Acknowledgement. We would like to thank Ben Chow and Peng Lu for continuously pressing us on a understandable proof of (1.4). We started to seriously work on it after the visit to Klaus Ecker in August and a stimulating discussion with him. We would like to thank him for that, as well as Dan Knopf and Peter Topping for discussions on a related issue. References [CY] J. Cheeger and S.-T. Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math. 34 (1981), no. 4, 465–480. [CH] B. Chow and R. Hamilton, Constrained and linear Harnack inqualities for parabolic equations , Invent. Math. 129 (1997), 213–238. [CLN] B. Chow, P. Lu and L. Ni, Hamilton’s Ricci flow, Vol 1 and 2, in preparation. [EKNT] K. Ecker, D. Knopf, L. Ni and P. Topping, Local monotonicity formulae for evolving metrics , preprint. [GL] N. Garofalo and E. Lanconelli, Asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients, Math. Ann. 283 (1989), no. 2, 211–239. [G] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083. [H] R. Hamilton, A matrix Harnack estimate for the heat equation , Comm. Anal. Geom. 1 (1993 ), no. 1, 113–126. [LY] P. Li and S.-T. Yau, On the parabolic kernel of the Schr¨ odinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. [N1] L. Ni, The entropy formula for linear heat equation, Jour. Geom. Anal. 14(2004), 85–98; Addenda, 14 (2004), 329–334. [N2] L. Ni, A monotonicity formula on complete K¨ ahler manifold with nonnegative bisectional curvature , J. Amer. Math. Soc. 17 ( 2004 ), 909–946. [N3] L. Ni, A new matrix Li-Yau-Hamilton inequality for K¨ ahler-Ricci flow, to appear in J. Differential Geom. [NT] L. Ni and L.-F. Tam, Plurisubharmonic functions and the K¨ ahler-Ricci flow, Amer. J. Math. 125 ( 2003 ), 623–654 . PROOF COPY



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[P] G. Perelman The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/ 0211159. [SY] R. Schoen and S.-T. Yau, Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994. v+235 pp. [STW] N. Sesum, G. Tian and X. Wang, Notes on Perelman’s paper . [Ye] R. Ye, Notes on reduced volume and asymptotic Ricci solitons of κ-solutions, http://www.math.lsa.umich.edu/research/ricciflow/perelman.html. Department of Mathematics, University of California at San Diego, La Jolla, CA 92093 E-mail address: [email protected]

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arXiv:math/0502494v2 [math.DG] 14 Mar 2005

¨ ANCIENT SOLUTIONS TO KAHLER-RICCI FLOW

Lei Ni1 University of California, San Diego August 2004 (Revised, March 2005) Abstract. In this paper, we prove that any non-flat ancient solution to K¨ ahler-Ricci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also prove that any gradient shrinking solitons with positive bisectional curvature must be compact. Both results generalize the corresponding earlier results of Perelman in [P1] and [P2]. The results can be applied to study the geometry and function theory of complete K¨ ahler manifolds with nonnegative bisectional curvature via K¨ ahler-Ricci flow. It also implies a compactness theorem on ancient solutions to K¨ ahler-Ricci flow.

§0 Introduction. The K¨ ahler-Ricci flow (0.1)

∂ g ¯(x, t) = −Rαβ¯(x, t) ∂t αβ

has been useful in the study of complex geometry in the work of [B, C1, M2], etc. The ancient solutions arise [H5] when one applies the parabolic blow-up to a finite time singularity or slowly forming (Type II) singularity as t approaches infinity. In [H5] Hamilton introduced some geometric invariants associated with ancient solutions. One of them is the so-called asymptotic volume ratio (also called cone angle at infinity), which is defined as V(M, g) := limr→∞ Vωon(r) rn , for a complete Riemannian manifold (M, g). Here n is the dimension (real) of M , Vo (r) is the volume of the ball of radius r centered at o (with respect to metric g) and ωn is the volume of unit ball in Rn . This asymptotic volume ratio is well-defined and independent of the choice of o in the case when M has non-negative Ricci curvature. One should consult [H5] for the condition (and the proof) under which V(M, g(t)) is independent of t for a family of metrics g(t) satisfying Ricci flow. When the meaning is clear in the context we simply denote V(M, g(t)) by V(g(t)). In [P1], Perelman studied the properties of ancient solutions with bounded nonnegative curvature operator, via his entropy and reduced volume (reduced distance) monotonicity. In particular, the following result is proved. 1 Research

partially supported by NSF grants and an Alfred P. Sloan Fellowship, USA.

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Theorem 1. Let (M, g(t)) be a complete non-flat ancient solution to Ricci flow, with bounded nonnegative curvature operator. Then V(g(t)) = 0. This result holds the key to the rest striking results on ancient solutions in [P1]. The assumption on the nonnegativity of the curvature operator is ensured in dimension three by Hamilton-Ivey’s pinching estimate [H5, Theorem 24.4], if the ancient solution is obtained as a blow-up limit of a finite time singularity. It is also needed to make effective uses of the reduced distance introduced in [P1, Section 7] in Perelman’s blow down procedure [P1. Section 11] in the study of the ancient solutions. For K¨ ahler-Ricci flow, one would like to replace the nonnegativity of the curvature operator by the nonnegativity of bisectional curvature since the nonnegativity of the sectional curvature is neither natural nor necessarily preserved under Kähler-Ricci flow. On the other hand, the argument of [P1, Section 11] made essential uses of the nonnegativity of sectional curvature and the properties of such Alexandrov spaces. There is no obvious way of adapting the proof of [P1, Section 11.4] to the K¨ ahler setting. Therefore one needs some new ingredients in order to generalize Perelman’s result to Kähler-Ricci flow assuming only the nonnegativity of the bisectional curvature. It turns out that this technical hurdle can be overcome by a result (Proposition 1.1) on the shrinking solitons, Perelman’s blow-down procedure and a splitting result for K¨ ahler manifolds with nonnegative bisectional curvature, proved recently by Luen-Fai Tam and the author in [NT2]. These will be the main focus of this paper. There exists another motivation of proving Theorem 1 for K¨ ahler-Ricci flow on K¨ ahler manifolds with nonnegative bisectional curvature. In a AMS meeting (November 2001, held at Irvine, California), Huai-dong Cao proposed the following conjecture (see also [C4] for a related problem on ancient solutions). Conjecture (Cao). Let (M m , g(t)) be a non-flat steady gradient K¨ ahler-Ricci soliton with bounded nonnegative bisectional curvature. Then V(g(t)) = 0. Proving Theorem 1 for Kähler-Ricci flow on K¨ ahler manifolds with nonnegative bisectional curvature will imply Cao’s conjecture. Combining the techniques from [P1] and [P2] with [NT2], we indeed can prove the following such generalization of Theorem 1. Theorem 2. Let (M m , g(t)) (m = dimC (M ), n=2m) be a non-flat ancient solution to K¨ ahler-Ricci flow (0.1). Assume that (M, g(t)) has bounded nonnegative bisectional curvature. Then V(g(t)) = 0. The only known result along this line, under the assumption of nonnegative bisectional curvature has been only proved for the case of m = 2, and only for the gradient K¨ ahler-Ricci solitons, which are special ancient solutions. Please see, for example [C4, CZ2]. Following [P1], an immediate consequence of Theorem 2 is the following compactness result. For an fixed κ > 0, the set of κ-solution to K¨ ahler-Ricci flow is compact module scaling. Please refer to Section 2 for the definitions of the κ-solutions. A compactness result for the elliptic case, namely the Kähler-Einstein metrics, was proved earlier in [T] for compact manifolds under extra assumptions on an integral bound of curvature, a volume lower bound and a diameter upper bound. Since for the Riemannian manifolds with non-negative Ricci curvature, the Bishop volo (r) ume comparison theorem asserts that ωV2m r2m is monotone non-increasing, the manifold M

ANCIENT SOLUTIONS

3

is called of maximum volume growth if V(M, g) > 0. The above result simply concludes that the ancient solutions with bounded nonnegative bisectional curvature (which is preserved under the Kähler-Ricci flow, by [B, M2]) can not be of maximum volume growth. Applying Shi’s short time existence result [Sh1], Hamilton’s singularity analysis argument in [H5, Theorem 16.2] (or Perelman’s result in [P1, Section 11]), as well as estimates from [NT1] and [N2] one can have the following corollary of Theorem 2. Corollary 1. Let (M m , g0 ) (m = dimC (M )) be a complete K¨ ahler manifold with bounded nonnegative bisectional curvature. Assume that M is of maximum volume growth. Then the K¨ ahler-Ricci flow (0.1), with g(x, 0) = g0 (x) has a long time solution. Moreover the solution has no slowly forming (Type II) singularity as t approaches ∞. In particular, there exists a C = C(M, g0 ) > 0 such that the scalar curvature R(y) satisfies Z C (0.2) R(y) dµ ≤ , (1 + r)2 Bx (r)

R where Bx (r) is the ball of radius r centered at x, Vx (r) = Vol(Bx (r)) and Bx (r) f (y) dµ = R 1 f (y) dµ. Furthermore, M is differmorphic (homeomorphic) to Cm , for m > 2 Vx (r) Bx (r) (m = 2), and is biholomorphic to a pseudoconvex domain in Cm .

The curvature decay statement of Corollary 1 confirms a conjecture of Yau in [Y, page 621], where he speculated that two assumptions in Shi’s main theorem of [Sh3] can be replaced by the maximum volume growth alone. See also the recent work of Wu and Zheng [WuZ] on various examples related to the above corollary. Corollary 2 also provides a partial answer to a question asked in [N2, Conjecture 3.1] on conditions equivalent to the existence of holomorphic functions of polynomial growth. In fact, there the author speculated that either the maximum volume growth or the average quadratic curvature decay is equivalent to the existence of nonconstant polynomial growth holomorphic functions provided the manifold has quasi-positive bisectional curvature. There also exists a related general conjecture of Yau on the non-existence of the bounded holomorphic functions, on which one can refer to [LW] for some recent progresses. Using Perelman’s Theorem 1 one can conclude the similar result for the Riemannian manifolds with bounded nonnegative curvature operator and maximum volume growth. For K¨ ahler-Ricci flow, under the stronger assumption that M is a Kähler manifold with bounded nonnegative curvature operator and maximum volume growth, a similar (but slightly weaker) estimate as (0.2) was proved earlier in [CZ4] by applying the dimension reduction argument of Hamilton. This is in fact a fairly easy consequence of Perelman’s Theorem 1. (See the proof of Corollary 1 for details. The main constraint of related results in [H5], for the application to K¨ ahler geometry, is that the argument of [H5] only works under the stronger assumption on nonnegativity of curvature operator/sectional curvature, which is sufficient for the study of three manifolds, but not for the study of Kähler manifolds of complex dimension ≥ 3.) In [Sh3-4], the long time existence result was proved under a uniform curvature decay assumption similar to (0.2), with/without the assumption of the maximum volume growth. In Corollary 1 we have the curvature decay as a consequence instead of an assumption, exactly as what Yau speculated in [Y]. Under certain further curvature average decay assumptions, the topological conclusion in Corollary 1 was proved earlier in [CZ3]. (See also [H6, Sh4] for related

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earlier fundamental works. The key observation on the improvement of injectivity radius lower bound was first observed in [Sh4].) The last statement of Corollary 1 was proved first in [Sh4] (see also [CZ3]) under the assumptions that the manifold has positive sectional curvature and a certain curvature average decay condition. Note also that Corollary 4.1 of [NT2] proved that M is Stein if M has maximum volume growth and nonnegative bisectional curvature (which is a conjecture of H. Wu). Corollary 2. Let (M m , g0 ) be a complete K¨ ahler manifold with bounded nonnegative holomorphic bisectional curvature and maximum volume growth. Then the transcendence degree of the rational function field of M (see [N2] for definition) is equal to m. Furthermore, M is biholomorphic to an affine quasi-algebraic variety. When m = 2, M is biholomorphic to C2 . In [M1, M3], a systematic scheme on embedding/compactifying complete K¨ ahler manifolds with positive curvature was developed. One can also consult the survey article [M4] for expositions on these methods and related results. Originally in [M1] under the assumption that M has positive bisectional curvature and certain curvature decay conditions, the conclusion that M is biholomorphic to C2 in the case m = 2, and that M is biholomorphic to an affine variety in the higher dimension were obtained. In fact in his fundamental work [M1] Mok laid down all the framework of the affine embedding and observed that one can apply the result of Ramanujam, which asserts that a quasi-projective surface homeomorphic to R4 must be biholomorphic to C2 , in the case m = 2. Later, for the case m = 2 only, in [CTZ], following Mok’s compactification scheme in [M1. M3], the authors improved the above result of Mok, replacing the assumptions of [M1] by that M has bounded positive bisectional curvature and of maximum volume growth. The result stated in [CTZ] also assumes a mild average curvature decay condition which replaces the stronger point-wise curvature decay assumption of [M1] via K¨ ahler-Ricci flow. This average curvature decay condition can be removed by combining with another later paper [CZ4], again assuming the positivity of the bisectional curvature and volume being of maximum growth. In this later improvement [CTZ], which is only restricted to the case of m = 2 (also the earlier paper [CZ2]), the proof also crucially relies on an observation only valid in complex dimension two, originally due to Ivey [I], that an ancient solution to K¨ ahler-Ricci flow with nonnegative bisectional curvature must have nonnegative curvature operator. Namely the method there relied crucially on the dimensional reduction results of Hamilton in [H5]. Hence it does not work under the assumption of the nonnegativity of bisectional curvature in higher dimensions. In Corollary 2, when m = 2, our statement assumes only nonnegativity instead of positivity of the bisectional curvature. Moreover, our new approach also works for the higher dimensional case. Note that the uniform multiplicity estimates recently proved in [N2] simplifies the steps in [M1] quite a bit. We should point out that the m = 2 case can also be obtained by combining Corollary 4.1 in [NT2] obtained by Luen-Fai Tam and the author with the argument of [CTZ]. However the method of current paper provides an unified direct approach which works for any dimensions. In the proof of Theorem 2 we need the following result on gradient shrinking solitons of K¨ ahler-Ricci flow, which is of independent interests.

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5

Theorem 3. Let (M m , g) be a non-flat gradient shrinking soliton to K¨ ahler-Ricci flow. (i) If the bisectional curvature of M is positive then M must be compact and isometricbiholomorphic to Pm . ˜ splits as (ii) If M has nonnegative bisectional curvature then the universal cover M ˜ = N1 × N2 × · · · × Nl × Ck isometric-biholomorphically, where Ni are compact M irreducible Hermitian Symmetric Spaces. In particular, V(M, g) = 0. Theorem 3 generalizes a corresponding recent result of [P2, Lemma 1.2], where Perelman shows that in dimension three, any κ-noncollapsed gradient shrinking soliton with bounded positive sectional curvature must be compact. When M is compact and m = 2, the classification in part (ii) was obtained in [I] under even weaker assumption on the nonnegativity of the isotropic curvature. For recent works on Kähler-Ricci flow on compact manifolds, one should refer to the survey articles [CC, Cn]. Perelman [P3] also has some important work on the conjecture (which arises from the related works of Hamilton and Tian) concerning the large time behavior of (normalized) Kähler-Ricci flow on compact K¨ ahler manifolds with c1 (M ) > 0 (cf. [N3]). In [CT1-2], very recent progresses have been made towards the uniformization problem addressed in Corollary 2. Finally, we should point out that when dimension m = 1 the above results are known from the work of Hamilton [H3] and [H5, Section 26]. (See also the work of Chow [Ch] and books [CK, CLN].) The method of this paper has other applications in the study of Ricci flow on Riemannian manifolds. Please see Section 3. Acknowledgement. The author would like to thank Professors H.-D. Cao, B. Chow, T. Ilmanen, R. Schoen and Jon Wolfson for helpful discussions, Professor P. Lu for pointing out the reference [Ye], Professors L.-F. Tam, M.-T. Wang, H. Wu and F.-Y. Zheng for their interests. Professor H.-D. Cao pointed out to the author that the earlier version of Corollary 2 has further complex geometric consequences relating to the uniformization of K¨ ahler manifolds with nonnegative bisectional curvature. It is a pleasure to record our gratitude to him. Finally we would like to thank the referee for pointing out a discrepancy in the earlier version of this paper (of which the author was also aware independently). §1 Proof to Theorem 2 and 3. Recall that a complete Riemannian manifold (M, g) is called a gradient shrinking soliton if there exists a smooth function f such that, for some positive constant a, (1.1)

∇i ∇j f + Rij − agij = 0.

Proposition 1.1. Let (M, g) be a Ricci non-flat gradient shrinking soliton. Assume that the Ricci curvature of M is nonnegative. Then there exists a δ = δ(M ) (1 ≥ δ > 0) such that (1.2)

R(x) ≥ δ > 0.

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Proof. It is well know from the strong maximum principle that the scalar curvature R(x) > 0. Differentiating (1.1) and applying the second Bianchi identity, we have that ∇i R = 2Rij fj .

(1.3) This implies that

∇i R + 2fij fj − 2afi = 2 (Rij + fij − agij ) fj =0 which further implies that there exists a constant C1 = C − 1(M ) such that R + |∇f |2 − 2af = C1 .

(1.4)

These equations are well known for the gradient shrinking solitons. Let o ∈ M be a fixed point. For any x ∈ M we denote the distance of x from o by r(x). Let γ(s) be minimal geodesic joining x from o, parametrized by the arc-length. For simplicity we often also denote r(x) by s0 . Let {Ei (s)} (0 ≤ i ≤ n − 1) be a parallel frame along γ(s) such that E0 (s) = γ ′ (s). If s0 ≥ 2, for s0 ≥ r0 ≥ 1, choose n − 1-variational vector fields Yi (s) (1 ≤ i ≤ n − 1) along γ(s) as    sEi (s), 0 ≤ s ≤ 1 Ei (s), 1 ≤ s ≤ s0 − r0 Yi (s) = .   s0 −s Ei (s), s0 − r0 ≤ s ≤ s0 r0

By the second variation consideration [P1, Lemma 8.3 (b)] (see also [H5, Theorem 17.4]), we have that n−1 X Z s0 |Yi′ (s)|2 − R(γ ′ (s), Yi (s), γ ′(s), Yi (s)) ds ≥ 0. 0

i=1

In particular we can find C(M ), which depends only the upper bound of the Ricci curvature of M on Bo (1), such that (1.5) Z

0

s0 −r0

n−1 − Ric(γ (s), γ (s)) ds ≤ C(M ) + r0 n−1 . ≤ C(M ) + r0 ′

′

Z

s0

s0 −r0

s0 − s r0

2

Ric(γ ′ (s), γ ′(s)) ds

Here we have used the fact the Ricci curvature is nonnegative. We claim that there exists a positive constant A = A(M ), if s0 ≥ A and R(x) ≤ 1, there exists another constant, still denoted by C(M ), such that (1.6)

Z

0

s0

Ric(γ ′ (s), γ ′ (s)) ds ≤

a s0 + C(M ). 2

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7

Assume that we have proved the claim (1.6). Then there exists C2 = C2 (M ) > 0 such that h∇f, γ

′

(s)i|γ(r(x)) o

Z

s0

d (h∇f, γ ′(s)i) ds ds 0 Z s0 dγ i (s) dγ j (s) = (∇i ∇j f ) ds ds ds 0 Z s0 = (a − Ric(γ ′ (s), γ ′ (s))) ds 0 Z s0 = ar(x) − Ric(γ ′ (s), γ ′ (s)) ds

=

0

a ≥ r(x) − C2 , 2

which implies that for every x ∈ M \ Bo (A) with R(x) ≤ 1, h∇f, ∇ri(x) ≥

a r(x) − C2 − |∇f |(o). 2

It in particular implies that for every such x, with r(x) ≥ a4 (C2 + |∇f |(o)) additionally, ∇f 6= 0. Now we can prove the proposition after the claim (1.6). For any x ∈ M \ (Bo (A) ∪ Bo ( a4 (C2 + |∇f |(o)))), without the loss of generality we can assume that R(x) ≤ 1, let σ(η) be the integral curves of ∇f , passing x with σ(0) = x. By (1.3) we have that −

(1.7)

d dσ i dσ j (R(σ(η)) = −2Rij ≤ 0. dη dη dη

This implies that R(x) ≥ R(σ(η)), for η < 0. On the other hand, (1.8)

−

d r(σ(η)) = −h∇r, ∇f i ≤ −(C2 + |∇f |(o)) ≤ 0 dη

as far as r(σ(η)) ≥ max(A, a4 (C2 + |∇f |(o))), noticing that we always have R(σ(η)) ≤ 1. This implies that the integral curve σ exists for all η < 0 since |∇f | is bounded inside the closed ball Bo (2r(x)). The estimate (1.8) also implies that there exists η1 < 0 such that r(σ(η1 )) = max(A, a4 (C2 + |∇f |(o))). Applying (1.7) we have that R(x) ≥

inf

y∈Bo (r(σ(η1 )))

R(y).

This proves the proposition assuming the claim (1.6). Now we prove the claim (1.6). First by equation (1.3) and the fact Rij ≥ 0 we have that fij ≤ agij . This implies that along any minimizing geodesic γ(s) from o, f ′′ (s) ≤ a. Hence there exists B = B(M ) such that f (x) ≤ (a + 1)r 2 (x)

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for r(x) ≥ B. Using (1.4) and the fact that R > 0 we have that |∇f |(x) ≤ 2(a + 1)r(x) for r(x) ≥ B. On the other hand, (1.3) also implies that |∇R|2 ≤ 4R2 |∇f |2 . The above two inequality implies the the estimate |∇ log R|(x) ≤ 2(a + 1)r(x) for r(x) ≥ B. Now we adapt the notations and situations right before (1.6) and choose r0 a in (1.5) such that n−1 r0 = ǫs0 with some fixed positive constant ǫ ≤ min(1, 2 ). Then (1.6) implies that Z

(1.9)

0

s0 −r0

Ric(γ ′ (s), γ ′ (s)) ds ≤ C(M ) + ǫs0 .

Notice that r0 = n−1 ≤ n−1 ≤ s20 if s0 ≥ A for some A = A(M ) ≥ max(1, 2B, 2 n−1 ). Now ǫs0 ǫ ǫ using the gradient estimate on log R above we have that Z s0 d R(γ(s1 )) =− log R(γ(s)) ds log R(γ(s0 )) s1 ds Z s0 ≤ |∇ log R| ds s1

≤ 2(a + 1)s0 (s0 − s1 )

for s1 ≤ s0 . Hence R(γ(s)) ≤ R(γ(s0 )) exp(

2(a + 1)(n − 1) 2(a + 1)(n − 1) ) ≤ exp( ) ǫ ǫ

for any s ≥ s0 − r0 . Here we have used the assumption R(x) = R(γ(s0 )) ≤ 1. This further implies that Z s0 Z s0 ′ ′ Ric(γ (s), γ (s)) ds ≤ R(γ(s)) ds s0 −r0

s0 −r0

2(a + 1)(n − 1) ) ǫ n−1 2(a + 1)(n − 1) = exp( ) ǫs0 ǫ ≤ C(ǫ, M ).

≤ r0 exp(

Together with (1.9), we prove our claim (1.6). Hence we complete the proof of the Proposition 1.1.

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Proof of Theorem 3. Case (i). By the strong maximum principle we can assume that R > 0 everywhere, otherwise one would have that M is Ricci flat, hence flat. By Proposition 1.1 , we know that R ≥ δ > 0, for some δ. In particular, it implies that for any x Z R(y) dµ ≥ δ. Bx (r)

R

Here is the average integral defined in Corollary 1 of the introduction. On the other hand, part (ii) of Theorem 4.2 in [NT2] implies that if M is not compact, then it must satisfies Z C2 (1.10) R(y) dµ ≤ 1+r Bx (r) for some C2 (x, M ) > 0. This is a contradiction! This implies that M must be compact. ˜ be the universal cover of M . Since M ˜ is still a shrinking soliton, we Case (ii). Let M ˜ . From part (i) of Theorem 4.2 in [NT2], we can split M ˜ can apply Proposition 1.1 to M as a product of two manifolds with one of the factor being compact and the other satisfies the curvature average decay (1.10). The classification result follows from the celebrated result of Siu-Yau [SiY], Mok [M2] as well as an observation of Koiso [Ko]. The conclusion V(M, g) = 0 now follows easily from (i) and (ii). The proof of Theorem 2 requires the blow-down procedure of Perelman in [P1, Proposition 11.2]. Since Proposition 11.2 of [P1] has only a sketched proof, in the following we present in a more detailed way the blow-down procedure of Perelman of [P1, 11.2] on the study of so-called bounded κ-solutions, complete ancient solutions with bounded nonnegative curvature operator and κ-non-collapsed in all scales. (See also [CLN], [KL], [STW] and [Ye] for various expositions on this part.) A solution (to Ricci flow) is called κ-non-collapsed if for any time t, for any ball Bx (r) (with respect to metric g(t)), satisfying |Rm|(y) ≤ r12 for all y ∈ Bx (r) one has Vx (r) ≥ κr n . One should refer to [P1, Section 4] for more discussions on κ-non-collapsing, [P1, Section 7, 11] for more details of the reduced distance and its properties. Now we adapt [P1, Section 7, 11] to K¨ ahler-Ricci flow and replace the nonnegativity of the curvature operator by the nonnegativity of the bisectional curvature. So in the following, we let (M, g(t)) be a complete non-flat ancient solution to Kähler-Ricci flow. We also assume that (M, g(t)) has bounded nonnegative bisectional curvature (the boundedness of curvature can be replaced by the differential Harnack inequality, (1.14) below) and it is κ-non-collapsed for some κ > 0. First we recall the definition of the reduced distance ℓ(y, τ ). Fix a space-time point (x0 , t0 ). Let τ = t0 − t. Define Z τ √ 1 √ ℓ(y, τ ) = inf η R + 4|γ ′ (η)|2 dη. γ,γ(0)=x0 ,γ(τ )=y 2 τ 0 We have factor 4 here since we study K¨ ahler-Ricci flow instead of Ricci flow. Since the R ≥ 0 and the metrics are shrinking (since Rαβ¯ ≥ 0) along the t direction it is easy to see that d2 (y, x0 ) . ℓ(y, τ ) ≥ t0 τ

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(Sometimes we also denote dt0 (y, x0 ) by d0 (y, x0 ) when we think in terms of τ .) As a consequence of above lower bound on ℓ we know that ℓ(y, τ ) achieves its minimum for each τ at some point finite distance away from x0 . Thus we can conclude that Claim 1. For each τ there exists a point x(τ ) such that ℓ(x(τ ), τ ) =

n . 2

Proof. By the equation (7.15) in [P1], using the maximum principle it was shown in [P1, Section 7] that miny∈M ℓ(y, τ ) ≤ n2 . Using the continuity and (1.10) we know the existence of x(τ ). Tom Ilmanen has a nice geometric explanation on this fact by making analogy with the mean curvature flow. Let us recall the equations satisfied by l(y, τ ) from [P1, Section 7]. First we have |∇ℓ|2 + R =

(1.11) Here

K=

Z

τ

1 1 ℓ − 3 K. τ τ2 3

η 2 H(X) dη

0

where H(X) = −Rτ − 2h∇R, Xi − 2hX, ∇Ri + 4Ric(X, X) − τ1 R is the trace different Harnack (also called Li-Yau-Hamilton) expression for shrinking solitons, X is (1, 0) component of the tangent vector of the minimizing L-geodesics. Notice that h·, ·i is the Hermitian product with respect to the Kähler metric. We also have that |∇ℓ|2 + ℓτ = −

(1.12)

1 3 K. 2τ 2

and ℓτ = R −

(1.13)

1 ℓ + 3 K. τ 2τ 2

Applying H.-D. Cao’s [C2] (in stead of Hamilton’s) trace differential Harnack (also called Li-Yau-Hamilton inequality) for ancient solutions to K¨ ahler-Ricci flow, (1.4)

−Rτ − 2h∇R, Xi − 2hX, ∇Ri + 4Ric(X, X) ≥ 0

we have that

1 H(X) ≥ − R. η

Therefore we have that K≥−

Z

0

τ

√ √ ηR dη ≥ −2 τ ℓ.

Applying the above lower bound to (1.11)–(1.13) we have that (1.15)

|∇ℓ|2 + R ≤

3 ℓ, τ

ANCIENT SOLUTIONS

|∇ℓ|2 + ℓτ ≤

(1.16)

11

ℓ τ

and ℓτ ≥ R −

(1.17)

2 2 ℓ ≥ − ℓ. τ τ

From (1.15), we have that

3 , 4τ

1

|∇ℓ 2 |2 ≤ which implies that 1 2

ℓ (y, τ ) ≤

(1.18)

r

n + dτ (x(τ ), y) 2

r

3 . 4τ

Now we can deduce the following results on the reduced distance ℓ(y, τ ). p Corollary 1.1. For y ∈ Bτ (x(τ ), τǫ ), r

ℓ(y, τ ) ≤

(1.19)

n + 2

r

3 4ǫ

!2

.

Hence ℓ(y, η) ≤ 4

(1.20) for all 2τ ≥ η ≥

τ 2

and y ∈ Bτ (x(τ ),

(1.20’)

ǫ

pτ

ǫ)

n + 2

r

3 4ǫ

!2

.

). And r

τ R(y, η) ≤ 12

(1.21) Moreover, on Bτ (x(τ ),

pτ

r

n + 2

r

3 4ǫ

× (δτ, 1δ τ ) one has that ℓ(y, η) ≤

1 ( δ

r

n + 2

r

3 ) 4ǫ

!2

!2

.

.

and (1.21’)

τ R(y, η) ≤ 3

1 ( δ

r

n + 2

r

3 ) 4ǫ

!2

.

The following corollary gives relation between the κ-constant in the definition of κnoncollapsing and the lower bound of the reduced volume. The converse is also true, even without any curvature sign assumptions. Please see [CLN] and author’s Dec, 2003 lecture at AIM for details.

12

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Corollary 1.2. There exists a constant C1 (n) > 0 such that V˜ (τ ) ≥ e−C1 (n) κ.

(1.22)

This in particular implies that limτ →∞ V˜ (τ ) > 0. Proof. Apply the (1.21) to Bτ (x(τ ), τ ) to see that it satisfies the curvature bound assumption on the ball in the non-collapsing assumption. Then the result follows from the estimate R −ℓ(y τ ) (1.20) and the definition of the reduced volume V˜ (τ ) = M e n2 dµτ . τ q p ˜ 1 (x(τ ), 1 ) = Bτ (x(τ ), τ ). The (1.21’) and the κ-nonLet g˜τ (s) = τ1 g¯(sτ ). Then B ǫ ǫ collapsing assumption implies, by Hamilton’s compactness result [H3], that ˜1 (x(τ ), B

r

1 1 ) × (δ, ), g˜τ (s) ǫ δ

!

→

B ∞ (x∞ ,

r

1 1 ) × (δ, ), g ∞ (s) ǫ δ

!

as solutions to Kähler-Ricci flow. This can be extended to an ancient solution (M∞ , g∞ (τ )). The estimates (1.15)–(1.17) ensure that ℓ(y, s) the reduced distance with respect to rescaled metric survives under the limit and converges to a function ℓ∞ (y, s). Let V∞ (s) =

Z

M∞

e−ℓ∞ (y,s) dµs . n s2

It was claimed by Perelman in [P1, 11.2] that Claim 2. lim V˜ (τ ) = V∞ (s).

(1.23)

τ →∞

In particular, one has that V∞ (s) is a constant and (M∞ , g ∞ ) is a non-flat gradient shrinking soliton. The claim follows if one can show that (1.24)

Z

(Bτ (x(τ ),

√τ

ǫ

))c

e−ℓ(y,τ ) dµτ ≤ C(ǫ) n τ2

with C(ǫ) → 0 as ǫ → 0. This can be proved easily if we can get an ‘effective’ lower bound d2 (x ,y) estimate of ℓ(y, τ ) in terms of τ τ0 . This last point was proved in [Ye, Lemma 2.2]. In fact it was proved that there exists positive constant C = C(n) such that for any y, z ∈ M . d2τ (z, y) ≤ ℓ(y, τ ). −ℓ(z, τ ) − 1 + C τ One should consult notes [KL, Ye] for more detailed exposition on the proof of Claim 2.

ANCIENT SOLUTIONS

13

Proof of Theorem 2. Assume that (M, g(t)) is an ancient solution (non-flat) defined on M × (∞, 0]. If M is compact, there is nothing to prove. So we assume that M is noncompact. We prove the theorem by the contradiction. So we assume that V(g(t0 )) > 0 for some t0 . By passing to it universal cover we can also assume that M is simply-connected. Then by part (ii) of Theorem 4.2 of [NT2] again (see also Corollary 4.1 of cited paper), we have that the scalar curvature has the average decay (1.10). Now apply Theorem 2.2 of [NT1] (and its proof for the time before t0 ) we conclude that for all t, V(g(t)) = V(g(t0 )) > 0. This in particular implies that (M, g(t)) is κ-non-collapsed in all scales (since the volume is non-collapsed even without assuming the curvature bound). Now we apply the above blow-down procedure of Perelman to obtain the limit (M∞ , g ∞), which is a gradient shrinking soliton. Since (M, g(t)) is assumed to have nonnegative bisectional curvature, the limit (M∞ , g ∞ ) also has nonnegative bisectional curvature. By the definition of (M∞ , g ∞ ), it is clear that the corresponding asymptotic volume ratio V(M∞ , g ∞ ) ≥ V(g(t0 )) > 0. On the other hand, since (M∞ , g ∞ ) is non-flat, its scalar curvature must be positive by the strong maximum principle. Now we can apply Theorem 3 (part (ii)) to the universal cover of M∞ and conclude that M∞ can not have maximum volume growth. This contradicts the fact that V(M∞ , g ∞ ) ≥ V(g(t0 )) > 0. The contradiction proves the theorem. §2 Applications to K¨ ahler manifolds and K¨ ahler-Ricci flow. Theorem 1 has nice applications to Ricci flow as shown in [P1]. Following [P1], we can derive the compactness result on ancient solutions to K¨ ahler-Ricci flow out of Theorem 2. We call an ancient non-flat solution (M, g(t)) defined on M × (−∞, 0] a κ-solution to K¨ ahler-Ricci flow, if (M, g(t)) has nonnegative bisectional curvature, satisfying the trace differential Harnack inequality (2.1)

Rt + h∇R, Xi + hX, ∇Ri + Ric(X, X) ≥ 0

for any (1, 0) vector field X, and (M, g(t)) is κ-non-collapsed on all scales for some fixed κ > 0. Recall that κ-non-collapsed on all scales means that for any time t and x0 ∈ M if for all y ∈ Bt (x0 , r), R(y, t) ≤ r12 , then V olt (Bt (x0 , r)) ≥ κr 2m . Notice that we do not require R being bounded. We formulate in such way to be able to make the result hold for any dimension. Theorem 2.1. The set of κ-solutions to K¨ ahler-Ricci flow is compact modulo scaling. The proof of the result follows the same line of the argument as Theorem 11.7 of [P1]. The argument in [P1] is quite robust. The key components of the argument in [P1] are Shi’s local derivative estimate, trace differential Harnack and the following consequence of Theorem 2. Corollary 2.1. For every ω > 0 there exist B = B(ω) < ∞, C = C(ω) < ∞, τ0 = τ0 (ω) > 0, with the following properties. (a) Suppose we have a (not necessarily complete) solution g(t) to the K¨ ahler-Ricci flow, defined on M ×[t0 , 0], so that at time t = 0 the metric ball B0 (x0 , r0 ) is compactly contained in M. Suppose that at each time t, t0 ≤ t ≤ 0, the metric g(t) has nonnegative bisectional

14

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curvature, and V olt (Bt (x0 , r0 )) ≥ ωr0n . Then we have an estimate R(x, t) ≤ Cr0−2 + B(t − t0 )−1 whenever dt (x, x0 ) ≤ 14 r0 . (b) If, rather than assuming a lower bound on volume for all t, we assume it only for t = 0, then the same conclusion holds with −τ0 r02 in place of t0 , provided that −t0 ≥ τ0 r02 .

The corollary above is exactly the same as Corollary 11.6 of [P1] with only the Ricci flow being replaced by Kähler-Ricci flow and curvature operator being replaced by bisectional curvature. The proof is the also the same if one replaces Theorem 1 of Perelman, whereever it is needed, by Theorem 2 of this paper. The robust scaling argument in the proof of Corollary 11.6 in [P1] also resembles, to some degree in the spirit, various curvature estimates proved by certain scaling argument in the study of the mean curvature flow and other PDEs. See for example, [S1, E, Wh], as well as Simon’s proof on Schauder’s estimates in [S2]. The detailed exposition on the proof of the above Theorem 2,1 and Corollary 2.1 can be found in [CLN, KL, STW] as well as author’s AIM lecture notes. In view of the compactness result above and the general tensor maximum principle proved in [NT2], we conjecture that H.-D. Cao’s matrix Li-Yau-Hamilton estimate holds on any complete Kähler manifolds with non-negative bisectional curvature. No assumption on curvature being bounded is needed. If confirmed, one has Theorem 2.1 for all κ non-collapsed ancient solutions with non-negative bisectional curvature. As in [P1], the following gradient estimate is a consequence of Theorem 2.1.

Corollary 2.2. There exists C = C(κ, m) such that for the κ-solution (M, g(t)) we have that 3 |Rt |(x, t) ≤ CR2 (x, t). |∇R|(x, t) ≤ CR 2 (x, t), Note that Theorem 2 also holds for ancient solutions with nonnegative bisectional curvature and differential Harnack (2.1). Also notice that the κ-solution here has different meaning from [P1. Sectiona 11]. The bounded κ-soltion defined last section corresponds to the κ-solution in [P1]. Proof of Corollary 1. By Shi’s short time existence result we know that the solution exists until the curvature blows up. But by Corollary 2.1 above (see also [P1, 11.5 and 11.6]) one has the estimate R(x, t) ≤

(2.2)

C1 , t+1

for some C1 = C1 (n, V(g(0))) > 0 Therefore, the solution exists for all time and is of Type III. Namely there is no slowly forming singularity at infinity. The result can also be shown using Hamilton’s blow-up argument in [H5]. In order to get the curvature decay estimate (0.2) we first apply Theorem 2.1 of [NT1] to conclude that there exists C2 = C2 (m) with Z √t sk(x, s) ds ≤ −C2 F (x, t) where k(x, r) =

R

Bx (r)

0

R(y, 0) dµ, with respect to the initial metric and det(gαβ¯(x, t)) . F (x, t) = log det(gαβ¯(x, 0))

ANCIENT SOLUTIONS

15

∂ (The above estimate follows easily from that facts −F (x, t) ≥ 0 and ∆0 − ∂t (−F )(x, t) ≤ −R(x, 0), where ∆0 is the Laplace operator with respect to g(x, 0). This is indeed the proof C1 ∂ F = R(x, t) ≤ 1+t , for 1 ≪ t, one has that on page 125 of [NT1].) Since − ∂t Z r (2.3) sk(x, s) ds ≤ C3 log(r + 2), 0

for some C3 = C3 (m, V(g(0))). (This implies what proved in [CZ4] under the stronger assumption on the nonnegativity of curvature operator.) Using the curvature decay estimate (2.2) together with the fact that the asymptotic volume ratio V(t) is a constant function of t, one can apply the local injectivity radius estimate of Cheeger-Gromov-Taylor [CGT, Theorem 4.3] (see also [CLY] for√earlier similar works) to conclude that the injectivity radius of (M, g(t)) has the size of C t, where C is a constant independent of t. This implies that M can be exhausted by open subdomains which diffeomorphic to Euclidean balls. From this one can conclude the topological type of M from by-now standard result from topology. The above is the observation in [CZ3], which follows the earlier construction in [Sh4, Section 9] and [H6]. Note that the Steinness of M has been proved for any complete K¨ ahler manifolds with the maximum volume growth and nonnegative bisectional curvature in [NT2, Corollary 4.2] (Please see also [WZ] for the even easier positive case.) One then can adapt the construction of [Sh4, Section 9] to conclude that M is biholomorphic to a pseudoconvex domain in Cm . To obtain (0.2), by [NT2, Theorem 6.1], we first solve the Poincaré-Lelong equation to obtain u such that ∂α ∂¯β¯u = Rαβ¯, and u is at most of logarithmic growth. By considering its heat equation deformation and adding a function of the form log(|z|2 + 1) in the case M splits some factors of C, noticing that M is diffeomorphic to Rn , we can obtain a strictly plurisubharmonic function of logarithmic growth on M. More precisely, let v(x, t) be the heat equation deformation of u(x) (without K¨ ahler-Ricci flow). By [NT2], v(x, t) has the same growth as u(x) and M splits as M = M1 × M2 , where on M1 the complex Hessian vαβ¯ (x, t) is positive definite and on M2 , vαβ¯ (x, t) ≡ 0. By the result of Cheng-Yau we know that v(x, t) must be a constant on M2 , which then implies that u(x) is a constant on M2 . This implies that M2 = Ck for some 0 ≤ k ≤ m. On M2 one can construct a strictly plurisubharmonic function of logarithmic growth by taking φ(z) = log(|z|2 + 1). Adding this to v(x, t) we have a strictly plurisubharmonic function of logarithmic growth on M . Now (0.2) follows from the proof of Corollary 3.2 in [N2]. Proof of Corollary 2. By the above proof of Corollary 1 we know that M supports a strictly plurisubharmonic function of logarithmic growth. The conclusion on the transcendence degree of rational function field follows from the L2 -estimate of ∂¯ and the dimension estimate proved in Theorem 3.1 of [N2]. See, for example, [N2, Theorem 5.2] for the constructions of holomorphic functions of polynomial growth, forming a local holomorphic coordinate for any given point in M . The construction via the well-known H¨ ormander’s L2 -estimates provides the existence of plenty holomorphic functions of polynomial growth, which provides a lower bound on the transcendence degree. The multiplicity estimates in Theorem 3.1 of [N2] gives the upper bound on the transcendental elements of the rational functions. The assertion that M is an affine quasi-algebraic variety follows from the construction in [M1]. See also [De] as well as [CTZ]. Notice that we now also have uniform multiplicity

16

LEI NI

estimates, thanks to the new monotonicity formula and Theorem 3.1 in [N2], which simplifies the steps of [M1] quite a bit. Here by an affine quasi-algebraic variety we mean an affine algebraic variety with some codimension one algebraic varieties removed. The part M is biholomorphic to C2 also follows as in [M1, page 256] by appealing to the result of Ramanujam that any quasi-projective surface homeomorphic to R4 is biholomorphic to C2 . Remark 2.1. The fact that maximum volume growth implies the ampleness of holomorphic functions of polynomial growth was conjectured in [N2]. It has been shown in [NT2] that the positivity of Ricci curvature together with some average curvature decay assumption also implies the same result on holomorphic function of polynomial growth. In fact, this also implies that M is an affine quasi-algebraic variety as in Corollary 2. The existence of harmonic functions of polynomial growth was obtained in [D] under the assumption of nonnegative Ricci curvature, maximum volume growth and the uniqueness of the tangent cone at infinity. The result here seems to indicate that the assumption on the uniqueness of the tangent cone may not be needed. After the completion of current paper, H.-D. Cao informed the author that the case of m = 2 of Corollary 2, assuming the maximum volume growth and nonnegativity of bisectional curvature, was also obtained by B.-L. Chen earlier. §3 Applications to Ricci flow. The proof of Proposition 1.1 can be used in some other situations. For example we can prove the following results. Proposition 3.1. (i) Let (M, g) be a compete steady gradient soliton. Assume that the Ricci curvature is pinched in the sense that Rij ≥ ǫRgij , for some ǫ > 0 with the scaler curvature R(x) > 0. Then there exist C > 0, a > 0 such that R(x) ≤ C exp(−a(r(x) + 1)).

(3.1)

Here r(x) is the distance function to some fixed point in M . (ii) Let (M, g) be a compete expanding gradient soliton. Assume that the Ricci curvature is pinched as above. Then (3.1) holds. Corollary 3.1. Assume that n ≥ 3.

(i) There is no steady gradient soliton with pinched Ricci curvature as in Proposition 3.1; (ii) There is no expanding gradient soliton with pinched Ricci curvature as in Proposition 3.1 and nonnegative sectional curvature.

In particular, any complete three manifolds with pinched Ricci curvature must be compact, therefore spherical. Same result holds for any higher dimensional complete Riemannian manifolds with pinched curvature operator in the sense that (3.2)

o

|Rm|2 = |W |2 + |V |2 ≤ δn (1 − ǫ)2 |U |2 = δn (1 − ǫ)2

2 R2 n(n − 1)

ANCIENT SOLUTIONS

17

1 2 where ǫ > 0, δ3 > 0, δ4 = 51 , δ5 = 10 and δn = (n−2)(n+1) , and W , V and U denote the Weyl curvature tensor, traceless Ricci part and the scalar curvature part, respectively, according to the curvature operator decomposition in [Hu].

Proof. The first part follows from Theorem 20.2 of [H5] and (i) of Proposition 3.1. Notice that the proof there works under the weaker assumption Rij > 0. For part (ii), one just need to recall the gap theorem of Greene-Wu [GW] (see also [PT]) asserting that a simply-connected complete Riemannian manifold with nonnegative sectional curvature and (3.1) must be flat, noticing that under the assumption of expanding gradient soliton and nonnegativity of the Ricci curvature, M is diffeomorphic to Rn (which in particular implies that the new expanders examples constructed in [FIK] can not have nonnegative Ricci curvature), since f is a strictly convex function with only one critical point. By (i) and (ii) and the discussion above, we conclude that curvature pinched manifolds must be compact. This is the main result proved in [CZ1]. Now the last claim in the corollary just restates the fundamental results of Hamilton [H1] and Huisken [Hu]. Note that when n = 2, (3.1) is automatic and the examples of Hamilton’s cigar and the expanding soliton exhibited in [CLN] show that the exponential decay proved in Proposition 3.1 is sharp. Remark 3.1. We speculate that there is no complete noncompact Riemannian manifold with the pinched Ricci as in Proposition 3.1 (we found out later that, according to Ben Chow, this was asked in case n = 3 by Hamilton earlier). If true, this is really a new Bonnet-Meyers type theorem since (3.2) is too strong to allow other topology. But we do not have any workable scheme to prove such general result at this moment. The details on the proof of results in this section will appear somewhere else. References [B] [C1] [C2] [C3] [C4] [CC] [CT1] [CT2] [CGT]

[Cn]

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[LW] [M1] [M2]

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19

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Department of Mathematics, University of California, San Diego, La Jolla, CA 92093 E-mail address: [email protected]

arXiv:math/0404165v1 [math.DG] 7 Apr 2004

Gaussian densities and stability for some Ricci solitons Huai-Dong Cao∗ Lehigh University

Richard Hamilton Columbia University

Tom Ilmanen† ETH-Z¨ urich and Columbia University April 7, 2004

Perelman [Pe02] has discovered a remarkable variational structure for the Ricci flow: it can be viewed as the gradient flow of the entropy functional λ. There are also two monotonicity formulas of shrinking or localizing type: the shrinking entropy ν, and the reduced volume. Either of these can be seen as the analogue of Huisken’s monotonicity formula for mean curvature flow [Hu90]. In various settings, they can be used to show that centered rescalings converge subsequentially to shrinking solitons, which function as idealized models for singularity formation. In this note, we exhibit the second variation of the λ and ν functionals, and investigate the linear stability of examples. We also define the “central density” of a shrinking Ricci soliton (shrinker) and compute its value for certain examples in dimension 4. Using these tools, one can sometimes predict or limit the formation of singularities in the Ricci flow. In particular, we show that certain Einstein manifolds are unstable for the Ricci flow in the sense that generic perturbations acquire higher entropy and thus can never return near the original metric. A detailed version of the calculations summarized in this announcement will follow in [CHI]. In §1, we investigate the stability of Perelman’s λ-functional. Its critical points are steady solitons (Ricci flat in the compact case). We compute the second variation D2 λ; the corresponding Jacobi field operator L is a degenerate negative elliptic integro-differential operator. In fact, L equals half the Lichnerowicz Laplacian ∆L on divergence-free symmetric tensors, and zero on Lie derivatives. This fact and further investigations of the second variation have been reported by Perelman [Pe03]. We call a steady soliton linearly stable if L ≤ 0, otherwise linearly unstable. If g is linearly unstable, then g can be perturbed so that λ(g) > 0, which will destabilize it utterly: it will decay into a ∗ With † With

partial support from NSF Grant DMS-0206847. partial support from Schweizerische Nationalfonds Grant 21-66743.01.

1

cacophony of shrinkers and disappear in finite time. One observes that λ(g) ≤ 0 for any metric on the torus T n , n ≤ 7; in fact this is equivalent to the positive mass theorem. By Guenther, Isenberg and Knopf [GIK02] every K3 surface is linearly stable; more generally, by Dai, Wang and Wei [DWW04] any manifold with a parallel spinor is linearly stable. Other cases are open. In §2, we investigate the stability of the ν-functional, whose critical points are shrinkers. The Jacobi field operator N of ν is like L but with lower order terms. We call a shrinker linearly stable if N ≤ 0. Again, N is closely related to the Lichnerowicz Laplacian. We observe that CPN (with the standard metric) is linearly stable, but all other compact complex surfaces with c1 > 0 are linearly unstable. Using results of Gasqui and Goldschmidt [GG96, GG91] the complex hyperquadric Q3 (a Hermitian symmetric space) is linearly unstable. This implies that Q3 is irremediably unstable in the sense that a generic (non-K¨ ahler!) perturbation of Q3 will never approach the original geometry of Q3 at any scale or time. On the other hand, the hyperquadric Q4 is linearly stable. Other cases are open. A notion of central density (or gaussian density) of a shrinker can be defined from either of Perelman’s monotonicity formulas; we call these notions Θ and ν. On a shrinker, the two definitions are equivalent via Θ = eν . For a general solution, eν is a lower bound for the central density of any shrinker that arises later as a singularity model, which restricts the shrinkers that may occur later. This is presented in §3. The central density of certain standard 4-dimensional examples are exhibited in a table in §4. We are grateful to Hugh Bray, Robert Bryant, Hubert Goldschmidt, Dan Knopf, and John Morgan for illuminating conversations.

1

Second Variation of the Entropy λ

The second variation of the Einstein functional is positive in the conformal direction but negative in all other directions. The glory of Perelman’s entropy is that there is a preliminary minimization over scalar functions that absorbs nearly all the positive directions: the Jacobi field operator is a linear integrodifferential operator with nonpositive symbol.1 Fix a compact manifold (M, g). Define [Pe02] Z F (g, f ) := e−f (|Df |2 + R). Define the entropy λ(g) := inf{F (g, f ) : f ∈

Cc∞ (M ),

Z

e−f = 1}

1 It is convenient that this happens without changing the metric via a conformal change. So the scalar variations do not affect the background geometry. Instead they satisfy a linear PDE.

2

The infimum is achieved by a function f solving −2∆f + |Df |2 − R = λ(g). Now consider variations g(s) = g + sh. Following Perelman, the first variation Dg λ(h) of λ is given by Z d λ(g(s)) = e−f (−Rc − D2 f ) : h, ds s=0

where f is the minimizer. A stationary point satisfies Rc + D2 f = 0,

which implies that g is a (gradient) steady soliton, that is, the Ricci flow with initial condition g satisfies g(t) = φ∗t (g) where φt is a family of diffeomorphism generated by the gradient vector field Df . In fact, any compact steady is Ricci flat with f = 0, λ = 0. Note by diffeomorphism invariance of λ that Dg λ vanishes on any Lie derivative h = Lx g. From this, by inserting h = −2(Rc+D2 f ) one recovers Perelman’s wonderful result that λ(g(t)) is nondecreasing on any Ricci a Ricci flow, and is constant if and only if g(t) is a steady soliton. We prove the following. Write Rm(h, h) := Rijkl hik hjl , div ω := Di ωi , (div h)i := Dj hji , (div∗ ω)ij = −(Di ωj + Dj ωi )/2 = −(1/2)Lω# gij . 1.1 Theorem The second variation Dg2 λ(h, h) of λ on a compact Ricci flat manifold is given by Z d2 1 1 λ(g(s)) = − |Dh|2 + | div h|2 − |Dvh |2 + Rm(h, h) ds2 s=0 2 2 Z = Lh : h, where Lh :=

1 1 ∆h + div∗ div h + D2 vh + Rm(h, ·), 2 2

and vh satisfies ∆vh = div div h. The symbol of L in the direction ξ ∈ Tx∗ M is σξ (h) = −πξ⊥ (h), where πξ⊥ (h) restricts h to the hyperplane ξ ⊥ . So the operator L is degenerate negative elliptic, and has a discrete spectrum with at most a finite-dimensional space of positive eigenfunctions. 3

Decompose C ∞ (Sym2 (T ∗ M )) as ker div ⊕ im div∗ . One verifies that L vanishes on im div∗ , that is, on Lie derivatives. On ker div one has 1 L = ∆L 2 where ∆L h := ∆h + 2Rm(h, ·) − Rc · h − h · Rc is the Lichnerowicz Laplacian on symmetric 2-tensors. We call a critical point g of λ linearly stable if L ≤ 0, and a maximizer if λ(g1 ) ≤ λ(g) for all g1 . A compact Ricci flat metric is a maximizer if and only if it admits no metric of positive scalar curvature. (This follows from Schoen’s solution of the Yamabe problem [S84].) Evidently a maximizer is stable. If g is not stable, then a slight perturbation will develop λ > 0 and R > 0 and (in principle) disappear in finite time as positive manifolds do. A good question is whether any Ricci-flat manifold is unstable. We call this the positive mass problem for Ricci flat manifolds. 1.2 Example T n admits no metric of positive scalar curvature by the positive mass theorem, so λ(g) ≤ 0 for all g on T n . 1.3 Example A Calabi-Yau K3 surface and more generally, any manifold with a parallel spinor has ∆L ≤ 0 [GIK02, DWW04]. So these manifolds are linearly stable in the sense presented here. 1.4 Example Let g be compact and Ricci flat. Following [B84, GIK02] we examine conformal variations. It is convenient to replace ug by h = Su := (∆u)g − D2 u which differs from the conformal direction only by a Lie derivative and is divergence free. We have ∆L Su = (S∆u)g, so ∆L has the same eigenvalues as ∆. In particular, N ≤ 0 in the conformal direction. This contrasts with the Einstein functional.

2

Second Variation of the Shrinker Entropy ν

Fix a complete manifold (M, g). Define Z 1 e−f τ (|Df |2 + R) + f − n dV. W(g, f, τ ) := n/2 (4πτ ) 4

Define the shrinker entropy by ν(g) := inf{W(g, f, τ ) : f ∈

Cc∞ (M ), τ

1 > 0, (4πτ )n/2

Z

e−f = 1}

Assume that M is compact or is asymptotic at infinity to a metric cone over a smooth, compact Riemannian manifold. One checks that ν(g) is realized by a pair (f, τ ) that solve the equations Z 1 n τ (−2∆f + |Df |2 − R) − f + n + ν = 0, f e−f = + ν, 2 (4πτ )n/2 and f grows quadratically. Consider variations g(s) = g + sh where h is smooth of compact support. Following Perelman, one calculates the first variation Dg ν(h) to be Z 1 d e−f (τ (−Rc − D2 f ) + g/2) : h. ν(g(s)) = ds s=0 (4πτ )n/2

A stationary point of ν satisfies

D2 f + Rc −

g =0 2τ

(1)

which says that g is a (gradient) shrinker, that is, its Ricci flow g(t) has the form g(t) := (T − t)ψt∗ (g)),

t < T,

where ψt are the diffeomorphisms generated by −Df , and τ = T − t. As before, Dg ν vanishes on Lie derivatives. By scale invariance it vanishes on multiplies of the metric. Inserting h = −2(Rc + D2 f − g/2τ ), one recovers Perelman’s brilliant formula that finds that ν(g(t)) is monotone on a Ricci flow, and constant if and only if g(t) is a gradient shrinker. A positive Einstein manifold is a shrinker with f ≡ n/2, normalized by Rc = g/2τ . We compute: 2.1 Theorem Let (M, g) be a positive Einstein manifold. The second variation Dg2 ν(h, h) is given by Z d2 τ 1 ν(g(s)) = − |Dh|2 + | div ds2 s=0 vol(g) 2 =

τ vol(g)

Z

N h : h,

5

1 v2 h|2 − |Dvh |2 + Rm(h, h) + h 2 4τ 2 Z 1 1 − trg h , 2n vol(g)

where N h :=

1 g 1 ∆h + div∗ div h + D2 vh + Rm(h, ·) − 2 2 2nτ vol(g)

Z

trg h.

and vh is the unique solution of vh = div div h, ∆vh + 2τ

Z

vh = 0.

There is a strictly more complicated formula in the case of non-Einstein shrinkers. As in the previous case, N is degenerate negative elliptic and vanishes on im div∗ . Write ker div = (ker div)0 ⊕ Rg R where (ker div)0 is defined by trg h = 0. Then on (ker div)0 we have 1 1 ∆L − N= 2 τ where ∆L is the Lichnerowicz Laplacian. So the linear stability of a shrinker comes down to the (divergence free) eigenvalues of the Lichnerowicz Laplacian. Let us write µL for the maximum eigenvalue of ∆L on symmetric 2-tensors and µN for the maximum eigenvalue of N on (ker div)0 , 2.2 Example The round sphere is geometrically stable (i.e. nearby metrics are attracted to it up to scale and gauge) by the results of Hamilton [Ha82, Ha86, Ha88] and Huisken [Hu88]. In particular it is linearly stable: µN = −2/(n − 1)τ < 0. 2.3 Example For CPN , the maximum eigenvalue of ∆L on (ker div)0 is µL = 1/τ by work of Goldschmidt [G04], so CPN is neutrally linearly stable, i.e. the maximum eigenvalue of N on (ker div)0 is µN = 0. Any product of two nonflat shrinkers N1n1 × N2n2 is linearly unstable, with µN = 1/2τ . The destabilizing direction h = g1 /n1 − g2 /n2 corresponds to a growing discrepancy in the size of the factors. More generally, any compact Kähler shrinker with dim H 1,1 (M ) ≥ 2 is linearly unstable. Again, this can be seen directly: a small perturbation into a non-canonical K¨ ahler class will move in a straight line nearly toward the vertex of the K¨ ahler cone, hence away from the canonical class (in a scale invariant sense). If M is K¨ ahler-Einstein, we compute µN as follows. Let σ be a harmonic 2-form and h be the corresponding metric perturbation; then ∆L h = 0, and if σ is chosen perpendicular to the Kähler form, then as above we obtain µN = 1/2τ . A complete list of compact complex surfaces with c1 > 0 is CP2 , CP1 × CP1 , and CP2 #k(−CP2 ), k = 1, . . . 8. Each of these has a unique Kähler shrinker metric (K¨ ahler-Einstein unless k = 1, 2). By the above, all are linearly unstable except CP2 . 6

Let QN denote the complex hyperquadric in CPN +1 defined by N +1 X

zi2 = 0,

i=0

a Hermitian symmetric space of compact type, hence a positive Kähler-Einstein manifold. Then Q2 is isometric to CP1 ×CP1 , the simplest example of the above instability phenomenon. 2.4 Example Consider Q3 . It has dim H 1,1 (Q3 ) = 1, so the above discussion does not apply. But the maximum eigenvalue of ∆L on (ker div)0 is µL = −2/3τ by work of Gasqui and Goldschmidt [GG96] (or see [GG04]). The proximate cause is a representation that appears in the sections of the symmetric tensors but not in scalars or vectors. Therefore, Q3 is linearly unstable with 1 . 6τ Since ν increases along the Ricci flow, this implies that a generic small perturbation of the Einstein metric g¯ will grow and g(t) will never return near g¯ at any scale or time. We say that g¯ is geometrically unstable. Now imagine that we start with a random metric on Q3 and propose to use the Ricci flow to find an Einstein metric or other canonical geometry for Q3 . Assuming there are no other critical points, we find that the flow combusts in singularities of more elementary type and the topology of the underlying manifold simplifies drastically, unless it happens to get hung up at the Einstein metric ν. So the Ricci flow has fundamentally more complicated behavior than in dimension three, as one expects. Further exploration of this example will appear in [BGIM]. µN =

2.5 Example Let Q4 be the 4-dimensional hyperquadric. The maximum eigenvalue of ∆L on symmetric tensors is µL = −1/τ by work of Gasqui and Goldschmidt [GG91] (or see [GG04]). So Q4 is neutrally linearly stable: µN = 0. Let g be a positive Einstein metric, and let us examine conformal variations. As before, without loss replace ug by the divergence-free variation ug h = Su := (∆u)g − D2 u + . 2τ As before, ∆L Su = (S∆u)g. Thus ∆L has the same eigenvalues as ∆fns |(ker S)⊥ . But ker S is empty except on round S n , which is linearly stable. Note that µfns ≤ −n/2(n − 1)τ with equality only on round S n . So we have: 2.6 Proposition A positive Einstein metric is linearly unstable for conformal variations if and only if the maximum eigenvalue of ∆ on functions satisfies −

1 n < µfns < − . τ 2(n − 1)τ

We do not know whether this inequality can ever be satisfied on a positive Einstein manifold. 7

3

The Central Density of a Shrinker

Our aim in this section is to define the central density of a gradient shrinker. First we define a suitable class of gradient shrinkers, then we review the two Perelman monotonicity formulas of shrinking type and apply them by taking the center point to be the parabolic vertex of the shrinker. Our principal result is that the two notions of density coincide. A gradient shrinker solves ∂g/∂t = −2Rc,

g(t) := −tψt∗ (g(−1)),

t < 0,

where ψt are the diffeomorphisms generated by the gradient of a function F (x, t). Differentiating the above expression yields D2 F + Rc −

g = 0, 2τ

(2)

where τ = −t. Normalizing F by adding a time-dependent constant, we obtain ∂F + |DF |2 = 0. ∂τ

(3)

Differentiating (2), taking the trace two ways, and applying Bianchi II and commutation rules yields D(|DF |2 + R − F/τ ) = 0. Adding a further global constant to F leads to the classical auxiliary equation |DF |2 + R −

F = 0. τ

(4)

Equations (2)-(4) are the fundamental equations for a gradient shrinker. Combining (2) and (3) yields the backward heat equation ∂F n = ∆F − |DF |2 + R − . ∂τ 2τ

(5)

In order to prove our results we need some analytic hypotheses on the metric of M . We assume that M is complete, connected, and κ-noncollapsed at all scales. We also assume that the curvature decays quadratically as x → ∞. (This is satisfied, for example, by the blowdown shrinker L(N, −1) [FIK04].) However, many of our results hold under the weaker hypothesis of bounded curvature. Under the quadratic decay hypothesis, g(t) converges in the Gromov-Hausdorff sense as t ր 0 to a metric cone C which is smooth except at the vertex, which we call 0. The convergence is smooth except on a compact set, which falls into the vertex, which we call 0. For a proof and further details, see [I]. We now wish to define a gaussian density centered at the parabolic vertex (y, s) = (0, 0) of the spacetime M := (M × (−∞, 0)) ∪ (C × {0}). We may do this in two ways: via the shrinking entropy or the reduced volume, both due to Perelman [Pe02].

8

The reduced volume generalizes Bishop volume monotonicity to the spacetime setting. For a smooth point (y, s) in any Ricci flow, define the reduced distance ℓ = ℓy,s by ! 2 Z τ dγ √ 1 t < s, x ∈ M, (6) ℓ(x, t) := √ inf σ + R dσ, 2 τ γ 0 du

where the infimum is taken over all paths (γ(u), u), t ≤ u ≤ s that connect (x, t) to (y, s). The reduced volume centered at (y, s) is defined by Z 1 θy,s (t) := e−ℓ(x,t) dVt (x). (4πτ )n/2 M

Perelman wonderfully shows that θy,s (t) is increasing in t and is constant precisely on a gradient shrinker. Now define ℓ0,0 by passing smooth points (yi , si ) to (0, 0). We have: 3.1 Proposition ℓ0,0 is well-defined and is locally Lipschitz on M. For t < 0, θ0,0 (t) is well defined, constant, and contained in (0,1]. This constant value we call the central density of (M,g(t)) and denote Θ(M ) = Θ(M, g(·)) := θ0,0 (t),

t < 0.

Next we turn to the shrinking entropy ν. Let (M, g(t)) be a smooth Ricci flow existing up to t = s and set τ := s − t. Let f solve the heat equation n ∂f = ∆f − |Df |2 + R − , ∂t 2τ that came up for the soliton potential of a shrinker (5). Define u by u :=

e−f . (4πτ )n/2

Remarkably, u solves the adjoint heat equation ∂u = ∆u − Ru. ∂t This leads to the conservation law Z Z 1 −f e = u=1 (4πτ )n/2

(7)

for t < s.

Perelman has shown [Pe02] ∂ 1 W(s − t, f (t), g(t)) = ∂t (4πτ )n/2

Z 9

g 2 2e−f D2 f + Rc − dVt ≥ 0. 2τ

and the right hand side vanishes precisely when g(t) is a gradient shrinker and f is its soliton potential. (This shows, as mentioned above, that ν increases in general and is constant on a shrinker.) If u emerges from a dirac source at a smooth point (y, s), we write u = uy,s , f = fy,s and define the shrinker entropy centered at (y, s) by φy,s (t) := W(s − t, fy,s (t), g(t)) Passing smooth points (yi , si ) to (0, 0), we prove: 3.2 Proposition u0,0 is well-defined, smooth, and positive on M × (−∞, 0) and solves equation (7). It satisfies Z Z 1 u0,0 ≡ e−f0,0 = 1, t < 0. (8) (4πτ )n/2 M Also, φ0,0 (t) is well-defined and lies in (−∞, 0] for all t < 0. In fact, it is constant, with φ0,0 (t) = ν(M ), t < 0. Since φ0,0 (t) is constant, f0,0 is a soliton potential and so f0,0 = F + C

(9)

for some constant C depending only on M. We now wish to relate Θ(M ) and ν(M ) via F . In the process we determine the value of C, and sharpen on a shrinker the general Perelman relation ℓ0,0 ≤ f0,0 [Pe02]. We begin with Θ(M ). Using the symmetry of the shrinker and a simple comparison argument, one checks: 3.3 Proposition The integral curves of F are minimizing L-geodesics emanating from (0, 0). Then using the homothetic time-symmetry of the shrinker, one obtains after a straightforward computation: ℓ0,0 = τ (|DF |2 + R) = F = f0,0 − C, and thus by the definition and (8), one gets: Θ(M ) = eC . Next, we evaluate ν. Compute ν(M ) = φ0,0 (t) Z 1 e−f0,0 τ (|Df0,0 |2 + R) + f0,0 − n dVt = n/2 (4πτ ) Z 1 e−f0,0 [τ (∆f0,0 + R) + f0,0 − n] dVt = (4πτ )n/2 Z 1 e−f0,0 [f0,0 − n/2] dVt = (4πτ )n/2 10

by integrating by parts and the trace of (2). On the other hand, by (4) and (9), the first integral expression also equals Z 1 e−f0,0 [f0,0 − C + f0,0 − n] dVt ν(M ) = (4πτ )n/2 We conclude that ν(M ) = 2ν(M ) − C, so ν(M ) = C. We summarize these results in a theorem. 3.4 Theorem On a shrinker satisfying the above assumptions, we have ℓ0,0 = F = f0,0 − ν(M ) and Θ(M ) = eν(M) . Details will appear in [CHI].

4

Table of Values

In this section we calculate Θ for some standard shrinkers. The computations are simplified by several observations. Normalize positive Einstein manifolds by Rc = g/2τ , τ = 1/2(n − 1), so that S n has radius 1. (1) For any shrinker M , Θ(M ) ≤ 1 with equality if and only if M = Rn . (2) Let M be a Ricci flat cone with g = dr2 + r2 gΣ where Σ is positive Einstein. Then M is a shrinker (with interior singularity), and Θ(M ) =

vol(Σ) . vol(S n )

(3) If M is a positive Einstein manifold, then (for any τ ) Θ(M ) =

1 4πτ e

n/2

volτ (M ) ≤ Θ(S n ),

with equality if and only if M = S n . n/2 n−1 n (4) Θ(S ) = vol(S n ). By way of comparison, note that for 2πe n n/2 vol(S n ).2 mean curvature flow, ΘMCF (S n ) = 2πe N N +1 vol(S 2N +1 ) (5) Θ(CPN ) = . πe 2π (6) The positive K¨ ahler-Einstein manifold M = CP2 #k(−CP2 ), k = 0, 3, . . . , 8, has Θ(M ) = (9 − k)/2e2 . 2 Following an observation of White, we note (tantalizingly) that the respective limits as p √ n → ∞ are 2/e and 2.

11

(7) Θ(M × N ) = Θ(M )Θ(N ). We say that one shrinker decays to another if there is a small perturbation of the first whose Ricci flow develops a singularity modelled on the second. Because the ν-invariant is monotone during the flow, decay can only occur from a shrinker of lower density to one of higher density. This creates a “decay lowerarchy”. (It should be a partial order.) We have computed the following density values in dimension 4. Note that the conclusion of Theorem 3.4 holds for all our examples, though not all are smooth enough to satisfy the hypotheses.

12

Shrinker

Type

Θ

Θ

R4

flat

1

1.000

S4

positive Einstein

6/e2

S3 × R

product

2 π/e3

S 2 × R2

product

2/e

L(2, −1)

blowdown shrinker [FIK04]

e

CP2

positive Einstein

9/2e2

.609

S2 × S2

product

4/e2

.541

CP2 #(−CP2 )

Koiso metric [K90, C94]

3.826/e2

.518

CP2 #(−CP2 )

Page metric [Pa78]

3.821/e2

.517

C(RP3 )

Ricci flat cone

1/2

.500

C(RP2 ) × R

product

1/2

.500

RP4

positive Einstein

3/e2

.406

CP2 #3(−CP2 )

positive Einstein

3/e2

.406

RP3 × R

product

RP2 × R2

product

1/e

CP2 #4(−CP2 )

positive Einstein

5/2e2

.338

C(S 3 /Z3 )

Ricci flat cone

1/3

.333

CP2 #5(−CP2 )

positive Einstein

2/e2

.271

CP2 #6(−CP2 )

positive Einstein

3/2e2

.203

CP2 #7(−CP2 )

positive Einstein

1/e2

.135

CP2 #8(−CP2 )

positive Einstein

1/2e2

.068

√

2−2

π/e3

13

.812 1/2

(1 +

1/2

√ 2)/2

.791 .736 .672

.396 .368

All manifolds in the table are created from Einstein manifolds except for L(2, −1) and the Koiso metric. The computations for these metrics will be detailed in [CHI]. The volume of the Page metric is computed in [Pa78]. The blowdown shrinker L(n, −1) is a Kähler shrinker defined on the total space of the tautological holomorphic line bundle oN −1 (−1) over CPN −1 , that is, on CN blown up at z = 0. The metric of L(N, −1) is U (N ) invariant, complete, and conelike at infinity, satisfying quadratic decay for the curvature. As t ր 0, the exceptional divisor CPN −1 shrinks to a point and elsewhere the metric converges smoothly to a cone metric on CN \{0} whose metric completion has a vertex at 0. For positive time, the flow can continue by a smooth, U (N )invariant K¨ ahler expander on CN . See [FIK]. The Koiso metric [K90, C94] and the Page metric [Pa78] are both U (2)invariant metrics on CP2 #(−CP2 ). The former, but not the latter, is Kähler. The remarks following Example 2.3 show that the Koiso metric has one direction of instability (in a K¨ ahler direction). On the other hand, the Page metric may well decay to the Koiso metric. By the discussion in [FIK04], this leads us to conjecture that either metric decays to CP2 via a CP1 pinches off.

References [BGIM] R. Bryant, H. Goldschmidt, J. Morgan and T. Ilmanen, in preparation. [B84]

C. Buzzanca, The Lichnerowicz Laplacian on tensors, Boll. Un. Mat. Ital. B 3 (1984) 531–541 (Italian).

[C94]

H.-D. Cao, Existence of gradient K¨ ahler-Ricci solitons, Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, MA, 1996, 1–16.

[CHI]

H.-D. Cao, R. Hamilton and T. Ilmanen, in preparation.

[DWW04] X. Dai, X. Wang, G.Wei, On the stability of riemannian manifolds with parallel spinors, math.DG/0311253, March 2004. [FIK04] M. Feldman, T. Ilmanen and D. Knopf, Rotationally symmetric shrinking and expanding gradient K¨ ahler-Ricci solitons, JDG, to appear. [FIN]

M. Feldman, T. Ilmanen and L. Ni, in preparation.

[GG91] J. Gasqui and H. Goldschmidt, On the geometry of the complex quadric, Hokkaido Math. J. 20 (1991) 279–312. [GG96] J. Gasqui and H. Goldschmidt, Radon transforms and spectral rigidity on the complex quadrics and the real Grassmannians of rank two, J. Reine Angew. Math. 480 (1996), 1–69. [GG04] J. Gasqui and H. Goldschmidt, Radon transforms and the rigidity of the Grassmannians, Princeton University Press, 2004.

14

[G04]

H. Goldschmidt, private communication.

[GIK02] C. Guenther, J. Isenberg, and D. Knopf, Stability of the Ricci flow at Ricci-flat metrics, Comm. Anal. Geom. 10 (2002) 741–777. [Ha82]

R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982) 255–306.

[Ha86]

R. S. Hamilton, Four-manifolds with positive curvature operator, J. Diff. Geom. 24 (1986) 153–179.

[Ha88]

R. S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math. 71, Amer. Math. Soc., Providence, 1988, 237–262.

[Hu88]

G. Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Diff. Geom. 21 (1985) 47–62.

[Hu90]

G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990) 285–299.

[I]

T. Ilmanen, in preparation.

[K90]

N. Koiso, On rotationally symmetric Hamilton’s equation for K¨ ahlerEinstein metrics, Recent topics in differential and analytic geometry, Adv. Stud. Pure Math. 18-I, Academic Press, Boston, MA, 1990, 327– 337.

[Pa78]

D. Page, A compact rotating gravitational instanton, Phys. Lett. 79B (1978) 235–238.

[Pe02]

G. Perelman, The entropy formula for the Ricci flow and its geometric applications, http://arxiv.org/abs/math/0211159, 2002.

[Pe03]

G. Perelman, lecture series, Stonybrrok, 2003.

[S84]

R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (1984) 479–495.

15

arXiv:math/0402194v3 [math.DG] 20 May 2004

Limiting behaviour of the Ricci flow Natasa Sesum

Abstract We will consider a τ -flow, given by the equation 1 τ gij

d dt gij

= −2Rij +

on a closed manifold M , for all times t ∈ [0, ∞). We will prove that

if the curvature operator and the diameter of (M, g(t)) are uniformly

bounded along the flow, then we have a sequential convergence of the flow toward the solitons.

1

Introduction

The studies of singularities and the limiting behaviours of solutions of various geometric partial differential equations have been important in geometric analysis. One of these important geometric equations is so called Ricci flow equation, itroduced by Richard Hamilton in [6]. It is the equation

d dt gij (t)

=

−2Rij , for a Riemannian metric gij (t). The short time existence of this

equation was proved by Hamilton in [6] and somewhat later the proof was significantly simplified by DeTurck in [4]. Hamilton showed that the Ricci flow preserves the positivity of the Ricci tensor in dimension three and of the curvature operator in all dimensions. This observation helped him to prove the convergence results in dimensions three and four, towards metrics of constant positive curvatures (in the case of positive Ricci curvature and positive curvature operator respectively). Besides the short time existence we can also study a long time existence of the Ricci flow. There is a well known Hamilton’s result.

1

Theorem 1 (Hamilton). For any smooth initial metric on a compact manifold there exists a maximal time T on which there is a unique smooth solution to the ricci flow for 0 ≤ t < T . Either T = ∞ or else the curvature is unbounded as t → T .

One can ask what happens to a solution if it exists for all times and under which conditions it will converge to a metric that will have nice properties. In the case of dimension three with positive Ricci curvature and dimension four with positive curvature operator we know that a solution converges to an Einstein metric. In general, we can not expect to get an Einstein metric in the limit. We can expect to get a solution to an evolution equation which moves under a one-parameter subgroup of the symmetry group of the equation. These kinds of solutions are called solitons. Our goal in this paper is to prove the following theorem. Theorem 2 (Main Theorem). Consider the flow dgij 1 = −2Rij + gij dt τ

(1)

on a compact manifold M , where τ > 0 is fixed, |Rm| ≤ C and diam(M, g(t) ≤ C ∀t ∈ [0, ∞). Then for every sequence of times ti → ∞ there exists a subsequence, so that g(ti + t) → h(t) and h(t) is a Ricci soliton.

The organization of the paper is as follows. In section 3 we will prove some properties of µ(g, τ ) that has been introduced by Perelman in [10]. They will be useful in the later sections of the paper. In section 3 we will prove Theorem 2. Acknowledgements: The author would like to thank her advisor Gang Tian for bringing this problem to her attention and for constant help and support.

2

2

Preliminaries

Perelman’s functional W and its properties will play an important role in the proof of Theorem 2. M will always denote a compact manifold, and

(gij )t = −2Rij + τ1 gij will be a flow that we will be considering throughout

the whole paper. Perelman’s functional W has been introduced in [10]. − n2

W (g, f, τ ) = (4πτ )

Z

M

e−f [τ (|∇f |2 + R) + f − n]dVg .

We will consider this functional restricted to f satisfying Z

n

(4πτ )− 2 e−f dV = 1.

(2)

M

W is invariant under simultaneous scaling of τ and g. Perelman showed

that the Ricci flow can be viewed as a gradient flow of functional W. Let

µ(g, τ ) = inf W(g, f, τ ) over smooth f satisfying (2). It has been showed by Perelman that there always exists a smooth minimizer on a closed manifold

M , that µ(g, τ ) is negative for small τ > 0 and that it tends to zero as τ → 0. One of the most important properties of W is the monotonicity formula.

Theorem 3 (Perelman).

d dt W

=

R

n

M

1 2τ |Rij +∇i ∇j f − 2τ gij |2 (4πτ )− 2 e−f dV ≥

0 and therefore W is increasing along the Ricci flow.

One of the very important applications of the monotonicity formula is noncollapsing theorem for the Ricci flow that has been proved by Perelman in [10]. Definition 4. Let gij (t) be a smooth solution to the Ricci flow (gij )t = −2Rij (t) on [0, T ). We say that gij (t) is loacally collapsing at T , if there is a

sequence of times tk → T and a sequence of metric balls Bk = B(pk , rk )

at times tk , such that rk−n Vol(Bk ) → 0.

rk 2 tk

is bounded, |Rm|(gij (tk )) ≤ rk−2 in Bk and

Theorem 5. If M is closed and T < ∞, then gij (t) is not locally collapsing

at T .

3

Sequential convergence of a τ -flow

3

Definition 6. τ -flow is given by the equation 1 d gij = −2Rij + gij , dt τ

(3)

for τ > 0. We want to prove the Theorem 2 in this section.

3.1

Convergence toward the solutions of the Ricci flow

In order to prove Theorem 2 we will first show that it is reasonable to expect a convergence toward a smooth manifold, i.e. that a limit manifold will not collapse. Claim 7. Consider the flow as above. For every fixed τ > 0 there exists a constant C such that Volg(t) (M ) ≥ C for every t, i.e. we have a uniform

lower bound on the volumes.

Proof. Assume that the claim is not true, i.e. that there exists a sequence ti s.t. Volg(ti ) (M ) → 0 as i → ∞. Let g¯(s) = c(s)g(t(s)) be unnormalized

flow, for s ∈ [0, τ ), where:

s t(s) = −τ ln(1 − ). τ s c(s) = 1 − . τ R(g) R(¯ g) = . c(s) ti

Find si , such that t(si ) = ti . We get that si = τ (1 − e− τ ). si → τ as i → ∞.

Let

max |Rm|(¯ g (s)) = Qi ,

M ×[0,si ]

4

(4)

and assume that the maximum is achieved at pi . By the corollary of Perelman’s noncollapsing theorem we have that:

for r ≤ C

q

τ Qi

Volg¯(t) B(pi , r) ≥ C1 , rn q and t ∈ [0, si ). Choose r = C Qτ i and t = si .

r p √ τ ˜ ) ≥ (C τ )n C1 = C. ( Qi )n Volg¯(si ) B(pi , C Qi n

Since Volg¯(si ) B(pi , r) = c(si ) 2 Volg(ti ) B(pi , r˜), where r˜ might be a different radius as a matter of scaling and since Qi ≤

C c(si )

(because the curvature of

g(t) is uniformly bounded), we get that:

˜ Volg(ti ) (M ) ≥ C/C, where C˜ and C do not depend on i. Let i → ∞ in the previous inequality to

get a contradiction. Therefore we have a uniform lower bound on volumes.

Remark 8. The assumptions of the Theorem 2 and the result of Claim 7 imply the uniform bounds on the curvature tensors, uniform upper bound on the diameters and uniform lower bounds on the volumes. Similarly like in the case of unnormalized flow, uniform bounds on the curvatures gives us uniform bounds on all covariant derivatives, so by Hamilton’s compactness theorem, for every sequence ti ր ∞ as i → ∞, there exists a subsequence

(call it again ti ), such that (M, g(ti + t)) converges to (M, h(t)), in the sense

that there exist diffeomorphisms φi : M → M , so that φ∗i g(ti + t) converge uniformly together with their covariant derivatives to metrics h(t) on com-

pact subsets of M × [0, ∞). Moreover, h(t) is a solution of a τ -flow as well.

3.2

Continuity of the minimizers for W n

We will recall a definition of Perelman’s functional W = (4πτ )− 2

R

e−f [τ (R+ n R |∇f |2 )+f −n]dV . The constraint on f for this functional is (*) (4πτ )− 2 e−f dV = 5

M

1. Let µ(g, τ ) = inf W(g, f, τ ) under the constraint (*). This infinimum has been achieved by some smooth minimizer f . Perelman has also proved that for a fixed metric g, limτ →0 µ(g, τ ) = 0 and µ(g, τ ) < 0 for a small value of τ > 0. In the case of a τ -flow g(t), τ > 0 is being fixed in time, and by the monotonicity formula for W we have that µ(g(t), τ ) is increasing along the

flow. Therefore, there exists limt→∞ µ(g(t), τ ). Claim 9. limt→∞ µ(g(t), τ ) is finite.

Proof. Assume that limt→∞ µ(g(t), τ ) = ∞. Then, ∀i, ∃ti s.t. µ(g(ti ), τ ) ≥ i. There exists a subsequence (call it ti ) such that (M, gi ) converges to (M, h),

for some metric h. From the first part of Lemma 10 we get that µ(g(ti ), τ ) < µ(h, τ ) + ǫ, for i big enough. Letting i → ∞ we get a contradiction. Lemma 10. If (M, gi ) tend to (M, h) when i → ∞, where gi = g(ti ) and ti ր ∞, then limi→∞ µ(gi , τ ) = µ(h, τ ).

Proof. µ(h, τ ) =

Z

M

Since

φ∗i gi

n

(τ (|∇f |2 + R(h)) + f − n)(4πτ )− 2 dVh .

→ ∞ uniformly with their covariant derivatives, if ǫ > 0 is fixed,

there exists some big i0 , so that for i ≥ i0 Z n ǫ µ(h, τ ) ≥ (τ (|∇f |2 + R(g˜i )) + f − n)(4πτ )− 2 dVg˜i − , 2 M

where g˜i = φ∗ gi . Change the variables in the above integral by diffeomorphism φi . µ(h, τ ) ≥

Z

n ǫ (τ (|∇i fi |2 + R(gi )) + fi − n)(4πτ )− 2 dVgi − , 2 M

where fi = φ∗ f . Perturb a little bit fi to get f˜i , by a quantity that tends to R n ˜ zero, so that M e−fi (4πτ )− 2 dVgi = 1. Since our geometries are uniformly

bounded, for big enough i0 we will have

µ(h, τ ) ≥ W(gi , f˜i , τ ) − ǫ ≥ µ(gi , τ ) − ǫ. 6

(5)

fi

Let ui = e− 2 . We have seen that minimizing µ(gi , τ ) by fi is equivalent to minimizing the following expression in ui : Z n τ (4|∇i ui |2 + Ri u2i ) − 2u2i ln ui − nu2i )(4πτ )− 2 dVgi . M

The minimizer ui has to satisfy the following elliptic differential equation τ (−4∆i ui + Ri ui ) − 2ui ln ui − nui = µi,τ ui .

(6)

µi,τ is uniformly bounded, since there is a finite limt→∞ µ(g(t), τ ). Now we can easily get:

τ

Z

Z

n

M

M

i.e. ui ∈ W 1,2 with

u2i (4πτ )− 2 dVi ≤ C, n

|∇i ui |2 (4πτ )− 2 dVi ≤ C,

(7)

(8)

||ui ||W 1,2 ≤ C ∀i.

From (6), by standard regularity theory of partial differential equations and Sobolev embedding theorems, we get that ui ∈ W k,p with uniformly bounded

W k,p norms, where p
0 so that ui ≥ C > 0 ∀i and ∀x ∈ M Proof. Assume that there exists a sequence ui and pi ∈ M , such that 0 < ui (pi )
0,

if µ(gi , τ ) = W(gi , fi , τ ), then fi → f in C 2,α norm, where µ(h, τ ) =

W(h, f, τ ).

3.3

Further estimates on the minimizers

In this subsection we want to use the minimizers ft for W at different times to

construct the functions ft (s) for s ∈ [0, t]. By using the parabolic regularity

we will be able to get the uniform estimates on C k,α norms of ft (s). This will

enable us to take a limit of this functions along the sequences. This limits are the functions that will turn out to be the potential functions that come into the equations describing the soliton type solutions arising in a limit. For any t we can find ft such that W(g(t), ft , τ ) = µ(g(t), τ ). If we flow

ft backward, we will get functions ft (s) that satisfy

dft (s) n = −R(s) − ∆ft (s) + |∇ft (s)|2 + , ds 2τ ft (t) = ft . 9

We know that minimizing W in f is equivalent to minimizing the correft

˜2t (s). The equation sponding functional in u ˜, where u ˜t = e− 2 . Let ut (s) = u

for ut (s) is dut n = −∆ut + (− + R(s))ut (s), ds 2τ ut (t) = ut . By the monotonicity of W along the flow (1) we have that µ(g(s), τ ) ≤ W(g(s), ft (s), τ ) ≤ W(g(t), ft , τ ) = µ(g(t), τ ). First of all, there exists limt→∞ µ(g(t), τ ). It is finite, since for every sequence ti → ∞ there exists a subsequence such that g(ti ) → h(0) and by Lemma 10

from the previous section, we have that µ(g(ti ), τ ) → µ(h(0), τ ).

Instead of functional W(g(s), ft (s), τ ) we can consider the equivalent

functional which depends on u ˜t (s) = e−ft (s)/2 . Z W(ut (s)) = [τ (4|∇˜ ut (s)|2 +R˜ ut (s)2 )−˜ ut (s)2 log u ˜t (s)2 −n˜ ut (s)2 ](4πτ )−n/2 dV, M

(10)

where u ˜t satisfy τ (−4∆˜ ut + R˜ ut ) − 2˜ ut ln u ˜t − n˜ ut = µ(g(t), τ )˜ ut , since ft is a minimizer for W. Since µ(g(t), τ ) is uniformly bounded, as in

the previous section we can get that C 2,α norms of u ˜t are uniformly bounded. This implies that C 2,α norms of ut are uniformly bounded. Before we proceed with further discussion notice the following. R n Remark 13. M (4πτ )− 2 e−ft (s) dVg(s) = 1. This is a simple consequence of R n the fact that M (4πτ )− 2 e−ft dVg(t) = 1, since ft is a minimizer for W with respect to g(t), and the following backward parabolic equation n d ft (s) = −∆ft (s) + |∇ft (s)|2 − R + . ds 2τ

10

Namely, Z Z n n d −ft (s) e dVg(s) ) = e−ft (s) (∆ft (s) − |∇ft (s)|2 + R − ( − R + )dVg(s) ds M 2τ 2τ ZM = ∆(e−ft (s) dVg(s) ) = 0 M

Since log is a concave function and u ˜t (s)2 (4πτ )−n/2 dV is a probability measure, we have by Jensen and Sobolev inequalities Z Z n−2 2 2 −n/2 u ˜t (s) log u ˜t (s) (4πτ ) dV = u ˜t (s)2 log u ˜t (s)4/(n−2) (4πτ )−n/2 dV 2 M MZ n−2 ≤ log u ˜t (s)2n/(n−2) (4πτ )−n/2 dV 2 MZ n−2 ≤ log[C (|∇˜ ut (s)|2 + u ˜t (s)2 )dV ](n−2)/n + 2 M n−2 log(4πτ )−n/2 + 2 Z n = log C τ (|∇˜ ut (s)|2 + u ˜t (s)2 )(4πτ )−n/2 dV. 2 M This inequality shows that Z τ |∇˜ ut (s)|2 (4πτ )−n/2 dV ≤ C.

(11)

M

The constant C does not depend either on t or s ∈ [0, t]. To conclude, we have the following estimates Z n |˜ ut (s)|2 (4πτ )− 2 dVs ≤ C1 M

n

τ (4πτ )− 2

Z

M

|∇s u ˜t (s)|2 dVs ≤ C2 ,

that is we have that |˜ ut |W1,2 ≤ C for a uniform constant C.

Take a sequence ti → ∞. There exists a subsequence such that g(ti +t) →

h(t) when i → ∞, where h(t) is a Ricci flow on M . This follows from

Hamilton’s compactness theorem ([7]). Fix A > 0. ft will be a minimizer for W with respect to g(t), which we flow backward, for every t. Let s ∈ [0, A]. 11

Lemma 14. For every A > 0 there exists δ = δ(A) > 0 such that ut+A (t + s) ≥ δ > 0 for all t and all s ∈ [0, A]. Proof. Assume that the statement of the lemma is not true. In that case there would exist a sequence si such that minM usi +A (si +ai ) → 0 as i → ∞,

for some ai ∈ [0, A]. Consider the equation

n d usi +A (si + t) = −∆usi +A (si + t) + (R − )usi +A (si + t), dt 2τ usi +A (si + A) = usi +A , for t ∈ [0, A]. Let u î (si + t) = minM usi +A (si + t). Then ∆ˆ usi +A (si + t) ≥ 0

and

d u î (si + t) ≤ C u î (si + t), dt

where C is a uniform constant. If we integrate it with respect to t, we get u î (si + A) ≤ eCA u î (si + t). Since u î (si + A) = minM usi +A and since by Lemma 11 we know that there exists a constant δ such that usi +A ≥ δ > 0, we have that usi +A (si + t) ≥

δ(A) > 0 for all i and all t ∈ [0, A]. This contradicts our assumption that u î (si + ai ) → 0 as i → ∞.

Lemma 15. For every A > 0 there exists C(A) such that 1. 2.

R

R

M

ut (s)2 dVg(s) ≤ C(A).

M

|∇ut (s)|2 dVg(s) ≤ C(A),

for all t ≥ A, s ∈ [t − A, t]. Proof. We will consider the equation d n ut (s) = −∆ut (s) + (R − )ut (s) ds 2τ ut (t) = ut ,

12

where ut = e−ft and ft is a minimizer for W with respect to metric g(t). Let u ˆt (s) = maxM ut (s). Then

d u ˆt (s) ≥ −C u ˆt (s), ds where C > 0 is a uniform constant that does not depend either on s or t, but on the uniform bounds on geometries g(t). If we integrate it with respect to s we get u ˆt = u ˆt (t) ≥ e−CA u ˆt (s), for any s ∈ [t − A, t]. On the other hand, we have already proved in the previous section that C 2,α norms of ut are uniformly bounded in t ∈ [0, ∞).

Therefore we get that 0 ≤ ut (s) ≤ C(A) on M for all t ∈ [A, ∞) and all

s ∈ [t − A, t]. Now we immediately get part 1 of our claim. For part 2 notice

that

since

R

Z

M

M

|∇˜ ut

|∇ut (s)|2 dVg(s) = 4 (s)|2

Z

M

˜ ut (s)|∇˜ ut (s)|2 dVg(s) ≤ C(A),

is uniformly bounded for all t ≥ A and s ∈ [t − A, t].

The previous two lemmas tell us that in order to find the uniform estimates on fti +A (ti + s) for s ∈ [0, A], it is enough to find the uniform C k,α

estimates on uti +A (ti + s). Our main goal in this section is to prove the following theorem. Theorem 16. Under the assumptions of the main theorem, with the notations as above, for every A > 0 there exists a uniform constant C, depending on A such that |ut (s)|C 2,α ≤ C for all t ≥ A, ∀s ∈ [t − A, t]. Proof. Consider the equation d n ut (s) = −∆ut (s) + (R(s) − )ut (s), ds 2τ for t ∈ [A, ∞) and s ∈ [t − A, t]. All our further estimates will depend on

A. We will use C to denote different absolute constants that depend on A 13

and the uniform bounds on our geometries g(t). Denote by h = ht (s) = n (− 2τ + R(s))ut (s). Omit the subscript t.

d u + ∆u = h. ds Z

Z Z d d 2 u) + 2 u∆u + (∆u)2 , ds ds M M M in mind that the metric depends on s. Z d − gij ∇i ( u)∇j udVs ds M Z Z 1 d n 2 − |∇u| dVs − |∇u|2 ( − R)dVs + 2 ds M 2τ M Z 1 gpi gqj Di uDj u(2Rpq − gpq )dVs , 2τ M

h2 =

M

where we should keep Z d u∆u = M ds = +

Z

(

(12)

(13)

where the second term on the right hand side of (13) comes from taking the derivative of the volume element and the third term appears from taking the derivative of gij . Denote the former one by J1 and the latter one by J2 . Z

2

(∆u)

=

M

Z

gij Di Dj ugkl Dk Dl u

M

= − = −

Z

gij gkl Dj uDi Dk Dl u

ZM

ij kl

g g Dj uDk Di Dl u +

M Z

= I+

Z

M

l gij gkl Dj uRiks Ds u

ij kl

g g Dk Dj uDi Dl u

M

= I+ where I =

R

Z

M

M

|∇2 u|2 ,

l D u. Let l ∈ (t − A, t) where A > 0. Integrating gij gkl Dj uRiks s

the equation (12) in s, from l to t gives Z t Z Z Z tZ d ( ( u)2 dVs )ds + |∇u|2 dVs |s=l + |∇2 u|2 dVs ds ds l M M l M Z tZ Z Z t = h2 + |∇u|2 dVs |s=t + (2J1 + 2J2 + I). l

M

M

l

14

Z

t

J1 ≤ AC

l

sup s∈(t−A,t)

Z

M

˜ |∇u|2 dVs ≤ C,

for every t. Similarly we get estimates for J2 and I. From all these estimates we can conclude the following Z t Z t−A

Z

t t−A

Z

sup

(14)

M

d ut (s))2 dVs ds ≤ C. ds

|∇2 ut (s)|2 dVs ds ≤ C.

(15)

M

Z

(16)

(

s∈(t−A,t) M

where C = C(A). Let u ˜t =

|∇u|2 dVs ≤ C,

d ds ut (s)

(we will not confuse this u ˜t with one

defined at the beginning of this section). Omit the subscript t. d n d u ˜ = −Ds ∆s u + [(R − )u]. ds ds 2τ Multiply the equation by u ˜ and integrate it along M . Z Z Z Z 1 d d 2 d d n 1 n d ij | u| dVs = − (g(s) Di Dj u)˜ u+ ( (R − ))u˜ u+ (R − )| u|2 dVs 2 ds M ds ds 2τ 2 M 2τ ds M ds ZM Z 1 d = 2 (−Rpq + gpq )gpi (s)gqj (s)Di Dj u˜ u− g(s)ij Di Dj ( u)˜ u+ 2τ ds M M Z Z Z n d ∂u 1 n d d u+ gjk ( Γkij ) u ˜+ (R − )| u|2 dVs . + ( (R − ))u˜ 2τ dt ∂xk 2 M 2τ ds M M ds R R d d Since M g(s)ij Di Dj ds u˜ u = − M |∇s ( ds u)|2 and since we are on the Ricci flow, metrics g(s) are uniformly bounded, after applying Cauchy-Schwartz

inequality and using the uniform boundedness of the curvature operator, we get Z d d u)|2 dVs ds + sup | u|2 ≤ ds t−A M s∈(t−A,t) M ds Z t Z Z t Z d 2 ≤ C | u| dVs ds + C |∇2 u|2 dVs ds + t−A M ds t−A M Z Z d 2 + | u| dVs |s=t + C |∇u|2 . ds M M Z

t

Z

|∇(

15

R

R

R

d 2 2 2 M | ds u(s)| dVs |s=t ≤ C( M |∆ut | + M h(t) ) where h(s) = Since ut = e−ft , where ft are the minimizers for W, like

n ( 2τ −R(s))u(s).

in the previous

section we can conclude that ut ∈ W k,p, with uniform bounds on W k,p norms R d u(s)dVs |s=t are uniformly (these bounds depend on k) and therefore, M | ds bounded in t. This estimate together with estimates (14) and (15) gives that Z

t t−A

Z

M

|∇(

d u)|2 dVs ds ≤ C. ds

Z

sup

s∈(t−A,t) M

If u ˜=

d ds u

˜= and h

d ds h

|

(17)

d 2 u| ≤ C. ds

(18)

then: d ˜ u ˜ = −Ds ∆u + h. ds

Ds ∆u =

d 1 (g(s)ij Di Dj u) = g(s)ip g(s)jq ( gpq − 2Rpq )Di Dj u + g(s)ij Di Dj u ˜ ds τ d + g(s)ij (Γkij )Dk u. ds

˜ − gip gjq ( 1 gpq − 2Rpq )Di Dj u − g(s)ij d (Γk )Dk u H = h (19) τ ds ij d = u ˜ + ∆˜ u. ds Rt R All the estimates that we have got so far tell that t−A M H 2 is uniformly bounded in t. The analogous estimates to the estimates (14), (15) and (16) for u, we can get for

d ds u

(by using the evolution equation for

d ds u

and all

the estimates that we have got so far by analyzing the evolution equation for u).

Z

t

t−A

Z

Z

(|∇2 (

M

t t−A

Z

( M

d u)|2 dVs ds ≤ C. ds

d2 2 u) dVs ds ≤ C. ds2 16

(20)

(21)

Z

sup

s∈(t−A,t) M

|∇(

d u)|2 dVs ≤ C. ds

(22)

To obtain these estimates we have used the fact that Z Z Z d n |∇ u|2 dVg(s) |s=t ≤ C( |∇∆ut |2 + |∇(R − )ut |2 , ds 2τ M M M where the right hand side is uniformly bounded in t, since ut = e−ft and ft are the minimizers for W.

d ut (s) + ht (s) By standard regularity theory, considering ∆ut (s) = − ds

as an elliptic equation whose right hand side has uniformly bounded W 1,2

norms for s ∈ (t − A, t) and all t ≥ A, we have that |ut (s)|W 3,2 ≤ C, for

a uniform constant C that depends on A. Take a derivative in s of the

equation

d ˜ ds u

d ds u.

= −∆˜ u + H, with u ˜=

Denote by u ¯=

d ˜. ds u

By using the

estimates that we have got for u ˜ it is easy to conclude that u ¯ satisfies the equation

where H1 =

d u ¯ = −∆¯ u + H1 , ds d d 1 ip jq ˜+g(s)ij ds (Γkij )Dk u ˜ ds H+g g (−2Rpq + τ )Di Dj u

and

Rt

R

t−A M

H12 dVg(s) ds

is uniformly bounded in t. As in the case of the previous estimates we can conclude that sup s∈(t−A,t)

sup

Z

Z

s∈(t−A,t) M

M

|

d 2 u ˜| dVs ≤ C, ds

|∇(

d u ˜)|2 dVs ≤ C. ds

d By regularity theory applied to the equation ∆˜ u = − ds u ˜ +H, we can get that d ds ut (s)

has uniformly bounded W 3,2 norms. If we go back to the parabolic

equation for ut (s) we can get that |ut (s)|W 5,2 ≤ C for all t ≥ A and all s ∈ (t − A, t). Continuing this process by taking more and more derivatives

in t of our original parabolic equation we can conclude that W p,2 norms of ut (s) are uniformly bounded for every p, by the constants that depend on A and p. Sobolev embedding theorem now gives that all C k,α norms of ut (s) 17

are uniformly bounded for all t > A and all s ∈ [t − A, t], by constants that depend on A and k.

Combining Theorem 16 and Lemma 14, we get that for every A there exist constants Ck = C(k, A) such that |ft (s)|C k,α ≤ Ck , for all t ≥ A and

all s ∈ [t − A, t].

3.4

Ricci soliton in the limit

In this subsection we want to finish the proof of Theorem 2. We have uniform curvature and diameter bounds for our flow g(t). We have already proved that we also have a volume noncollapsing condition along the flow, for all times t ≥ 0. This gives a uniform lower bound on the

injectivity radii. Hamilton’s compactness theorem (modified to the case of

our flow) gives that for every sequence ti → ∞ there exists a subsequence so

that g(ti + t) → h(t) uniformly on compact subsets of M × [0, ∞) and that h(t) is a solution to the Ricci flow (1). We will show below that for each t,

h(t) satisfies actually a Ricci soliton equation with the Hessian of function fh (t) involved, where fh (t) is a smooth one parameter family of functions. We will now see how we get the functions fh (t), using the estimates on ft (s) from the previous subsection and Perelman’s monotonicity formula. Take any t and let ft be a function so that µ(g(t), τ ) = W(g(t), ft , τ ).

Flow ft backward. Fix A > 0. Then:

I(t) = W(g(t+A), ft+A , τ )−W(g(t), ft+A (t), τ ) ≤ µ(g(t+A), τ )−µ(g(t), τ ) → 0(t → ∞). 0 ≤ I(t) =

Z

0

A

d W (g(t + s), ft+A (t + s), τ )ds → 0, du

as t → ∞. We will consider uti +A (ti + s) where s ∈ [0, A]. We will divide the proof of the theorem in a few steps. Step 16.1. ∀A > 0, limi→∞

d du W (g(s

+ ti ), fti +A (s + ti ), τ ) = 0 for almost

all s ∈ [0, A].

18

Proof. I(ti ) → 0 by Claim 9. On the other hand I(ti ) = W(g(ti +A), fti +A , τ )−W(g(ti ), fti +A (ti ), τ ) = Since by Perelman’s monotonicity formula 0, we have that

d W (g(ti limi→∞ du

d du W (g(ti

Z

0

A

d W (g(ti +s), fti +A (ti +s), τ )ds. du

+ s), fti +A (ti + s), τ ) ≥

+ s), fti + (ti + s), τ ) = 0 for almost all

s ∈ [0, A], for Z

A

lim

0

i→∞

d W (g(ti + s), fti +A (ti + s), τ )ds ≤ lim I(ti ), i→∞ du

by Fatuous lemma. Step 16.2. |˜ ut (s)|C 2,α ≤ C, ∀t, where u ˜t (s) =

d ds ut (s).

Proof. Following the notation of the previous subsection, we get that: d u ˜t (s) = −∆˜ ut (s) + Ht (s), ds where Ht (s) =

d ip jq 1 ds ht (s) + g g ( τ gpq

u ˜t (t) =

d − 2Rpq )Di Dj u + gij ds (Γkij )Dk u.

n d ut (s) = −∆ut + (− + R)ut . ds 2τ

In the previous subsection we have proved that there exist a uniform lower and an upper bound on ut (s) and that |ut (s)|W 3,p ≤ C(p, A) for all t ≥ A

and all s ∈ [t − A, t]. Similarly we can get that |ut (s)|W k,p ≤ C(k, p, A) and therefore |˜ ut (s)|W k−2,p ≤ C(k, p, A), ∀t ≥ A and all s ∈ [t − A, t]. We can get

that |˜ ut (s)|C 2,α ≤ C, for all t ≥ A and ∀s ∈ [t − A, t]. We can extend this to

all higher order time derivatives of ut (s).

Step 16.3. For every A > 0 there exists a subsequence ti , so that the limit metric h(s) of a sequence g(ti + s) is a Ricci soliton for s ∈ [0, A]. Proof. By step 16.1 we have that lim Rjk (ti + s) + ∇j ∇k fti +A (ti + s) −

i→∞

19

1 gjk (ti + s) = 0, 2τ

for almost all s ∈ [0, A] and almost all x ∈ M , since Z n 1 d 2τ |Rjk +∇j fti +A ∇k fti +A − gjk |2 dVg(ti +s) . W(g(ti +s), fti +A (ti +s), τ ) = (4πτ )− 2 ds 2τ M By Lemma 14 and Theorem 16, we have that 0 < C1 ≤ |uti +A (s + ti )| ≤ C2

for all i ≥ i0 and all s ∈ [0, A], for some constants C1 and C2 that depend

on A. By step 16.2 and Theorem 16 we can find a subsequence, say {ti } such that fti +A (ti + s) converges in C 2,α norm to fÃ (s) for all s ∈ [0, A]

and all x ∈ M . More precisely, for a countable dense subset {sj } of [0, A]

there exists a subsequence so that fti +A (ti + sj ) converges in C 2,α norm to fÃ (sj ) on M . For any s ∈ [0, A] there exists a subsequence tik so that ft +A (ti + s) converges to fÃ (s) in C 2,α norm. We want to show that ik

k

2,α

C actually fti +A (ti + s) → fÃ (s). For that we use the fact that

d ds fti +A (ti + s)

is uniformly bounded in C 2,α norm, and therefore

|fÃ (s) − fÃ (s0 )|C 2,α < ǫ, for some small ǫ > 0 and some s0 ∈ {sj } that is sufficiently close to s. We also have

|fÃ (s0 ) − fti +A (ti + s0 )|C 2,α < ǫ, for i ≥ i0 and

|fti +A (ti + s) − fti +A (ti + s0 )|C 2,α < ǫ,

d since | ds fti +A (ti +s)|C 2,α ≤ C(A), for all i ≥ i0 and all s ∈ [0, A]. By triangle

inequality, we now get that for every ǫ > 0 there exists i0 so that |fÃ (s) − fti +A (ti + s)|C 2,α < 3ǫ, for all i ≥ i0 and all s ∈ [0, A].

fti +A (ti + s) converges in C 2,α norm on M to fÃ (s), for all s ∈ [0, A].

Finally, we get that

Rjk + ∇j ∇k fÃ (s) − 20

1 hjk (s) = 0, 2τ

(23)

for all s ∈ [0, A], and for almost all x ∈ M . Because of the continuity it will hold for all x ∈ M . Since h(s) is a Ricci flow, all covariant derivatives of h

and the covariant derivatives of a curvature operator are uniformly bounded, and therefore |∇p fÃ (s)| ≤ C(p), ∀s ∈ [0, A] and all p ≥ 2. Also we have that p

d p˜ | ds k ∇ fA (s)| ≤ C(p, k) where C(p, k) does not depend on A, for p ≥ 2.

Step 16.4. We can glue all the functions fÃ that we get for different values of A, to get a function fh (s) defined on M × [0, ∞), which defines our metric h(s) as a soliton type solution for all times s ≥ 0.

Proof. Take any increasing sequence Aj → ∞. For every Aj , by the previC 2,α ous step we can extract a subsequence ti so that ft +A (ti + s) → fÃ (s) i

j

j

for all s ∈ [0, Aj ]. Diagonalization procedure gives a subsequence so that C 2,α ft +A (s) → fÃ (s) for all j and all s ∈ [0, Aj ]. For this subsequence ti i

j

j

we have that g(ti + t) → h(t), uniformly on compact subsets of ×[0, ∞). Compare the functions fÃj and fÃk for j < k, on the interval [0, Aj ]. We know that they both satisfy

n = 0, ∆h(s) fÃr + R(h(s)) − 2τ and therefore ∆h(s) (fÃj − fÃk ) = 0. Since M is compact, this implies that fÃ (s) = fÃ (s) + cAk (s), for s ∈ [0, Aj ], where cAk (s) is a constant function Aj

j

k

Aj

for every s ∈ [0, Aj ]. On the other hand, because of the integral normalization condition, we have

− n2

(4πτ ) − n2

(4πτ )

Z

Z

−fÃj (s)

e

M

−fÃk (s)

e M

A

dVh(s) = 1,

−cAk (s)

dVh(s) = 1 = e

j

−n 2

(4πτ )

Z

−fÃj (s)

e M

dVh(s) ,

Ak which implies that cA (s) = 0 for all s ∈ [0, Aj ] and all k ≥ j. Therefore j ˜ ˜ fAj (s) = fAk (s) for all s ∈ [0, Aj ]. Define a function fh (s) in the following

21

way. Let fh (s) = fÃj (s), for all s ∈ [0, Aj ] and all Aj → ∞. fh (s) is a well

defined function because of the previous discussion. We also have that R(h(s))pq + ∇p ∇q fh (s) −

1 h(s)pq = 0, 2τ

(24)

holds for all s ∈ [0, ∞). The definition of fh (s) does not depend on a choice

of an increasing sequence Aj . Namely, if Bj were another increasing sequence and if fh′ (s) were functions defined using the sequences Bj and ti (ti is the same sequence as above), then at each time both functions fh (s) and fh′ (s) would satisfy the same equation (24) and the same integral normalization condition. Therefore fh (s) = fh′ (s) for all s ∈ [0, ∞).

3.5

Some properties of the limit solitons

Let ti be any sequence converging to infinity. Then as we have seen earlier, there exists a subsequence such that g(ti + s) → h(s), where h(s) is a Ricci ˆ soliton. Let R(h(t)) = min R(h(t)). We will first state a theorem that R. Hamilton proved in his paper [9]. ˆ≤ Theorem 17 (Hamiton). Under the normalized Ricci flow, whenever R ˆ ≥ 0 it remains so forever. 0, it is increasing, whereas if ever R We will use the proof of Theorem 17 to prove the following lemma. ˆ Lemma 18. Under the assumptions of Theorem 2, R(h(t)) ≥ 0, ∀t, for the limit metric h(t) of any sequence of metrics g(ti ), where g(t) is a solution of d 1 gjk = −2Rjk (g(t)) + gjk (t). dt τ ˆ Proof. Assume that there exists t0 such that R(h(t 0 )) < 0. Without loss of generality assume that t0 = 0. Since g(ti ) → h(0) as i → ∞, there exists i0 , ˆ so that for all i ≥ i0 R(g(t i )) < 0. The evolution equation for R is 2 n d R = ∆R + 2|Ric|2 + R(R − ). dt n 2τ 22

This implies

d ˆ 2ˆ ˆ n R ≥ R( R − ). dt n 2τ ˆ ≤ 0, then R ˆ is increasing (since d R ˆ ≥ 0). If R ˆ ≥ 0 at some time it can If R dt ˆ not go negative at later times. If there existed t > ti such that R(g(t)) ≥ 0, 0

ˆ ≥ 0 would remain so forever, for all s ≥ t and therefore we could not then R ˆ ˆ have R(g(t i )) < 0 for ti > t. That contradicts the fact that R(g(ti )) < 0 for ˆ all i ≥ i0 . Therefore ∀t ≥ ti we have that R(g(t)) < 0. 0

ˆ 2ˆ ˆ n dR ≥ R( R − ) ≥ 0, dt n 2τ ˆ is increasing and therefore there exfor all t big enough. That implies R ˆ ˆ ists limt→∞ R(g(t)) = −C ≤ 0. Moreover R(h(s)) = −C for all s. Since

ˆ ˆ limi→∞ R(g(t i )) = R(h(0)) < 0, C > 0. We also have that

ˆ dR(h(s)) 2ˆ n 2 n ˆ ≥ − R(h(s))( − R(h(s))) = C( + C) ≥ 0. ds n 2τ n 2τ The left hand side of the above inequality is zero and therefore we get that n or C = 0. Since C > 0, we get a contradiction. Therefore C = − 2τ

R(h(t)) ≥ 0 for all t, what we wanted to prove.

Remark 19. Let (M, g) be a compact manifold and g(t) be a Ricci flow on M . Since d W= dt

Z

M

2τ |Rij + ∇i ∇j f −

n 1 gij |2 (4πτ )− 2 e−f dV, 2τ

W(g, f, τ ) = const along the flow, if g is a Ricci soliton satisfying the equa-

tion

Rij + ∇i ∇j f −

1 gij = 0. 2τ

Let ti → ∞ and si → ∞ be two sequences such that g(ti + t) → h(t) and

g(si + t) → h′ (t) where h(t) and h′ (t) are 2 Ricci solitons on M that have

been constructed earlier. We have proved that Rjk (h) + ∇j ∇k fh (t) − 23

1 hjk = 0, 2τ

Rjk (h′ ) + ∇j ∇k fh′ (t) −

1 ′ h = 0, 2τ jk

where fh (t) = lim lim fAj +ti (ti + t), j→∞ i→∞

fh′ (t) = lim lim fBj +si (si + t), j→∞ i→∞

for some increasing sequences Aj → ∞ and Bj → ∞. By Remark 19 we know that W(h(t), fh (t), τ ) = C1 and W (h′ (t), fh′ (t), τ ) = C2 are constant

along the flows h(t) and h′ (t) respectively.

Lemma 20. C1 = C2 , i.e. W(h(t), fh (t), τ ) is a same constant for all solitons h(t) that arise as limits of sequences of metrics of our original flow

g(t) (1) on a compact manifold M . Proof. W(g(ti + t), fti +Aj (ti + t), τ ) − W(g(si ), fsi +Bj (si ), τ ) ≤ ≤ W(g(ti + Aj ), fti +Aj (ti + Aj ), τ ) − W(g(si ), fsi (si ), τ ) = = µ(g(ti + Aj ), τ ) − µ(g(si ), τ ) → 0,

(25)

where we have used the fact that W(g(t), f (t), τ ) increases in t along the

flow (1) and the fact that fsi (si ) = fsi is a minimizer for W(g(si ), f, τ ) over R n all f belonging to a set {f | M (4πτ )− 2 e−f dVg(si ) }. Similarly, W(g(ti + t), fti +Aj (ti + t), τ ) − W(g(si ), fsi +Bj (si ), τ ) ≥

≥ W(g(ti + t), fti +t (ti + t), τ ) − W(g(si + Bj ), fsi +Bj (si + Bj ), τ ) = = µ(g(ti + t), τ ) − µ(g(si + Bj ), τ ) → 0, when i → ∞. From equations (25) and (26), letting i → ∞ we get W(h(t), fÃj (t), τ ) − W(h′ (0), f˜B′ j (0), τ ) ≤ 0. W (h(t), fÃj (t), τ ) − W (h′ (0), f˜B′ j (0), τ ) ≥ 0. 24

(26)

Let j → ∞ to get C1 = W(h(t), fh (t), τ ) = W(h′ (0), fh′ (0), τ ) = C2 .

Lemma 21. For every Ricci soliton h(t) that arises as a limit of some sequence of metrics of our original flow g(t), the corresponding function fh (t), that we have constructed before, is a minimizer for Perelman’s functional W

with respect to a metric h(t).

Proof. We will first proof the following claim. Claim 22. There exists a sequence ti → ∞ such that g(ti + t) → h(t) as

1 hjk = 0 i → ∞, where h(t) is a Ricci soliton satisfying Rjk (h) + ∇j ∇k fh − 2τ

and fh (t) is a minimizer for W(h(t), f, τ ).

Proof of the Claim. Let H(t) = (4πτ )−n/2

R

M

2τ |Ri j + ∇i ∇j ft −

1 2 2τ gij | dt,

where ft is a function such that µ(g(t), τ ) = W (g(t), ft , τ ). If we flow ft backward by the equation d n f = −∆f + |∇f |2 − R + , dt 2τ

starting at time t, for every t > 0 we get solutions ft (s). Look at Ft (s) = W(g(s), ft (s), τ ). We know that n d Ft (s) = (4πτ )− 2 ds

Z

M

2τ |Rjk + ∇j ∇k ft (s) −

Ft (s) is a continuous function in s ∈ [0, t] and lims→t

1 g(s)jk |2 dVg(s) . 2τ d ds Ft (s)

= H(t). There-

fore there exists a left derivative of Ft (s) at point t and (Ft )′− (t) = H(t) for

every t > 0. Moreover, g(t) and all the derivatives of ft up to the second order are Lipshitz functions in t (this follows from the estimates in the previous subsections) and therefore

25

µ(t) := µ(g(t), τ ) = {f |

R

inf

n

(4πτ )− 2 e−f =1} M

W(g(t), f, τ )

is a Lipshitz function in t as well, i.e. k(t) = Ft (t) = W(g(t), ft , τ ) is a Lipshitz function in t. This tells that k(t) is differentiable in t, almost

everywhere. Our discussion then implies that k′ (t) = H(t) in a sense of distributions. Z

∞

H(t)dt =

δ

= =

lim

Z

K→∞ δ

K

k′ (t)dt

lim W (g(K), fK , τ ) − W (g(δ), fδ , τ )

K→∞

lim (µ(g(K), τ ) − µ(g(δ), τ ) ≤ C,

(27)

K→∞

where δ > 0 and C is some uniform constant. We have that

R∞ δ

H(t) ≤ C.

This implies that there exists a sequence ti → ∞ such that H(ti ) → 0 as i → ∞, i.e.

1 gjk )(ti ) = 0. 2τ By what we have proved before, after extracting a subsequence we can assume that g(ti ) → h(0) smoothly and fti → f˜ in C 2,α norm, where by Theorem 12 f˜ is a minimizer for W with respect to metric h(0). Therefore, lim (Rjk + ∇j ∇k fti −

i→∞

1 Rjk (h(0)) + ∇j ∇k f˜ − hjk (0) = 0. 2τ

(28)

On the other hand g(ti + t) → h(t) as i → ∞ where h(t) is a Ricci soliton

and

1 hjk (t) = 0, (29) 2τ where fh (t) = limj→∞ limi→∞ fti +Aj (ti + t), for some sequence Aj → ∞. From equations (28) and (29) we have that ∆(fh (0) − f˜) = 0, i.e. fh (0) = R n ˜ f˜ + C for some constant C. We know that M (4πτ )− 2 e−f dVh(0) = 1, since f˜ is a minimizer. From the construction of fh (t) it follows that Rjk (h(t)) + ∇j ∇k fh (t) −

26

R

−n 2 e−fh (0) dV h(0) M (4πτ )

= 1 and therefore f˜ = fh (0). Since there exists

a finite limit, limt→∞ µ(g(t), τ ), we have that µ(h(0), τ ) = µ(h(t), τ ) for all t. This implies that µ(h(t), τ ) = µ(h(0), τ ) = W(h(0), f˜, τ ) = W(h(0), fh (0), τ ) = W(h(t), fh (t), τ ), where we have used the fact that W is constant along a soliton. This means that fh (t) is a minimizer for W with respect to a metric h(t), for every

t ≥ 0.

To continue the proof of Lemma 21 take any sequence si → ∞. By

a sequential convergence of our original flow g(t) to Ricci solitons, after extracting a subsequence we may assume that g(si + t) → h′ (t) as i → ∞

where h′ (t) is a Ricci soliton. Take a soliton h(t) with the properties as in Claim 22. From the convergence of µ(g(t), τ ) we know that µ(h′ (t), τ ) = µ(h(s), τ ) for all t and all s. µ(h′ (t), τ ) = µ(h(s), τ ) = W(h(s), fh (s), τ ).

(30)

By Lemma 20 we have that W(h(s), fh (s), τ ) = W(h′ (t), fh′ (t), τ ) for all s

and t. Combining this with (30) gives that µ(h′ (t), τ ) = W(h′ (t), fh′ (t), τ ),

i.e. fh′ (t) is a minimizer for h′ (t) for every t.

One useful property of the sequential soliton limits of our flow (1) is that all limit solitons are the solutions of the normalized flow equation d 2 hij = −2Rij + r(h(t))hij , dt n R 1 where r(h(t)) = Volh(t) M M R(h(t))dVh(t) . In the case of any of our soliton

n limits, we have that R(h(t))+∆fh (t)− 2τ = 0 and therefore r = r(h(t)) =

n 2τ

for all t ≥ 0. Remark 23. Let ti → ∞ and g(ti + t) → h(t), where h(t) is an Einstein

metric with an Einstein constant

1 2τ .

If Volh′ (M ) = Volh (M ), for any other

27

limit soliton h′ , then h′ is an Einstein metric with the same Einstein constant 1 2τ .

Proof. The fact that h is Einstein metric implies that ∇i ∇j fh = −2Rij + 1 τ hij

= 0, that is ∆fh = 0.

Since M is compact, fh = C such that

(4πτ )−n/2 e−C Volh (M ) = 1. An easy computation shows that µ(h, τ ) = W(h, C, τ ) = C −

n 2,

and therefore µ(h′ , τ ) = µ(h, τ ) = C −

n 2.

Then,

(4πτ )−n/2 e−C Volh′ (M ) = 1, implies that f = C is a minimizer for W with

respect to h′ as well. This yields

τ (2∆f − |∇f |2 + R(h′ )) + f − n = C − that is R(h′ ) = From ∆fh′ =

n , 2

n . 2τ

n − R(h′ ) = 0, 2τ

we get that fh′ = C and therefore Rij (h′ ) + ∇i ∇j fh′ − yields Rij (h′ ) =

1 ′ h = 0, 2τ ij

1 ′ 2τ hij .

In the discussion that follows we will use Moser’s weak maximum principle. We will state it below, for a reader’s convenience. Lemma 24 (Moser’s weak maximum principle). Let g = g(t), 0 ≤

t < T , be a smooth family of metrics, b a nonnegative constant and f a nonnegative function on M × [0, T ) which satisfies the partial differential

inequality

df ≤ ∆f + bf, dt on M × [0, T ], where ∆ refers to a Laplacian at time t. Then for any x ∈ M ,

t ∈ [0, T ),

28

n 1 1 1+n/2 |f (x, t)| ≤ c √ ecHd max(1, d 2 )(b + l + ) 2 ecbt ||f0 ||L2 , t V

where c is a positive constant depending only on n and d = max0≤t≤T diam(M, g(t)), p H = max0≤t≤T ||Ric||C 0 , f0 = f (·, 0), V = min0≤t≤T Volg(t) (M ). The following remark will give us a condition that will imply obtaining

the Einstein metrics in the limit. Remark 25. If g(t) is a solution to (gij )t = −2Rij + τ1 gij , for t ∈ [0, ∞)

such that

1. A curvature operator and a diameter are uniformly bounded along the flow. 2. 0 ≤ R(x, t) ≤

n 2τ

for all x ∈ M and all t ∈ [0, ∞).

Then all the solitons that arise as limits of the subsequences of our flow g(t) are Einstein metrics with scalar curvatures R = zero, uniformly on M as t → ∞. Tij = Rij −

n 2τ

R n gij

and Tij (t) converge to

is a traceless part of the

Ricci curvature.

Proof of the Remark. Notice that now we do not make an assumption that one of the metrics that we get in a limit is an Einstein metric. Look at the R evolution equation for r(t) = Volt1(M ) M RdVt , d 1 r(t) = (2 dt Volt (M )

R≤

n 2τ

implies r(t) ≤

n 2τ

Z

2 |T | + (1 − ) n M 2

Z

R( M

n n − R) + r(r − ). 2τ 2τ

and therefore

d 2 r(t) ≥ dt Volt (M )

Z

M

|T |2 + r(r −

n ). 2τ

(31)

We have proved that in the case of flow g(t), a volume noncollapsing condin d ln(Volt (M )) = 2τ − r and C1 ≤ Volt (M ) ≤ tion holds for all times t ≥ 0. dt R∞ n C2 give that 0 ( 2τ − r(t))dt < ∞. We can integrate the inequality (31) in

29

t ∈ [0, ∞). This, together with the uniform estimates on Volt (M ) and r(t)

give that

Z

0

∞Z

M

|T |2 dVt ≤ C.

(32)

Following the calculations in Hamilton’s paper [6], Rugang computed the evolution equation for T under a normalized Ricci flow ([12]). In the case of flow (1) we have d 4 n |T |2 = ∆|T |2 − 2|∇T |2 + 4Rm(T ) · T + (R − )|T |2 . dt n 2τ

(33)

Since the curvature operators of g(t) are uniformly bounded, we derive from equation (33) that d |T | ≤ ∆|T | + C|T |. dt Applying Lemma 24 to this differential inequality and intervals [t − 1, t + 1] where t > 1, we derive 2

2

|T | (x, t) ≤ ||T || (t)C 0 (M )

Z ≤ C(

Mt−1

|T |2 ),

where Mt = (M, g(t)). Integrate this inequality in t ∈ [k, k+1], for all k ≥ k0 and sum up all the inequalities that we get this way. We get Z ∞ X Z k+1 Z ||T ||2 dt ≤ C ( |T |2 )dt k0

k≥k0

Z

∞

k0

2

||T || dt ≤ C

Z

k

∞Z

k0

M

Mt−1

|T |2 dVt−1 dt,

where dVt−1 is a volume form for metric g(t−1). because

d dt

ln Volt =

n 2τ

R

M

|T |2 dVt−1 ≤ C

(34) R

M

|T |2 dVt ,

−R and the curvatures of g(t) are uniformly bounded.

The right hand side of inequality (34) is bounded by a uniform constant, beR∞ cause of the estimate (32). Therefore k0 ||T ||2 dVt ≤ C. If there exists (p, t0 ) such that |T |2 (p, t0 ) > ǫ, then there is a small

neighbourhood of (p, t0 ) in M ×[0, ∞), say Uδ (p, t0 ) = Bp (δ, t0 )×[t0 −δ, t0 +δ]

such that |T |2 (x, t) ≥

ǫ 2

for all (x, t) ∈ Uδ (p, t0 ). This follows from the fact

that in the case of a Ricci flow, a bound |Rm| ≤ C implies |Dk Dtl Rm| ≤ 30

C(k, l). Costant δ does not depend on a point (p, t0 ) ∈ M × [0, ∞), since all

our bounds and estimates are uniform.

If there existed ǫ > 0 and a sequence of points (pi , ti ) ∈ M × [0, ∞), with

ti → ∞ such that |T (pi , ti )| ≥ ǫ then we would have that ||T ||C 0 ≥ 2ǫ for R∞ all t ∈ [ti − δ, ti + δ] and for all i. This would imply C ≥ 0 ||T ||2 dVt ≥ P∞ i=0 ǫδ = ∞. This is impossible. Therefore, ||T ||C 0 (Mt ) → 0 as t → ∞. d dt

ln(Volt ) =

n 2τ

− R ≥ 0 for all t imply that there exists a finite

limt→∞ Volt for every x ∈ M (otherwise we can argue as in the previous

paragraph). If we integrate this equation in t ∈ [0, ∞), we will get that R∞ n 0 ( 2τ − R)dt < ∞. As in the case for a traceless part of the Ricci curvature

T , we can conclude that limt→∞ R =

n 2τ

uniformly on M .

We can conclude that under the assumptions given at the beginning of this remark, for every sequence ti → ∞ we can find a subsequence such that g(ti + t) → h(t), where h(t) is an Einstein soliton with scalar curvature

We also know that Rij − there exists limt→∞ Volt .

1 2τ gij

n 2τ .

→ 0 as t → ∞, uniformly on M and that

To conclude, we have proved a sequential convergence of a solution of a τ -flow towards solitons (generalizations of Einstein metrics), under uniform curvature and diameter assumptions. We still do not know whether we get a unique soliton (up to diffeomorphisms) in the limit or not. All observations in this subsection are in favour of the uniqueness of a soliton in the limit.

References [1] H.D.Cao: Deformation of Kahler metrics to Kahler-Einstein metrics on comapct Kahler manifolds; Invent. math. 81 (1985) 359–372. [2] J. Cheeger, T. Colding: On the structure of spaces with Ricci curvature bounded below.

31

[3] T. Colding: Ricci curvature and volume convergence; Annals of Mathematics, 145 (1997), 477–501. [4] D. Deturck: Deforming metrics in the direction of their ricci tensors; J. Diff. Geom. 18 (1983), 157–162. [5] D. Glickenstein: Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates; preprint arXiv:math.DG/0211191 v2. [6] R. Hamilton: Three-manifolds with positive Ricci curvature, Journal of Differential Geometry 17 (1982) 225–306. [7] A compactness property for solutions of the Ricci flow, Amer.J.Math. 117 (1995) 545–572. [8] The formation of singularities in the Ricci flow, Surveys in Differential Geometry, vol. 2, International Press, Cambridge, MA (1995) 7–136. [9] R. Hamilton:Non-singular solutions of the Ricci flow on 3 manifolds, Communications in analysis and geometry vol. 7 (1999) 695–729. [10] G. Perelman: The entropy formula for the Ricci flow and its geometric applications, preprint. [11] Rothaus: Logarithmic Sobolev inequality and spectrum, Journal of Dif. Anal. 42 (1981) 109–120. [12] Rugang Ye: Ricci flow, Einstein metrics and space forms, Transactions of the american mathematical society, volume 338, number 2 (1993) 871–895.

32

Ricci Flow with Surgery on Four-manifolds

arXiv:math/0504478v3 [math.DG] 4 Jun 2006

with Positive Isotropic Curvature Bing-Long Chen and Xi-Ping Zhu Department of Mathematics Zhongshan University Guangzhou, P.R.China (Revised version)

Abstract In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of the main theorem of Hamilton in [19]; the other is to extend some results of Perelman [27], [28] to four-manifolds. During the proof we have actually provided, parallel to the paper of the second author with H.-D. Cao [3], all necessary details for the part from Section 1 to Section 5 of Perelman’s second paper [28] on the Ricci flow.

1

1. Introduction Let M n be a compact n-dimensional Riemannian manifold with metric gij (x). The Ricci flow is the following evolution equation ∂ gij (x, t) = −2Rij (x, t), for x ∈ M and t > 0, ∂t

(1.1)

with gij (x, 0) = gij (x), where Rij (x, t) is the Ricci curvature tensor of the evolving metric gij (x, t). This evolution system was initially introduced by Hamilton in [13]. Now it has been found to be a powerful tool to understand the geometry, topology and complex structure of manifolds (see for example [13], [14], [15], [19], [20], [4], [22], [2] [9], [8], [27], [28], [3] etc.) One of the main topics in modern mathematics is to understand the topology of compact three dimensional and four dimensional manifolds. The idea to approach this problem via the Ricci flow is to evolve the initial metric by the evolution equation (1.1), and try to study the geometries under the evolution. The key point of this method is to get the long-time behavior of the solutions of the Ricci flow. For a compact three (or four) dimensional Riemannian manifold with positive Ricci curvature (or positive curvature operator, respectively) as initial data, Hamilton [13] (or [14] respectively) proved that the solution to the Ricci flow keeps shrinking and tends to a compact manifold with positive constant curvature before the solution vanishes. Consequently, a compact three-manifold with positive Ricci curvature or a compact four-manifold with positive curvature operator is diffeomorphic to the round sphere or a quotient of it by a finite group of fixed point free isometrics in the standard metric. In these classical cases, the singularities are formed everywhere simultaneously and with the same rates. Note that even though the Ricci flow may develop singularities everywhere at the same time, the singularities can still be formed with different rates. The general case is that the Ricci flow may develop singularities in some parts while keeps smooth in other parts for general initial metrics. This suggests that we have to consider the structures of all the singularities (fast or slow forming). For the general case, naturally one would like to cut off the singularities and to continue the Ricci flow. If the Ricci flow still develops singularity after a while, one can do the surgeries and run the Ricci flow 2

again. By repeating this procedure, one will get a kind of “weak” solution to the Ricci flow. Furthermore, if the “weak” solution has only a finite number of surgeries at any finite time interval and one can remember what had been cut during the surgeries, as well as the “weak” solution has a wellunderstood long-time behavior, then one will also get the topology structure of the initial manifold. This surgerically modified Ricci flow was initially developed by Hamilton [19] for compact four-manifolds. More recently, the idea of the Ricci flow with surgery was further developed by Perelman [28] for compact three-manifolds (see [3] for complete detail). Let us give a brief description for the arguments of Hamilton in [19]. Recall that a Riemannian four-manifold is said to have positive isotropic curvature if for every orthonormal four-frame the curvature tensor satisfies R1313 + R1414 + R2323 + R2424 > 2R1234 . An incompressible space form N 3 in a four-manifold M 4 is a threedimensional submanifold diffeomorphic to S3 /Γ (the quotient of the threesphere by a group of isometries without fixed point) such that the fundamental group π1 (N 3 ) injects into π1 (M 4 ). The space form is said to be essential unless Γ = {1}, or Γ = Z2 and the normal bundle is non-orientable. In [19], Hamilton considered a compact four-manifold M 4 with no essential incompressible space-form and with a metric of positive isotropic curvature. He used this metric as initial data, and evolved it by the Ricci flow. From the evolution equations of curvatures, one can easily see that the curvature will become unbounded in finite time. Under the positive isotropic curvature assumption, he proved that as the time tends to the first singular time, either the solution has positive curvature operator everywhere, or it contains a neck, a region where the metric is very close to the product metric on S3 × I, where I is an interval and S3 is a round three-sphere, or a quotient of this by a finite group acting freely. When the solution has positive curvature operator everywhere, it is diffeomorphic to S4 or RP4 by [14], so the topology of the manifold is understood and one can throw it away. When there is a neck in the solution, he used the no essential incompressible space form assumption to conclude that the neck must be S3 × I or S3 × I/Z2 where Z2 acts antipodally on S3 and by reflection on I. For the first case, one can 3

replace S3 × I with two caps (i.e. two copies of the differential four-ball B4 ) by cutting the neck and rounding off the neck. While for the second case, one can do the quotient surgery to eliminate an RP4 summand. In [19], Hamilton performed these cutting and gluing surgery arguments so carefully that the positive isotropic curvature assumption and the improved pinching estimates are preserved under the surgeries. It is not hard to show that, after surgery, the new manifold still has no essential incompressible space form. Then by using this new manifold as initial data, one can run the Ricci flow and do the surgeries again. These arguments were given in Section A-D of [19]. In the last section (Section E) of [19], Hamilton showed that after a finite number of surgeries in finite time, and discarding a finite number of pieces which are diffeomorphic to S4 , RP4 , the solution becomes extinct. This concludes that the four-manifold is diffeomorphic to S4 , RP4 , S3 × S1 , the twisted product e 1 ( i.e., S3 ×S ˜ 1 = S3 × S1 /Z2 , where Z2 flips S3 antipodally and rotates S3 ×S S1 by 1800 ), or a connected sum of them. The celebrated paper [28] tells us how to recognize the formation of singularities and how to perform the surgeries. One can see from Section A to D of [19] that every statement is accurate and every proof is complete, precise and detailed. Unfortunately, the last section (Section E) contains some unjustified statements, which have been known for the experts in this field for several years. For example one can see the comment of Perelman in [28] (Page 1, the second paragraph) and one can also check that the proof of Theorem E 3.3 of [19] is incomplete (in Proposition 3.4 of the present paper, we will prove a stronger version of Theorem E 3.3 of [19]). The key point is how to prevent the surgery times from accumulated (furthermore, it requires to perform only a finite number of surgeries in each finite time interval). By inspecting the last section of [19], it seems that surgeries were taken on the parts where the singularities are formed from the global maximum points of curvature. Intuitively, the other parts, where the curvatures go to infinity also but not be comparable to the global maximums, will still develop singularities shortly after surgery if one only performs the surgeries for the global maximum points of curvature. To prevent the surgery times from accumulated, one needs to cut off those singularities (not just the curvature maximum points) also. This says that one needs to perform surgeries for all 4

singularities. Another problem is that, when one performs the surgeries with a given accuracy at each surgery time, it is possible that the errors may add up to a certain amount which causes the surgery times to accumulate. To prevent this from happening, as time goes on, successive surgeries must be performed with increasing accuracy. Recently, Perelman [27], [28] presented the striking ideas how to understand the structures of all singularities of the three-dimensional Ricci flow, how to find “fine” necks, how to glue “fine” caps, and how to use rescaling to prove that the times of surgery are discrete. When using rescaling arguments for surgically modified solutions of the Ricci flow, one encounters the difficulty of how to apply Hamilton’s compactness theorem, which works only for smooth solutions. To overcome the difficulty, Perelman argued in [28] by choosing the cutoff radius in necklike regions small enough to push the surgical regions far away in space. But it still does not suffice to take a smooth limit since Shi’s interior derivative estimate is not available, and so one cannot be certain that Hamilton’s compactness result holds when only having the bound on curvatures. This is discussed in [3] and this paper. In this paper, inspired by Perelman’s works, we will study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space-form. We will give a complete proof for the main theorem of Hamilton in [19]. One of our major contribution in this paper is to establish several time-extension results for the surgical solutions in the proof of the discreteness of surgery times so that the surgical solutions are smooth on some uniform (small) time intervals (on compact subsets) and Hamilton’s compactness theorem is still applicable. In Perelman’s works [27, 28], the universal noncollapsing property of singularity models is a crucial fact to prove the surviving of noncollapsing property under surgery. Another feature of this paper is our proof on this crucial fact. In dimension three, one obtains this by using Perelman’s classification of three-dimensional shrinking Ricci solitons with nonnegative curvature (see [3] for the details). But in the present four dimension case, we are not able to obtain a complete classification for shrinking solitons. In the previous version, we presented an argument to obtain the universal noncollapsing for shrinking solitons. But, as pointed out to us by Joerg Enders, that argument contains a gap. Fortunately in the 5

present version, we find a new argument, without appealing a classification of shrinking Ricci solitons, to get the universal noncollapsing for all possible singularity models. During the proof we have actually provided, up to slight modifications, all necessary details for the part from Section 1 to Section 5 of Perelman’s second paper [28] on Ricci flow to approach the Poincaré conjecture. The complete details of the arguments in three-dimension can be found in the recent paper of H.-D. Cao and the second author in [3]. Furthermore, a complete proof to the Poincaré conjecture and Thurston’s geometrization conjecture has been given in [3]. The main result of this paper is the following Theorem 1.1 Let M 4 be a compact four-manifold with no essential incompressible space-form and with a metric gij of positive isotropic curvature. (k) Then we have a finite collection of smooth solutions gij (t), k = 0, 1, · · · , m, to the Ricci flow, defined on Mk4 × [tk , tk+1 ), (0 = t0 < · · · < tm+1 ) with (0) M04 = M 4 and gij (t0 ) = gij , which go singular as t → tk+1 , such that the following properties hold: (i) for each k = 0, 1, · · · , m − 1, the compact (possible disconnected) four(k) manifold Mk4 contains an open set Ωk such that the solution gij (t) can be smoothly extended to t = tk+1 over Ωk ; (k) (k+1) 4 (ii) for each k = 0, 1, · · · , m − 1, (Ωk , gij (tk+1 )) and (Mk+1 , gij (tk+1 )) contain compact (possible disconnected) four-dimensional submanifolds with smooth boundary, which are isometric and then can be denoted by Nk4 ; (iii) for each k = 0, 1, · · · , m − 1, Mk4 \ Nk4 consists of a finite number 4 of disjoint pieces diffeomorphic to S3 × I, B4 or RP4 \ B4 , while Mk+1 \ Nk4 consists of a finite number of disjoint pieces diffeomophic to B4 ; 4 (iv) for k = m, Mm is diffeomorphic to the disjoint union of a finite 4 4 e 1 , or RP4 #RP4 . number of S , or RP , or S3 × S1 , or S3 ×S As a direct consequence we have the following classification result of Hamilton [19].

Corollary 1.2( Hamilton [19]) A compact four-manifold with no essential incompressible space-form and with a metric of positive isotropic curvature

6

e 1 ,or a connected sum of is diffeomorphic to S4 , or RP4 , or S3 × S1 , or S3 ×S them. This paper contains five sections and an appendix. In Section 2 we recall the pinching estimates of Hamilton obtained in [19] and present two useful geometric properties for complete noncompact Riemannian manifolds with positive sectional curvature. The usual way to understand the singularities of the Ricci flow is to take a rescaling limit and to find the structure of the limiting models. In Section 3 we study the limiting models, so called ancient κ-solutions. We will establish the uniform κ-noncollapsing, compactness and canonical neighborhood structures for ancient κ-solutions. These generalize the analogs results of Perelman [27] from three-dimension to four-dimension. In Section 4 we will extend the canonical neighborhood characterization to any solution of the Ricci flow with positive isotropic curvature, and describe the structure of the solution at the singular time. In Section 5, we will define the Ricci flow with surgery as Perelman in [28]. By a long inductive argument, we will obtain a long-time existence result for the surgerically modified Ricci flow so that the solution becomes extinct in finite time and takes only a finite number of surgeries. This will give the proof of the main theorem. In the appendix we will prove the curvature estimates for the standard solutions and give the canonical neighborhood description of the standard solution in dimension four, which are used in Section 5 for the surgery arguments. Table of Contents 1. Introduction 2. Preliminaries 3. Ancient solutions 3.1 Splitting lemmas 3.2 Elliptic type estimate, canonical neighborhood decomposition for noncompact κ-solutions 3.3 Universal noncollapsing of ancient κ-solutions 3.4 Canonical neighborhood structures 4. The structure of the solutions at the singular time 7

5. Ricci flow with surgery for four manifolds Appendix. Standard solutions We are grateful to Professor H.-D. Cao for many helpful discussions and Professor S.-T. Yau for his interest and encouragement. We also thank Joerg Enders for telling us an error in the previous version. The first author is partially supported by FANEDD 200216 and NSFC 10401042 and the second author is partially supported by NSFC 10428102 and the IMS of The Chinese University of Hong Kong.

2. Preliminaries Consider a four-dimensional compact Riemannian manifold M 4 . The curvature tensor of M 4 may be regarded as a symmetric bilinear form Mαβ on the space of real forms Λ2 . It is well known that one can decompose Λ2 into Λ2+ ⊕Λ2− as eigen-spaces of the Hodge star operator with eigenvalues ±1. This gives a block decomposition of the curvature operate (Mαβ ) as A B t B C

(Mαβ ) =

!

.

It was shown in Lemma A2.1 of [19] that a four-manifold has positive isotropic curvature if and only if a1 + a2 > 0 and c1 + c2 > 0 where a3 ≥ a2 ≥ a1 , c3 ≥ c2 ≥ c1 are eigenvalues of the matrices A and C respectively. Let {X1 , X2 , X3 , X4 } be a positive oriented orthonormal basis of oneforms. Then ϕ1 = X1 ∧ X2 + X3 ∧ X4 , ϕ2 = X1 ∧ X3 + X4 ∧ X2 , ϕ3 = X1 ∧ X4 + X2 ∧ X3 is a basis of Λ2+ and ψ1 = X1 ∧ X2 − X3 ∧ X4 , ψ2 = X1 ∧ X3 − X4 ∧ X2 , ψ3 = X1 ∧ X4 − X2 ∧ X3 is a basis of Λ2− . It is easy to check trA = trC = 21 R by using this orthonormal basis and the Bianchi identity. Since B may not be symmetric, its eigenvalues need to be explained as follows. For an appropriate choice of orthonormal bases y1+ , y2+ , y3+ of Λ2+ and 8

y1− , y2−, y3− of Λ2− the matrix



 b1 0 0   B =  0 b2 0  . 0 0 b3

with 0 ≤ b1 ≤ b2 ≤ b3 . They are actually the eigenvalues of the symmetric √ √ matrices B t B or t BB. In [19] Hamilton proved that the Ricci flow on a compact four-manifold preserves positive isotropic curvature and obtained the following improving pinching estimate. Lemma 2.1 (Theorem B1.1 and Theorem B2.3 of [19]) Given an initial metric on a compact four-manifold with positive isotropic curvature, there exist positive constants ρ, Λ, P < +∞ depending only on the initial metric, such that the solution to the Ricci flow satisfies a1 + ρ > 0 and c1 + ρ > 0, max{a3 , b3 , c3 } ≤ Λ(a1 + ρ) and max{a3 , b3 , c3 } ≤ Λ(c1 + ρ), and p

b3 ΛeP t p ≤1+ (a1 + ρ)(c1 + ρ) max{log (a1 + ρ)(c1 + ρ), 2}

(2.1) (2.2)

(2.3)

at all points and all times.

This lemma tells us that as we consider the Ricci flow for a compact four-manifold with positive isotropic curvature, any rescaling limit along a sequence of points where the curvatures become unbounded must still have positive isotropic curvature and satisfies the following restricted isotropic curvature pinching condition a3 ≤ Λa1 , c3 ≤ Λc1 , b23 ≤ a1 c1 .

(2.4)

In the rest of this section, we will give two useful geometric properties for Riemannian manifolds with nonnegative sectional curvature. Let (M n , gij ) be an n-dimensional complete Riemannian manifold and let ε be a positive constant. We call an open subset N ⊂ M n to be an ε-neck 9

−1

of radius r if (N, r −2gij ) is ε-close, in C [ε ] topology, to a standard neck Sn−1 × I with I of the length 2ε−1 and Sn−1 of the scalar curvature 1. Proposition 2.2 There exists a constant ε0 = ε0 (n) > 0 such that every complete noncompact Riemannian manifold (M n , gij ) of nonnegative sectional curvature has a positive constant r0 such that any ε-neck of radius r on (M n , gij ) with ε ≤ ε0 must have r ≥ r0 . Proof . We argue by contradiction. Suppose there exists a sequence of positive constants εα → 0 and a sequence of n-dimensional complete noncompact Riemannian manifolds (M α , gijα ) with nonnegative sectional curvature such that for each fixed α, there exists a sequence of εα -necks Nk of radius at most 1/k on M α with centers Pk divergent to infinity. Fix a point P on the manifold M α and connect each Pk to P by a minimizing geodesic γk . By passing to subsequence we may assume the angle θkl between geodesic γk and γl at P is very small and tends to zero as k, l → +∞, and the length of γk+1 is much bigger than the length of γk . Let us connect Pk to Pl by a minimizing geodesic ηkl . For each fixed l > k, let P˜k be a point on the geodesic γl such that the geodesic segment from P to P˜k has the same length as γk and consider the triangle ∆P Pk P˜k in M α with vertices P , Pk and P˜k . By comparing with the corresponding triangle in the Euclidean plane R2 whose sides have the same corresponding lengths, Toponogov comparison theorem implies 1 d(Pk , P˜k ) ≤ 2 sin( θkl ) · d(Pk , P ). 2 Since θkl is very small, the distance from Pk to the geodesic γl can be realized by a geodesic ζkl which connects Pk to a point Pk′ on the interior of the geodesic γl and has length at most 2 sin( 12 θkl ) · d(Pk , P ). Clearly the angle between ζkl and γl at the intersection point Pk′ is π2 . Consider α to be fixed and sufficiently large. We claim that as k large enough, each minimizing geodesic γl with l > k, connecting P to Pl , goes through the neck Nk . Suppose not, then the angle between γk and ζkl at Pk is close to either zero or π since Pk is in the center of an εα -neck and α is sufficiently large. If the angle between γk and ζkl at Pk is close to zero, we consider the triangle ∆P Pk Pk′ in M α with vertices P , Pk , and Pk′ . By applying Toponogov comparison theorem to compare the angles of this triangle with those of the 10

corresponding triangle in the Euclidean plane with the same corresponding lengths, we find that it is impossible. Thus the angle between γk and ζkl at Pk is close to π. We now consider the triangle ∆Pk Pk′ Pl in M α with the three sides ζkl , ηkl and the geodesic segment from Pk′ to Pl on γl . We have seen that the angle of ∆Pk Pk′ Pl at Pk is close to zero and the angle at Pk′ is π2 . ¯ P¯k P¯′ P¯l in the Euclidean plane By comparing with corresponding triangle ∆ k R2 whose sides have the same corresponding lengths, Toponogov comparison theorem implies 3 ∠P¯l P¯k P¯k′ + ∠P¯l P¯k′ P¯k ≤ ∠Pl Pk Pk′ + ∠Pl Pk′ Pk < π. 4 This is impossible since the length between P¯k and P¯k′ is much smaller than the length from P¯l to either P¯k or P¯′ . So we have proved each γl with l > k k

passes through the neck Nk . Hence by taking a limit, we get a geodesic ray γ emanating from P which passes through all the necks Nk , k = 1, 2, · · · , except a finite number of them. Throwing these finite number of necks, we may assume γ passes through all necks Nk , k = 1, 2, · · · . Denote the center sphere of Nk by Sk , and their intersection points with γ by pk ∈ Sk ∩ γ, for k = 1, 2, · · · . Take a sequence points γ(m) with m = 1, 2, · · · . For each fixed neck Nk , arbitrarily choose a point qk ∈ Nk near the center sphere Sk , draw a geodesic segment γ km from qk to γ(m). Now we claim that for any fixed neck Nl with l > k, γ km will pass through Nl for all sufficiently large m. We argue by contradiction. Let us place the all necks Ni horizontally so that the geodesic γ passes through each Ni from the left to the right. We observe that the geodesic segment γ km must pass through the right half of Nk ; otherwise γ km can not be minimal. Then as m large enough, the distance from pl to the geodesic segment γ km must be achieved by the distance from pl to some interior point pk ′ of γ km . Let us draw a minimal geodesic η from pl to the interior point pk ′ with the angle at the intersection point pk ′ ∈ η ∩ γ km to be π2 . Suppose the claim is false. Then the angle between η and γ at pl is close to 0 or π since εα is small. If the angle between η and γ at pl is close to 0, we consider the triangle ¯ pl p¯k ′ γ¯ (m) in the plane ∆pl pk ′ γ(m) and construct a comparison triangle ∆¯ with the same corresponding length. Then by Toponogov comparison, we 11

¯ pl p¯k ′ γ¯ (m) is less see the sum of the inner angles of the comparison triangle ∆¯ than 3π/4, which is impossible. If the angle between η and γ at pl is close to π, by drawing a minimal geodesic from ξ from qk to pl , we see that ξ must pass through the right half of Nk and the left half of Nl ; otherwise ξ can not be minimal. Thus the three inner angles of the triangle ∆pl pk ′ qk are almost 0, π/2, 0 respectively. This is also impossible by Toponogov comparison theorem. Hence we have proved that the geodesic segment γ km passes through Nl as m large enough. Consider the triangle ∆pk qk γ(m) with two long sides pk γ(m)(⊂ γ) and qk γ(m)(= γ km ). For any s > 0, choose two points p˜k on pk γ(m) and q˜k on qk γ(m) with d(pk , p˜k ) = d(qk , q˜k ) = s. By Toponogov comparison theorem, we have (

d(˜ pk , q˜k ) 2 ) d(pk , qk )

=

¯ pk γ(m)˜ d(˜ pk , γ(m))2 + d(˜ qk , γ(m))2 − 2d(˜ pk , γ(m))d(˜ qk , γ(m)) cos ∡(˜ qk ) 2 2 ¯ d(pk , γ(m)) + d(qk , γ(m)) − 2d(pk , γ(m))d(qk , γ(m)) cos ∡(pk γ(m)qk )

≥

¯ pk γ(m)˜ d(˜ pk , γ(m))2 + d(˜ qk , γ(m))2 − 2d(˜ pk , γ(m))d(˜ qk , γ(m)) cos ∡(˜ qk ) 2 2 ¯ d(pk , γ(m)) + d(qk , γ(m)) − 2d(pk , γ(m))d(qk , γ(m)) cos ∡(˜ pk γ(m)˜ qk )

=

¯ pk γ(m)˜ (d(˜ pk , γ(m)) − d(˜ qk , γ(m)))2 + 2d(˜ pk , γ(m))d(˜ qk , γ(m))(1 − cos ∡(˜ qk )) 2 ¯ (d(˜ pk , γ(m)) − d(˜ qk , γ(m))) + 2d(pk , γ(m))d(qk , γ(m))(1 − cos ∡(˜ pk γ(m)˜ qk ))

≥

d(˜ pk , γ(m))d(˜ qk , γ(m)) d(pk , γ(m))d(qk , γ(m))

→ 1 ¯ k γ(m)qk ) and ∡(˜ ¯ pk γ(m)˜ as m → ∞, where ∡(p qk ) are the the corresponding angles in the corresponding comparison triangles. Letting m → ∞, we see that γ km has a convergent subsequence whose limit γ k is a geodesic ray passing through all Nl with l > k. Denote by pj = γ(tj ), j = 1, 2, · · · . From the above computation, we deduce that d(pk , qk ) ≤ d(γ(tk + s), γ k (s)). 12

for all s > 0. Let ϕ(x) = limt→+∞ (t−d(x, γ(t))) be the Busemann function constructed from the ray γ. Note that the level set ϕ−1 (ϕ(pj )) ∩ Nj is close to the center sphere Sj for any j = 1, 2, · · · . Now let qk be any fixed point in ϕ−1 (ϕ(pk )) ∩ Nk . By the definition of Busemann function ϕ associated to the ray γ, we see that ϕ(γ k (s1 )) − ϕ(γ k (s2 )) = s1 − s2 for any s1 , s2 ≥ 0. Consequently, for each l > k, by choosing s = tl − tk , we see γ k (tl − tk ) ∈ ϕ−1 (ϕ(pl )) ∩ Nl . Since γ(tk + tl − tk ) = pl , it follows that d(pk , qk ) ≤ d(pl , γ k (s)). with s = tl − tk > 0. This implies that the diameter of ϕ−1 (ϕ(pk )) ∩ Nk is not greater the diameter of ϕ−1 (ϕ(pl ))∩Nl for any l > k, which is a contradiction as l much larger than k. Therefore we have proved the proposition. # In [18], Hamilton discovered an interesting result, called finite bump theorem, about the influence of a bump of strictly positive curvature in a complete noncompact Riemannian manifold with nonnegative sectional curvature. Namely, minimal geodesic paths that go past the bump have to avoid it. The following result is in the same spirit as Hamilton’s finite bump theorem. Proposition 2.3 Suppose (M n , g) is a complete n-dimensional Riemanian manifold with nonnegative sectional curvature. Let P ∈ M n be fixed, and Pj ∈ M n a sequence of points and Rj a sequence of positive numbers with d(P, Pj ) → +∞ and Rj d(P, Pj )2 → +∞. If the sequence of marked ∞ manifolds (M n , Rj g, Pj ) converges in Cloc topology (in Cheeger sense) to a ˜ n , g˜), then the limit (M ˜ n , g˜) splits as the metric product smooth manifold (M of the form R × N, where N is a nonnegatively curved manifold of dimension n − 1. Proof: Let us denote by |OQ| = d(O, Q) for the distance of two points

13

O, Q ∈ M n . Without loss of generality, we may assume that for each j 1 + 2|P Pj | ≤ |P Pj+1|.

(2.5)

Draw a minimal geodesic γj from P to Pj and a minimal geodesic σj from Pj to Pj+1, both parameterized by the arclength. By the compactness of unit sphere of the tangent space at P , {γj′ (0)} has a convergent subsequence. We may further assume 1 ′ θj = |∡(γj′ (0), γj+1 (0))| < . j

(2.6)

∞ Since (M n , Rj g, Pj ) converges in Cloc topology (in Cheeger sense) to a n ˜ ˜ smooth marked manifold (M , g˜, P ), by further choices of subsequences, we may also assume γj and σj converge to two geodesic rays e γ and σ e starting at n e ˜ P . We claim that that γ˜ ∪ σ ˜ forms a line in M . Since the sectional curvature n ˜ ñ of M is nonnegative, then by Toponogov splitting theorem [6] the limit M must split as R × N isometrically. To prove the claim, we argue by contradiction. Suppose e γ∪σ e is not a line, then for each j, there exist two points Aj ∈ γj and Bj ∈ σj such that as j → +∞,  Rj d(Pj , Aj ) → A > 0,     R d(P , B ) → B > 0, j j j  Rj d(Aj , Bj ) → C > 0, (2.7)    but A + B > C.

Pj σj (((( ( ( Bj ( ( ( δj (((( Pj+1 ( A ( j ( γj (((( (((( ( ( (( (((( (((((( (( ( ( P Now draw a minimal geodesic δj from Aj to Bj . Consider comparison ¯ P¯j P¯ P¯j+1 and △ ¯ P¯j A¯j B ¯j in R2 with triangle △ |P¯j P¯ | = |Pj P |, |P¯j P¯j+1| = |Pj Pj+1 |, |P¯ P¯j+1 | = |P Pj+1|, 14

¯j | = |Pj Bj |, |A¯j B ¯j | = |Aj Bj |. and |P¯j A¯j | = |Pj Aj |, |P¯j B By Toponogov comparison theorem [6], we have ¯j ≥ ∡P¯ P¯j P¯j+1 . ∡A¯j P¯j B

(2.8)

On the other hand, by (2.6) and using the Toponogov comparison theorem again, we have 1 (2.9) ∡P¯j P¯ P¯j+1 ≤ ∡Pj P Pj+1 < , j and since |P¯j P¯j+1 | > |P¯ P¯j | by (2.5), we further have 1 ∡P¯j P¯j+1 P¯ ≤ ∡P¯j P¯ P¯j+1 < . j

(2.10)

Thus the above inequalities (2.8)-(2.10) imply that ¯j > π − 2 . ∡A¯j P¯j B j Hence ¯j |2 − 2|A¯j P¯j | · |P¯j B ¯j | cos(π − 2 ). ¯j |2 ≥ |A¯j P¯j |2 + |P¯j B |A¯j B j

(2.11)

Multiplying the above inequality by Rj and letting j → +∞, we get C ≥A+B which contradicts with (2.7). Therefore we have proved the proposition. # Corollary 2.4 Suppose (X, d) is a complete n-dimensional Alexandrov space with nonnegative curvature. Let P ∈ X be fixed, and Pj ∈ X a sequence of points and Rj a sequence of positive numbers with d(P, Pj ) → +∞ and 1

Rj d2 (P, Pj ) → +∞. Then the marked spaces (X, Rj2 d, Pj ) have a (GromovHausdorff ) convergent subsequence such that the limit splits as the metric product of the form R × N, where N is a nonnegatively curved Alexandrov space. 15

Proof: By the compactness theorem of Alexandrov spaces (see [1]), there 1

is a subsequence of (X, Rj2 d, Pj ), which converges (in the sense of Gromove Pe) of dimene d, Hausdorff) to a nonnegatively curved Alexandrov space (X, sion ≤ n. By Toponogov splitting theorem [25] for Alexandrov spaces, we e contains a line. Note that the same only need to show that the limit X inequality (2.6) now follows from the compactness of the space of directions at a fixed point [1]. Since the Toponogov triangle comparison theorem still holds on Alexsandrov Spaces (in fact, the notion of the curvature of general metric spaces is defined by Toponogov triangle comparison), the same argument of the Proposition 2.3 proves the corollary. #

3. Ancient Solutions A solution to the Ricci flow on a compact four-manifold with positive isotropic curvature develops singularities in finite time. The usual way to understand the formations of the singularities is to rescale the solution along the singularities and to try to take a limit for the rescaled sequences. According to Lemma 2.1, a rescaled limit will be a complete non-flat solution to the Ricci flow ∂ gij = −2Rij , ∂t on an ancient time interval −∞ < t ≤ 0, called an ancient solution, which has positive isotropic curvature and satisfies the restricted isotropic curvature pinching condition (2.4). We remark that as we consider the general singularities (not be necessarily those points coming from the maximum of the curvature ), we don’t know whether at a priori, the rescaled limit exists, and even assuming the existence, whether the limit has bounded curvature for each t. Nevertheless, in this section we will take the attention to those rescaled limits with bounded curvature. According to Perelman [27], a solution to the Ricci flow is κ-noncollapsed for scale r0 > 0 if we have the following statement: whenever we have |Rm|(x, t) ≤ r0−2 , 16

for all t ∈ [t0 − r02 , t0 ], x ∈ Bt (x0 , r0 ), for some (x0 , t0 ), then there holds V olt0 (Bt0 (x0 , r0 )) ≥ κr04 . Here we denote by Bt (x0 , r0 ) and V olt0 the geodesic ball centered at x0 of radius r0 with respect to the metric gij (t) and the volume with respect to the metric gij (t0 ) respectively. It was shown by Perelman [27] that any rescaled limit obtained by blowing up a smooth solution to the Ricci flow on a compact manifold in finite time is κ-noncollapsed on all scales for some κ > 0. We say a solution to the Ricci flow on a four-manifold is an ancient κ -solution with restricted isotropic curvature pinching (for some κ > 0) if it is a smooth solution to the Ricci flow on the ancient time interval t ∈ (−∞, 0] which is complete, has positive isotropic curvature and bounded curvature, and satisfies the restricted isotropic curvature pinching condition (2.4), as well as is κ-noncollapsed on all scales.

3.1 Splitting lemmas To understand the structures of the solutions to the Ricci flow on a compact four-manifold with positive isotropic curvature, we are naturally led to investigate the ancient solutions which have positive isotropic curvature, satisfy the restricted isotropic curvature pinching condition (2.4) and are κ-noncollapsed for all scales. Note that the restricted isotropic curvature pinching condition (2.4) implies the curvature operator is nonnegative. In this subsection we will derive two useful splitting results without assuming bounded curvature condition. Lemma 3.1 Let (M 4 , gij ) be a complete noncompact Riemannian manifold which satisfies the restricted isotropic curvature pinching condition (2.4) and has positive curvature operator. And let P be a fixed point in M 4 , {Pl }1≤l 0 4 large enough, since ϕ is an exhausting on M . Now fix such a large positive constant c. Among all spheres which are Z2 invariant, contained in the c4 |ϕ(x) manifold {x ∈ M b ≤ c}, and intersect the S3 /Γ in the homotopy class [γ] 6= 0, there will be one of least area since the boundary of the manifold c4 |ϕ(x) {x ∈ M b ≤ c} is strictly convex. This sphere must even have least area among all nearby spheres. For if a nearby sphere of less area divides in two parts bounding [γ] in S3 /Γ, one side or the other has less than half the area of the original sphere. We could then double this half to get a sphere of less area which is Z2 invariant, contradicting the assumption that ours was of least area among this class. But the hypothesis of positive isotropic curvature implies there are no stable minimal two-spheres as was shown in [24]. Hence we get a contradiction unless X = S3 /Γ is incompressible in M 4 . Therefore we have proved the lemma. # Lemma 3.2 Let (M 4 , gij (t)) be an ancient solution which has positive isotropic curvature and satisfies the restricted isotropic curvature pinching condition (2.4). If its curvature operator has a nontrivial null eigenvector somewhere at some time, then the solution is, up to a scaling, the evolving round cylinder R × S3 or a metric quotient of the round cylinder R × S3 . Proof. Recall that the solution gij (t) has nonnegative curvature operator everywhere and every time. Because the curvature operator of the ancient solution gij (t) has a nontivial null eigenvector somewhere at some time, it follows from [14] (by using Hamilton’s strong maximum principle) that at any earlier time the solution has null eigenvector everywhere and the Lie algebra of the holonomy group is restricted a proper subalgebra of so(4). Since the 19

ancient solution is nonflat and has positive isotropic curvature, we rule out the subalgebras {1}, u(2), so(2)×so(2), so(2)×{1} as on R4 , CP2 , S2 ×S2 , S2 × R2 or a metric quotient of them. The only remaining possibility for the Lie subalgebra of the holonomy is so(3). Now the only way we get holonomy so(3) is when in some basis we have A = B = C in the curvature operator matrix, so that ! A A (Mαβ ) = , A A which corresponds to the fact that the metric gij (t) is locally a product of R× X for some smooth three-dimensional manifold X with curvature operator A. Then the inequality b23 ≤ a1 c1 in the restricted isotropic curvature pinching condition (2.4) implies that A is a multiple of the identity. Moreover this is true at every point, it follows from the contracted second Bianchi identity that the factor X has (positive) constant curvature. Consequently, X is compact and then for each t, the metric gij (t) is isometric to (up to a scaling) the evolving metric of the round cylinder R × S3 or a metric quotient of it. #

3.2 Elliptic type estimate, canonical neighborhood decomposition for noncompact κ-solutions The following elliptic type Harnack property for four-dimensional ancient κ-solutions with restricted isotropic curvature pinching will be crucial for the analysis of the structure of singularities of the Ricci flow on four-manifold with positive isotropic curvature. The analogous result for three-dimensional ancient κ-solutions was implicitely given by Perelman in Section 11.7 of [27] and Section 1.5 of [28]. Proposition 3.3 For any κ > 0, there exist a positive constant η and a positive function ω : [0, +∞) → (0, +∞) with the following properties. Suppose we have a four-dimensional ancient κ-solution (M 4 , gij (t)), −∞ < t ≤ 0, with restricted isotropic curvature pinching. Then 20

(i) for every x, y ∈ M 4 and t ∈ (−∞, 0], there holds R(x, t) ≤ R(y, t) · ω(R(y, t)d2t (x, y)); (ii) for all x ∈ M 4 and t ∈ (−∞, 0], there hold 3

|∇R|(x, t) ≤ ηR 2 (x, t) and |

∂R |(x, t) ≤ ηR2 (x, t). ∂t

Proof. Obviously we may assume the ancient κ-solution is not a metric quotient of the round neck R × S3 . (i) We only need to establish the estimate at t = 0. Let y be fixed in M 4 . By rescaling, we can assume R(y, 0) = 1. Let us first consider the case that sup{R(x, 0)d20 (x, y)|x ∈ M 4 } > 1. Define z to be the closest point to y (at time t = 0) satisfying R(z, 0)d20 (z, y) = 1. 1 We want to bound R(x, 0)/R(z, 0) from above for x ∈ B0 (z, 2R(z, 0)− 2 ). Connect y to z by a shortest geodesic and choose a point ze lying on the 1 geodesic satisfying d0 (e z , z) = 14 R(z, 0)− 2 . Denote by B the ball centered at 1 ze and with radius 41 R(z, 0)− 2 (with respect to the metric at t = 0). Clearly 1 1 the ball B lies in B0 (y, R(z, 0)− 2 ) and lies outside B0 (y, 12 R(z, 0)− 2 ). Thus as x ∈ B, we have 1 1 R(x, 0)d20 (x, y) ≤ 1 and d0 (x, y) ≥ R(z, 0)− 2 2

which imply R(x, 0) ≤

1 1

( 12 R(z, 0)− 2 )2

, on B.

Then by Li-Yau-Hamilton inequality [16] and the κ-noncollapsing, we have 1 1 V ol0 (B) ≥ κ( R(z, 0)− 2 )4 4

and then

1 κ (8R(z, 0)− 2 )4 . 20 2 So by Corollary 11.6 of [27], there exists a positive constant A1 depending only on κ such that 1

V ol0 (B0 (z, 8R(z, 0)− 2 ) ≥

1

R(x, 0) ≤ A1 R(z, 0), for x ∈ B0 (z, 2R(z, 0)− 2 ). 21

(3.1)

We now consider the remaining case: R(x, 0)d20 (x, y) ≤ 1 for all x ∈ M 4 . We choose a point z ∈ M 4 satisfying R(z, 0) ≥ 21 sup{R(x, 0)|x ∈ M 4 }. Obviously we also have the estimate (3.1) in the remaining case. After having the estimate (3.1), we next want to bound R(z, 0) for the chosen z ∈ M 4 . By combining with Li-Yau-Hamilton inequality [16], we have R(x, t) ≤ A1 R(z, 0), 1

for all x ∈ B0 (z, 2R(z, 0)− 2 ) and all t ≤ 0. It then follows from Shi’s local derivative estimate [32] that ∂ R(z, t) ≤ A2 R(z, 0)2 , ∂t

for all − R−1 (z, 0) ≤ t ≤ 0,

where A2 is some constant depending only on κ. This implies R(z, −cR−1 (z, 0)) ≥ cR(z, 0) for some small positive constant c depending only on κ. On the other hand, by using the Harnack estimate [16] (as a consequence of Li-Yau-Hamilton inequality), we have 1 = R(y, 0) ≥ e cR(z, −cR−1 (z, 0))

for some small positive constant e c depending only on κ. Thus we obtain R(z, 0) ≤ A3

(3.2)

for some positive constant A3 depending only on κ. The combination of (3.1) and (3.2) gives −1

R(x, 0) ≤ A1 A3 , on B0 (y, A3 2 ). Thus by the κ-noncollapsing there exists a positive constant r0 depending only on κ such that V ol0 (B0 (y, r0)) ≥ κr04 . For any fixed R0 ≥ r0 , we have V ol0 (B0 (y, R0 )) ≥ κ( 22

r0 4 ) · R04 . R0

By applying Corollary 11.6 of [27] again, there exists a positive constant ω(R02 ) depending only on R0 and κ such that 1 R(x, 0) ≤ ω(R02 ), on B0 (y, R0 ). 4 This gives the desired estimate. (ii) It immediately follows from the above assertion (i), the Li-Yau-Hamilton inequality [16] and Shi’s local derivative estimates [32]. # Remark. The argument in the last paragraph of the above proof for (i) implies the following assertion: For any ζ > 0, there is a positive function ω depending only on ζ such that if there holds 1 V olt0 (Bt0 (y, R(y, t0)− 2 )) ≥ ζ, 4 R(y, t0 )− 2 for some fixed point y and some t0 ∈ (−∞, 0], then we have the following the elliptic type estimate R(x, t0 ) ≤ R(y, t0 ) · ω(R(y, t0)d2t0 (x, y)) for all x ∈ M. This estimate will play a key role in deriving the universal noncollapsing property in the next subsection. Let gij (t), −∞ < t ≤ 0, be a nonflat solution to the Ricci flow on a fourmanifold M 4 . Fix a small ε > 0. We say that a point x0 ∈ M 4 is the center of an evolving ε-neck, if the solution gij (t) in the set {(x, t)| − ε−2 Q−1 < t ≤ 0, d20(x, x0 ) < ε−2 Q−1 }(with Q = R(x0 , 0)) is, after scaling with factor −1 Q, ε-close (in C [ε ] topology) to the corresponding subset of the evolving round cylinder R × S3 , having scalar curvature one at t = 0. The following result generalizes Corollary 11.8 of Perelman [27] to fourdimension and verifies Theorem E 3.3 of Hamilton [19]. The crucial information in the following Proposition is that the constant C = C(ε) > 0 depends only on ε. 23

Proposition 3.4 For any ε > 0, there exists C = C(ε) > 0 such that if gij (t) is a nonflat ancient κ-solution with restricted isotropic curvature pinching on a noncompact four-manifold M 4 for some κ > 0, and Mε4 denotes the set of points of M 4 , which are not centers of evolving ε-necks, then either the whole M 4 is a metric quotient of the round cylinder R × S3 or Mε4 satisfies the following properties (i) Mε4 is compact, and 1 (ii) diam(Mε4 ) ≤ CQ− 2 and C −1 Q ≤ R(x, 0) ≤ CQ, whenever x ∈ Mε4 , where Q = R(x0 , 0) for some x0 ∈ ∂Mε4 and diam(Mε4 ) is the diameter of the set Mε4 with respect to the metric gij (0). Proof. Note that the curvature operator of the ancient κ-solution is nonnegative. We first consider the easy case that the curvature operator has a nontrivial null vector somewhere at some time. By Lemma 3.2, we know that the ancient κ-solution is a metric quotient of the round cylinder R × S3 . We then assume the curvature operator of the ancient κ-solution is positive everywhere. Firstly we want to show Mε4 is compact. Argue by contradiction. Suppose there exists a sequence of points zk , k = 1, 2, · · · , going to infinity (with respect to the metric gij (0)) such that each zk is not the center of any evolving ε-neck. For arbitrarily fixed point z0 ∈ M 4 , it follows from Proposition 3.3 (i) that 0 < R(z0 , 0) ≤ R(zk , 0) · ω(R(zk , 0)d20(zk , z0 )) which implies that lim R(zk , 0)d20(zk , z0 ) = +∞.

k→∞

By Lemma 3.1 and Proposition 3.3 and Hamilton’s compactness theorem, we conclude that zk is the center of an evolving ε-neck as k sufficiently large. This is a contradiction, so we have proved that Mε4 is compact. Note that M 4 is diffeomorphic to R4 since the curvature operator is positive. We may assume ε > 0 so small that Hamilton’s replacement for Schoenflies conjecture and its proof (Theorem G1.1 and Lemma G1.3 of [19]) are available. Since every point outside the compact set Mε4 is the center of an evolving ε-neck, it follows that the approximate round three-sphere crosssection through the center divides M 4 into two connected components such 24

that one of them is diffeomorphic to the four-ball B4 . Let ϕ be a Busemann function on M 4 (constructed from all geodesic rays emanating from a given point), it is a standard fact that ϕ is convex and proper. Since Mε4 is compact, Mε4 is contained in a compact set K = ϕ−1 ((−∞, A]) for some large A. We note that each point x ∈ M 4 \ Mε is the center of an ε-neck. It is clear that there is an ε-neck N lying entirely outside K. Consider a point x on one of its boundary components of the ε-neck N. Since x ∈ M 4 \ Mε4 , there is an ε-neck adjacent to the initial ε-neck, producing a longer neck. We then take a point on the boundary of the second ε-neck and continue. This procedure can either terminate when we get into Mε or go on infinitely to produce a semi-infinite (topological) cylinder. The same procedure can be repeated for the other boundary component of the initial ε-neck. This procedure will give ˜ If N ˜ never touch M 4 , the manifold will be a maximal extended neck N. ε diffeomorphic to the standard infinite cylinder, which is a contradiction. If ˜ touch M 4 , then there is a geodesic connecting two both of the two ends of N ε points of Mε4 and passing through N. This is impossible since the function ϕ ˜ will touch M 4 and the other end is convex. So we conclude that one end of N ε will tend to infinity to produce a semi-infinite (topological) cylinder. Then one can find an approximate round three-sphere cross-section which encloses the whole set Mε4 and touches some point x0 ∈ ∂Mε4 . We now want to show 1 that R(x0 , 0) 2 · diam(Mε4 ) is bounded from above by some positive constant C = C(ε) depending only on ε. Suppose not, there exist a sequence of nonflat noncompact ancient κj solutions with restricted isotropic curvature pinching and with positive curvature operator, for some sequence of positive constants κj , such that for above chosen points x0 ∈ Mε4 there would hold 1

R(x0 , 0) 2 · diam(Mε4 ) → +∞.

(3.3)

Since the point x0 lies in some 2ε-neck, clearly, there is a universal positive ))/( √ 1 )4 . By the remark after the lower bound for V ol0 (B0 (x0 , √ 1 R(x0 ,0)

R(x0 ,0)

proof of the previous Proposition 3.3, we see that there is a universal positive function ω : [0, +∞) → [0, +∞) such that the elliptic type estimate R(x, 0) ≤ R(x0 , 0) · ω(R(x0 , 0)d20 (x, x0 )) 25

(3.4)

holds for all x ∈ M. Let us scale the ancient solutions around the points x0 with the factors R(x0 , 0). By (3.4), Hamilton’s compactness theorem (Theorem 16.1 of [18]) and the universal noncollasing property at x0 , we can extract a convergent subsequence. From the choice of the points x0 and (3.3), the limit contains a line. Actually we may draw a geodesic ray from some point x1 ∈ Mε4 which is far from x0 (in the normalized distance). This geodesic ray must across some vertical three-sphere containing x0 . The limit of these rays gives us a line. Then by Toponogov splitting theorem the limit is isometric to R × X 3 for some smooth three-manifold X 3 . As before, by using the restricted isotropic curvature pinching condition (2.4) and the contracted second Bianchi identity, we see that X 3 = S3 /Γ for some group Γ of isometrics without fixed points. Then we apply the same argument as in the proof of Lemma 3.1 to conclude that Γ = {1}. This says that the limit must be the evolving round cylinder R × S3 . This contradicts with the fact that each chosen points x0 is not the center of any evolving ε-neck. Therefore we have proved 1

diam(Mε4 ) ≤ CQ− 2 for some positive constant C = C(ε) depending only on ε, where Q = R(x0 , 0). Finally by combining this diameter estimate and the remark after proposition 3.3, we directly deduce e−1 Q ≤ R(x, 0) ≤ CQ, e C whenever x ∈ Mε4 ,

e depending only on ε. for some positive constant C

#

Consequently, by applying the standard volume comparison to Proposition 3.4, we conclude that all complete noncompact four-dimensional ancient κ-solutions with restricted isotropic curvature pinching and positive curvature operator are κ0 -noncollapsing on all scales for some universal constant κ0 > 0. In the next subsection, we will prove this universal noncollapsing property for both compact and noncompact cases. 26

3.3 Universal noncollapsing of ancient κ-solutions First we note that the universal noncollapsing is not true for all metric quotients of round R × S3 . The main result of this section is to establish the universal noncollapsing property for all ancient κ-solutions with restricted isotropic curvature pinching which are not metric quotients of round R × S3 . The analogous result for three-dimensional ancient κ-solutions was claimed by Perelman in Remark 11.9 of [27] and Section 1.5 of [28]. Theorem 3.5 There exists a positive constant κ0 with the following property. Suppose we have a four-dimensional (compact or noncompact) ancient κ-solution with restricted isotropic curvature pinching for some κ > 0. Then either the solution is κ0 -noncollapsed for all scales, or it is a metric quotient of the round cylinder R × S3 . Proof. Let gij (x, t), x ∈ M 4 and t ∈ (−∞, 0], be an ancient κ-solution with restricted isotropic curvature pinching for some κ > 0. We had known that the curvature operator of the solution gij (x, t) is nonnegative everywhere and every time. If the curvature operator of the solution gij (x, t) has a nontrivial null eigenvector somewhere at some time, then we know from Lemma 3.2 that the solution is a metric quotient of the round neck R × S3 . We now assume the solution gij (x, t) has positive curvature operator everywhere and every time. We want to apply backward limit argument of Perelman to take a sequence points qk and a sequence of times tk → −∞ such that the scalings of gij (·, t) around qk with factors |tk |−1 (and shifting ∞ the times tk to zero) converge in Cloc topology to a non-flat gradient shrinking soliton. Clearly, we may assume the nonflat ancient κ-solution is not a gradient shrinking Ricci soliton. For arbitrary point (p, t0 ) ∈ M 4 × (−∞, 0], we define as in [27] that τ = t0 − t, for t < t0 , Rτ √ 1 2 l(q, τ ) = √ inf{ 0 s(R(γ(s), t0 − s) + |γ(s)| ˙ gij (t0 −s) )ds| 2 τ γ : [0, τ ] → M 4 with γ(0) = p, γ(τ ) = q}, Z and Ve (τ ) = (4πτ )−2 exp(−l(q, τ ))dVt0 −τ (q), M4

27

where | · |gij (t0 −s) is the norm with respect to the metric gij (t0 − s) and dVt0 −τ is the volume element with respect to the metric gij (t0 − τ ). According to [27], l is called the reduced distance and Ve (τ ) is called the reduced volume. Since the manifold M 4 may be noncompact, one would ask whether the reduced volume is finite. Since the scalar curvature is nonnegative and the curvature is bounded, it is no hard to see that the reduced distance is quadratically growth and then the reduced volume is always finite. (Actually, by using Perelman’s Jacobian comparison theorem [27] one can show that the reduced volume is always finite for any complete solution of the Ricci flow (see [3] for the detail)). In [27], Perelman proved that the reduced volume Ve (τ ) is nonincreasing in τ , and the monotonicity is strict unless the solution is a gradient shrinking Ricci soliton. From [27] (Section 7 of [27]), the function L(q, τ ) = 4τ l(q, τ ) satisfies ∂ L + △L ≤ 8. ∂τ It is clear that L(·, τ ) achieves its minimum on M 4 for each τ > 0 since the scalar curvature is nonnegative. Then the minimum of L(·, τ ) − 8τ is nonincreasing, so in particular, the minimum of l(·, τ ) does not exceed 2 for each τ > 0. Thus for each τ > 0 we can find q = q(τ ) such that l(q(τ ), τ ) ≤ 2. We can apply Perelman’s Proposition 11.2 in [27] to conclude that the scalings ∞ of gij (·, t0 − τ ) around q(τ ) with factors τ −1 converge in Cloc topology along a subsequence τ → +∞ to a non-flat gradient shrinking soliton. Because the proof of this proposition in [27] is just a sketch, we would like to give its detail in the following for the completeness. We first claim that for any A ≥ 1, one can find B = B(A) < +∞ such that for every τ > 1 there holds l(q, τ ) < B and τ R(q, t0 − τ ) ≤ B

(3.5)

whenever 21 τ ≤ τ ≤ Aτ and d2t − τ (q, q( τ2 )) ≤ Aτ . 0 2 Indeed, by Section 7 of [27], the reduced distance l satisfies the following  ∂ l K  (3.6)  ∂τ l = − τ + R + 2τ 3/2 , K l 2 |∇l| = −R + τ − τ 3/2 , (3.7)   K 2 △l ≤ −R + τ − 2τ 3/2 , (3.8) 28

in the sense of distributions, and the equality holds everywhere if and only if Rτ we are on a gradient shrinking soliton, where K = 0 s3/2 Q(X)ds and Q(X) is the trace Li-Yau-Hamilton quadratic given by Q(X) = −

R ∂ R − − 2 < ∇R, X > +2Ric(X, X) ∂τ τ

and X is the tangential (velocity) vector of a L-shortest curve γ : [0, τ ] → M 4 connecting p to q. By applying the trace Li-Yau-Hamilton inequality [16] to the ancient κsolution, we have R Q(X) ≥ − τ and then Z τ √ √ K≥− sRds ≥ −2 τ l. 0

Thus by (3.7) we get

|∇l|2 + R ≤

3l . τ

(3.9)

At τ = τ2¯ , we have r

√ √ τ τ l(q, ) ≤ 2 + sup{|∇ l|} · dt0 − τ (q, q( )) 2 2 2 r √ 3A ≤ 2+ , 2

and

τ 6 √ R(q, t0 − ) ≤ ( 2 + 2 τ

r

3A 2 ), 2

(3.10)

(3.11)

√ for q ∈ Bt0 − τ (q( τ2 ), Aτ ). Since the scalar curvature of an ancient solution 2 with nonnegative curvature operator is pointwisely nondecreasing in time (by the trace Li-Yau-Hamilton inequality [16]), we further have r √ 3A 2 τ R(q, t0 − τ ) ≤ 6A( 2 + ) (3.12) 2 whenever 21 τ ≤ τ ≤ Aτ and d2t0 − τ (q, q( τ2 )) ≤ Aτ . 2 By (3.6), (3.7) and (3.12), we have r ∂ l 3A √ 3A 2 l≤− + ( 2+ ) ∂τ 2τ τ 2 29

and by integrating this inequality and using the estimate (3.10), we obtain r √ 3A 2 ) (3.13) l(q, τ ) ≤ 7A( 2 + 2 whenever 12 τ ≤ τ ≤ Aτ and d2t − τ (q, q( τ2 )) ≤ Aτ . So we have proved the 0 2 assertion (3.5). The scaling of the ancient κ-solution around q( τ2 ) with factor ( τ2 )−1 is gij (s) = e

τ 2 gij (·, t0 − s ) τ 2

for s ∈ [0, +∞). The assertion (3.5) implies that for all s ∈ [1, 2A] and all q e s) ≤ B where R e is the scalar curvawith dist2egij (1) (q, q( τ2 )) ≤ A, we have R(q, ture of the rescaled metric e gij . Then we can use Hamilton’s compactness theorem ([17] or more precisely Theorem 16.1 of [18]) and the κ-noncollapsing assumption to obtain a sequence τ k → +∞ such that the marked evolv(k) (k) ing manifolds (M 4 , e gij (s), q( τ2k )), with geij (s) = τ2k gij (·, t0 − s τ2k ) and s ∈ 4

∞ [1, +∞), converge in Cloc topology to an evolving manifold (M , g ij (s), q) ∂ g ij = 2Rij on M × [1, +∞). with s ∈ [1, +∞), where g ij (s) satisfies ∂s (k) Denote by e lk the corresponding reduced distance of e gij (s). It is easy to see that e lk (q, s) = l(q, τ2k s) for s ∈ [1, +∞). After rescaling we still have

e(k) ≤ 6e |∇e lk |2eg(k) + R lk ij

and by (3.5), e lk are uniformly bounded at finite distances. Thus the above gradient estimate implies that the functions e lk tend (up to a subsequence) to a function l which is a locally Lipschitz function on M . From (3.6)-(3.8), we have ∂ e e(k) + 2 ≥ 0, (lk ) − △e lk + |∇e lk |2 − R ∂s s

e e(k) + lk − 4 ≤ 0, 2△e lk − |∇e lk |2 + R s which can be rewritten as (

∂ e(k) )((4πs)−2 exp(−e −△+R lk )) ≤ 0, ∂s 30

(3.14)

˜ e e l l e(k) )e− 2k + lk − 4 e− 2k ≤ 0, (3.15) −(4△ − R s in the sense of distribution. Clearly, these two inequalities imply that the limit l satisfies ∂ ( − △ + R)((4πs)−2 exp(−l)) ≤ 0, (3.16) ∂s ¯l − 4 l l (3.17) −(4△ − R)e− 2 + e− 2 ≤ 0, s in the sense of distributions. (k) Denote by Ve (k) (s) the reduced volume of the rescaled metric e gij (s). Since e lk (q, s) = l(q, τ2k s), we see that Ve (k) (s) = Ve ( τ2k s). The monotonicity of the reduced volume Ve (τ ) (see [27]) then implies that lim Ve (k) (s) = V , for s ∈ [1, 2]

k→+∞

for some positive constant V . But we are not sure whether the limiting V is 4 exactly the Perelman’s reduced volume of the limiting manifold (M , g ij (s)), because the points q( τ2k ) may diverge to infinity. Nevertheless, we can insure that V is not less than the Perelman’s reduced volume of the limit. Note that Z 2 d e (k) (k) (k) e e V (2) − V (1) = (V (s))ds 1 ds Z 2 Z ∂ e(k) )((4πs)−2 exp(−e = ds ( −△+R lk ))dVeg(k) (s) . ij ∂s 4 1 M Thus we deduce that in the sense of distributions, (

∂ − △ + R)((4πs)−2 exp(−l)) = 0, ∂s l

−(4△ − R)e− 2 +

¯l − 4 l e− 2 = 0, s

(3.18) (3.19)

and then the standard parabolic equation theory implies that l is actually smooth. Here we used (3.6)-(3.8) to show that the equality in (3.16) implies the equality in (3.17). Set υ = [s(2△l − |∇¯l|2 + R) + l − 4] · (4πs)−2 e−l .

31

A direct computation gives (

1 ∂ − △ + R)υ = −2s|Rij + ∇i ∇j l − g ij |2 · (4πs)−2 e−ℓ . ∂s 2s

(3.20)

Since the equation (3.18) implies υ ≡ 0, the limit metric g ij satisfies Rij + ∇i ∇j l −

1 g = 0. 2s ij

(3.21)

Thus the limit is a gradient shrinking Ricci soliton. To show the limiting gradient shrinking Ricci soliton to be nonflat, we first show that constant V is strictly less than 1. Indeed, by considering the reduced volume Ve (τ ) of the ancient κ-solution, we get from Perelman’s Jacobian comparison theorem [27] that Z Ve (τ ) = (4πτ )−2 e−l dVt0 −τ 4 ZM 2 ≤ (4π)−2 e−|X| dX Tp M 4

= 1.

Recall that we assumed the nonflat ancient κ-solution is not a gradient shrinking Ricci soliton. Thus by the monotonicity of the reduced volume [27], we have Ve (τ ) < 1 for τ > 0. This implies that V < 1. We now argue by contradiction. Suppose the limit g ij (s) is flat. Then by (3.21) we have 1 2 ∇i ∇j l = g ij and △l = . 2s s And then by (3.19), we get l |∇l|2 = . s √ Since the function l is strictly convex, it follows that 4sl is a distance function (from some point) on the complete flat manifold M. From the smoothness of the function l, we conclude that the flat manifold M must be R4 . In this case we would have its reduced distance to be ¯l and its reduced volume to be 1. Since V is not less than the reduced volume of the limit, this is a contradiction. Therefore the limiting gradient shrinking soliton g ij is nonflat. 32

4

Now we consider the nonflat gradient shrinking Ricci soliton (M , g ij ). Of course it is still κ-noncollapsed for all scales and satisfies the restricted 4 isotropic curvature pinching condition (2.4). We first show that (M , g ij (s)) has bounded curvature at each s > 0. Clearly it suffices to consider s = 1. By 4 Lemma 3.2, we may assume the soliton (M , g ij (1)) has positive curvature operator everywhere. Let us argue by contradiction. Suppose not, then we claim that for each positive integer k, there exists a point xk such that  ¯ k , 1) ≥ k,   R(x ¯ 1) ≤ 4R(x ¯ k , 1), for x ∈ Bg¯(·,1) (xk , p k  ).  R(x, ¯ k , 1) R(x

Indeed, xk can be constructed as a limit of a finite sequence {yi}, defined as ¯ 0 , 1) ≥ k. Inductively, if yi cannot follows. Let y0 be any fixed point with R(y be taken as xk , then there is a yi+1 such that  ¯ i+1 , 1) > 4R(y ¯ i , 1),  R(y  k  .  dg¯(·,1) (yi , yi+1) ≤ p ¯ R(yi, 1) Thus we have

¯ i , 1) > 4i R(y ¯ 0 , 1) ≥ 4ik, R(y dg¯(·,1) (yi , y0) ≤ k

i X j=1

√

1 4j−1 k

√ < 2 k.

Since the soliton is smooth, the sequence {yi } must be finite. The last element fits. Note that the limiting soliton still satisfies the Li-Yau-Hamilton inequality. Then we have ¯ s) ≤ R(x, ¯ 1) ≤ 4R(x ¯ k , 1) R(x, for x ∈ Bg¯(·,1) (xk , √ ¯ k

) and 1 ≤ s ≤ 1 +

1 ¯ k ,1) . R(x

By the κ-noncollapsing ¯ 4 , R(x ¯ k , 1)¯ and the Hamilton’s compactness theorem [17], a sequence of (M g (·, 1+ (·) ¯ 4 ¯) at least on ¯ k ,1) , xk ) will converge to a complete smooth solution (M , g R(x ¯ k , 1) → ∞, it follows the interval [0, 1). Since dg¯(·,1) (xk , x0 ) → ∞ and R(x ¯ 4 = R × S3 . This contradicts with Proposition 2.2. from Lemma 3.1 that M R(xk ,1)

4

So we have proved that (M , g ij (s)) has bounded curvature at each s > 0. 33

4

We next show that the soliton (M , g ij ) is κ′0 -noncollapsed on all scales 4 for some universal positive constant κ′0 . If the soliton (M , g ij ) has positive curvature operator, we know from Hamilton’s result [14] and Proposition 3.4 4 that either the soliton (M , g ij ) is the round S4 or RP4 when it is compact, or it is κ′0 -noncollapsed for all scales for some universal positive constant 4 κ′0 when it is noncompact. (Furthermore, when the soliton (M , g ij ) is the round S4 or RP4 , it follows from Hamilton’s pinching estimates in [14] that the original ancient κ-solution (M 4 , gij (t)) is also the round S4 or RP4 ). While 4 if the soliton (M , g ij ) has a nontrivial null eigenvector somewhere at some 4 time, we know from Lemma 3.2 that the soliton (M , g ij ) is R × S3 /Γ, a metric quotient of the round neck R × S3 . For each σ ∈ Γ, (s, x) ∈ R × S3 , write σ(s, x) = (σ1 (s, x), σ2 (s, x)) ∈ R×S3 . Since σ sends lines to lines, and σ sends cross spheres to cross spheres, we have σ1 (s, x) = σ1 (s, y),∀ x, y ∈ S3 . This says that σ1 reduces to a function of s alone on R. Moreover, for any (s, x), (s′ , x′ ) ∈ R × S3 , since σ preserves the distances between cross spheres {s} × S3 and {s′ } × S3 , we have |σ1 (s, x) − σ1 (s′ , x′ )| = |s − s′ |. So the projection Γ1 of Γ to the factor R is an isometric subgroup of R. We ¯ 4 , g¯ij ) = R × S3 /Γ was compact, it, as an ancient solution, know that if (M ¯ 4 , g¯ij ) = could not be κ-noncollapsed on all scale as t → −∞. Thus (M R × S3 /Γ is noncompact. It follows that Γ1 = {1} or Z2 . We conclude that, in both cases, there is a Γ-invariant cross sphere S3 in R × S3 . Denote it by {0} × S3 . Γ acts on {0} × S3 without fixed points. Recall that we have assumed that the ancient solution (M 4 , gij ) has positive curvature operator. Then we apply Hamilton’s argument in Theorem C4.1 of [19] when M 4 is compact and apply the modified argument in the proof of Lemma 3.1 when M 4 is noncompact to conclude that ({0} × S3 )/Γ is incompressible in M 4 (i.e., π1 (({0} × S3 )/Γ) injects into π1 (M 4 )). By Synge theorem and the Soul theorem [6], the fundamental group π1 (M 4 ) is either {1} or Z2 . This 4 implies that Γ is either {1} or Z2 . Thus the limiting soliton (M , g ij ) is also κ′0 -noncollapsed on all scales for some universal positive constant κ′0 . We next use the κ′0 -noncollapsing of the limiting soliton to derive a κ0 noncollapsing for the original ancient κ-solution. By rescaling, we may assume that R(x, t) ≤ 1 for all (x, t) satisfying dt0 (x, p) ≤ 2 and t0 −1 ≤ t ≤ t0 . We only need to bound the volume V olt0 (Bt0 (p, 1)) from below by a universal 34

positive constant. 1 Denote by ǫ = V olt0 (Bt0 (p, 1)) 4 . For any υ ∈ Tp M 4 , it was known from √ [27] that one can find a L-geodesic γ(τ ), starting at p, with limτ →0+ τ γ(τ ˙ )= υ, which satisfies the following L-geodesic equation √ d √ 1√ ˙ − τ ∇R + 2Ric( τ γ, ˙ ·) = 0. ( τ γ) dτ 2

(3.22)

Note from Shi’s local derivative estimate (see [32]) that |∇R| is also uniformly bounded. By integrating the L-geodesic equation we see that as τ ≤ ǫ with the property that γ(σ) ∈ Bt0 (p, 1) for σ ∈ (0, τ ], there holds √ ˙ ) − υ| ≤ Cǫ(|υ| + 1) | τ γ(τ

(3.23)

for some universal positive constant C. Here we implicitly used the fact that the metrics gij (t) are equivalent to each other on Bt0 (p, 1) × [t0 − 1, t0 ], which is a easy consequence of the boundedness of the curvature there. Without 1 loss of generally, we may assume Cǫ ≤ 14 and ǫ ≤ 100 . Then for υ ∈ Tp M 4 1 with |υ| ≤ 14 ǫ− 2 and for τ ≤ ǫ with the property that γ(σ) ∈ Bt0 (p, 1) for σ ∈ (0, τ ], we have Z τ dt0 (p, γ(τ )) ≤ |γ(σ)|dσ ˙ 0 Z 1 − 1 τ dσ √ ǫ 2 < 2 σ 0 = 1. This shows

1 1 L exp{|υ| ≤ ǫ− 2 }(ǫ) ⊂ Bt0 (p, 1) (3.24) 4 where L exp(·)(ǫ) denotes the exponential map of the L distance with parameter ǫ (see [27] or [3] for details). We decompose the reduced volume Ve (ǫ) as Ve (ǫ) =

Z

(4πǫ)−2 exp(−l)dVt0 −ǫ 4 Z ZM + ≤ 1 L exp{|υ|≤ 14 ǫ− 2 }(ǫ)

1 M 4 \L exp{|υ|≤ 14 ǫ− 2 }(ǫ)

35

(4πǫ)−2 exp(−l)dVt0 −ǫ . (3.25)

The first term on RHS of (3.25) can be estimated by Z (4πǫ)−2 exp(−l)dVt0 −ǫ 1 1 −2 }(ǫ) L exp{|υ|≤ 4 ǫ Z (4πǫ)−2 exp(−l)dVt0 ≤ e4ǫ Bt0 (p,1)

(3.26)

≤ e4ǫ (4π)−2ǫ−2 V olt0 (Bt0 (p, 1)) = e4ǫ (4π)−2 ǫ2 .

where we used (3.24) and the equivalence of the evolving metric over Bt0 (p, 1). While the second term on the RHS of (3.25) can be estimated as follows Z (4πǫ)−2 exp(−l)dVt0 −ǫ 1 1 −2 4 M \L exp{|υ|≤ 4 ǫ }(ǫ) Z (3.27) −2 ≤ (4πτ ) exp(−l)J(τ )|τ =0 dυ 1 {|υ|> 14 ǫ− 2 }

by Perelman’s Jacobian comparison theorem [27], where J(τ ) is the Jacobian of the L-exponential map. For any υ ∈ Tp M, we consider a L-geodesic γ(τ ) starting at p with √ ˙ ) = υ. To evaluate the Jacobian of the L exponential map at limτ →0+ τ γ(τ τ = 0 we choose linear independent vectors υ1 , · · · , υ4 in Tp M and let Vi (τ ) = (L expυ (τ ))∗ (υi) =

d |s=0 L exp(υ+sυi ) (τ ), i = 1, · · · , 4. ds

The L-Jacobian J(τ ) is given by J(τ ) = |V1 (τ ) ∧ · · · ∧ V4 (τ )|gij (τ ) /|v1 ∧ · · · ∧ v4 |. By the L-geodesic equation (3.22) and the deriving of (3.23), we see that as τ > 0 small enough, √ d | τ L exp(υ+sυi ) (τ ) − (υ + sυi )| ≤ o(1) dτ for s ∈ (−ǫ, ǫ) and i = 1, · · · , 4, where o(1) tends to zero as τ → 0+ uniformly in s. This implies that lim+

τ →0

√

τ V˙ i (τ ) = υi , i = 1, · · · , 4, 36

so we get lim+ τ −2 J(τ ) = 1.

τ →0

(3.28)

To evaluate l(·, τ ) at τ = 0, we use (3.23) again to get Z τ √ 1 2 s(R + |γ(s)| ˙ )ds l(·, τ ) = √ 2 τ 0 → |υ|2,

as τ → 0+ ,

thus l(·, 0) = |υ|2. Hence by combining (3.27)-(3.29) we have Z (4πǫ)−2 exp(−l)dVt0 −ǫ 1 1 −2 4 M \L exp{|υ|≤ 4 ǫ }(ǫ) Z ≤ (4π)−2 exp(−|υ|2)dυ 1

(3.29)

(3.30)

{|υ|> 14 ǫ− 2 }

< ǫ2 . By summing up (3.25), (3.26) and (3.30), we obtain Ve (ǫ) < 2ǫ2 .

(3.31)

On the other hand we recall that there are sequences τk → +∞ and q(τk ) ∈ M 4 with l(q(τk ), τk ) ≤ 2 so that the rescalings of the ancient κsolution around q(τk ) with factor τk−1 converge to a gradient shrinking Ricci soliton which is κ′0 -noncollapsing on all scales for some universal positive constant κ′0 . For sufficiently large k, we construct a path γ : [0, 2τk ] → M 4 , connecting p to any given point q ∈ M 4 , as follows: the first half path γ|[0,τk ] connects p to q(τk ) such that Z τk √ 1 l(q(τk ), τk ) = √ τ (R + |γ(τ ˙ )|2 )dτ ≤ 3, 2 τk 0 and the second half path γ|[τk ,2τk ] is a shortest geodesic connecting q(τk ) to q with respect to the metric gij (t0 − τk ). Note that the rescaled metric

37

√ τk−1 gij (t0 − τ ) over the domain Bt0 −τk (q(τk ), τk ) × [t0 − 2τk , t0 − τk ] is sufficiently close to the gradient shrinking Ricci soliton. Then by the estimates (3.5) and the κ′0 -noncollapsing of the shrinking soliton, we get Z e V (2τk ) = (4π(2τk ))−2 exp(−l(q, 2τk ))dVt0 −2τk (q) ZM (4π(2τk ))−2 exp(−l(q, 2τk ))dVt0 −2τk (q) ≥ √ Bt0 −τk (q(τk ), τk )

≥ β

for some universal positive constant β. By applying the monotonicity of the reduced volume [27] and (3.31), we deduce that

This proves

β ≤ Ve (2τk ) ≤ Ve (ǫ) < 2ǫ2 . V olt0 (Bt0 (p, 1)) ≥ κ0 > 0

for some universal positive constant κ0 . Therefore we have proved the Theorem. # Once the universal noncollapsing of ancient κ-solution with restricted isotropic curvature pinching is established, we can also strengthen the elliptic type estimates in Proposition 3.3 to the following form. Proposition 3.6 There exist a positive constant η and a positive function ω : [0, +∞) → (0, +∞) with the following properties. Suppose we have a four-dimensional ancient κ-solution (M 4 , gij (t)), −∞ < t ≤ 0, with restricted isotropic curvature pinching for some κ > 0. Then (i) for every x, y ∈ M 4 and t ∈ (−∞, 0], there holds R(x, t) ≤ R(y, t) · ω(R(y, t)d2t (x, y)); (ii) for all x ∈ M 4 and t ∈ (−∞, 0], there hold 3

|∇R|(x, t) ≤ ηR 2 (x, t) and | 38

∂R |(x, t) ≤ ηR2 (x, t). ∂t

The following result generalizes Theorem 11.7 of Perelman [27] to fourdimension. Corollary 3.7 The set of four-dimensional ancient κ-solutions with restricted isotropic curvature pinching and positive curvature operator is precompact modulo scaling in the sense that for any sequence of such solutions ∞ and marked points (xk , 0) with R(xk , 0) = 1, we can extract a Cloc converging subsequence, and the limit is also an ancient κ0 -solution with restricted isotropic curvature pinching. Proof: Consider any sequence of four-dimensional ancient κ-solutions with restricted isotropic curvature pinching and positive curvature operator and marked points (xk , 0) with R(xk , 0) = 1. By the Proposition 3.6 (i), Li-Yau-Hamilton inequality [16] and Hamilton’s compactness theorem ∞ (Theorem 16.1 of [18]), we can extract a Cloc converging subsequence such 4 that the limit (M , g ij (t)) is an ancient solution to the Ricci flow and satisfies the restricted isotropic curvature pinching condition (2.4), as well as is κ-noncollapsed for all scales. Moreover, the limit still satisfies the Li-YauHamilton inequality and the assertions (i) and (ii) of Proposition 3.6. To show the limit is an ancient κ-solution, we remain to show the limit has bounded curvature at the time t = 0. By the virtue of Lemma 3.2, we may assume the limit has positive curvature operator everywhere. We now argue by contradiction. Suppose the ¯ 4 , g¯ij (0)) is unbounded, then there curvature of the limit (at t = 0) (M is a sequence of points Pl divergent to infinity at the time t = 0 with the scalar curvature R(Pl , 0) → +∞. By Hamilton’s compactness theorem (Theorem 16.1 of [18]) and the estimates in the assertions (i) and (ii) of Proposition 3.6, we know that a subsequence of the rescaled solutions 4 ¯ l , 0)g ij (·, ¯ t ), Pl ) converges in C ∞ to a smooth nonflat limit. (M , R(P loc R(Pl ,0) And by Lemma 3.1, the limit must be the round neck R × S3 . This contradicts Proposition 2.2. Therefore we have proved the corollary. #

39

3.4 Canonical neighborhood structures We now examine the structures of four-dimensional nonflat ancient κsolutions with restricted isotropic curvature pinching. As before by Lemma 3.2, we have seen a four-dimensional nonflat ancient κ-solution with restricted isotropic curvature pinching, whose curvature operator has a nontrivial null vector somewhere at some time, must be a metric quotient of the round cylinder R × S3 . So we only need to consider the ancient κ-solutions with positive curvature operator. The following theorem gives their canonical neighborhood structures. The analogous result in three-dimensional case was given by Perelman in Section 1.5 of [28]. Theorem 3.8 For every ε > 0 one can find positive constants C1 = C1 (ε), C2 = C2 (ε) such that for each point (x, t) in every four-dimensional ancient κ-solution (for some κ > 0) with restricted isotropic curvature pinching and with positive curvature operator, there is a radius r, 0 < r < 1 C1 (R(x, t))− 2 , so that some open neighborhood Bt (x, r) ⊂ B ⊂ Bt (x, 2r) falls into one of the following three categories: (a) B is an evolving ε-neck (in the sense that it is the time slice at time t of the parabolic region {(x′ , t′ )|x′ ∈ B, t′ ∈ [t − ε−2 R(x, t)−1 , t]} which is, after scaling with factor R(x, t) and shifting the time t to 0, ε-close (in −1 C [ε ] topology) to the subset (I×S3 )×[−ε−2 , 0] of the evolving round cylinder R × S3 , having scalar curvature one and length 2ε−1 to I at time zero, or (b) B is an evolving ε-cap (in the sense that it is the time slice at the time t of an evolving metric on open B4 or RP4 \ B4 such that the region outside some suitable compact subset of B4 or RP4 \B4 is an evolving ε-neck), or (c) B is a compact manifold (without boundary) with positive curvature operator (thus it is diffeomorphic to S4 or RP4 ); furthermore, the scalar curvature of the ancient κ-solution in B at time t is between C2−1 R(x, t) and C2 R(x, t), and the volume of B in case (a) and case (b) satisfies (C2 R(x, t))−2 ≤ V olt (B). .

40

Proof. If the nonflat ancient κ-solution is noncompact, the conclusions follow immediately from (the proof of) Proposition 3.4. We thus assume the nonflat ancient κ-solution is compact. By Theorem 3.5 we see that such ancient κ-solution is κ0 -noncollapsed for all scales for some universal positive constant κ0 . We argue by contradiction. Suppose for some ε > 0, there exists a sequence of compact ancient κ0 -solutions (Mk4 , gk ) with restricted isotropic curvature pinching and with positive curvature operator, a sequence of points xk ∈ Mk4 , and sequences of positive constants C1k with C1k → +∞ as 2 k → +∞ and C2k = ω(4C1k ) with the function ω given in Proposition 1 3.6 such that at time t, for every radius r, 0 < r < C1k R(xk , t)− 2 , any open neighborhood B with Bt (xk , r) ⊂ B ⊂ Bt (xk , 2r) can not fall into any one of the three categories (a), (b) and (c). Clearly, the diameter of each 1 Mk4 at time t is at least C1k R(xk , t)− 2 ; otherwise one can choose suitable 1 r ∈ (0, C1k R(xk , t)− 2 ) and B = Mk4 , which falls into the category (c), so −1 that the scalar curvature in B at t is between C2k R(xk , t) and C2k R(xk , t) by using Proposition 3.6 (i). Now by scaling the ancient κ0 -solutions along the points (xk , t) with factors R(xk , t) and shifting the time t to 0, it follows from Corollary 3.7 that a subsequence of these rescaled ancient κ0 -solutions ∞ converge in Cloc topology to a noncompact nonflat ancient κ0 -solution with restricted isotropic curvature pinching. If the noncompact limit has a nontrivial null curvature eigenvector somewhere, then by Lemma 3.2 we conclude that the limit is round cylinder R×S3 or a metric quotient R × S3 /Γ. By the same reason as in the proof of Theorem 3.5, the projection Γ1 of Γ to the factor R is an isometric subgroup of R. Since the limit R × S3 /Γ is noncompact, Γ1 must be {1} or Z2 . Thus we have a Γ-invariant cross-sphere S3 in R ×S3 /Γ, and Γ acts on it without fixed points. Denote this cross sphere by {0} × S3 . Since each (Mk4 , gk ) is compact and has positive curvature operator, we know from [14] that each Mk4 is diffeomorphic to S4 or RP4 . Then by the proof of Theorem 3.5 and applying theorem C 4.1 of [19], we conclude that the limit is R × S3 or R × S3 /Γ with Γ = Z2 . If Γ = Z2 , we claim that Γ must act on R × S3 /Γ by flipping both R and S3 . Indeed, as shown before, Γ1 = {1} or Z2 . If Γ1 = {1}, then R × S3 /Γ = 41

R × RP3 . Let Γ+ be the normal subgroup of Γ preserving the orientation of the cylinder, and π1 (Mk4 )+ be the normal subgroup of π1 (Mk4 ) preserving the orientation of the universal cover of Mk4 . Since the manifold Mk4 is diffeomorphic to RP4 , this induces an absurd commutative diagram: Z2 Z2 0 k k k 0 −→ Γ+ −→ Γ −→ Γ/Γ+ −→ 0 ↓ ↓≀ ↓ 4 + 4 4 0 −→ π1 (Mk ) −→ π1 (Mk ) −→ π1 (Mk )/π1 (Mk4 )+ −→ 0 k k k 0 Z2 Z2 where the vertical morphisms are induced by the inclusion R × S3 /Γ ⊂ M. Therefore Γ1 = Z2 . Denote by σ1 be the isometry of R ×S3 acting by flipping both R and S3 around {0} × S 3 . Clearly, for any σ ∈ Γ with σ 6= 1, σ ◦ σ1 is an isometry of R × S3 whose projection on the factor R is the identity map. Then σ ◦ σ1 is only a rotation of the factor S3 in R × S3 . Note that σ ◦ σ1 |{0}×S3 is identity. We conclude that σ = σ1 and the claim holds. When the limit is the round cylinder R × S3 , a suitable neighborhood B (for suitable r) of xk would fall into the category (a) for sufficiently large k; while when the limit is the Z2 quotient of the round cylinder R × S3 with the antipodal map flipping both S3 and R, a suitable neighborhood B (for suitable r) of xk would fall into the category (b) (over RP4 \ B4 ) or into the category (a) for sufficiently large k. This is a contradiction. If the noncompact limit has positive curvature operator everywhere, then by Proposition 3.4, a suitable neighborhood B (for suitable r) of xk would fall into the category (b) (over B4 ) for sufficiently large k. We also get a contradiction. Finally, the statements on the curvature estimate and volume estimate for the neighborhood B follows directly from Theorem 3.6 and Proposition 3.4. Therefore we have proved the theorem. #

42

4. The Structure of Solutions at the Singular Time Let (M 4 , gij (x)) be a four-dimensional compact Riemannian manifold with positive isotropic curvature and let gij (x, t), x ∈ M 4 and t ∈ [0, T ), be a maximal solution to the Ricci flow (1.1) with gij (x, 0) = gij (x) on M 4 . Since the initial metric gij (x) has positive scalar curvature, it is easy to see that the maximal time T must be finite and the curvature tensor becomes unbounded as t → T . According to Perelman’s noncollapsing theorem I (Theorem 4.1 of √ [27]), the solution gij (x, t) is κ-noncollapsed on the scale T for all t ∈ [0, T ) for some κ > 0. Now let us take a sequence of times tk → T , and a sequence of points pk ∈ M 4 such that for some positive constant C, |Rm |(x, t) ≤ CQk with Qk = |Rm(pk , tk )| whenever x ∈ M 4 and t ∈ [0, tk ], called a sequence of (almost) maximal points. Then by Hamilton’s compactness theorem [17], a sequence of the scalings of the solution gij (x, t) along the points pk with factors Qk converges to a complete ancient κ-solution with restricted isotropic curvature pinching. This says, for any ε > 0, there exists a positive number k0 such that as k ≥ k0 , the solution in the parabolic region −2 −1 {(x, t) ∈ M 4 × [0, T ) | d2tk (x, xk ) < ε−2 Q−1 k , tk − ε Qk < t ≤ tk } is, after −1 scaling with the factor Qk , ε-close (in C [ε ] -topology) to the corresponding subset of the ancient κ-solution with restricted isotropic curvature pinching. Let us describe the structure of any ancient κ-solution (with restricted isotropic curvature pinching). If the curvature operator is positive everywhere, then each point of the ancient κ-solution has a canonical neighborhood described in Theorem 3.8. While if the curvature operator has a nontrivial null eigenvector somewhere, then by Hamilton’s strong maximum principle and the pinching condition (2.4) the ancient κ-solution is isometric to R × S3 /Γ, a metric quotient of the round cylinder R × S3 . Since it is κnoncollapsed for all scales, the metric quotient R × S3 /Γ can not be compact. Suppose we make an additional assumption that the compact four manifold M 4 has no essential incompressible space form. Then by the proofs of Theorems 3.5 and 3.8 and applying Theorem C 4.1 of [19], we have Γ = {1}, or Γ = Z2 acting antipodally on S3 and by reflection on R. Thus in both cases, each point of the ancient κ solution has also a canonical neighborhood described in Theorem 3.8. 43

Hence we see that each such (almost) maximal point (xk , tk ) has a canonical neighborhood which is either an evolving ε-neck or an evolving ε-cap, or a compact manifold (without boundary) with positive curvature operator. This gives the structure of the singularities coming from a sequence of (almost) maximal points (xk , tk ). However this argument does not work for the singularities coming from a sequence of points (yk , τk ) with τk → T and |Rm(yk , τk )| → +∞ when |Rm(yk , τk )| is not comparable with the maximum of the curvature at the time τk , since we can not take a limit directly. We now follow a refined rescaling argument of Perelman (Theorem 12.1 of [27]) to obtain a uniform canonical neighborhood structure theorem for fourdimensional solutions at any point where its curvature is suitable large. Theorem 4.1 Given ε > 0, κ > 0, 0 < θ, ρ, Λ, P < +∞, one can find r0 > 0 with the following property. If gij (x, t), t ∈ [0, T ) with T > 1, is a solution to the Ricci flow on a four-dimensional manifold M 4 with no essential incompressible space form, which has positive isotropic curvature, is κ-noncollapsed on the scales ≤ θ and satisfies (2.1), (2.2) and (2.3) in Lemma 2.1, then for any point (x0 , t0 ) with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , the solution in the parabolic region {(x, t) ∈ M 4 × [0, T )|d2t0 (x, x0 ) < ε−2 Q−1 , t0 − ε−2 Q−1 < t ≤ t0 } is, after scaling by the factor Q, ε-close (in −1 C [ε ] -topology) to the corresponding subset of some ancient κ-solution with restricted isotropic curvature pinching. Consequently each point (x0 , t0 ), with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , satisfies the gradient estimates 3

|∇R(x0 , t0 )| < 2ηR 2 (x0 , t0 ) and |

∂ R(x0 , t0 )| < 2ηR2 (x0 , t0 ), ∂t

(4.1)

and has a canonical neighborhood B with Bt0 (x0 , r) ⊂ B ⊂ Bt0 (x0 , 2r) for 1 some 0 < r < C1 (ε)(R(x0 , t0 ))− 2 , which is either an evolving ε-neck, or an evolving ε-cap, or a compact four-manifold with positive curvature operator. Here η is the universal constant in Proposition 3.6 and C1 (ε) is the positive constant in Theorem 3.8. Proof. Let C(ε) be a positive constant depending only on ε such that C(ε) → +∞ as ε → 0+ . It suffices to prove that there exists r0 > 0 such that for any point (x0 , t0 ) with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , the solution in the 44

parabolic region {(x, t) ∈ M 4 × [0, T ) | d2t0 (x, x0 ) < C(ε)Q−1 , t0 − C(ε)Q−1 < t ≤ t0 } is, after scaling by the factor Q, ε-close to the corresponding subset of some ancient κ-solution with restricted isotropic curvature pinching. The constant C(ε) will be determined later. We argue by contradiction. Suppose for some ε > 0, κ > 0, 0 < (k) θ, ρ, Λ, P < +∞, there exists a sequence of solutions (Mk4 , gij (·, t)) to the Ricci flow on compact four-manifolds with no essential incompressible space form, having positive isotropic curvature and satisfying (2.1), (2.2) and (2.3), defined on the time interval [0, Tk ) with Tk > 1, and a sequence of positive (k) numbers rk → 0 such that each solution (Mk4 , gij (·, t)) is κ-noncollapsed on the scales ≤ θ; but there exists a sequence of points xk ∈ Mk4 and times tk ≥ 1 with Qk = Rk (xk , tk ) ≥ rk−2 such that the solution in the parabolic region −1 {(x, t) ∈ Mk4 ×[0, Tk ) | d2tk (x, xk ) < C(ε)Q−1 k , tk −C(ε)Qk < t ≤ tk } is not, after scaling by the factor Qk , ε-close to the corresponding subset of any ancient κ-solution with restricted isotropic curvature pinching, where Rk denotes the (k) (k) scalar curvature of (Mk4 , gij (·, t)). For each solution (Mk4 , gij (·, t)), we may adjust the point (xk , tk ) with tk ≥ 21 and with Qk = Rk (xk , tk ) as large as possible so that the conclusion of the theorem fails at (xk , tk ), but holds for any 1 −2 (x, t) ∈ Mk4 × [tk − Hk Q−1 k , tk ] satisfying R(x, t) ≥ 2Qk , where Hk = 4 rk → (1) (1) +∞ as k → +∞. Indeed, suppose not, by setting (xk , tk ) = (xk , tk ), we can (ℓ) (ℓ) (ℓ−1) (ℓ−1) (ℓ−1) (ℓ−1) inductively choose (xk , tk ) ∈ Mk4 × [tk − Hk (Rk (xk , tk ))−1 , tk ] (ℓ) (ℓ) (ℓ−1) (ℓ−1) satisfying Rk (xk , tk ) ≥ 2Rk (xk , tk ), but the conclusion of the theo(ℓ) (ℓ) rem fails at (xk , tk ) for each ℓ = 2, 3, · · · . Since the solution is smooth and (ℓ)

(ℓ)

(ℓ−1)

Rk (xk , tk ) ≥ 2Rk (xk

(ℓ−1)

, tk

)

≥ 2ℓ−1Rk (xk , tk ),

(ℓ)

tk

(ℓ−1)

(ℓ−1)

(ℓ−1)

− Hk (Rk (xk , tk ))−1 ℓ−1 X (2i−1 Rk (xk , tk ))−1 ≥ tk − Hk ≥ tk

i=1

≥ tk − 2Hk (R(xk , tk ))−1 1 ≥ , 2 45

the above choosing process must terminate in finite step and the last element fits. (k) Let (Mk4 , e gij (·, t), xk ) be the rescaled solutions obtained by rescaling the (k) manifolds (Mk4 , gij (·, t)) with factors Qk = Rk (xk , tk ) and shifting the time ek the rescaled scalar curvature. We will show that tk to 0. Denote by R (k) a subsequence of the rescaled solutions (Mk4 , e gij (·, t), xk ) converges to an ancient κ-solution with restricted isotropic curvature pinching, which is a contradiction. In the followings we divide the argument into four steps. Step 1 We want to prove a local curvature estimate in the following assertion. Claim: For each (x, t) with tk − H2k Q−1 k < t ≤ tk , we have Rk (x, t) ≤ 4Qk −1 −1 2 whenever t − cQk ≤ t ≤ t and dt (x, x) ≤ cQk , where Qk = Qk + Rk (x, t) and c > 0 is a small universal constant. −1 1 −1 To prove this, we consider any point (x, t) ∈ Bt (x, (cQk ) 2 ) × [t − cQk , t] with c > 0 to be determined. If Rk (x, t) ≤ 2Qk , there is nothing to show. If Rk (x, t) > 2Qk , consider a space time curve γ that goes straightly from (x, t) to (x, t) and goes from (x, t) to (x, t) along a minimizing geodesic (with (k) respect to the metric gij (·, t)). If there is a point on γ with the scalar curvature 2Qk , let p be the nearest such point to (x, t); if not, put p = (x, t). On the segment of γ from (x, t) to p, the scalar curvature is not less than 2Qk . According to the choice of the point (xk , tk ), the solution along the segment is ε-close to that of some ancient κ-solutions with restricted isotropic curvature pinching. Of course we may assume ε > 0 is very small. It follows from Proposition 3.6 (ii) that −1

|∇(Rk 2 )| ≤ 2η and |

∂ −1 (R )| ≤ 2η ∂t k

on the segment for some universal constant η > 0. Then by choosing c > 0 (depending only on η) small enough we get the desired curvature bound by integrating the above derivative estimate along the segment. This proves the assertion. Step 2 We next want to show that the curvatures of the rescaled solu(k) tions e gij (·, t) at the new time t = 0 (i.e., the original time tk ) stay uniformly bounded at bounded distances from xk . 46

For all σ ≥ 0, set

and

ek (x, 0) | k ≥ 1, x ∈ M 4 with d0 (x, xk ) ≤ σ} M(σ) = sup{R k σ0 = sup{σ ≥ 0 | M(σ) < +∞}.

Note that σ0 > 0 by Step 1. By the assumptions (2.1) and (2.2), it suffices to show σ0 = +∞. We now argue by contradiction to show σ0 = +∞. Suppose not, we may find (after passing to a subsequence if necessary) a sequence of ek (yk , 0) → +∞ as k → points yk ∈ Mk4 with d0 (yk , xk ) → σ0 < +∞ and R +∞. Let γk (⊂ Mk4 ) be a minimizing geodesic segment from xk to yk , zk ∈ γk ek (zk , 0) = 2 and βk the subsegment the point on γk closest to yk at which R of γk running from yk to zk . By Step 1 the length of βk is uniformly bounded away from zero for all k. And by the assumptions (2.1) and (2.2), we have (k) a uniform curvature bound on the open balls B0 (xk , σ) ⊂ (Mk4 , e gij (·, 0)) for each fixed σ < σ0 . Note that the κ-noncollapsing assumption implies the (k) uniform injectivity radius bound for (Mk4 , e gij (·, 0)) at the marked points xk . Then by the virtue of Hamilton’s compactness theorem 16.1 in [18] ( see [3] for the details on generalizing Hamilton’s compactness theorem to finite balls), (k) we can extract a subsequence of the marked (B0 (xk , σ0 ), e gij (·, 0), xk ) which ∞ converges in Cloc topology to a marked (noncomplete) manifold (B∞ , e gij∞, x∞ ), so that the segments γk converge to a geodesic segment (missing an endpoint) γ∞ ⊂ B∞ emanating from x∞ , and βk converge to a subsegment β∞ of γ∞ . (∞) gij ), and y∞ ∈ B ∞ the limit point Let B ∞ denote the completion of (B∞ , e of γ∞ . (∞) e∞ the scalar curvature of (B∞ , e Denote by R gij ). Since the rescaled ek along βk are at least 2, it follows from the choice scalar curvatures of R (∞) of the points (xk , tk ) that for any q0 ∈ β∞ , the manifold (B∞ , e gij ) in e∞ (q0 ))−1 } is 2ε-close to the corresponding {q ∈ B∞ |dist2(∞) (q, q0 ) < C(ε)(R gij e

subset of (a time slice of) of some ancient κ-solution with restricted isotropic curvature pinching. From the argument in the second paragraph of this section, we know that such an ancient κ-solution with restricted isotropic cur1 vature pinching at each point (x, t) has a radius r, 0 < r < C1 (2ε)R(x, t)− 2 , such that its canonical neighborhood B with Bt (x, r) ⊂ B ⊂ Bt (x, 2r), is 47

either an evolving 2ε-neck, or an evolving 2ε-cap, or a compact manifold (without boundary) with positive curvature operator, moreover the scalar curvature on the ball is between C2 (2ε)−1R(x, t) and C2 (2ε)R(x, t), where C1 (2ε) and C2 (2ε) are the positive constants in Theorem 3.8. We now choose C(ε) = max{2C1 (2ε)2 , ε−2}. By the local curvature estimate in Step 1, we e∞ becomes unbounded along γ∞ going to y∞ . see that the scalar curvature R This implies that the canonical neighborhood around q0 can not be a compact manifold (without boundary) with positive curvature operator. Note that γ∞ is shortest since it is the limit of a sequence of shortest geodesics. Without loss of generality, we may assume ε is suitably small. These imply that as q0 sufficiently close to y∞ , the canonical neighborhood around q0 can not be a 2ε-cap. Thus we conclude that each q0 ∈ γ∞ , which is sufficiently close to y∞ , is the center of a 2ε-neck. Denote by [ (∞) e∞ (q0 ))− 12 ) (⊂ (B∞ , e gij )) B(q0 , 24π(R U= q0 ∈γ∞

e∞ (q0 ))− 21 . e∞ (q0 ))− 21 ) is the ball centered at q0 of radius 24π(R where B(q0 , 24π(R Clearly, it follows from the assumptions (2.1), (2.2) and (2.3) that U has (∞) nonnegative curvature operator. Since the metric e gij is cylindrical at any point q0 ∈ γ∞ which is sufficiently close to y∞ , we see that the metric space U = U ∪{y∞ } by adding the point y∞ , is locally complete and strictly intrinsic near y∞ . Here strictly intrinsic means that the distance between any two points can be realized by shortest geodesics. Furthermore y∞ cannot be an interior point of any geodesic segment in U. This implies that the curvature of U at y∞ is nonnegative in Alexandrov sense. Note that for any very small radius σ, the geodesic sphere ∂B(y∞ , σ) is an almost round sphere of radius ≤ 3εσπ. By [1] or [5] we have a four-dimensional tangent cone at y∞ with aperture ≤ 20ε. Moreover, by [1] or [5], any four-dimensional tangent cone Cy∞ U at y∞ must be a metric cone. For each tangent cone, pick z ∈ Cy∞ U such that the distance between the vertex y∞ and z is one. Then the ball B(z, 21 ) ⊂ Cy∞ U is the Gromov-Hausdorff limit of the scalings of a sequence (k) of balls B0 (zk , sk ) ⊂ (Mk4 , e gij (·, 0)) by some factors ak , where sk → 0+ . Since the tangent cone is four-dimensional and has aperture ≤ 20ε, the factors ak 48

˜ k (zk , 0). By using the local curvature estimate in must be comparable with R ∞ Step 1, we actually have the convergence in the Cloc topology for the solutions (k) geij (·, t) over the balls B0 (zk , sk ) and over some time interval t ∈ [−δ, 0] for some sufficiently small δ > 0. The limiting B(z, 21 ) ⊂ Cy∞ U is a piece of the nonnegative (operator) curved and nonflat metric cone. On the other hand, since the radial direction of the cone is flat, by Hamilton’s strong maximum principle [14] and the pinching condition (2.4) as in the proof of Lemma 3.2, the limiting B(z, 12 ) would be a piece of R × S3 or R × S3 /Γ (a metric quotient). This is a contradiction. So we have proved that the curvatures of (k) the rescaled metrics geij (·, 0) stay uniformly bounded at bounded distances from xk . By the local curvature estimate in Step 1, we can locally extend the above curvature control backward in time a little. Then by the κ-noncollapsing as∞ sumption and Shi’s derivative estimates [32], we can take a Cloc limit from (k) 4 the sequence of marked rescaled solutions (Mk , e gij (·, t), xk ). The limit, de(∞) 4 noted by (M∞ , e gij (·, t), x∞ ), is κ-noncollapsing on all scales, is defined on a 4 4 space-time open subset of M∞ × (−∞, 0] containing the time slice M∞ × {0}, and satisfies the restricted isotropic curvature pinching condition (2.4) by the assumptions (2.1), (2.2) and (2.3). (∞)

4 ,e gij (·, t)) at the time slice Step 3 We further claim that the limit (M∞ t = 0 has bounded curvature. (∞) 4 We have known that the curvature operator of the limit (M∞ ,e gij (·, t)) is nonnegative everywhere. If the curvature operator has a nontrivial null eigenvector somewhere, we can argue as in the proof of Lemma 3.2 by using Hamilton’s strong maximum principle [14] and the restricted isotropic curvature pinching condition (2.4) to deduce that the universal cover of the limit is isometric to the standard R × S3 . Thus the curvature of the limit is bounded in this case. (∞) 4 Assume that the curvature operator of the limit (M∞ ,e gij (·, t)) at the time slice t = 0 is positive everywhere. Suppose there exists a sequence of 4 e∞ (pj , 0) → +∞ as j → points pj ∈ M∞ such that their scalar curvatures R +∞. By the local curvature estimate in Step 1 and the assertion of the above Step 2 (for the marked points pj ) as well as the κ-noncollapsed assumption, a (∞) 4 e subsequence of the rescaled and marked manifolds (M∞ , R∞ (pj , 0)e gij (·), pj )

49

∞ converges in Cloc topology to a smooth nonflat limit Y . Then by Lemma 3.1 we conclude that Y is isometric to R×S3 with the standard metric. This con(∞) 4 tradicts with Proposition 2.2. So the curvature of the limit (M∞ ,e gij (·, t)) at the time slice t = 0 must be bounded.

Step 4 Finally we want to extend the limit backward in time to −∞. By the local curvature estimate in Step 1, we now know that the limit (∞) 4 (M∞ ,e gij (·, t)) is defined on [−a, 0] for some a > 0. Denote by

t′ = inf{ t˜ | we can take a smooth limit on (t˜, 0](with bounded curvature at

each time slice) from a subsequence of the rescaled solutions g˜k }.

We first claim that there is a subsequence of the rescaled solutions g˜k which ∞ converges in Cloc topology to a smooth limit (M∞ , g˜∞ (·, t)) on the maximal ′ time interval (t , 0]. Indeed, let tk be a sequence of negative numbers such that tk → t′ and k there exist smooth limits (M∞ , g˜∞ (·, t)) defined on (tk , 0]. For each k, the limit has nonnegative and bounded curvature operator at each time slice. Moreover by the Claim in Step 1, the limit has bounded curvature on each ˜ the scalar curvature upper bound subinterval [−b, 0] ⊂ (tk , 0]. Denote by Q ˜ is the same for all k). Then we can apply of the limit at time zero (where Q Li-Yau-Hamilton inequality [16] to get k ˜∞ ˜ −tk ), R (x, t) ≤ Q( t − tk

˜ k (x, t) are the scalar curvatures of the limits (M∞ , g˜k (·, t)). Hence where R ∞ ∞ by the definition of convergence and the above curvature estimates, we can ∞ find a subsequence of the rescaled solutions g˜k which converges in Cloc topol′ ogy to a smooth limit (M∞ , g˜∞ (·, t)) on the maximal time interval (t , 0]. We next claim that t′ = −∞. (∞) 4 Suppose not, then the curvature of the limit (M∞ ,e gij (·, t)) becomes unbounded as t → t′ > −∞. Since the minimum of the scalar curvature is 4 e∞ (x∞ , 0) = 1, we see that there is a y∞ ∈ M∞ nondecreasing in time and R such that e∞ (y∞ , t′ + c ) < 3 0 0 is the universal constant in the assertion of Step 1. By using (∞) 4 Step 1 again we see that the limit (M∞ ,e gij (·, t)) in a small neighborhood of the point y∞ at the time slice t = t′ + 3c can be extended backward to the time interval [t′ − 3c , t′ + 3c ]. We remark that the distances at the time t and the time 0 are roughly equivalent in the following sense dt (x, y) ≥ d0 (x, y) ≥ dt (x, y) − const.

(4.2)

4 for any x, y ∈ M∞ and t ∈ (t′ , 0]. Indeed from the Li-Yau-Hamilton inequality [16] we have the estimate

∂ e e∞ (x, t) · (t − t′ )−1 , for (x, t) ∈ M 4 × (t′ , 0]. R∞ (x, t) ≥ −R ∞ ∂t

e denotes the supermum of the scalar curvature R e∞ at t = 0, then If Q ′ 4 e∞ (x, t) ≤ Q( e −t ), on M∞ × (t′ , 0]. R t − t′

By applying Lemma 8.3 (b) of [27], we have

q e dt (x, y) ≤ d0 (x, y) + 30(−t ) Q ′

4 for any x, y ∈ M∞ and t ∈ (t′ , 0]. On the other hand, since the curvature operator of the limit geij∞ (·, t) is nonnegative, we have

dt (x, y) ≥ d0 (x, y)

4 for any x, y ∈ M∞ and t ∈ (t′ , 0]. Thus we obtain the estimate (4.2). The estimate (4.2) insures that the limit around the point y∞ at any time t ∈ (t′ , 0] is exactly the original limit around x∞ at the time t = 0. Consider (k) the rescaled sequence of (Mk4 , e gij (·, t)) with the marked points replaced by the associated sequence yk → y∞ . By applying the same arguments as the (k) above Step 2 and Step 3 to the new marked sequence (Mk4 , e gij (·, t), yk ), we (∞) 4 conclude the original limit (M∞ ,e gij (·, t)) is actually well defined on the time 4 slice M∞ × {t′ } and also has uniformly bounded curvature for all t ∈ [t′ , 0]. By taking a subsequence from the original subsequence and combining Step ′′ 1, we can extend the limit backward to a larger interval [t , 0] ) (t′ , 0]. This is a contradiction with the definition of t′ .

51

Therefore we have proved a subsequence of the rescaled solutions (Mk4 , (k) geij (·, t), xk ) converges to an ancient κ-solution with restricted isotropic curvature pinching. This is a contradiction. We finish the proof of the theorem. # From now on, we always assume that the initial datum is a compact fourmanifold M 4 with no essential incompressible space form and with positive isotropic curvature. Let gij (x, t), x ∈ M 4 and t ∈ [0, T ), be a maximal solution to the Ricci flow with T < +∞. Without loss of generality, after a scaling on the initial metric, we may assume T > 1. It was shown in [19] that the solution gij (x, t) remains positive isotropic curvature. By Lemma 2.1, there hold (2.1), (2.2) and (2.3) for some positive constants 0 < ρ, Λ, P < +∞ (depending only on the initial datum). And by Perelman’s no local collapsed √ theorem I [27] the solution is κ-noncollapsed on the scale T for some κ > 0 (depending only on the initial datum). Then for any sufficiently small ε > 0, we can find r0 > 0 with the property described in Theorem 4.1. Let Ω denote the set of all points in M 4 , where curvature stays bounded as t → T . The estimates (4.1) imply that Ω is open and R(x, t) → +∞ as t → T for each x ∈ M 4 \Ω. If Ω is empty, then the solution becomes extinct at time T and the manifold is either diffeomorphic to S4 or RP4 , or entirely covered by evolving ε-necks or evolving ε-caps shortly before the maximal time T , so M 4 is diffeomorphic to S4 , or RP4 , or RP4 #RP4 or S3 × S1 , or e 1 . The reason is as follows. We only need to consider the situation that S3 ×S the manifold M 4 is entirely covered by evolving ε-necks and evolving ε-caps shortly before the maximal time T . If M 4 contains a cap C, then there is a cap or a neck adjacent to the neck like end of C. The former case implies that M 4 is diffeomorphic to S4 , RP4 , or RP4 #RP4 . In the latter case, we get a new longer cap and continue the procedure. Finally, we must end up with a cap, producing a S4 , RP4 , or RP4 #RP4 . If M 4 contains no caps, we start with a neck N, consider the other necks adjacent to the boundary of N, this gives a longer neck and we continue the procedure. After a finite number of steps, the neck must repeat itself. By considering the orientation of M 4 , we e 1. conclude that M 4 is diffeomorphic to S3 × S1 or S3 ×S 52

We can now assume that Ω is not empty. By using the local derivative estimates of Shi [32] (or see [18]), we see that as t → T , the solution gij (·, t) has a smooth limit g ij (·) on Ω. Let R(x) denote the scalar curvature of g ij . By the positive isotropic curvature assumption on the initial metric, we know that the metric g ij (·) also has positive isotropic curvature; in particular, R(x) is positive. For any σ < r0 , let us consider the set Ωσ = {x ∈ Ω | R(x) ≤ σ −2 }. Note that for any fixed x ∈ ∂Ω, as xj ∈ Ω and xj → x with respect to the initial metric gij (·, 0), we have R(xj ) → +∞. In fact, if there was a subsequence xjk so that the limit limk→∞ R(xjk ) exists and is finite, then it would follow from the gradient estimates (4.1) that R is uniformly bounded in some small neighborhood of x ∈ ∂Ω (with respect to the induced topology of the initial metric gij (·, 0)); this is a contradiction. From this observation and the compactness of the initial manifold, we see that Ωσ is compact (with respect to the metric g ij (·)). For the further discussion, we follow [28] to introduce the following terminologies. Denote by I a (finite or infinite) interval. Recall that an ε-neck (of radius r) is an open set with a Riemannian −1 metric, which is, after scaling the metric with factor r −2 , ε-close (in C [ε ] topology) to the standard neck S3 × I with the product metric, where S3 has constant scalar curvature one and I has length 2ε−1 . A metric on S3 × I, such that each point is contained in some ε-neck, is called an ε-tube, or an ε-horn, or a double ε-horn, if the scalar curvature stays bounded on both ends, or stays bounded on one end and tends to infinity on the other, or tends to infinity on both ends, respectively. A metric on B4 or RP4 \B4 , such that each point outside some compact subset is contained in an ε-neck, is called an ε-cap or a capped ε-horn, if the scalar curvature stays bounded or tends to infinity on the end, respectively. Now take any ε-neck in (Ω, g ij ) and consider a point x on one of its boundary components. If x ∈ Ω\Ωσ , then there is either an ε-cap or an εneck, adjacent to the initial ε-neck. In the latter case we can take a point on the boundary of the second ε-neck and continue. This procedure can either terminate when we get into Ωσ or an ε-cap, or go on infinitely, producing 53

an ε-horn. The same procedure can be repeated for the other boundary component of the initial ε-neck. Therefore, we conclude that each ε-neck of (Ω, g ij ) is contained in a subset of Ω of one of the following types: (a) an ε-tube with boundary components in Ωσ , or (b) an ε-cap with boundary in Ωσ , or (c) an ε-horn with boundary in Ωσ , or (d) a capped ε-horn, or (e) a double ε-horn.

(4.3)

Similarly, each ε-cap of (Ω, g ij ) is contained in a subset of Ω of either type (b) or type (d). It is clear that there is a definite lower bound (depending on σ) for the volume of subsets of types (a), (b), (c), so there can be only finite number of them. Thus we conclude that there is only a finite number of components of Ω, containing points of Ωσ , and every such component has a finite number of ends, each being an ε-horn. While by taking into account that Ω has no compact components, every component of Ω, containing no points of Ωσ , is either a capped ε-horn, or a double ε-horn. Nevertheless, if we look at the solution for a slightly earlier time t, each ε-neck or ε-cap of (M, gij (·, t)) is contained in a subset of types (a) and (b); while the ε-horns, capped ε-horns and double ε-horns, observed at the maximal time T , are connected together to form ε-tubes and ε-caps at the slightly earlier time t. Hence, by looking at the solution for times just before T , we see that the topology of M 4 can be reconstructed as follows: take the all components Ωj , 1 ≤ j ≤ k, of Ω which contains points of Ωσ , truncate their ε-horns, and glue a finite collection of tubes S3 ×I and caps B4 or RP4 \B4 to the boundary components of truncated Ωj . Thus M 4 is diffeomorphic to a connected sum e 1 (which correspond of Ωj , 1 ≤ j ≤ k, with a finite number of S3 × S1 or S3 ×S to glue a tube to two boundary components of the same Ωj ), and a finite number of RP4 . Here Ωj denotes Ωj with each ε-horn one point compactified. (One might wonder why we do not also cut other ε-tubes or ε-caps so that we can remove more volumes; we will explain it a bit later.) More geometrically, one can get Ωj in the following way: in every ε-horn of Ωj one can find an ε-neck, cut it along the middle three-sphere, remove 54

the horn-shaped end, and glue back a cap (i.e., a differentiable four-ball). Thus to understand the topology of M 4 , one only need to understand the topologies of the compact four-manifolds Ωj , 1 ≤ j ≤ k. Recall that the four-manifold M 4 has no essential incompressible space form, we now claim that each Ωj still has no essential incompressible space form. Clearly, we only need to check the assertion that if N is an essential incompressible space form in Ωj , then N will be also incompressible in M 4 . After moving N slightly, we can choose N such that N ⊂ Ωj . Then N can be regarded as a submanifold in M 4 (unaffected by the surgery). We now argue by contradiction. Suppose γ ⊂ N is a homotopically nontrivial curve which bounds a disk D in M 4 . We want to modify the map of disk D so that γ bound a new disk in Ωj , which will gives the desired contradiction. Let E1 , E2 , · · · , Em be all the ε-horn ends of Ωj , S1 , S2 , · · · , Sm ⊂ Ωj be the corresponding cross spheres lying inside the ε-horn ends Ej respectively. Let us perturb the spheres S1 , S2 , · · · , Sm slightly so that they meet D transversely in a finite number of simple closed curves (we only consider those Sj with Sj ∩ D 6= φ). After removing those curves which are contained in larger ones in D, we are left with a finite number of disjoint simple closed curves, denoted by C1 , C2 , · · · , Cl . We denote the enclosed disks of C1 , C2 , · · · , Cl in D by D1 , D2 , · · · , Dl . Since S3 is simply-connected, each intersection curves in S1 , S2 , · · · , Sm can be shrunk to a point. So by filling the holes D1 , D2 , · · · , Dl , we obtain a new continuous map from D to M 4 such that the image of D1 ∪D2 · · ·∪Dl is contained in S1 ∪S2 · · ·∪Sm ⊂ Ωj . On the other hand, since D\(D1 ∪D2 · · ·∪Dl ) is connected, γ(the image of ∂D) ⊂ N, we know that the image of D\(D1 ∪ D2 · · · ∪ Dl ) must be contained in Ωj . Therefore, γ bounds a new disk in Ωj . This proves that after the surgery, each Ωj still has no essential incompressible space form. As shown by Hamilton in Section D of [19], provided ε > 0 small enough, one can perform the above surgery procedure carefully so that the compact four-manifolds Ωj , 1 ≤ j ≤ k, also have positive isotropic curvature. Naturally, one can evolve each Ωj by the Ricci flow again and carry out the same surgery procedure to produce a finite collection of new compact four-manifolds with no essential incompressible space form and with positive isotropic curvature. By repeating this procedure indefinitely, it will be likely 55

to give us the long time existence of a kind of “weak” solution to Ricci flow.

5. Ricci Flow with Surgery for Four-manifolds We begin with an abstract definition of the solution to the Ricci flow with surgery which is adapted from [28]. Definition 5.1 Suppose we have a collection of compact four-dimensional (k) + smooth solutions gij (t) to the Ricci flow on Mk4 × [t− k , tk ) with no essential incompressible space form and with positive isotropic curvature, which go 4 singular as t → t+ k and where each manifold Mk may be disconnected with (k) only a finite number of connected components. Let (Ωk , g ij ) be the limits (k) of the corresponding solutions gij (t) as t → t+ k . Suppose also that for each (k−1) (k) − + k we have tk = tk−1 , and (Ωk−1 , g ij ) and (Mk4 , gij (t− k )) contain compact (possibly disconnected) four-dimensional submanifolds with smooth boundary which are isometric. Then by identifying these isometric submanifolds, we say it is a solution to the Ricci flow with surgery on the time interval + + which is the union of all [t− k , tk ), and say the times tk are surgery times. The procedure described in the last paragraph of the previous section gives us a solution to the Ricci flow with surgery. However, in order to understand the topology of the initial manifold from the solution to the Ricci flow with surgery, one encounters the following two difficulties: (i) how to prevent the surgery times from accumulation? (ii) how to get the long time behavior of the solution to the Ricci flow with surgery? In view of this, it is natural to consider those solutions having ”good” properties. Let ε be a fixed small positive number. We will only consider those solutions to the Ricci flow with surgery which satisfy the following a priori assumptions (with accuracy ε): Pinching assumption: There exist positive constants ρ, Λ, P < +∞ such that there hold a1 + ρ > 0 and c1 + ρ > 0, (5.1) max{a3 , b3 , c3 } ≤ Λ(a1 + ρ) and max{a3 , b3 , c3 } ≤ Λ(c1 + ρ), 56

(5.2)

and p

everywhere.

ΛeP t b3 p ≤ 1+ , (a1 + ρ)(c1 + ρ) max{log (a1 + ρ)(c1 + ρ), 2}

(5.3)

Canonical neighborhood assumption (with accuracy ε): For the given ε > 0, there exist two constants C1 (ε), C2 (ε) and a non-increasing positive function r on [0, +∞) such that for every point (x, t) where the scalar curvature R(x, t) is at least r −2 (t), there is an open neighborhood B, 1 Bt (x, σ) ⊂ B ⊂ Bt (x, 2σ) with 0 < σ < C1 (ε)R(x, t)− 2 , which falls into one of the following three categories: (a) B is a strong ε-neck (in the sense that B is an ε-neck and it is the slice at time t of the parabolic neighborhood {(x′ , t′ ) | x′ ∈ B, t′ ∈ [t − R(x, t)−1 , t]}, where the solution is well defined on the whole parabolic neighborhood and is, after scaling with factor R(x, t) and shifting the time to −1 zero, ε-close (in C [ε ] topology) to the corresponding subset of the evolving standard round cylinder S3 × R with scalar curvature 1 at the time zero), or (b) B is an ε-cap, or (c) B is a compact four-manifold with positive curvature operator; furthermore, the scalar curvature in B at time t is between C2−1 R(x, t) and C2 R(x, t), and satisfies the gradient estimate 3

|∇R| < ηR 2 and |

∂R | < ηR2 , ∂t

(5.4)

and the volume of B in case (a) and case (b) satisfies (C2 R(x, t))−2 ≤ V olt (B). Here C1 and C2 are some positive constants depending only on ε, and η is a universal positive constant. Clearly, we may always assume the above C1 and C2 are twice bigger than the corresponding constants C1 ( 2ε ) and C2 ( 2ε ) in Theorem 3.8 with the accuracy 2ε . The main purpose of this section is to construct a long-time solution to the Ricci flow with surgery which starts with an arbitrarily given compact four-manifold with no essential incompressible space form and with positive 57

isotropic curvature, so that the a priori assumptions are satisfied and there are only a finite number of surgery times at each finite time interval. The construction will be given by an induction argument. Firstly, for an arbitrarily given compact four-manifold (M 4 , gij (x)) with no essential incompressible space form and with positive isotropic curvature, the Ricci flow with it as initial data has a maximal solution gij (x, t) on [0, T0 ) with T0 < +∞. Without loss of generality, after a scaling on the initial metric, we may assume T0 > 1. It follows from Lemma 2.1 and Theorem 4.1 that the a priori assumptions above hold for the smooth solution on [0, T0 ). Suppose that we have a solution to the Ricci flow with surgery, with the given compact four-manifold (M 4 , gij (x)) as initial datum, which is defined on [0, T ) with T < +∞, going singular at the time T , satisfies the a priori assumptions and has only a finite number of surgery times on [0, T ). Let Ω denote the set of all points where the curvature stays bounded as t → T . As shown before, the gradient estimate (5.4) in the canonical neighborhood assumption implies that Ω is open and that R(x, t) → +∞ as t → T for x lying outside Ω. Moreover, as t → T , the solution gij (x, t) has a smooth limit g ij (x) on Ω. For δ > 0 to be chosen much smaller than ε, we let σ = δr(T ) where r(t) is the positive nonincreasing function in the definition of the canonical neighborhood assumption. We consider the corresponding compact set Ωσ = {x ∈ Ω | R(x) ≤ σ −2 } where R(x) is the scalar curvature of g ij . If Ωσ is empty, the manifold (near the maximal time T ) is entirely covered by ε-tubes, ε-caps and compact components with positive curvature operator. Clearly, the number of compact components is finite. Then in this case the manifold (near the maximal time T ) is diffeomorphic to the union of a finite number of S4 , or RP4 , or S3 × S1 , e 1 , or a connected sum of them. Thus when Ωσ is empty, the proceor S3 ×S dure stops here, and we say that the solution becomes extinct. We now assume Ωσ is not empty. Every point x ∈ Ω\Ωσ lies in one of subsets of listing in (4.3), or in a compact component with positive curvature operator or in a compact component which is contained in Ω\Ωσ and is diffeomorphic e 1 . Note again that the number of compact to S4 , or RP4 , or S3 × S1 , or S3 ×S 58

components is finite. Let us throw away all the compact components lying Ω\Ωσ or with positive curvature operator, and then consider the all components Ωj , 1 ≤ j ≤ k, of Ω which contains points of Ωσ . (We will consider the components of Ω\Ωσ consisting of capped ε-horns and double ε-horns later). We could perform Hamilton’s surgerical procedure in Section D of [19] at every horn of Ωj , 1 ≤ j ≤ k, so that the positive isotropic curvature condition and the pinching assumption is preserved. Note that if we perform the surgeries at the necks with certain fixed accuracy ε on the high curvature region at each surgery time, then it is possible that the errors of surgeries may accumulate to a certain amount so that for some later time we can not recognize the structure of very high curvature region. This prevents us to carry out the process in finite time with finite steps. Hence in order to maintain the a priori assumptions with the same accuracy after surgery, we need to find sufficient “fine” necks in the ε-horns and to glue sufficient “fine” caps in the procedure of surgery. Note that δ > 0 will be chosen much smaller than ε > 0. The following lemma gives us the “fine” necks in the ε-horns. (The corresponding result in three-dimension is Lemma 4.3 in [28]). Now we explain that why we only perform the surgeries in the horns with boundary in Ωσ . At the first sight, we should also cut off all those ε-tubes and ε-caps in the surgery procedure. But in general, we are not able to find a “finer” neck in an ε-tube or in ε-cap, and such surgeries at “rough” ε-necks will certainly loss some accuracy. This is the reason why we will only perform the surgeries in the ε-horns with boundary in Ωσ . 1 , 0 < δ < ε and 0 < T < +∞, there Lemma 5.2 Given 0 < ε < 100 exists a radius 0 < h < δσ, depending only on δ, r(T ) and the pinching assumption, such that if we have a solution to the Ricci flow with surgery, with a compact four-manifold (M 4 , gij (x)) with no essential incompressible space form and with positive isotropic curvature as initial data, defined on [0, T ), going singular at the time T , satisfies the a priori assumptions and has only a finite number of surgery times on [0, T ), then for each point x −1

with h(x) = R 2 (x) ≤ h in an ε-horn of (Ω, g ij ) with boundary in Ωσ , the neighborhood BT (x, δ −1 h(x)) = {y ∈ Ω|distgij (y, x) ≤ δ −1 h(x)} is a strong 59

δ-neck (i.e., {(y, t) | y ∈ BT (x, δ −1 h(x)), t ∈ [T − h2 (x), T ]} is, after scaling −1 with factor h−2 (x), δ-close (in C [δ ] topology) to the corresponding subset of the evolving standard round cylinder S3 × R over the time interval [−1, 0] with scalar curvature 1 at the time zero). Proof. We argue as in [28] by contradiction. Suppose that there exists (k) a sequence of solution gij (·, t), k = 1, 2, · · · , to the Ricci flow with surgery, (k) satisfying the a priori assumptions, defined on [0, T ) with limits (Ωk , g ij ), k = 1, 2, · · · , as t → T , and exist points xk , lying inside an ε-horn of Ωk , which contains the points of Ωkσ , and having h(xk ) → 0 as k → +∞ such that the neighborhood BT (xk , δ −1 h(xk )) are not strong δ-necks. (k) Let e gij (·, t) be the rescaled solutions by the factor R(xk ) = h−2 (xk ) (k) around (xk , T ). We will show that a sequence of e gij (·, t) converges to the evolving round R × S3 , which gives the desired contradiction. (k) Note that e gij (·, t), k = 1, 2, · · · , are modified by surgery. We can not apply Hamilton’s compactness theorem directly since it states only for smooth (k) solutions. For each (unrescaled) surgical solution geij (·, t), we pick a point ¯ k ) = 2C 2 (ε)σ −2 , in the ε-horn of (Ωk , g¯(k) ) with boundary in zk , with R(z 2 ij Ωkσ , where C2 (ε) is the positive constant in the canonical neighborhood assumption. From the definition of ε-horn and the canonical neighborhood (k) assumption, we know that each point x lying inside the ε-horn of (Ωk , g¯ij ) with dg¯(k) (x, Ωkσ ) ≥ dg¯(k) (zk , Ωkσ ) has a strong ε-neck as its canonical neighij

ij

borhood. Since h(xk ) → 0, each xk lies deeply inside the ε-horn. Thus for (k) each positive A < +∞, the rescaled (surgical) solutions e gij (·, t) with the marked origins xk over the geodesic balls Beg(k) (·,0) (xk , A), centered at xk of ij

(k) gij (·, 0)), e

radii A (with respect to the metrics will be smooth on some uniform (size) small time intervals for all sufficiently large k, if the curvatures of (k) the rescaled solutions e gij at t = 0 in Beg(k) (·,0) (xk , A) are uniformly bounded. ij In such situation, the Hamilton’s compactness theorem is applicable. Then we can now apply the same argument in Step 2 of the proof Theorem 4.1 to (k) conclude that the curvatures of the rescaled solutions e gij (·, t) at the time T stay uniformly bounded at bounded distances from xk ; otherwise we get a piece of a non-flat nonnegative curved metric cone as a blow-up limit, which would contradict with Hamilton strong maximum principle [14]. Hence as

60

(∞)

∞ before we can get a Cloc limit geij (·, t), defined on a space-time set which is relatively open in the half space-time {t ≤ T } and contains the time slice (k) {t = T }, from the rescaled solutions e gij (·, t). By the pinching assumption, the limit is a complete manifold with the restricted pinching condition (2.4) and with nonnegative curvature operator. Since xk was contained in an ε-horn with boundary in Ωkσ , and h(xk )/σ → 0, the limiting manifold has two ends. Thus by Toponogov splitting theorem, it admits a (maybe not round at this moment) metric splitting R × S3 because xk was the center of a strong ε-neck. We further apply the restricted isotropic curvature pinching condition (2.4) and contracted second Bianchi identity as before to conclude that the factor S3 must be round at time 0. By combining with the canonical neighborhood assumption, we see that the limit is defined on the time interval [−1, 0]. By Toponogov splitting theorem, the splitting R × S3 is at each time t ∈ [−1, 0]; so the limiting solution is just the standard evolving round cylinder. This is a contradiction. We finish the proof of Lemma 5.2. #

The property in the above lemma that the radius h depends only on δ, the time T and the pinching assumption, independent of the surgical solution, is crucial; otherwise we will not be able to cut off enough volume at each surgery to guarantee the number of surgeries being finite in each finite time interval. Remark. The proof of Lemma 5.2 actually proves a more stronger result: for any δ > 0, there exists a radius 0 < h < δσ, depending only on δ, r(T ) and −1

the pinching assumption, such that for each point x with h(x) = R 2 (x) ≤ h in an ε-horn of (Ω, g ij ) with boundary in Ωσ , {(y, t) | y ∈ BT (x, δ −1 h(x)), t ∈ −1 [T −δ −2 h2 (x), T ]} is, after scaling with factor h−2 (x), δ-close (in C [δ ] topology) to the corresponding subset of the evolving standard round cylinder S3 ×R over the time interval [−δ −2 , 0] with scalar curvature 1 at the time zero. This fact will be used in the proof of Proposition 5.4. The reason is as follows. Let us use the notation in the proof the Lemma 5.2 and argue by contradiction. Note that the scalar curvature of the limit at

61

time t = −1 is 1− 21(−1) . Since h(xk )/ρ → 0, each point in the limiting mani3 fold at time t = −1 has also a strong ε-neck as its canonical neighborhood. Thus the limit is defined at least on the time interval [−2, 0]. Inductively, suppose the limit is defined on the time interval [−m, 0] with bounded curvature for some positive integer −m, then by the isotropic pinching condition, Toponogov splitting theorem and evolution equation of the scalar curvature on the round R × S3 , we see that R = 1+12 m at time −m. Since h(xk )/ρ → 0, 3 each point in the limiting manifold at time t = −m has also a strong ε-neck as its canonical neighborhood, we see that the limit is defined at least on the time interval [−(m + 1), 0] with bounded curvature. So by induction we prove that the limit exists on the ancient time interval (−∞, 0]. Therefore the limit is the evolving round cylinder S3 ×R over the time interval (−∞, 0], which gives the desired contradiction. To specialize our surgery, we now fix a standard capped infinite cylinder for n = 4 as follows. Consider the semi-infinite standard round cylinder N0 = S3 × (−∞, 4) with the metric g0 of scalar curvature 1. Denote by z the coordinate of the second factor (−∞, 4). Let f be a smooth nondecreasing convex function on (−∞, 4) defined by   f (z) = 0, z ≤ 0,      D   f (z) = ce− z , z ∈ (0, 3],   f (z) is strictly convex on z ∈ [3, 3.9],       f (z) = − 12 log(16 − z 2 ), z ∈ [3.9, 4),

where the small (positive) constant c and big (positive) constant D will be determined later. Let us replace the standard metric g0 on the portion S3 × [0, 4) of the semi-infinite cylinder by gˆ = e−2f g0 . Then the resulting metric gˆ will be smooth on R4 obtained by adding a point to S3 × (−∞, 4) at z = 4. We denote by C(c, D) = (R4 , gˆ). Clearly, C(c, D) is a standard capped infinite cylinder. We next use a compact portion of the standard capped infinite cylinder C(c, D) and the δ-neck obtained in Lemma 5.2 to perform the following surgery due to Hamilton [19]. 62

Consider the solution metric g¯ at the maximal time T < +∞. Take an ε-horn with boundary in Ωρ . By Lemma 5.2, there exists a δ-neck N of radius 0 < h < δρ in the ε-horn. By definition, (N, h−2 g¯) is δ-close (in −1 C [δ ] topology) to the standard round neck S3 × I of scalar curvature 1 with I = (−δ −1 , δ −1 ). The parameter z ∈ I induces a function on the δ-neck N. Let us cut the δ-neck N along the middle (topological) three-sphere T N {z = 0}. Without loss of generality, we may assume that the right T hand half portion N {z ≥ 0} is contained in the horn-shaped end. Let ϕ be a smooth bump function with ϕ = 1 for z ≤ 2, and ϕ = 0 for z ≥ 3. Construct a new metric g˜ on a (topological) four-ball B4 as follows  g¯, z = 0,          e−2f g¯, z ∈ [0, 2], g˜ =   ϕe−2f g¯ + (1 − ϕ)e−2f h2 g0 , z ∈ [2, 3],        2 −2f h e g0 , z ∈ [3, 4]. The surgery is to replace the horn-shaped end by the cap (B4 , g˜). The following lemma, due to Hamilton [19], determines the constants c and D in the δ-cutoff surgery so that the pinching assumption is preserved under the surgery.

Lemma 5.3 ( Hamilton [19] D3.1) (Justification of the pinching assumption) There are universal positive constants δ0 , c0 and D0 such that for any T˜ there is a constant h0 > 0 depending on the initial metric and T˜ such that if we take a δ-cutoff surgery at a δ-neck of radius h at time T ≤ T˜ with δ < δ0 and h−2 ≥ h−2 0 , then we can choose c = c0 and D = D0 in the definition of f (z) such that after the surgery, there still holds the pinching condition (2.1) (2.2) (2.3): a1 + ρ > 0 and c1 + ρ > 0, max{a3 , b3 , c3 } ≤ Λ(a1 + ρ) and max{a3 , b3 , c3 } ≤ Λ(c1 + ρ), and p

ΛeP t b3 p ≤1+ (a1 + ρ)(c1 + ρ) max{log (a1 + ρ)(c1 + ρ), 2} 63

at all points at time T . Moreover, after the surgery, any metric ball of radius 1 δ − 2 h with center near the tip (i.e. the origin of the attached cap) is, after 1 scaling with factor h−2 , δ 2 -close the corresponding ball of the standard capped infinite cylinder C(c0 , D0 ). # We call the above procedure as a δ-cutoff surgery. Since there are only finite number of horns with their other ends connected to Ωσ , we only need to perform a finite number of such δ-cutoff surgeries at the time T . Besides those horns, there could be capped horns, double horns and compact components lying Ω\Ωσ or with positive curvature operator. As explained before, capped horns and double horns are connected with horns to form tubes or capped tubes at any time slightly before T . Thus when we truncated the horns at the δ-cutoff surgeries, we actually had removed these together with the hornshaped ends away. So we can regard the capped horns and double horns (of Ω \ Ωσ ) to be extinct and throw them away at the time T . Remember that we have thrown away all the compact components lying in Ω \ Ωσ or with positive curvature operator. Each of such compact components is diffeomorphic to S4 , e 1 , and the number of compact components is finite. or RP4 , or S3 ×S1 , or S3 ×S e 1 at the Thus we actually throw a finite number of S4 , RP4 , S3 × S1 or S3 ×S time T also. (Note that we allow that the manifold may be disconnected before and after the surgeries). Let us agree to declare extinct every compact component with positive curvature operator or lying in Ω \ Ωσ ; in particular, that allows to exclude the components with positive curvature operator from the list of canonical neighborhoods. Summarily, our surgery at the time T consists of the following four procedures: (1) perform δ-cutoff surgeries for all ε-horns which have the other ends connected to Ωσ , (2) declare extinct every compact component which has positive curvature operator, (3) throw away all capped horns and double horns lying in Ω \ Ωσ , (4) declare extinct every compact components lying in Ω \ Ωσ . 64

After the surgery at the time T , the pinching assumption still holds for the surgically modified manifolds. With this (maybe disconnected) surgically modified manifold as initial data, we now continue our solution until it becomes singular for the next time T ′ (> T ). Therefore we have extended the solution to the Ricci flow with surgery, originally defined on [0, T ), to the new time interval [0, T ′) (with T ′ > T ). Moreover, as long as 0 < δ ≤ δ0 , the solution with δ-cutoff surgeries on the new time interval [0, T ′) still has positive isotropic curvature and no essential incompressible space form, and from [19] and Lemma 5.3 it still satisfies the pinching assumption. Denote the minimum of the scalar curvature at time t by Rmin (t) > 0. Since the δ-cutoff surgeries occur at the points lying deeply in the ε-horns, the minimum of the scalar curvature Rmin (t) of the solution at each time-slice is achieved in the region unaffected by the surgeries. Thus we know from the evolution equation of the scalar curvature that d 1 2 Rmin (t) ≥ Rmin (t). dt 2 By integrating this inequality, we conclude that the maximal time T of any solution to the Ricci flow with δ-cutoff surgeries must be bounded by 2/Rmin (0) < +∞. Let T˜ = 2/Rmin (0) in Lemma 5.3, then there is a constant 1 h0 determined by T˜ . Set δ¯ = 12 Rmin (0) 2 h0 . We know that if we perform ¯ δ0 }, then the pinching assumptions the δ-cutoff surgery with δ < min{δ, (5.1),(5.2),(5.3) are satisfied for the solution to the Ricci flow with δ-cutoff surgery. Next we make further restrictions on δ to justify the canonical neighborhood assumption. Clearly, we only need to check the following assertion. Proposition 5.4 (Justification of the canonical neighborhood assumption) Given a compact four-manifold with positive isotropic curvature and no essential incompressible space form and given ε > 0, there exist decreasing ¯ > δej > 0, j = 1, 2, · · · , with the sequences ε > rej > 0, κj > 0, min{ε2 , δ0 , δ} e on [0, +∞) by δ(t) e = δej following property. Define a positive function δ(t) when t ∈ [(j − 1)ε2, jε2 ). Suppose we have a solution to the Ricci flow with surgery, with the given four-manifold as initial datum defined on the time interval [0, T ) and with a finite number of δ-cutoff surgeries such that any e δ-cutoff surgery at a time t ∈ (0, T ) with δ = δ(t) satisfies 0 < δ(t) ≤ δ(t). 65

T Then on each the time interval [(j − 1)ε2 , jε2 ] [0, T ), the solution satisfies the κj -noncollapsing condition on all scales less than ε and the canonical neighborhood assumption (with accuracy ε) with r = rej . Here and in the followings, we call a (four-dimensional) solution gij (t), 0 ≤ t < T , to the Ricci flow with surgery is κ-noncollapsed at a point (x0 , t0 ) on the scales less than ρ (for some κ > 0, ρ > 0) if it satisfies the following property: whenever r < ρ and |Rm(x, t)| ≤ r −2 for all those (x, t) ∈ P (x0 , t0 , r, −r 2 ) = {(x′ , t′ ) | x′ ∈ Bt′ (x0 , r), t′ ∈ [t0 − r 2 , t0 ]}, for which the solution is defined, we have V olt0 (Bt0 (x0 , r)) ≥ κr4 . Before we give the proof of the proposition, we need to check κ-noncollapsing condition. Lemma 5.5 For a given compact four-manifold with positive isotropic curvature and no essential incompressible space form and given ε > 0, suppose we have constructed the sequences, satisfying the above proposition for 1 ≤ j ≤ ℓ. Then there exists κ > 0, such that for any r, 0 < r < ε, one can ¯ which depends on r, ε and may also depend find δe with 0 < δe < min{ε2 , δ0 , δ}, on the already constructed sequences, with the following property. Suppose we have a solution, with the given four-manifold as initial data, to the Ricci flow with surgery defined on a time interval [0, T ] with ℓε2 ≤ T < (ℓ + 1)ε2 such that the assumptions and conclusions of Proposition 5.4 hold on [0, ℓε2), the canonical neighborhood assumption (with accuracy ε) with r holds on [ℓε2 , T ], and each δ(t)-cutoff surgery in the time interval t ∈ [(ℓ − 1)ε2 , T ] ˜ Then the solution is κ-noncollapsed on [0, T ] for all scales has 0 < δ(t) < δ. less than ε. Proof. Consider a parabolic neighborhood P (x0 , t0 , r0 , −r02 ) = {(x, t)|x ∈ Bt (x0 , r0 ), t ∈ [t0 − r02 , t0 ]}, with ℓε2 ≤ t0 ≤ T , and 0 < r0 ≤ ε, where the solution satisfies |Rm| ≤ r0−2 whenever it is defined. We will prove that V olt0 (Bt0 (x0 , r0 )) ≥ κr04 . 66

Let η be the universal positive constant in the definition of the canonical neighborhood assumption. Without loss of generality, we always assume 1 r. η ≥ 10. Firstly, we want to show that one may assume r0 ≥ 2η Obviously, the curvature satisfies the estimate |Rm(x, t)| ≤ 20r0−2, 1 1 2 1 for those (x, t) ∈ P (x0 , t0 , 2η r0 , − 8η r0 ) = {(x, t) | x ∈ Bt (x0 , 2η r0 ), t ∈ [t0 − 1 2 1 r , t ]}, for which the solution is defined. When r0 < 2η r, we can enlarge 8η 0 0 r0 to some r0′ ∈ [r0 , r] so that

|Rm| ≤ 20r0′−2 1 ′2 1 ′ r0 , − 8η r0 ) (whenever it is defined), and either the equality on P (x0 , t0 , 2η holds somewhere or r0′ = r. In the case that the equality holds somewhere, it follows from the pinching assumption that we have R > 10r0′−2 1 ′2 1 ′ r0 , − 8η r0 ). Here, without loss of generality, we have somewhere in P (x0 , t0 , 2η assumed r is suitably small. Then by the gradient estimates in the definition of the canonical neighborhood assumption, we know

R(x0 , t0 ) > r0′−2 ≥ r −2 . Hence the desired noncollapsing estimate in this case follows directly from the canonical neighborhood assumption. (Recall that we have excluded every component which has positive sectional curvature in the surgery procedure and then we have excluded them from the list of canonical neighborhoods. Here we also used the standard volume comparison when the canonical neighborhood is an ε-cap). While in the case that r0′ = r, we have the curvature bound |Rm(x, t)| ≤ (

1 −2 r) , 2η

1 1 1 for those (x, t) ∈ P (x0 , t0 , 2η r, −( 2η r)2 ) = {(x, t) | x ∈ Bt (x0 , 2η r), t ∈ [t0 − 1 2 ( 2η r) , t0 ]}, for which the solution is defined. It follows from the standard

67

volume comparison that we only need to verify the noncollapsing estimate 1 1 r. Thus we have reduced the proof to the case r0 ≥ 2η r. for r0 = 2η The reduced distance from (x0 , t0 ) is Z τ √ 1 2 l(q, τ ) = √ inf{ s(R(γ(s), t0 −s)+|γ(s)| ˙ gij (t0 −s) )ds | γ(0) = x0 , γ(τ ) = q} 2 τ 0 where τ = t0 − t with t < t0 . Firstly, we need to check that the minimum of the reduced distance is achieved by curves unaffected by surgery. According to Perelman [28], we call a space-time curve in the solution track is admissible if it stays in the space-time region unaffected by surgery, and we call a space-time curve in the solution track is a barely admissible curve if it is on the boundary of the set of admissible curves. The following assertion gives a big lower bound for the reduced lengths of barely admissible curves. e r, reℓ , ε) > 0 with Claim 1. For any L < +∞ one can find δe = δ(L, the following property. Suppose that we have a curve γ, parametrized by t ∈ [T0 , t0 ], (ℓ − 1)ε2 ≤ T0 < t0 , such that γ(t0 ) = x0 , T0 is a surgery time and γ(T0 ) lies in a 4h-collar of the middle three-sphere of a δ-neck with the radius h obtained in Lemma 5.2, where the δ-cutoff surgery was taken. Suppose also ˜ each δ(t)-cutoff surgery in the time interval t ∈ [(ℓ−1)ε2 , T ] has 0 < δ(t) < δ. Then we have an estimate Z t0 −T0 √ τ (R(γ(t0 − τ ), t0 − τ ) + |γ(t ˙ 0 − τ )|2gij (t0 −τ ) )dτ ≥ L, (5.5) 0

where τ = t0 − t ∈ [0, t0 − T0 ]. Before we can verify this assertion, we need to do some premilary works. Let O be the point near γ(T0 ) which corresponds to the center of the (rotationally symmetric) capped infinite round cylinder. Recall from Lemma 1 5.3 that a metric ball of radius δ − 2 h at time T0 centered at O is, after scaling −1 1 with factor h−2 , δ 2 -close (in C [δ 2 ] topology) to the corresponding ball in the capped infinite round cylinder. We need to consider the solutions to the Ricci flow with the capped infinite round cylinder (with scalar curvature 1 outsider some compact set) as initial data and we require the solutions have also bounded curvature; we call such a solution a standard solution as in [28]. From Shi [32], we know such a solution exists. The uniqueness of 68

the Ricci flow for compact manifolds is well-known (see for example, Section 6 of [18]). In [10], we prove a uniqueness theorem which states that if the initial data is a complete noncompact Riemannian manifold with bounded curvature, then the solution to the Ricci flow in the class of complete solutions with bounded curvature is unique. Thus the standard solution with a capped infinite round cylinder as initial data is unique. In the appendix, we will show that the standard solution exists on the time interval [0, 32 ) and has nonnegative curvature operator, and its scalar curvature satisfies R(x, t) ≥

C −1 , 3 −t 2

(5.6)

everywhere for some positive constant C. For any 0 < θ < 23 , let Q be the maximum of the scalar curvature of the standard solution in the time interval [0, θ] and let △t = (T1 − T0 )/N < εη −1 Q−1 h2 with T1 = min{t0 , T0 + θh2 } and η given in the canonical neighborhood assumption. Set tk = T0 + k△t, k = 1, · · · , N. 1 Note that the ball BT0 (O, A0h) at time T0 with A0 = δ − 2 is, after scal1 ing with factor h−2 , δ 2 -close to the corresponding ball in the capped infinite round cylinder. Assume first that for each point in BT0 (O, A0h), the solution is defined on [T0 , t1 ]. By the gradient estimate (5.4) in the canonical neighborhood assumption and the choice of △t we have a uniform curvature bound on this set for h−2 -scaled metric. Then by the uniqueness theorem in [10], 1 1 if δ 2 → 0 (i.e., A0 = δ − 2 → +∞), the solution with h−2 -scaled metric will ∞ converge to the standard solution in Cloc topology. Therefore we can find A1 , depending only on A0 and tending to infinity with A0 , such that the solution in the parabolic region P (O, T0, A1 h, t1 − T0 ) = {(x, t)|x ∈ Bt (O, A1h), t ∈ [T0 , T0 + (t1 − T0 )]} is, after scaling with factor h−2 and shifting time T0 to zero, A−1 1 -close to the corresponding subset in the standard solution. In particular, the scalar curvature on this subset does not exceed 2Qh−2 . Now if each point in BT0 (O, A1 h) the solution is defined on [T0 , t2 ], then we can repeat the procedure, defining A2 , such that the solution in the parabolic region P (O, T0, A2 h, t2 − T0 ) = {(x, t)|x ∈ Bt (p, A2 h), t ∈ [T0 , T0 + (t2 − T0 )]} is, after scaling with factor h−2 and shifting time T0 to zero, A−1 2 -close to the corresponding subset in the standard solution. Again, the scalar curvature on this subset still does not exceed 2Qh−2 . Continuing this way, we 69

eventually define AN . Note that N is depending only on θ. Thus for arbie θ, ε) > 0 such trarily given A > 0 (to be determined), we can choose δ(A, e θ, ε), and assuming that for each point in BT (O, A(N −1) h) that as δ < δ(A, 0 the solution is defined on [T0 , T1 ], we have A0 > A1 > · · · > AN > A, and the solution in P (O, T0, Ah, T1 − T0 ) = {(x, t)|x ∈ Bt (O, Ah), t ∈ [T0 , T1 ]} is, after scaling with factor h−2 and shifting time T0 to zero, A−1 -close to the corresponding subset in the standard solution. Now assume that for some k (1 ≤ k ≤ N − 1) and a surgery time t+ ∈ (tk , tk+1 ](or t+ ∈ (T0 , t1 ]) such that on BT0 (O, Ak h) the solution is defined on [T0 , t+ ), but for some point of this ball it is not defined past t+ . Clearly the above argument also shows that the parabolic region P (O, T0, Ak+1h, t+ − T0 ) = {(x, t)|x ∈ Bt (x, Ak+1 h), t ∈ [T0 , t+ )} is, after scaling with factor h−2 and shifting time T0 to zero, A−1 k+1 -close to the corresponding subset in the standard solution. In particular, as the time tends to t+ , the ball BT0 (O, Ak+1h) keeps on looking like a cap. Since the scalar curvature on the set BT0 (O, Ak h) × [T0 , tk ] does not exceed 2Qh−2 , it follows from the pinching assumption, the gradient estimates in the canonical neighborhood assumption and the evolution equation of the metric that the diameter of the 1 set BT0 (O, Ak h) at any time t ∈ [T0 , t+ ) is bounded from above by 4δ − 2 h. These imply that no point of the ball BT0 (O, Ak h) at any time near t+ can e θ, ε) with δ(A, e θ, ε) > 0 small be the center of a δ-neck for any 0 < δ < δ(A, 1 enough, since 4δ − 2 h L. T0

Indeed, we know from the estimate (5.6) that on the standard solution, Z θ Z θ 3 Rdt ≥ const. ( − t)−1 dt 0 0 2 2θ = −const. log(1 − ). 3 e sufficiently close to 3 , we have the desired estimate on By choosing θ = θ(L) 2 the standard solution. If Tγ = T1 < t0 and the solution on BT0 (O, Ah) exists up to the time interval [T0 , T1 ], the solution in the parabolic region P (O, T0, Ah, T1 − T0 ) = {(x, t)|x ∈ Bt (O, Ah), t ∈ [T0 , T1 ]} is, after scaling with factor h−2 and shifting time T0 to zero, A−1 -close to the corresponding subset in the standard solution. Then we have Z Tγ Z θ 3 2 (R(γ(t), t) + |γ(t)| ˙ ( − t)−1 dt gij (t) )dt ≥ const. T0 0 2 = −const. log(1 −

2θ ), 3

which gives the desired estimate in this case. While if Tγ < T1 and γ(Tγ ) ∈ ∂BT0 (O, Ah), we see that the solution on BT0 (O, A0 h) exists up to the time interval [T0 , Tγ ] and is, after scaling, A−1 e be chosen close to corresponding set in the standard solution. Let θ = θ(L) e to be the maximum of the scalar curvature of as above and set Q = Q(L) the standard solution in the time interval [0, θ]. On the standard solution, e so large that for each t ∈ [0, θ], we can choose A = A(L) dt (O, ∂B0 (O, A)) ≥ d0 (O, ∂B0 (O, A)) − 4(Q + 1)t ≥ A − 4(Q + 1)θ 3 A, ≥ 5

and dt (O, ∂B0 (O, 71

A A )) ≤ , 2 2

where we used Lemma 8.3 of [27] in the first inequality. Now our solution in the subset BT0 (O, Ah) up to the time interval [T0 , Tγ ] is (after scaling) A−1 -close to the corresponding subset in the standard solution. This implies Z Tγ Z Tγ 1 1 1 2 2 2 Ah ≤ |γ(t)| ˙ |γ(t)| ˙ gij (t) ≤ ( gij (t) dt) (Tγ − T0 ) 5 T0 T0 and then

Z

Tγ T0

2 (R(γ(t), t) + |γ(t)| ˙ gij (t) )dt ≥

A2 e > L, 25θ

e large enough. This proves the Claim 2. by choosing A = A(L) 1 We now use the above Claim 2 to verify Claim 1. Since r0 ≥ 2η r and −2 |Rm | ≤ r0 on P (x0 , t0 , r0 , −r02 ) = {(x, t)|x ∈ Bt (x0 , r0 ), t ∈ [t0 − r02 , t0 ]} (whenever it is defined), we can require δe > 0, depending on r and reℓ , so that γ(T0 ) does not lie in the region P (x0 , t0 , r0 , −r02 ). Let △t be maximal such that γ|[t0 −△t,t0 ] ⊂ P (x0 , t0 , r0 , −△t) (i.e., t = t0 − △t is the first time for γ escaping the parabolic region P (x0 , t0 , r0 , −r02 )). Obviously we may assume that Z △t √ τ (R(γ(t0 − τ ), t0 − τ ) + |γ(t ˙ 0 − τ )|2gij (t0 −τ ) )dτ < L. 0

If △t < it follows from the curvature bound |Rm| ≤ r0−2 on P (x0 , t0 , r0 , −r02 ) and the Ricci flow equation that Z △t |γ(t ˙ 0 − τ )|dτ ≥ cr0 r02 ,

0

for some universal positive constant c. On the other hand, by CauchySchwartz inequality, we have Z △t Z △t Z △t √ 1 1 1 2 2 √ dτ ) 2 τ (R + |γ| ˙ )dτ ) · ( |γ(t ˙ 0 − τ )|dτ ≤ ( τ 0 0 0 1

1

≤ 2L 2 (△t) 4

which yields 1

(△t) 2 ≥

c2 r02 . 4L

Thus we always have 1

(△t) 2 ≥ min{r0 , 72

c2 r02 }. 4L

Then Z

0

t0 −T0

√

2

τ (R + |γ| ˙ )dτ ≥ (△t)

1 2

Z

≥ min{r0 ,

t0 −T0

△t c2 r02

4L

(R + |γ| ˙ 2 )dτ

}

Z

t0 −T0

△t

(R + |γ| ˙ 2 )dτ.

e r, reℓ ) > By applying Claim 2, we can require the above δe further to find δe = δ(L, e there holds 0 so small that as 0 < δ < δ, Z t0 −T0 c2 r02 −1 (R + |γ| ˙ 2 )dτ ≥ L(min{r0 , }) . 4L △t Hence we have verified the desired assertion (5.5). Now choose L = 100 in (5.5), then it follows from Claim 1 that there exists δe > 0, depending on r and reℓ , such that as each δ-cutoff surgery at e every barely admissible curve γ the time interval t ∈ [(ℓ − 1)ε2, T ] has δ < δ, with endpoints (x0 , t0 ) and (x, t), where t ∈ [(ℓ − 1)ε2 , t0 ), has Z t0 −t √ L(γ) = τ (R(γ(τ ), t0 − τ ) + |γ(τ ˙ )|2gij (t0 −τ ) )dτ ≥ 100, 0

which implies the reduced distance from (x0 , t0 ) to (x, t) satisfies l ≥ 25ε−1 .

(5.8)

We also observe that the absolute value of l(x0 , τ ) is very small as τ closes to zero. We can then apply a maximum principle argument as in Section 7.1 of [27] to conclude lmin (τ ) = min{l(x, τ )| x lies on the solution manifold at time t0 − τ } ≤ 2, for τ ∈ (0, t0 − (ℓ − 1)ε2 ], because barely admissible curves do not carry minimum. In particular, there exists a minimizing curve γ of lmin (t0 − (ℓ − 1)ε2 ), defined on τ ∈ [0, t0 − (ℓ − 1)ε2 ] with γ(0) = x0 , such that √ (5.9) L(γ) ≤ 2 · (2 2ε) < 10ε. Consequently, there exists a point (x, t) on the minimizing curve γ with t ∈ [(ℓ − 1)ε2 + 14 ε2 , (ℓ − 1)ε2 + 43 ε2 ] such that R(x, t) ≤ 50e rℓ−2. 73

(5.10)

Otherwise, we would have L(γ) ≥

Z

t0 −(ℓ−1)ε2 − 14 ε2 t0 −(ℓ−1)ε2 − 43 ε2

√ τ R(γ(τ ), t0 − τ )dτ

2 1 3 ≥ 50e rℓ−2 · ( ε2 ) 2 3 2 > 10ε since 0 < reℓ < ε; this contradicts (5.9). Next we want to get a lower bound for the reduced volume of a ball around x of radius about reℓ at some time-slice slightly before t. Since the solution satisfies the canonical neighborhood assumption on the time interval [(ℓ − 1)ε2 , ℓε2 ), it follows from the gradient estimate (5.4) that R(x, t) ≤ 400e rℓ−2

(5.11)

1 −1 1 −1 2 for those (x, t) ∈ P (x, t, 16 η reℓ , − 64 η reℓ ) for which the solution is defined. And since the points where occur the δ-cutoff surgeries in the time interval [(ℓ − 1)ε2 , ℓε2) have their scalar curvature at least δ −2 reℓ−2 , the solu1 −1 1 −1 2 η reℓ , − 64 η reℓ ) (this tion is defined on the whole parabolic region P (x, t, 16 says, this parabolic region is unaffected by surgery). Thus by combining (5.9) and (5.11), the reduced distance from (x0 , t0 ) to each point of the ball 1 −1 Bt− 1 η−1 reℓ2 (x, 16 η reℓ ) is uniformly bounded by some universal constant. Let 64 1 −1 us define the reduced volume of the ball Bt− 1 η−1 reℓ2 (x, 16 η reℓ ) by 64

=

Z

Bt−

1 Vet0 −t+ 1 η−1 reℓ2 (Bt− 1 η−1 reℓ2 (x, η −1 reℓ )) 64 64 16

1 −1 (x, 16 η reℓ ) 1 −1 r e2 64 η ℓ

(4π(t0 −t+

1 −1 2 −2 1 η reℓ )) exp(−l(q, t0 −t+ η −1 reℓ2 ))dVt− 1 η−1 reℓ2 (q). 64 64 64

Hence by the κℓ -noncollapsing assumption on the time interval [(ℓ−1)ε2, ℓε2 ), 1 −1 we conclude that the reduced volume of the ball Bt− 1 η−1 reℓ2 (x, 16 η reℓ ) is 64 bounded from below by a positive constant depending only on κℓ and reℓ . Finally we want to get a lower bound estimate for the volume of the ball Bt0 (x0 , r0 ). We have seen the reduced distance from (x0 , t0 ) to each point of 1 −1 the ball Bt− 1 η−1 reℓ2 (x, 16 η reℓ ) is uniformly bounded by some universal con64 stant. Without loss of generality, we may assume ε > 0 is very small. Then 74

1 −1 η reℓ ) can be it follows from (5.8) that the points in the ball Bt− 1 η−1 reℓ2 (x, 16 64 connected to (x0 , t0 ) by shortest L-geodesics, and all of these L-geodesics are admissible (i.e., they stay in the region unaffected by surgery). The union 1 −1 of all shortest L-geodesics from (x0 , t0 ) to the ball Bt− 1 η−1 reℓ2 (x, 16 η reℓ ), de64 1 −1 noted by CBt− 1 η−1 reℓ2 (x, 16 η reℓ ), forms a cone-like subset in space-time with 64 1 −1 the vertex (x0 , t0 ). Denote B(t) by the intersection of CBt− 1 η−1 reℓ2 (x, 16 η reℓ ) 64 with the time-slice at t. The reduced volume of the subset B(t) is defined by Z Vet0 −t (B(t)) = (4π(t0 − t))−2 exp(−l(q, t0 − t))dVt (q). B(t)

1 −1 η reℓ ) lies entirely in the region Since the cone-like subset CBt− 1 η−1 reℓ2 (x, 16 64 unaffected by surgery, we can apply Perelman’s Jacobian comparison [27] to conclude that

1 Vet0 −t (B(t)) ≥ Vet0 −t+ 1 η−1 reℓ2 (Bt− 1 η−1 reℓ2 (x, η −1 reℓ )) 64 64 16 ≥ c(κℓ , reℓ )

(5.12)

1 −1 2 for all t ∈ [t− 64 η reℓ , t0 ], where c(κℓ , reℓ ) is some positive constant depending only on κℓ and reℓ . 1 Denote by ξ = r0−1 V0 ℓt0 (Bt0 (x0 , r0 )) 4 . Our purpose is to give a positive lower bound for ξ. Without loss of generality, we may assume ξ < 41 , thus 1 −1 2 e 0 −ξr02 ) the subset of the points 0 < ξr02 < t0 − t¯+ 64 η reℓ . And denote by B(t at the time-slice {t = t0 − ξr02 } where every point can be connected to (x0 , t0 ) e 0 − ξr02 ). by an admissible shortest L-geodesic. Clearly B(t0 − ξr02) ⊂ B(t 1 e reℓ , ε) sufficiently small, the region P (x0 , t0 , r0 , −r 2 ) Since r0 ≥ 2η r and δe = δ(r, 0 is unaffected by surgery. Then by the exactly same argument as deriving (3.24) in the proof of Theorem 3.5, we see that there exists a universal positive constant ξ0 such that as 0 < ξ ≤ ξ0 , there holds

L exp{|v|≤ 1 ξ− 12 } (ξr02) ⊂ Bt0 (x0 , r0 ). 4

75

(5.13)

e 0 − ξr02 ) is given by The reduced volume B(t

e 0 − ξr 2)) Veξr02 (B(t 0 Z = (4πξr02)−2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q) e −ξr 2 ) B(t Z 0 0 (4πξr02 )−2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q) = e 0 −ξr 2 )∩L exp B(t 0

+

Z

{|v|≤ 1 4ξ

e 0 −ξr 2 )\L exp B(t 0

−1 2

}

(5.14)

(ξr02 )

2 1 (ξr0 ) 1 ξ− 2 } {|v|≤ 4

(4πξr02)−2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q).

By (5.13), the first term on the RHS of (5.14) can be estimated by Z (4πξr02)−2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q) e 0 −ξr 2 )∩L exp B(t 0 4ξ

≤e

4ξ

Z

{|v|≤ 1 4ξ

Bt0 (x0 ,r0 )

1 −2

}

(ξr02 )

(4πξr02 )−2 exp(−l)dVt0 (q)

(5.15)

≤ e (4π)−2 ξ 2 . And the second term on the RHS of (5.14) can be estimated by Z (4πξr02)−2 exp(−l(q, ξr02))dVt0 −ξr02 (q) ≤

e 0 −ξr 2 )\L exp B(t 0

{|v|≤ 1 4ξ

Z

1 {|v|> 41 ξ − 2 }

= (4π)

−2

Z

−1 2

}

(ξr02 )

(4πτ )−2 exp(−l)J(τ )|τ =0 dv

1

{|v|> 14 ξ − 2 }

(5.16)

exp(−|v|2 )dv,

by using Perelman’s Jacobian comparison theorem [27] (as deriving (3.30) in the proof of Theorem 3.5). Hence the combination of (5.12), (5.14), (5.15) and (5.16) bounds ξ from blow by a positive constant depending only on κℓ and reℓ . Therefore we have completed the proof of the lemma. # Now we can prove the proposition. 76

Proof of Proposition 5.4. The proof of the proposition is by induction: having constructed our sequences for 1 ≤ j ≤ ℓ, we make one more step, defining reℓ+1 , κℓ+1 , δeℓ+1 , and redefining δeℓ = δeℓ+1 . In views of the previous lemma, we only need to define reℓ+1 and δeℓ+1 . In Theorem 4.1 we have obtained the canonical neighborhood structure for smooth solutions. When adapting the arguments in the proof of Theorem 4.1 to the present surgical solutions, we will encounter two new difficulties. The first new difficulty is how to take a limit for the surgerically modified solutions. The idea to overcome the first difficulty consists of two parts. The first part, due to Perelman [28], is to choose δeℓ and δeℓ+1 small enough to push the surgical regions to infinity in space. (This is the reason why we need to redefine δeℓ = δeℓ+1 .) The second part is to show that solutions are smooth on some uniform small time intervals (on compact subsets) so that we can apply Hamilton’s compactness theorem, since we only have curvature bounds; otherwise Shi’s interior derivative estimate may not be applicable. In fact, the second part idea is more crucial. That is just concerned with the question whether the surgery times accumulate or not. Unfortunately, as written down in the third paragraph of section 5.4 of [28], the second part was not addressed. The second new difficulty is that, when extending the limiting surgically modified solution backward in time, it is possible to meet the surgical regions in finite time. This also indicates the surgery times may accumulate. The idea to overcome this difficulty is somewhat similar to the above second part idea for the first difficulty. We will use the canonical neighborhood charaterization of the standard solution in Corollary A.2 in Appendix to exclude this possibility. We now start to prove the proposition by contradiction. Suppose for sequence of positive numbers r α and δeαβ , satisfying r α → 0 as α → ∞ and 1 (→ 0), there exist sequences of solutions gijαβ to the Ricci flow with δeαβ ≤ αβ surgery, where each of them has only a finite number of cutoff surgeries and has the given compact four-manifold as initial datum, so that the following two assertions hold: (i) each δ-cutoff at a time t ∈ [(ℓ − 1)ε2 , (ℓ + 1)ε2 ] satisfies δ ≤ δeαβ ; and (ii) the solutions satisfy the statement of the proposition on [0, ℓε2 ], but 77

violate the canonical neighborhood assumption (with accuracy ε) with r = r α on [ℓε2 , (ℓ + 1)ε2]. For each solution gijαβ , we choose t¯ (depending on α, β) to be the nearly first time for which the canonical neighborhood assumption (with accuracy ε) is violated. More precisely, we choose t¯ ∈ [ℓε2 , (ℓ + 1)ε2] so that the canonical neighborhood assumption with r = r α and with accuracy parameter ε is violated at some (¯ x, t¯), however the canonical neighborhood assumption with accuracy parameter 2ε holds on t ∈ [ℓε2 , t¯]. After passing to subsequences, we may assume each δeαβ is less than the δe in Lemma 5.5 with r = r α when α is fixed. Then by Lemma 5.5 we have uniform κ-noncollapsing for all scales less than ε on [0, ¯t] with some κ > 0 independent of α, β. Slightly abusing notation, we will often drop the indices α, β. Let e gijαβ be the rescaled solutions along (¯ x, t¯) with factors R(¯ x, t¯)(≥ r −2 → +∞) and shift the times t¯ to zero. We hope to take a limit of the rescaled solutions for subsequences of α, β → ∞ and show the limit is an ancient κ-solution, which will give the desired contradiction. We divide the following arguments into six steps. e ˆt) ≤ A Step 1. Let (y, tˆ) be a point on the rescaled solution geijαβ with R(y, (A ≥ 1) and tˆ ∈ [−(t¯ − (ℓ − 1)ε2 )R(¯ x, t¯), 0], then we have estimate e t) ≤ 10A R(x,

(5.17)

1 for those (x, t) in the parabolic neighborhood P (y, ˆt, 12 η −1 A− 2 , − 18 η −1 A−1 ) et′ (y, 1 η −1 A− 12 ), t′ ∈ [tˆ− 1 η −1 A−1 , tˆ]}, for which the rescaled , {(x′ , t′ ) | x′ ∈ B 2 8 solution is defined. Indeed, as in the first step of the proof of Theorem 4.1, this follows directly from the gradient estimates (5.4) in the canonical neighborhood assumption with parameter 2ε.

Step 2. In this step, we will prove three time extending results. Assertion 1. For arbitrarily fixed α, 0 < A < +∞, 1 ≤ C < +∞ and 0 ≤ B < 12 ε2 (r α )−2 − 18 η −1 C −1 , there is a β0 = β0 (ε, A, B, C) (independent e0 (¯ of α) such that if β ≥ β0 and the rescaled solution geijαβ on the ball B x, A) is defined on a time interval [−b, 0] with 0 ≤ b ≤ B and the scalar curvature 78

satisfies e t) ≤ C, R(x,

e0 (¯ on B x, A) × [−b, 0],

e0 (¯ then the rescaled solution e gijαβ on the ball B x, A) is also defined on the 1 −1 −1 extended time interval [−b − 8 η C , 0].

Before the proof, we need a simple observation: once a space point in the Ricci flow with surgery is removed by surgery at some time, then it never appears for later time; if a space point at some time t can not be defined before the time t , then either the point lies in a gluing cap of the surgery at time t or the time t is the initial time of the Ricci flow. Proof of Assertion 1. Firstly we claim that there exists β0 = β0 (ε, A, B, C) e0 (¯ such that as β ≥ β0 , the rescaled solution e gijαβ on the ball B x, A) can be defined before the time −b (i.e., there are no surgeries interfering in e0 (¯ B x, A) × [−b − ǫ′ , −b] for some ǫ′ > 0). We argue by contradiction. Suppose not, then there is some point x˜ ∈ e B0 (¯ x, A) such that the rescaled solution e gijαβ at x˜ can not be defined before the time −b. By the above observation, there is a surgery at the time −b such that the point x˜ lies in the instant gluing cap. 1 ˜ (= R(¯ Let h x, t¯) 2 h) be the cut-off radius at the time −b for the rescaled solution. Clearly, there is a universal constant D such that 1

˜ ≤ R(˜ ˜ e x, −b)− 2 ≤ Dh. D−1 h

By Lemma 5.3 and looking at the rescaled solution at the time −b, the 1 ˜ constitute a (δeαβ ) 2 -cap K. gluing cap and the adjacent δ-neck, of radius h,

For any fixed small positive constant δ ′ (much smaller than ε), we see 1 −1 e e(−b) (˜ B x, (δ ′ ) R(˜ x, −b)− 2 ) ⊂ K

as β large enough. We first verify the following

Claim 1. For any small constants 0 < θ˜ < 23 , δ ′ > 0, there exists a ˜ > 0 such that as β ≥ β(δ ′ , ε, θ), ˜ we have β(δ ′ , ε, θ) αβ ˜ is defined on the time e(−b) (˜ (i) the rescaled solution e gij over B x, (δ ′ )−1 h) ˜h ˜ 2 ]; interval [−b, 0] ∩ [−b, −b + ( 32 − θ) 1 ˜ in the (δeαβ ) 2 -cap K evolved by the Ricci flow e(−b) (˜ (ii) the ball B x, (δ ′ )−1 h) ˜h ˜ 2 ] is, after scaling with factor on the time interval [−b, 0] ∩ [−b, −b + ( 3 − θ) 2

79

˜ −2 , δ ′ -close ( in C [δ′−1 ] topology) to the corresponding subset of the standard h solution. This claim is somewhat known in the first claim in the proof of Lemma 5.5. Indeed, suppose there is a surgery at some time t˜˜ ∈ [−b, 0]∩(−b, −b+( 32 − ˜h ˜ 2 ] which removes some point x˜˜ ∈ B ˜ We assume t˜˜ ∈ (−b, 0] e(−b) (˜ θ) x, (δ ′ )−1 h). be the first time with that property. ¯ ′ , ε, θ) ˜ Then by the proof of the first claim in Lemma 5.5, there is a δ¯ = δ(δ 1 ¯ then the ball B ˜ in the (δeαβ ) 2 -cap K e(−b) (˜ such that if δeαβ < δ, x, (δ ′ )−1 h) evolved by the Ricci flow on the time interval [−b, t˜˜) is, after scaling with

˜ −2 , δ ′ -close to the corresponding subset of the standard solution. Note factor h ˜ are equivalent. By e(−b) (˜ that the metrics for times in [−b, t˜˜) on B x, (δ ′ )−1 h) ˜ e(−b) (˜ the proof of the first claim in Lemma 5.5, the solution on B x, (δ ′ )−1 h) keeps looking like a cap for t ∈ [−b, t˜˜). On the other hand, by definition, the surgery is always performed along the middle three-sphere of a δ-neck ˜ are removed e(−b) (˜ x, (δ ′ )−1 h) with δ < δeαβ . Then as β large, all the points in B (as a part of a capped horn) at the time t˜˜. But x˜ (near the tip of the cap) exists past the time t˜˜. This is a contradiction. Hence we have proved that ˜ is defined on the time interval [−b, 0] ∩ [−b, −b + ( 3 − θ) ˜h ˜ 2 ]. e(−b) (˜ B x, (δ ′ )−1 h) 2 e(−b) (˜ The δ ′ -closeness of the solution on B x, (δ ′ )−1 h) × ([−b, 0] ∩ [−b, −b + ˜h ˜ 2 ]) with the corresponding subset of the standard solution follows ( 23 − θ) by the uniqueness theorem and the canonical neighborhood assumption with parameter 2ε as in the proof of the first claim in Lemma 5.5. Then we have proved Claim 1. We next verify the following Claim 2. large.

˜h ˜ 2 as β ˜ There is θ˜ = θ(CB), 0 < θ˜ < 32 , such that b ≤ ( 23 − θ)

Note from Theorem A.1 in Appendix, there is a universal constant D′ > 0 such that the standard solution ( of dimension four) satisfies the following curvature estimate 2D′ R(y, s) ≥ 3 . −s 2

We choose θ˜ = 3D′ /2(2D′ + 2CB). Then as β large enough, the rescaled

80

solution satisfies e t) ≥ R(x,

3 2

D′ ˜ −2 h −2 ˜ − (t + b)h

(5.18)

˜h ˜ 2 ]). ˜ × ([−b, 0] ∩ [−b, −b + ( 3 − θ) e(−b) (˜ on B x, (δ ′ )−1 h) 2 ˜h ˜ 2 . Then by combining with the assumption R(˜ e x, t) ≤ Suppose b ≥ ( 32 − θ) ˜h ˜ 2 − b, we have C for t = ( 3 − θ) 2

C≥

3 2

D′ ˜ −2 , h ˜ −2 − (t + b)h

and then θ˜ ≥

1

3D ′ 2CB D′ + CB

.

This is a contradiction. Hence we have proved Claim 2. The combination of the above two claims shows that there is a positive ˜ constant 0 < θ˜ = θ(CB) < 32 such that for any δ ′ > 0, there is a positive ˜ such that as β ≥ β(δ ′, ε, θ), ˜ we have b ≤ ( 3 − θ) ˜h ˜ 2 and the rescaled β(δ ′ , ε, θ) 2 ˜ on the time interval [−b, 0] is, after e(−b) (˜ solution in the ball B x, (δ ′ )−1 h) ˜ −2 , δ ′ -close ( in C [(δ′ )−1 ] topology) to the corresponding scaling with factor h subset of the standard solution. e ≤ C on B e0 (¯ By (5.18) and the assumption R x, A) × [−b, 0], we know that ˜ the cut-off radius h at the time −b for the rescaled solution satisfies r ′ ˜ ≥ 2D . h 3C Let δ ′ > 0 be much smaller than ε and min{A−1 , A}. Since d˜0 (˜ x, x¯) ≤ ˜ depending only on θ˜ such that A, it follows that there is constant C(θ) ˜ ≪ (δ ′ )−1 h. ˜ We now apply Corollary A.2 in Appendix d˜(−b) (˜ x, x ¯) ≤ C(θ)A with the accuracy parameter ε/2. Let C(ε/2) be the positive constant in Corollary A.2. Without loss of generality, we may assume the positive constant C1 (ε) in the canonical neighborhood assumption is larger than 4C(ε/2). As δ ′ > 0 is much smaller that ε and min{A−1 , A}, the point x¯ at the time t¯ has a neighborhood which is either a 43 ε-cap or a 34 ε-neck. Since the canonical neighborhood assumption with accuracy parameter ε is violated at (¯ x, t¯), the neighborhood of the point x¯ at the new time zero for 81

the rescaled solution must be a 34 ε-neck. By Corollary A.2 (b), we know the neighborhood is the slice at the time zero of the parabolic neighborhood 4 e x, 0)−1 , b}) e x, 0)− 21 , − min{R(¯ P (¯ x, 0, ε−1 R(¯ 3

e x, 0) = 1) which is 3 ε-close (in C [ 3 ε ] topology) to the correspond(with R(¯ 4 ing subset of the evolving standard cylinder S3 × R over the time interval [− min{b, 1}, 0] with scalar curvature 1 at the time zero. If b ≥ 1, the 43 ε-neck is strong, which is a contradiction. While if b < 1, the 34 ε-neck at time −b is contained in the union of the gluing cap and the adjacent δ-neck where the δ-cutoff surgery was taken. Since ε is small (say ε < 1/100), it is clear that the point x¯ at time −b is the center of an ε-neck which is entirely contained in the adjacent δ-neck. By the remark after Lemma 5.2, the adjacent δ-neck approximates an ancient κ-solution. This implies the point x¯ at the time t¯ 4 −1

has a strong ε-neck, which is also a contradiction. Hence we have proved that there exists β0 = β0 (ε, A, B, C) such that as e0 (¯ β ≥ β0 , the rescaled solution on the ball B x, A) can be defined before the time −b. Let [tαβ A , 0] ⊃ [−b, 0] be the largest time interval so that the rescaled e0 (¯ solution e gijαβ can be defined on B x, A) × [tαβ A , 0]. We finally claim that αβ tA ≤ −b − 18 η −1 C −1 as β large enough. Indeed, suppose not, by the gradient estimates as in Step 1, we have the curvature estimate e t) ≤ 10C R(x, e0 (¯ on B x, A) × [tαβ A , −b]. Hence we have the curvature estimate e t) ≤ 10C R(x,

e0 (¯ on B x, A) × [tαβ A , 0]. By the above argument there is a β0 = β0 (ε, A, B + 1 −1 −1 e0 (¯ η C , 10C) such that as β ≥ β0 , the solution in the ball B x, A) can be 8 αβ defined before the time tA . This is a contradiction. Therefore we have proved Assertion 1. Assertion 2. For arbitrarily fixed α, 0 < A < +∞, 1 ≤ C < +∞ and 1 −1 0 < B < 12 ε2 (r α )−2 − 50 η , there is a β0 = β0 (ε, A, B, C) (independent of e0 (¯ α) such that if β ≥ β0 and the rescaled solution e gijαβ on the ball B x, A) is 82

defined on a time interval [−b + ǫ′ , 0] with 0 < b ≤ B and 0 < ǫ′ < the scalar curvature satisfies e t) ≤ C R(x,

1 −1 η 50

and

e0 (¯ on B x, A) × [−b + ǫ′ , 0],

e0 (¯ e −b+ǫ′ ) ≤ 3 , then the rescaled and there is a point y ∈ B x, A) such that R(y, 2 αβ 1 −1 solution e gij at y is also defined on the extended time interval [−b − 50 η , 0] and satisfies the estimate e t) ≤ 15 R(y, for t ∈ [−b −

1 −1 η , −b 50

+ ǫ′ ].

Proof of Assertion 2. We imitate the proof of Assertion 1. If the rescaled 1 −1 η , −b + ǫ′ ), solution e gijαβ at y can not be defined for some time in [−b − 50 1 −1 then there is a surgery at some time t˜˜ ∈ [−b − 50 η , −b + ǫ′ ] such that y 1 ˜ (= R(¯ lies in the instant gluing cap. Let h x, t¯) 2 h) be the cutoff radius at the ˜ time t˜ for the rescaled solution. Clearly, there is a universal constant D > 1 ˜ ≤ R(y, ˜ By the gradient estimates as in Step 1, e ˜˜t)− 21 ≤ Dh. such that D−1 h the cutoff radius satisfies

˜ ≥ D−1 15− 21 . h

As in Claim 1 (i) in the proof of Assertion 1, for any small constants ˜ > 0 such that as β ≥ β(δ ′ , ε, θ), ˜ 0 < θ˜ < 32 , δ ′ > 0, there exists a β(δ ′ , ε, θ) ˜ × ([t˜˜, ( 3 − θ) ˜h ˜ 2 + t˜˜] ∩ (t˜˜, 0]). e˜(y, (δ ′)−1 h) there is no surgery interfering in B 2 t˜ Without loss of generality, we may assume that the universal constant η is ˜h ˜ 2 + t˜˜ > −b + 1 η −1 . As in Claim much larger than D. Then we have ( 32 − θ) 50 ˜ 2, we can use the curvature bound assumption to choose θ˜ = θ(B, C) such ˜ 3 2 ˜h ˜ + t˜ ≥ 0; otherwise that ( 2 − θ) C≥

D′ ˜2 θ˜h

for some universal constant D′ , and |t˜˜ + b| ≤ which implies θ˜ ≥

1 −1 η , 50

3D ′ 1 −1 2C(B+ 50 η )

1+

D′ 1 −1 C(B+ 50 η )

83

.

1 −1 η )). This is a contradiction if we choose θ˜ = 3D′ /2(2D′ + 2C(B + 50 3 ˜ So there is a positive constant 0 < θ˜ = θ(B, C) < 2 such that for any ′ ′ ˜ ˜ we have δ > 0, there is a positive β(δ , ε, θ) such that as β ≥ β(δ ′ , ε, θ), ˜h ˜ 2 and the solution in the ball B ˜ on the time interval e˜(˜ −t˜˜ ≤ ( 23 − θ) x, (δ ′ )−1 h) t˜ ˜ −2 , δ ′ -close (in C [δ′−1 ] topology) to the [t˜˜, 0] is, after scaling with factor h

corresponding subset of the standard solution. Then exactly as in the proof of Assertion 1, by using the canonical neighborhood structure of the standard solution in Corollary A.2, this gives the desired contradiction with the hypothesis that the canonical neighborhood assumption with accuracy parameter ε is violated at (¯ x, t¯), as β sufficiently large. The curvature estimate at the point y follows from Step 1. Therefore we complete the proof of Assertion 2. Note that the standard solution satisfies R(x1 , t) ≤ D′′ R(x2 , t) for any t ∈ [0, 12 ] and any two points x1 , x2 , where D′′ ≥ 1 is a universal constant.

Assertion 3. For arbitrarily fixed α, 0 < A < +∞, 1 ≤ C < +∞ , there is 1 a β0 = β0 (ε, AC 2 ) such that if any point (y0 , t0 ) with 0 ≤ −t0 < 12 ε2 (r α )−2 − 1 −1 −1 e 0 , t0 ) ≤ C , then η C of the rescaled solution e gijαβ for β ≥ β0 satisfies R(y 8 1 −1 −1 either the rescaled solution at y0 can be defined at least on [t0 − 16 η C , t0 ] and the rescaled scalar curvature satisfies

or we have

e 0 , t) ≤ 10C R(y

for t ∈ [t0 −

1 −1 −1 η C , t0 ], 16

e 1 , t0 ) ≤ 2D′′ R(x e 2 , t0 ) R(x

et0 (y0 , A), where D′′ is the above universal confor any two points x1 , x2 ∈ B stant.

Proof of Assertion 3. Suppose the rescaled solution e gijαβ at y0 can not 1 −1 −1 η C , t0 ), then there is a surgery at some be defined for some t ∈ [t0 − 16 1 −1 −1 ˜ ˜ time t ∈ [t0 − 16 η C , t0 ] such that y0 lies in the instant gluing cap. Let h 1 (= R(¯ x, t¯) 2 h) be the cutoff radius at the time t˜ for the rescaled solution geijαβ . By the gradient estimates as in Step 1, the cutoff radius satisfies ˜ ≥ D−1 10− 21 C − 21 , h 84

where D is the universal constant in the proof of the Assertion 1. Since we ˜ 2 + t˜ > t0 . As in assume η is suitable larger than D as before, we have 12 h Claim 1 (ii) in Assertion 1, for arbitrarily small δ ′ > 0, we know that as ˜ on the time et˜(y0 , (δ ′ )−1 h) β large enough the rescaled solution on the ball B ˜ −2 , δ ′ -close (in C [(δ′ )−1 ] topology) interval [t˜, t0 ] is, after scaling with factor h ˜ ≫ A as to the corresponding subset of the standard solution. Since (δ ′ )−1 h β large enough, Assertion 3 follows from the curvature estimate of standard solution in the time interval [0, 21 ]. Step 3. For any subsequence (αm , βm ) of (α, β) with r αm → 0 and δ αm βm → 0 as m → ∞, we next argue as in the second step of the proof of Theorem 4.1 to show that the curvatures of the rescaled solutions g˜αm βm at new times zero (after shifting) stay uniformly bounded at bounded distances from x¯ for all sufficiently large m. More precisely, we will prove the following assertion: Assertion 4. Given any subsequence of the rescaled solutions g˜ijαm βm with r αm → 0 and δ αm βm → 0 as m → ∞, then for any L > 0, there are constants C(L) > 0 and m(L) such that the rescaled solutions g˜ijαm βm satisfy ˜ 0) ≤ C(L) for all points x with d˜0 (x, x¯) ≤ L and all m ≥ 1; (i) R(x, ˜0 (¯ (ii) the rescaled solutions over the ball B x, L) are defined at least on 1 −1 −1 the time interval [− 16 η C(L) , 0] for all m ≥ m(L). Proof of Assertion 4. For all ρ > 0, set ˜ 0) | m ≥ 1 and d˜0 (x, x¯) ≤ ρ in the rescaled solutions g˜αm βm } M(ρ) = sup{R(x, ij and ρ0 = sup{ρ > 0 | M(ρ) < +∞}. Note that the estimate (5.17) implies that ρ0 > 0. For (i), it suffices to prove ρ0 = +∞. We argue by contradiction. Suppose ρ0 < +∞. Then there are a sequence of points y in the rescaled solutions g˜ijαm βm with d˜0 (¯ x, y) → ρ0 < +∞ and ˜ 0) → +∞. Denote by γ a minimizing geodesic segment from x¯ to y and R(y, ˜0 (¯ denote by B x, ρ0 ) the geodesic open ball centered at x¯ of radius ρ0 on the rescaled solution g˜ijαm βm . 85

First, we claim that for any 0 < ρ < ρ0 with ρ near ρ0 , the rescaled solu1 −1 ˜0 (¯ η M(ρ)−1 , 0] tions on the balls B x, ρ) are defined on the time interval [− 16 for all large m. Indeed, this follows from Assertion 3 or Assertion 1. For the later purpose in Step 6, we now present an argument by using Assertion 3. If the claim is not true, then there is a surgery at some time 1 −1 ˜0 (¯ t˜ ∈ [− 16 η M(ρ)−1 , 0] such that some point y˜ ∈ B x, ρ) lies in the instant 1 ′ ˜ gluing cap. We can choose sufficiently small δ > 0 such that 2ρ0 < (δ ′ )− 2 h, ˜ ≥ D−1 20− 21 M(ρ)− 12 are the cutoff radius of the rescaled solutions where h at t˜. By applying Assertion 3 with (˜ y , 0) = (y0 , t0 ), we see that there is a m(ρ0 , M(ρ)) > 0 such that as m ≥ m(ρ0 , M(ρ)), e 0) ≤ 2D′′ R(x,

e0 (¯ for all x ∈ B x, ρ). This is a contradiction as ρ → ρ0 . Since for each fixed 0 < ρ < ρ0 with ρ near ρ0 , the rescaled solutions on 1 −1 ˜0 (¯ η M(ρ)−1 , 0] for all the ball B x, ρ) are defined on the time interval [− 16 large m, by Step 1 and Shi’s derivative estimate, we know that the covariant ˜0 (¯ derivatives and higher order derivatives of the curvatures on B x, ρ− (ρ02−ρ) )× 1 −1 [− 32 η M(ρ)−1 , 0] are also uniformly bounded. By the uniform κ-noncollapsing and the virtue of Hamilton’s compactness theorem 16.1 in [18] (see [3] for the details on generalizing Hamilton’s compactness theorem to finite balls), after passing to a subsequence, we ∞ ˜0 (¯ can assume that the marked sequence (B x, ρ0 ), e gijαm βm , x¯) converges in Cloc topology to a marked (noncomplete) manifold (B∞ , e gij∞ , x¯) and the geodesic segments γ converge to a geodesic segment (missing an endpoint) γ∞ ⊂ B∞ emanating from x¯. Clearly, the limit has restricted isotropic curvature pinching (2.4) by the pinching assumption. Consider a tubular neighborhood along γ∞ defined by [ e∞ (q0 ))− 21 ), V = B∞ (q0 , 4π(R q0 ∈γ∞

e∞ denotes the scalar curvature of the limit and B∞ (q0 , 4π(R e∞ (q0 ))− 2 ) where R e∞ (q0 ))− 21 . Let B ¯∞ denote is the ball centered at q0 ∈ B∞ with the radius 4π(R ¯∞ the limit point of γ∞ . Exactly as the completion of (B∞ , e gij∞ ), and y∞ ∈ B in the second step of the proof of Theorem 4.1, it follows from the canonical neighborhood assumption with accuracy parameter 2ε that the limiting 1

86

metric e gij∞ is cylindrical at any point q0 ∈ γ∞ which is sufficiently close to y∞ and then the metric space V¯ = V ∪ {y∞ } by adding the point y∞ has nonnegative curvature in Alexandrov sense. Consequently we have a fourdimensional non-flat tangent cone Cy∞ V¯ at y∞ which is a metric cone with aperture ≤ 20ε. On the other hand, note that by the canonical neighborhood assumption, the canonical 2ε-neck neighborhoods are strong. Thus at each point q ∈ V near y∞ , the limiting metric e gij∞ actually exists on the whole parabolic neighborhood \ 1 e∞ (q))− 21 , − 1 η −1 (R e∞ (q))−1 ), V P (q, 0, η −1 (R 3 10 and is a smooth solution of the Ricci flow there. Pick z ∈ Cy∞ V¯ with distance one from the vertex y∞ and it is nonflat around z. By definition the ball B(z, 12 ) ⊂ Cy∞ V¯ is the Gromov-Hausdorff convergent limit of the scalings of a sequence of balls B∞ (zk , σk )(⊂ (V, e gij∞ )) where σk → 0. Since the estimate (5.17) survives on (V, e gij∞) for all A < +∞, and the tangent cone is four-dimensional and nonflat around z, we see that this convergence ∞ is actually in Cloc topology and over some ancient time interval. Since the 1 limiting B∞ (z, 2 )(⊂ Cy∞ V¯ ) is a piece of nonnegatively (operator) curved nonflat metric cone, we get a contradiction with Hamilton’s strong maximum principle [14] as before. So we have proved ρ0 = ∞. This proves (i). By the same proof of Assertion 1 in Step 2, we can further show that for ˜0 (¯ any L, the rescaled solutions on the balls B x, L) are defined at least on the 1 −1 −1 time interval [− 16 η C(L) , 0] for all sufficiently large m. This proves (ii). Step 4. For any subsequence (αm , βm ) of (α, β) with r αm → 0 and δeαm βm → 0 as m → ∞, by Step 3, the κ-noncollapsing and Hamilton’s compactness ∞ theorem, we can extract a Cloc convergent subsequence of g˜ijαm βm over some space time open subsets containing t = 0. We now want to show any such limit has bounded curvature at t = 0. We prove by contradiction. Suppose not, then there is a sequence of points zk divergent to infinity in the limiting metric at time zero with curvature divergent to infinity. Since the curvature at zk is large (comparable to one), zk has canonical neighborhood which is a 2ε-cap or strong 2ε-neck. Note that the boundary of 2ε-cap lies in some 2ε-neck. So we get a sequence of 2ε-necks with radius going to zero. Note 87

also that the limit has nonnegative sectional curvature. Without loss of the generality, we may assume 2ε < ε0 , where ε0 is the positive constant in Proposition 2.2. Thus this arrives a contradiction with Proposition 2.2. Step 5. In this step, we will choose some subsequence (αm , βm ) of (α, β) so that we can extract a complete smooth limit on a time interval [−a, 0] for some a > 0 from the rescaled solutions e gijαm βm of the Ricci flow with surgery. Choose αm , βm → ∞ so that r αm → 0, δeαm βm → 0, and Assertion 1, 2, 3 hold with α = αm , β = βm for all A ∈ {p/q | p, q = 1, 2 · · · , m}, and B, C ∈ {1, 2, · · · , m}. By Step 3, we may assume the rescaled solutions ∞ topology at the time t = 0. Since the curvature of geijαm βm converge in Cloc the limit at t = 0 is bounded by Step 4, it follows from Assertion 1 in Step 2 and the choice of the subsequence (αm , βm ) that the limiting (M∞ , e gij∞ (·, t)) is defined at least on a backward time interval [−a, 0] for some positive constant a and is a smooth solution to the Ricci flow there. Step 6. We further want to extend the limit of Step 5 backward in time to infinity to get an ancient κ-solution. Let e gijαm βm be the convergent sequence obtained in the above Step 5. Denote by tmax = sup{ t′ | we can take a smooth limit on (−t′ , 0] (with bounded curvature at each time slice) from a subsequence of the rescaled solutions geijαm βm }.

We first claim that there is a subsequence of the rescaled solutions e gijαm βm ∞ which converges in Cloc topology to a smooth limit (M∞ , e gij∞ (·, t)) on the maximal time interval (−tmax , 0]. Indeed, let tk be a sequence of positive numbers such that tk → tmax and there exist smooth limits (M∞ , e gk∞ (·, t)) defined on (−tk , 0]. For each k, the limit has nonnegative curvature operator and has bounded curvature at each time slice. Moreover by the gradient estimate in canonical neighborhood assumption with accuracy parameter 2ε, the limit has bounded curvature on e the scalar curvature upper each subinterval [−b, 0] ⊂ (−tk , 0]. Denote by Q e independent of k). Then we can apply bound of the limit at time zero (Q 88

Li-Yau-Hamilton inequality [16] to get e∞ (x, t) ≤ R k

tk e Q, t + tk

e∞ (x, t) are the scalar curvatures of the limits (M∞ , e where R gk∞(·, t)). Hence k by the definition of convergence and the above curvature estimates, we can ∞ find a subsequence of the rescaled solutions e gijαm βm which converges in Cloc topology to a smooth limit (M∞ , e gij∞(·, t)) on the maximal time interval (−tmax , 0]. We need to show −tmax = −∞. Suppose −tmax > −∞, there are only the following two possibilities: either (1) The curvature of the limiting solution (M∞ , e gij∞ (·, t)) becomes unbounded as t ց −tmax ; or (2) For each small constant θ > 0 and each large integer m0 > 0, there is some m ≥ m0 such that the rescaled solution e gijαm βm has a surgery time Tm ∈ [−tmax − θ, 0] and a surgery point xm lying in a gluing cap at the times Tm so that d2Tm (x, x¯) is uniformly bounded from above by a constant independent of θ and m0 . We next claim that the possibility (1) always occurs. Suppose not, then the curvature of the limiting solution (M∞ , e gij∞ (·, t)) is uniformly bounded by (some positive constant) Cˆ on (−tmax , 0]. In particular, for any A > 0, there is a sufficiently large integer m1 > 0 such that any rescaled solution e gijαm βm e0 (¯ with m ≥ m1 on the geodesic ball B x, A) is defined on the time interval 1 −1 ˆ −1 [−tmax + 50 η C , 0] and its scalar curvature is bounded by 2Cˆ there. (Here, without loss of generality, we may assume that the upper bound Cˆ is so large 1 −1 ˆ −1 that −tmax + 50 η C < 0.) By Assertion 1 in Step 2, as m large enough, e0 (¯ x, A) can be defined on the extended the rescaled solution e gijαm βm over B 1 −1 ˆ −1 e ≤ 10Cˆ time interval [−tmax − 50 η C , 0] and have the scalar curvature R e0 (¯ on B x, A) × [−tmax − 1 η −1 Cˆ −1 , 0]. So we can extract a smooth limit from 50

the sequence to get the limiting solution which is defined on a larger time 1 −1 ˆ −1 interval [−tmax − 50 η C , 0]. This contradicts with the definition of the maximal time −tmax . We now remain to exclude the possibility (1).

89

By using Li-Yau-Hamilton inequality [16] again, we have e∞ (x, t) ≤ R

tmax e Q. t + tmax

So we only need to control the curvature near −tmax . Exactly as in the Step 4 of proof of Theorem 4.1, it follows from Li-Yau-Hamilton inequality that q e (5.19) d0 (x, y) ≤ dt (x, y) ≤ d0 (x, y) + 30tmax Q for any x, y ∈ M∞ and t ∈ (−tmax , 0]. Since the infimum of the scalar curvature is nondecreasing in time, we 1 −1 have some point y∞ ∈ M∞ and some time −tmax < t∞ < −tmax + 50 η e e such that R∞ (y∞ , t∞ ) < 5/4. By (5.19), there is a constant A > 0 such that e for all t ∈ (−tmax , 0]. dt (¯ x, y∞ ) ≤ A/2 Now we return back to the rescaled solution e gijαm βm . Clearly, for arbitrarily given small ǫ′ > 0, as m large enough, there is a point ym in the underlying manifold of e gijαm βm at time 0 satisfying the following properties

e e m , t∞ ) < 3 , det (¯ x, ym ) ≤ A (5.20) R(y 2 for t ∈ [−tmax + ǫ′ , 0]. By the definition of convergence, we know that for e as m large enough, the rescaled solution over B e0 (¯ any fixed A ≥ 2A, x, A) is defined on the time interval [t∞ , 0] and satisfies e e t) ≤ 2tmax Q R(x, t + tmax

e0 (¯ on B x, A) × [t∞ , 0]. Then by Assertion 2 of Step 2, we have proved there is a sufficiently large m ¯ 0 such that as m ≥ m ¯ 0 , the rescaled solutions e gijαm βm 1 −1 η , 0], and satisfy at ym can be defined on [−tmax − 50 e m , t) ≤ 15 R(y

1 −1 for t ∈ [−tmax − 50 η , t∞ ]. We now prove a statement analogous to Assertion 4 (i) of Step 3.

Assertion 5. For the above rescaled solutions e gijαm βm and m ¯ 0 , we have that for any L > 0, there is a positive constant ω(L) such that the rescaled solutions e gijαm βm satisfy e t) ≤ ω(L) R(x, 90

for all (x, t) with d˜t (x, ym ) ≤ L and t ∈ [−tmax − m≥m ¯ 0.

1 −1 η , t∞ ] 50

and for all

Proof of Assertion 5. We slightly modify the argument in the proof of Assertion 4 (i). Let

and

e t) | d˜t (x, ym ) ≤ ρ and t ∈ [−tmax − 1 η −1 , t∞ ] M(ρ) = sup{R(x, 50 αm βm ,m ≥ m ¯ 0} in the rescaled solutions geij ρ0 = sup{ρ > 0 | M(ρ) < +∞}.

Note that the estimate (5.17) implies that ρ0 > 0. We only need to show ρ0 = +∞. We argue by contradiction. Suppose ρ0 < +∞. Then, after passing to subsequence, there are a sequence of (˜ ym , tm ) in the rescaled solutions αm βm 1 −1 geij with tm ∈ [−tmax − 50 η , t∞ ] and d˜tm (ym , y˜m ) → ρ0 < +∞ such that e ym , tm ) → +∞. Denote by γm a minimizing geodesic segment from ym to R(˜ etm (ym , ρ0 ) the geodesic open ball centered y˜m at the time tm and denote by B at ym of radius ρ0 on the rescaled solution geijαm βm (·, tm ). For any 0 < ρ < ρ0 with ρ near ρ0 , by applying Assertion 3 as before, etm (ym , ρ) are defined on the we get that the rescaled solutions on the balls B 1 −1 time interval [tm − 16 η M(ρ)−1 , tm ] for all large m. And by Step 1 and Shi’s derivative estimate, we further know that the covariant derivatives of the etm (ym , ρ − (ρ0 −ρ) ) × [tm − 1 η −1 M(ρ)−1 , tm ] are curvatures of all order on B 2 32 also uniformly bounded. Then by the uniform κ-noncollapsing and Hamilton’s compactness theorem, after passing to a subsequence, we can assume ∞ ˜tm (ym , ρ0 ), e that the marked sequence (B gijαm βm (·, tm ), ym ) converges in Cloc topology to a marked (noncomplete) manifold (B∞ , e gij∞ , y∞ ) and the geodesic segments γm converge to a geodesic segment (missing an endpoint) γ∞ ⊂ B∞ emanating from y∞ . Clearly, the limit also has restrictive isotropic curvature pinching (2.4). Then by repeating the same argument as in the proof of Assertion 4 (i) in the rest, we derive a contradiction with Hamilton’s strong maximum principle. This proves Assertion 5. 91

We then apply the second estimate of (5.20) and Assertion 5 to conclude that for any large constant 0 < A < +∞, there is a positive constant C(A) such that for any small ǫ′ > 0, the rescaled solutions e gijαm βm satisfy e t) ≤ C(A), R(x,

(5.21)

e0 (¯ for all x ∈ B x, A) and t ∈ [−tmax + ǫ′ , 0], and for all sufficiently large m. Then by applying Assertion 1 in Step 2, we conclude that the rescaled e0 (¯ solutions e gijαm βm on the geodesic balls B x, A) are also defined on the ex1 tended time interval [−tmax + ǫ′ − 8 η −1 C(A)−1 , 0] for all sufficiently large m. Furthermore, by the gradient estimates as in Step 1, we have e t) ≤ 10C(A), R(x,

e0 (¯ for x ∈ B x, A) and t ∈ [−tmax + ǫ′ − 81 η −1 C(A)−1 , 0]. Since ǫ′ > 0 is e0 (¯ arbitrarily small, the rescaled solutions e gijαm βm on B x, A) are defined on the 1 −1 −1 extended time interval [−tmax − 16 η C(A) , 0] and satisfy e t) ≤ 10C(A), R(x,

(5.22)

1 −1 e0 (¯ η C(A)−1 , 0], and for all sufficiently for x ∈ B x, A) and t ∈ [−tmax − 16 large m. Now, by taking convergent subsequences from the rescaled solutions e gijαm βm , we see that the limit solution is defined smoothly on a space-time open subset of M∞ × (−∞, 0] containing M∞ × [−tmax , 0]. By Step 4, we see that the limiting metric e gij∞ (·, −tmax ) at time −tmax has bounded curvature. Then by combining with the 2ε-canonical neighborhood assumption we conclude that the curvature of the limit is uniformly bounded on the time interval [−tmax , 0]. So we have excluded the possibility (1). Hence we have proved a subsequence of the rescaled solutions converges to an ancient κ-solution. Finally by combining with the canonical neighborhood theorem of ancient κ-solutions with restricted isotropic curvature pinching condition (Theorem 3.8) and the same argument in the second paragraph of Section 4, we see that (¯ x, t¯) has a canonical neighborhood with parameter ε, which is a contradiction. Therefore we have completed the proof of the proposition.

# 92

Summing up, we have proved that for an arbitrarily given compact fourmanifold with positive isotropic curvature and with no essential incompressible space form, there exist non-increasing positive (continuous) functions e and re(t), defined on [0, +∞), such that for arbitrarily given positive δ(t) e on [0, +∞), the Ricci flow with (continuous) function δ(t) with δ(t) < δ(t) surgery, with the given four-manifold as initial datum, has a solution on a maximal time interval [0, T ), with T ≤ 2/Rmin (0) < +∞, obtained by evolving the Ricci flow and by performing δ-cutoff surgeries at a sequence of times e i ) at each time ti , so 0 < t1 < t2 < · · · < ti < · · · < T with δ(ti ) ≤ δ ≤ δ(t that the pinching assumption and the canonical neighborhood assumption with r = re(t) are satisfied. (At this moment we still do not know whether the surgery times ti are discrete). Clearly, the upper derivative of the volume in time satisfies d V (t) ≤ 0 dt since the scalar curvature is nonnegative. Thus V (t) ≤ V (0) for all t ∈ [0, T ). Also note that at each time ti , the volume which is cut down by δ(ti )-cutoff surgery is at least an amount of h4 (ti ) with h(ti ) depending only on δ(ti ) and re(ti ) (by Lemma 5.2). Thus the set of the surgery times {ti } must be finite. So we have proved the following long-time existence result.

Theorem 5.6 Given a compact four-dimensional Riemannian manifold with positive isotropic curvature and with no essential incompressible space form, and given any fixed small constant ε > 0, there exist non-increasing e and re(t), defined on [0, +∞), such that positive (continuous) functions δ(t) e on for arbitrarily given positive (continuous) function δ(t) with δ(t) ≤ δ(t) [0, +∞), the Ricci flow with surgery, with the given four-manifold as initial datum, has a solution satisfying the the pinching assumption and the canonical neighborhood assumption (with accuracy ε) with r = re(t) on a maximal time interval [0, T ) with T < +∞ and becoming extinct at T , which is obtained by evolving the Ricci flow and by performing a finite number of cutoff surgeries with each δ-cutoff at time t ∈ (0, T ) having δ = δ(t). Consequently, 93

the initial manifold is diffeomorphic to a connected sum of a finite copies of e 1. S4 , RP4 , S3 × S1 , and S3 ×S # Finally, the main theorem (Theorem 1.1) stated in Section 1 is a direct consequence of the above theorem.

Appendix. Standard Solutions In this appendix, we will prove the curvature estimates for the standard solutions, and give a canonical neighborhood description for the standard solution in dimension four. We have used these estimates and the description in Section 5 for the surgery arguments. The curvature estimate for the special case that the dimension is three and the initial metric is rotationally symmetric, was earlier claimed by Perelman in [28]. Theorem A.1 Let gij be a complete Riemannian metric on Rn (n > 2) with nonnegative curvature operator and with positive scalar curvature which is asymptotic to a round cylinder of scalar curvature 1 at infinity. Then there is a complete solution gij (·, t) to the Ricci flow, with gij as initial metric, ), has bounded curvature in each which exists on the time interval [0, n−1 2 n−1 closed time interval [0, t] ⊂ [0, 2 ), and satisfies the estimate R(x, t) ≥

C −1 n−1 −t 2

for some C depending only on the initial metric gij . Proof. Since the initial metric has bounded curvature operator and has a positive lower bound on its scalar curvature, by [32] and the maximum principle, the Ricci flow has a solution g(·, t) on a maximal time interval [0, T ) with T < ∞. By Hamilton’s maximum principle, the solution g(x, t) has nonnegative curvature operator for t > 0. Note that the injectivity radius of the initial metric has a positive lower bound, so by the same proof of Perelman’s no local collapsing theorem I (in the section 7.3 of [27], or see the proof of Theorem 3.5 of this paper), there is a κ = κ(T, gij ) > 0 such √ that gij (·, t) is κ-noncollapsed on the scale T . 94

We will firstly prove the following assertion. Claim 1 There is a positive function ω : [0, ∞) −→ [0, ∞) depending only on the initial metric and κ such that R(x, t) ≤ R(y, t)ω(R(y, t)d2t (x, y)) for all x, y ∈ M n = Rn , t ∈ [0, T ). The proof is similar to that of Proposition 3.3. Notice that the initial metric has nonnegative curvature operator and its scalar caurvature satisfies C −1 ≤ R(x) ≤ C

(A.1)

1 for some positive constant C > 1. By maximum principle, we know T ≥ 2nC 1 1 and R(x, t) ≤ 2C for t ∈ [0, 4nC ]. The assertion is clearly true for t ∈ [0, 4nC ]. 1 n Now fix (y, t0) ∈ M × [0, T ) with t0 ≥ 4nC . Let z be the closest point to y with the property R(z, t0 )d2t0 (z, y) = 1 (at time t0 ). Draw a shortest geodesic from y to z and choose a point z˜ on the geodesic satisfying dt0 (z, z˜) = 1 1 R(z, t0 )− 2 , then we have 4

R(x, t0 ) ≤

1 ( 12 R(z, t0 )

− 21

1 1 on Bt0 (˜ z , R(z, t0 )− 2 ) 4

, )2

Note that R(x, t) ≥ C −1 everywhere by the evolution equation of the scalar curvature. Then by Li-Yau-Hamilton inequality [16], for all (x, t) ∈ 1 1 1 1 R(z, t0 )− 2 ) × [t0 − ( 8nC R(z, t0 )− 2 )2 , t0 ], we have Bt0 (˜ z , 8nC t0 1 ) 1 , 1 1 2 √ t0 − ( 8n C ) ( 2 R(z, t0 )− 2 )2 1 1 ≤[ R(z, t0 )− 2 ]−2 8nC

R(x, t) ≤ (

Combining this with the κ-noncollapsing, we have V ol(Bt0 (˜ z,

1 1 1 1 R(z, t0 )− 2 )) ≥ κ( R(z, t0 )− 2 )n 8nC 8nC

and then 1

V ol(Bt0 (z, 8R(z, t0 )− 2 )) ≥ κ( 95

1 1 n ) (8R(z, t0 )− 2 )n 64nC

So by Corollary 11.6 (b) of [27], there hold 1

for all x ∈ Bt0 (z, 2R(z, t0 )− 2 ).

R(x, t0 ) ≤ C(κ)R(z, t0 ),

Here in the following we denote by C(κ) various positive constants depending only on κ, n and the initial metric. Now by Li-Yau-Hamilton inequality [16] and local gradient estimate of Shi [32], we obtain R(x, t) ≤ C(κ)R(z, t0 ), and |

∂ R|(x, t) ≤ C(κ)(R(z, t0 ))2 ∂t

1

1

1 for all (x, t) ∈ Bt0 (z, 2R(z, t0 )− 2 )) × [t0 − ( 8nC R(z, t0 )− 2 )2 , t0 ]. Therefore by combining with the Harnack estimate [16], we obtain

R(y, t0 ) ≥ C(κ)−1 R(z, t0 − C(κ)−1 R(z, t0 )−1 ) ≥ C(κ)−2 R(z, t0 )

Consequently, we have showed that there is a constant C(κ) such that 1

1

V ol(Bt0 (y, R(y, t0)− 2 )) ≥ C(κ)−1 (R(y, t0)− 2 )n and 1

R(x, t0 ) ≤ C(κ)R(y, t0) for all x ∈ Bt0 (y, R(y, t0)− 2 ). 1

In general, for any r ≥ R(y, t0 )− 2 , we have n

V ol(Bt0 (y, r)) ≥ C(κ)−1 (r 2 R(y, t0 ))− 2 r n . By applying Corollary 11.6 of [27] again, there exists a positive constant ω(r 2 R(y, t0)) depending only on the constant r 2 R(y, t0) and κ such that R(x, t0 ) ≤ R(y, t0)ω(r 2 R(y, t0)),

1 for all x ∈ Bt0 (y, r). 4

This proves the desired Claim 1. Now we study the asymptotic behavior of the solution at infinity. For any 0 < t0 < T , we know that the metrics gij (x, t) with t ∈ [0, t0 ] has uniformly bounded curvature by the definition of T . Let xk be a sequence of points with d0 (x0 , xk ) 7→ ∞. By Hamilton’s compactness theorem [17], after taking 96

a subsequence, gij (x, t) around xk will converge to a solution to the Ricci flow on R × Sn−1 with round cylinder metric of scalar curvature 1 as initial data. Denote the limit by g˜ij . Then by the uniqueness theorem in [10], we have n−1 2 ˜ t) = , for all t ∈ [0, t0 ]. R(x, n−1 −t 2 It follows that T ≤ following assertion

n−1 . 2

In order to show T =

n−1 , 2

it suffices to prove the

. Fix a point x0 ∈ M n , then there is a δ > 0, Claim 2. Suppose T < n−1 2 such that for any x ∈ M with d0 (x, x0 ) ≥ δ −1 , we have R(x, t) ≤ 2C +

n−1 −t

n−1 2

for all

t ∈ [0, T )

where C is the constant in (A.1). In view of Claim 1, if Claim 2 holds, then sup M n ×[0,T )

R(y, t) ≤ ω(δ −2 (2C +

n−1 n−1 ))(2C + n−1 ) −T −T 2

n−1 2

0, there is a (xδ , tδ ) with 0 < tδ < T such that R(xδ , tδ ) > 2C +

n−1 and d0 (xδ , x0 ) ≥ δ −1 . − tδ

n−1 2

Let t¯δ = sup{t | Since

lim

R(y, t) =

d0 (y,x0 )→∞ 1 ≤ t¯δ 4nC

sup

R(y, t) < 2C +

M n \B0 (x0 ,δ−1 ) n−1 /( n−1 2 2

n−1 }. −t

n−1 2

− t) and supM ×[0,

1 ] 4nC

R(y, t) ≤ 2C, we

know ≤ tδ and there is a x¯δ such that d0 (x0 , x ¯δ ) ≥ δ −1 and R(¯ xδ , t¯δ ) = 2C + n − 1/( n−1 − t¯δ ). By Claim 1 and Hamilton’s compactness 2 theorem [17], as δ → 0 and after taking subsequence, the metrics gij (x, t) −1 on B0 (¯ xδ , δ 2 ) over the time interval [0, ¯tδ ] will converge to a solution g˜ on

97

˜ = R × Sn−1 with standard metric of scalar curvature 1 as initial datum M over the time interval [0, t¯∞ ], and its scalar curvature satisfies ˜ x∞ , t¯∞ ) = 2C + n − 1 , R(¯ n−1 − t¯∞ 2 ˜ t) ≤ 2C + n − 1 , for all t ∈ [0, t¯∞ ], R(x, n−1 − t¯∞ 2

where (¯ x∞ , t¯∞ ) is the limit of (¯ xδ , t¯δ ). On the other hand, by the uniqueness theorem in [10] again, we know ˜ x∞ , t¯∞ ) = R(¯

n−1 2 n−1 − t¯∞ 2

which is a contradiction. Hence we have proved Claim 2 and then have verified T = n−1 . 2 Now we are ready to show C˜ −1 , R(x, t) ≥ n−1 −t 2

for all (x, t) ∈ M n × [0,

n−1 ), 2

(A.2)

for some positive constant C˜ depending only on the initial metric. ), by Claim 1 and κ-noncollapsing, there is For any (x, t) ∈ M n × [0, n−1 2 a constant C(κ) > 0 such that 1

1

V olt (Bt (x, R(x, t)− 2 )) ≥ C(κ)−1 (R(x, t)− 2 )n . Then by the volume estimate of Calabi-Yau [30] on manifolds with nonnegative Ricci curvature, for any a ≥ 1, we have 1 1 a V olt (Bt (x, aR(x, t)− 2 )) ≥ C(κ)−1 (R(x, t)− 2 )n . 8n n On the other hand, since (M , gij (·, t)) is asymptotic to a cylinder of scalar /( n−1 − t) , for sufficiently large a > 0, we have curvature n−1 2 2 r n n−1 n−1 − t)) ≤ C(n)a( − t) 2 . V olt (Bt (x, a 2 2 Combining these two inequalities, we have for all sufficiently large a: q n−1 −t n n−1 2 − 12 − t) 2 ≥ V olt (Bt (x, a( )) C(n)a( 1 )R(x, t) 2 R(x, t)− 2 q n−1 −t 2 − 12 n −1 a ( ≥ C(κ) ) , 1 )(R(x, t) 8n R(x, t)− 2 98

which gives the desired estimate (A.2). Therefore we complete the proof of the theorem. # We now fix a standard capped infinite cylinder metric on R4 as follows. Consider the semi-infinite standard round cylinder N0 = S3 × (−∞, 4) with the metric g0 of scalar curvature 1. Denote by z the coordinate of the second factor (−∞, 4). Let f be a smooth nondecreasing convex function on (−∞, 4) defined by   f (z) = 0, z ≤ 0,      D   f (z) = ce− z , z ∈ (0, 3],   f (z) is strictly convex on z ∈ [3, 3.9],       f (z) = − 12 log(16 − z 2 ), z ∈ [3.9, 4),

where the small (positive) constant c = c0 and big (positive) constant D = D0 are fixed as in Lemma 5.3. Let us replace the standard metric g0 on the portion S3 × [0, 4) of the semi-infinite cylinder by gˆ = e−2f g0 . Then the resulting metric gˆ will be smooth on R4 obtained by adding a point to S3 × (−∞, 4) at z = 4. We denote the manifold by (R4 , gˆ). Next we will consider the “canonical neighborhood” decomposition of the fixed standard solution with (R4 , gˆ) as initial metric. Corollary A.2. Let gij (x, t) be the above fixed standard solution to the Ricci flow on R4 × [0, 32 ). Then for any ε > 0, there is a positive constant C(ε) such that each point (x, t) ∈ R4 × [0, 32 ) has an open neighborhood B, 1 with Bt (x, r) ⊂ B ⊂ Bt (x, 2r) for some 0 < r < C(ε)R(x, t)− 2 , which falls into one of the following two categories: either (a) B is an ε-cap, or (b) B is an ε-neck and it is the slice at the time t of the parabolic neigh1 borhood P (x, t, ε−1R(x, t)− 2 , − min{R(x, t)−1 , t}), on which the standard solution is, after scaling with the factor R(x, t) and shifting the time t to zero, −1 ε-close (in C [ε ] topology) to the corresponding subset of the evolving standard cylinder S3 × R over the time interval [− min{tR(x, t), 1}, 0] with scalar curvature 1 at the time zero. 99

Proof. First, we discuss the curvature pinching of this fixed standard solution. Because the initial metric is asymptotic to a cylinder, we have a uniform isotropic curvature pinching at initial, that is to say, there is a universal constant Λ′ > 0 such that max{a3 , b3 , c3 } ≤ Λ′ a1 and max{a3 , b3 , c3 } ≤ Λ′ c1 . Moreover since the initial metric has nonnegative curvature operator, we have b23 ≤ a1 c1 . By the pinching estimates of Hamilton [14] [19], b23 ≤ a1 c1 is preserved, and the following two estimates are also preserved max{a3 , b3 , c3 } ≤ max{Λ′ , 5}a1 and max{a3 , b3 , c3 } ≤ max{Λ′ , 5}c1, under the Ricci flow. The proof of the lemma is reduced to two assertions. We now state and prove the first assertion which takes care of those points with times close to 3 . 2 Assertion 1. For any ε > 0, there is a positive number θ = θ(ε) with 0 < θ < 32 such that for any (x0 , t0 ) ∈ R4 × [θ, 32 ), the standard solution on the parabolic neighborhood 1

P (x0 , t0 , ε−1R(x0 , t0 )− 2 , −ε−2 R(x0 , t0 )−1 ) −1

is well-defined and is, after scaling with the factor R(x0 , t0 ), ε-close (in C [ε ] topology) to the corresponding subset of some oriented ancient-κ solution with restricted isotropic curvature pinching (2.4). We argue by contradiction. Suppose the Assertion 1 is not true, then there exists ε0 > 0 and a sequence of points (xk , tk ) with tk → 32 , such that the standard solution on the parabolic neighborhoods 1

−2 −1 P (xk , tk , ε−1 , −ε−2 0 R(xk , tk ) 0 R(xk , tk ) )

is not, after scaling by the factor R(xk , tk ), ε0 -close to the corresponding subset of any ancient κ-solution. Note that by Theorem A.1, there is a constant C > 0 (depending only on the initial metric, hence it is universal) such that R(x, t) ≥ C −1 /( 23 − t). This implies 3 −1 ε−2 ≤ Cε−2 0 R(xk , tk ) 0 ( − tk ) < tk , 2 100

and then the standard solution on the parabolic neighborhoods 1

−2 −1 P (xk , tk , ε−1 , −ε−2 0 R(xk , tk ) 0 R(xk , tk ) )

is well-defined as k large. By Claim 1 in Theorem A.1, there is a positive function ω : [0, ∞) → [0, ∞) such that R(x, tk ) ≤ R(xk , tk )ω(R(xk , tk )d2tk (x, xk )) for all x ∈ R4 . Now by scaling the standard solution gij (·, t) around (xk , tk ) with the factor R(xk , tk ) and shifting the time tk to zero, we get a sequence of the rescaled solutions to the Ricci flow g˜ijk (x, t˜) = R(xk , tk )gij (x, tk + t˜/R(xk , tk )) defined on R4 with t˜ ∈ [−R(xk , tk )tk , 0]. We denote the scalar ˜ By com˜ k and d. curvature and the distance of the rescaled metric g˜ijk by R bining with the Claim 1 in Theorem A.1 and the Li-Yau-Hamilton inequality, we get ˜ k (x, 0) ≤ ω(d˜2 (x, xk )) R 0 R(xk , tk )tk ˜ k (x, t˜) ≤ ω(d˜20(x, xk )) R ˜ t + R(xk , tk )tk for any x ∈ R4 and t˜ ∈ (−R(xk , tk )tk , 0]. Note that R(xk , tk )tk → ∞ by Theorem A.1. We have shown in the proof of Theorem A.1 that the standard solution is κ-noncollapsed on all scales less than 1 for some κ > 0. Then from the κ-noncollapsing, the above curvature estimates and Hamilton’s compactness theorem (Theorem 16.1 of [18]), we know g˜ijk (x, t˜) has a convergent subsequence (as k → ∞) whose limit is an ancient, κ-noncollapsed, complete and oriented solution with nonnegative curvature operator. This limit must has bounded curvature by the same proof of Step 3 in Theorem 4.1. It also satisfies the restricted isotropic pinching condition (2.4). This gives a contradiction. The Assertion 1 is proved. We now fix the constant θ(ε) obtained in Assertion 1. Let O be the tip of the manifold R4 (it is rotationally symmetric about O at time 0, it remains so as t > 0 by the uniqueness Theorem [10]). Assertion 2 There are constants B1 (ε), B2 (ε) depending only on ε, such that if (x0 , t0 ) ∈ M × [0, θ) with dt0 (x0 , O) ≤ B1 (ε), then there is a 0 < 101

r < B2 (ε) such that Bt0 (x0 , r) is an ε-cap; if (x0 , t0 ) ∈ M × [0, θ) with dt0 (x0 , O) ≥ B1 (ε), then the parabolic neighborhood 1

P (x0 , t0 , ε−1 R(x0 , t0 )− 2 , − min{R(x0 , t0 )−1 , t0 }) is after scaling with the factor R(x0 , t0 ) and shifting the time t0 to zero, ε−1 close (in C [ε ] topology) to the corresponding subset of the evolving standard cylinder S3 × R over the time interval [− min{t0 R(x0 , t0 ), 1}, 0] with scalar curvature 1 at the time zero. Since the standard solution exists on the time interval [0, 32 ), there is a constant B0 (ε) such that the curvatures on [0, θ(ε)] are uniformly bounded by B0 (ε). This implies that the metrics in [0, θ(ε)] are equivalent. Note that the initial metric is asymptotic to a standard cylinder. For any sequence of points xk with d0 (O, xk ) → ∞, after taking a subsequence, gij (x, t) around xk will converge to a solution to the Ricci flow on R × S3 with round cylinder metric of scalar curvature 1 as initial data. By the uniqueness theorem [10], the limit solution must be the standard evolving round cylinder. This implies that there is a constant B1 (ε) > 0 depending on ε such that for any (x0 , t0 ) with t0 ≤ θ(ε) and dt0 (x, O) ≥ B1 (ε), the standard solution on 1 the parabolic neighborhood P (x0 , t0 , ε−1 R(x0 , t0 )− 2 , − min{R(x0 , t0 )−1 , t0 }) is, after scaling with the factor R(x0 , t0 ), ε-close to the corresponding subset of the evolving round cylinder. Since the solution is rotationally symmetric around O, the cap neighborhood structures of those points x0 with dt0 (x0 , O) ≤ B1 (ε) follows directly. The Assertion 2 is proved. Therefor we finish the proof of Corollary A.2. #

References [1] Burago,Y. Gromov, M. and Perelman, G., A. D. Alexandrov spaces with curvatures bounded below, Russian Math. Surveys 47 (1992), 1-58. [2] Cao, H.-D., Deformation of K¨ ahler metrics to K¨ ahler-Einstein metrics on compact K¨ ahler manifolds, Invent. Math., 81 (1985), no. 2, 359–372. 102

[3] Cao, H. D. and Zhu, X. P., A complete proof of the Poincaré and geometrization conjectures – application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math., 10 (2006), no. 2, 165-492. [4] Chow, B., The Ricci flow on 2-sphere, J. Diff. Geom. 33 (1991), 325-334. [5] Cheeger, J. and Colding, T. H., On the structure of the spaces with Ricci curvature bounded below I. J. Diff. Geom. 46 (1997), 406-480. [6] Cheeger, J. and Ebin, D., Comparison theorems in Riemannian geometry, North-Holland (1975). [7] Cheeger, J. and Gromoll, D., On the structure of complete manifolds of nonnegative curvature, Ann. of Math., 46 (1972), 413-433. [8] Chen, B. L., Tang, S. H. and Zhu, X. P., A uniformization theorem of complete noncompact Kähler surfaces with positive bisectional curvature, J. Diff. Geom. 67 (2004),519-570. [9] Chen, B. L. and Zhu, X. P., On complete noncompact Kähler manifolds with positive bisectional curvature, Math. Ann. 327 (2003), 1-23. [10] Chen, B. L. and Zhu, X. P., Uniqueness of the Ricci flow on complete noncompact manifolds, arXiv:math. DG/0505447 v3 May 2005, preprint. [11] De Turck, D., Deforming metrics in the direction of their Ricci tensors J. Diff. Geom. 18 (1983), 157-162. [12] Ding, Y., Notes on Perelman’s second paper http://www.math. lsa.umich.edu/research/ricciflow/perelman.html [13] Hamilton, R. S., Three manifolds with positive Ricci curvature , J. Diff. Geom. 17 (1982), 255-306. [14] Hamilton, R. S., Four–manifolds with positive curvature operator, J. Diff. Geom. 24 (1986), 153-179. [15] Hamilton, R. S., The Ricci flow on surfaces, Contemporary Mathematics 71 (1988) 237-261. 103

[16] Hamilton, R. S., The Harnack estimate for the Ricci flow, J. Diff. Geom. 37 (1993), 225-243. [17] Hamilton, R. S., A compactness property for solution of the Ricci flow, Amer. J. Math. 117 (1995), 545-572. [18] Hamilton, R. S., The formation of singularities in the Ricci flow, Surveys in Diff. Geom. (Cambridge, MA, 1993), 2, 7-136, International Press, Combridge, MA,1995. [19] Hamilton, R. S., Four manifolds with positive isotropic curvature, Comm. Anal. Geom.,5(1997),1-92. (or see, Collected Papers on Ricci Flow, Edited by H. D. Cao, B. Chow, S. C. Chu and S. T. Yau, International Press 2002). [20] Hamilton, R. S., Non-singular solutions to the Ricci flow on three manifolds, Comm. Anal. Geom. 1 (1999), 695-729. [21] Hirsch, M. W., Differential Topology, Springer-Verlag, 1976. [22] Huisken, G., Ricci deformation of the metric on a Riemanian mnifold J. Diff. Geom. 21 (1985), 47-62. [23] Kleiner, B. and Lott, J., Note on Perelman’s paper, http://www.math.lsa.umich.edu/research/ricciflow/perelman.html. [24] Micallef, M. and Moore, J. D., Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), 199-227. [25] Milka, A. D., Metric structure of some class of spaces containing straight lines, Ukrain. Geometrical. Sbornik, 4, 1967, 43-48. [26] Morgan, J. W., Recent progress on the Poincar´ e conjecture and the classification of 3-manifolds, Bull. of the A. M. S., 42 (2004) no. 1, 57-78. [27] Perelman, G., The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159 v1 November 11, 2002, preprint. 104

[28] Perelman, G., Ricci flow with surgery on three arXiv:math.DG/0303109 v1 March 10, 2003, preprint.

manifolds

[29] Perelman, G., Finite extinction time to the solutions to the Ricci flow on certain three manifolds, arXiv: math. DG/0307245 July 17, 2003, preprint. [30] Schoen, R. and Yau, S. T., Lectures on differential geometry, in conference proceedings and Lecture Notes in Geometry and Topology, Volume 1, International Press Publications, 1994. [31] Sesum, N., Tian, G. and Wang, X. D., Notes on Perelman’s paper on the entropy formula for the Ricci flow and its geometric applications. [32] Shi, W. X., Deforming the metric on complete Riemannian manifold, J. Diff. Geom., 30 (1989), 223-301.

105

arXiv:math/0505447v3 [math.DG] 27 May 2005

Uniqueness of the Ricci Flow on Complete Noncompact Manifolds Bing-Long Chen and Xi-Ping Zhu Department of Mathematics Zhongshan University Guangzhou, P.R.China May, 2005

Abstract The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80’s, Shi [20] generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ricci flow with surgery. In this paper, we give an affirmative answer for the uniqueness question. More precisely, we prove that the solution of the Ricci flow with bounded curvature on a complete noncompact manifold is unique.

1

1

Introduction

Let (M n , gij ) be a complete Riemannian (compact or noncompact) manifold. The Ricci flow ∂ gij (x, t) = −2Rij (x, t), ∂t

for x ∈ M n and t ≥ 0,

(1.1)

with gij (x, 0) = gij (x), is a weakly parabolic system on metrics. This evolution system was introduced by Hamilton in [8]. Now it has proved to be powerful in the research of differential geometry and lower dimensional topology (see for example Hamilton’s works [8], [9], [10], [13] and the recent works of Perelman [16], [17]). The first matter for the Ricci flow (1.1) is the short time existence and uniqueness of the solutions. When the manifold M n is compact, Hamilton proved in [8] that the Ricci flow (1.1) has a unique solution for a short time. So the problem has been well settled on compact manifolds. In [4], De Turck introduced an elegant trick to give a simplified proof. Later on, Shi [20] extended the short time existence result to noncompact manifolds. More precisely, Shi [20] proved that if (M n , gij ) is complete noncompact with bounded curvature, then the Ricci flow (1.1) has a solution with bounded curvature on a short time interval. In this paper, we will deal with the uniqueness of the Ricci flow on complete noncompact manifolds. The uniqueness of the Ricci flow is important in the theory of the Ricci flow with surgery (see for example [16], [17] and [2]). When we consider the Ricci flow on a compact manifold, the Ricci flow will generally develop singularities in finite time. In the theory of the Ricci flow with surgery, one eliminates the singularities by Hamilton’s geometric surgeries (cut off the high curvature part and glue back a standard cap, then run the Ricci flow again). An important question in this theory is to control the curvature of the glued cap after surgery. The uniqueness theorem of the Ricci flow insures that the solution on glued cap is sufficiently close to a (complete noncompact) standard solution, which is the evolution of capped round cylinder. Then we can apply the estimate of the standard solutions [17] and [2] to get the desired control on curvature. The employing of the uniqueness theorem is essential. So even if we consider the Ricci flow on compact manifolds, we still have to encounter the problem of uniqueness on noncompact manifolds. It is well-known that the uniqueness of the solution of a parabolic system on a complete noncompact manifold does not always hold if one does not impose any growth condition of the solutions. For example, even the simplest linear heat equation on R with zero as initial data has a nontrivial solution which grows faster than 2 ea|x| for any a > 0 whenever t > 0. This says, for the standard linear heat equation, 2 the most growth rate for the uniqueness is ea|x| . Note that in a K¨ ahler manifold, the Ricci curvature is given by Ri¯j = −

∂2 logdet(gk¯l). ∂z i ∂ z¯j 2

Thus the reasonable growth rate that we can expect for the uniqueness of the Ricci flow is the solution with bounded curvature. In this paper, we will prove the following uniqueness theorem of the Ricci flow. Theorem 1.1 Let (M n , gij (x)) be a complete noncompact Riemannian manifold of dimension n with bounded curvature. Let gij (x, t) and g¯ij (x, t) be two solutions to the Ricci flow on M n × [0, T ] with the same gij (x) as initial data and with bounded curvatures. Then gij (x, t) = g¯ij (x, t) for all (x, t) ∈ M n × [0, T ]. Since the Ricci flow is not a strictly parabolic system, our argument will apply the De Turck trick. This is to consider the composition of the Ricci flow with a family of diffeomorphisms generated by the harmonic map flow. By pulling back the Ricci flow by this family of diffeomorphisms, the evolution equations become strictly parabolic. In order to use the uniqueness theorem of a strict parabolic system on a noncompact manifold, we have to overcome two difficulties. The first one is to establish a short time existence for the harmonic map flow between noncompact manifolds. The second one is to get a priori estimates for the harmonic map flow so that after pulling backs, the solutions to the strictly parabolic system still satisfy suitable growth conditions. To the best of our knowledge, one can only get short time existence of harmonic map flow by imposing negative curvature or convex condition on the target manifolds (see for example, [6] and [5]) or by simply assuming the image of initial data lying in a compact domain on the target manifold (see for example [15]). In [2], we observed that the condition of injectivity radius bounded from below ensures certain uniform (local) convexity and showed that this is sufficient to give the short time existence and the a priori estimates for the harmonic map flow. Thus in [2], we obtained the uniqueness under an additional assumption that the initial metric has a positive lower bound on injectivity radius. The main purpose of this paper is to remove this additional assumption. Note from [3] or [1] that the injectivity radius of the initial manifold decays at worst exponentially. This allows us to conformally straighten the initial manifold at infinity. Our idea is to study the evolution equations coming from the composition of the Ricci flow and harmonic map flow, as well as a conformal change. This new approach has the advantage of transforming the Ricci flow equation to a strictly parabolic system on a manifold with uniform geometry at infinity. We expect that it could also give new short time existence for the Ricci flow without assuming the boundedness of the curvature of the initial metric. As a direct consequence, we have the following result. Corollary 1.2 Suppose (M n , gij (x)) is a complete Riemannian manifold, and suppose gij (x, t) is a solution to the Ricci flow with bounded curvature on M n × [0, T ] and with gij (x) as initial data. If G is the isometry group of (M n , gij (x)), then G remains to be an isometric subgroup of (M n , gij (x, t)) for each t ∈ [0, T ]. This paper is organized as follows. In Section 2, we study the harmonic map 3

flow coupled with the Ricci flow. In Section 3, we study the Ricci-De Turck flow and prove the uniqueness theorem. We are grateful to Professor S. T. Yau for many helpful discussions and encouragement. The second author is partially supported by the IMS of The Chinese University of Hong Kong and the first author is supported by FANEDD 200216 and NSFC 10401042.

2

Harmonic map flow coupled with the Ricci flow

Let (M n , gij (x)) and (N m , hij (y)) be two Riemannian manifolds, f : M n → N m be a map. The harmonic map flow is the following evolution equation for maps from M n to N m ,   ∂ f (x, t) = △f (x, t), for x ∈ M n , t > 0, (2.1) ∂t  n f (x, 0) = f (x), for x ∈ M , where △ is defined by using the metric gij (x) and hαβ (y) as follows △f α (x, t) = g ij (x)∇i ∇j f α (x, t),

and

α β γ ∂2f α k ∂f α ∂f ∂f − Γ + Γ . (2.2) ij βγ ∂xi ∂xj ∂xk ∂xi ∂xj Here we use {xi } and {y α} to denote the local coordinates of M n and N m respectively, Γkij and Γαβγ the corresponding Christoffel symbols of gij and hαβ . Let gij (x, t) be a complete smooth solutions of the Ricci flow with gij (x) as initial data, then the harmonic map flow coupled with Ricci flow is the following equation   ∂ f (x, t) = △ f (x, t), for x ∈ M n , t > 0, t ∂t  f (x, 0) = f (x), for x ∈ M n ,

∇i ∇j f α =

where △t is defined as above by using the metric gij (x, t) and hαβ (y). Suppose gij (x, t) is a solution to the Ricci flow on M n × [0, T ] with bounded curvature |Rm|(x, t) ≤ k0

for all (x, t) ∈ M n × [0, T ]. Let (N n , hαβ ) = (M n , gij (·, T )) be the target manifold. The purpose of this section is to prove the following theorem

Theorem 2.1 There exists 0 < T0 < T , depending only on k0 , T and n such that the harmonic map flow coupled with the Ricci flow   ∂ F (x, t) = △ F (x, t), t (2.3) ∂t  F (·, 0) = identity, 4

has a solution on M n × [0, T0 ] satisfying the following estimates |∇F | ≤ C˜1 , k−2 |∇k F | ≤ C˜k t− 2 ,

(2.4)

for all k ≥ 2,

for some constants C˜k depending only on k0 , T , k and n. The proof will occupy the rest of this section.

2.1

Expanding base and target metrics at infinity

We will construct appropriate auxiliary functions on M n and N n and do conformal deformations for the base and the target metrics. Firstly, we construct the function on (N n , hαβ ). The function can be obtained by solve certain equations [18] or smoothing certain functions by convolution [7]. Lemma 2.2 Fix p ∈ N n . Then for any a > 1, function ϕa on N n such that  on   ϕa (y) ≡ 0 d(y, p) 6 ϕa (y) 6 C0 d(y, p) on   k |∇ ϕa | 6 Ck on

there exists a C ∞ nonnegative B(p, a), N n \B(p, 2a), n

(2.5)

for k > 1,

N ,

where Ck , i = 0, 1, 2, · · · , are constants depending only on k0 and T ; the distance d(y, p), the covariant derivatives ∇k ϕa and the norms |∇k ϕa | are computed by using the metric hαβ . Proof. Let ξ be a smooth nonnegative increasing function on R such that ξ(s) = 0 for s ∈ (−∞, 45 ], and ξ = 1 for s ∈ [ 74 , ∞). For each y ∈ N n , by averaging the ) and d(p, y) over a suitable ball of the tangent space Ty N n (see for functions ξ( d(p,y) a example [7]), we obtain two smooth functions ξa and ρ. Notice that (N n , hαβ ) = (M n , gij (·, T )), thus all the covariant derivatives of the curvatures of hαβ are bounded by using Shi’s gradient estimates [20]. Then ϕa = Cξa ρ, for some constant C depending only on k0 and T , is the desired function. # Recall from [3] and [1] that on a complete manifold with bounded curvature, the injectivity radius decays at worst exponentially; more precisely, there exists a ˜ constant C(n) > 0 depending only on the dimension, and there exists a constant δ > 0 depending on n, k0 and the injectivity radius at p such that ˜

√ k0 d(y,p)

inj(N n , hαβ , y) > δe−C(n) √ ˜ k0 ϕa and set Fix a > 1, let ϕa = 4C(n) a

haαβ = eϕ hαβ . 5

.

(2.6)

(2.7)

Clearly, haαβ = hαβ on B(p, a). Note that (N n , hαβ ) = (M n , gij (·, T )), so the function ϕa is also a function on M n . Let a

gija (x, t) = eϕ gij (x, t)

(2.8)

be the new family of metrics on M n . Instead of (2.3), we will consider a new harmonic map flow  a a a   ∂ F (x, t) = △ F (x, t), t ∂t (2.3)a a   F (·, 0) = identity, a

a

where △t F is defined by using the metric gija (x, t) and haαβ (y). Before we can solve (2.3)a , we have to discuss the geometry of the new metrics haαβ (y) and gija (x, t). Let us first compute the curvature and its covariant derivatives and injectivity radius of (N n , haαβ ) as follows. By a direct computation, we get a

a

Rαβγδ

eϕ =e Rαβγδ + {|∇ϕa |2 (hαδ hβγ − hαγ hβδ ) 4 + (2∇α ∇δ ϕa − ∇α ϕa ∇δ ϕa )hβγ + (2∇β ∇γ ϕa − ∇β ϕa ∇γ ϕa )hαδ − (2∇β ∇δ ϕa − ∇β ϕa ∇δ ϕa )hαγ − (2∇α ∇γ ϕa − ∇α ϕa ∇γ ϕa )hβδ } ϕa

(2.9)

a

where Rαβγδ is the curvature of haαβ , ∇α ϕa ,∇α ∇δ ϕa and |∇α ϕa | are computed by the metric hαβ . Therefore, by combining with (2.5), we have a

a

|Rm |ha 6 e−ϕ (k0 + C(n)(C2 + C12 )) < ∞.

(2.10)

For higher derivatives, we rewrite (2.9) in a simple form a

a

Rm = eϕ {Rm + ∇ϕa ∗ ∇ϕa ∗ h2 ∗ h−1 + ∇2 ϕa ∗ h} where we use A ∗ B to express some linear combinations of tensors formed by contractions of tensor product of A and B. Note that a 1 Γαβγ − Γαβγ = [∇β ϕa δγα + ∇γ ϕa δβα − hαη hβγ ∇η ϕa ] 2 = (∇ϕa ∗ h ∗ h−1 )αβγ ,

so by induction, we have a a

a

a

k X

∇l Rm ∗

a

a

a

∇k Rm = ∇∇k−1Rm + (Γ − Γ) ∗ ∇k−1 Rm ϕa

=e {

l=0

X

i1 +···+ip =k−l

∇i1 ϕa ∗ · · · ∗ ∇ip ϕa +

6

X

i1 +···+ip =k+2

∇i1 ϕa ∗ · · · ∗ ∇ip ϕa }, (2.11)

where we denote ∇0 ϕa = 1. By combining with (2.5) and gradient estimate of Shi [20], we get a a k

− k+2 ϕa 2

|∇ R m |ha 6 e

k X C(n, k0 , k, C1, · · · , Ck+2)( |∇l Rm | + 1) l=0

− k+2 ϕa 2

(2.12)

6 C(n, k0 , T, k)e 6 C(n, k0 , T, k).

For the injectivity radius of haαβ , we know from (2.5) and (2.7) that for any y ∈ N n \B(p, 2a + 1), a

˜

√

B(y, 1) ⊃ B(y, e−2C(n)

k0 (ϕa +C1 )

)

and a

V olha (B(y, 1)) =

Z

√

˜

(e4C(n)

a

k 0 ϕa

B(y,1) √ ˜ 2C(n) k0 (ϕa −C1 )

>e

n

)2

√ ˜ −2C(n) k0 (ϕa +C1 )

V olh (B(y, e

(2.13) ))

a

where we denote by B(y, 1) the ball centered at y and of radius 1 with respect to a

metric haαβ , and V olha (B(y, 1)) its volume. Since ϕa (y) > d(y, p), for y ∈ N n \B(p, 2a + 1), there holds ˜

√

e−2C(n)

˜

k0 (ϕa +C1 )

√

6 δe−C(n)

k0 d(y,p)

(2.14)

−1

logδ √ |). By (2.6), (2.10), (2.13), (2.14) and volume for y ∈ N n \B(p, 2a + 1 + | C(n) ˜ k0 comparison theorem, we have a

˜

√

V olha (B(y, 1)) > c(n, k0 )e2nC(n) > c(n, k0 ),

k0 (ϕa −C1 )

˜

√

(e−2C(n)

k0 (ϕa +C1 ) n

)

(2.13)

By combining this with the local injectivity radius estimate in [3] or [1], we get ˜ k0 ) > 0, inj(N n , ha , y) > C(n,

for y ∈ N n \B(p, 2a + 1 + |

Consequently, we have proved the following lemma.

7

logδ −1 √ |). ˜ C(n) k0

Lemma 2.3 There exists a sequence of constants C¯0 , C¯1 , · · · , with the following a property. For all a > 1, there exists ia > 0, such that the metrics haαβ = eϕ hαβ on N n satisfy a a

k+2 a |∇k Rm |ha 6 C¯k e− 2 ϕ 6 C¯k inj(N n , haαβ ) > ia > 0

(2.15)

for k = 0, 1, · · · .

# We next estimate the curvature and the its covariant derivatives of gija (x, t) = a eϕ gij (x, t). By the Ricci flow equation, we have Z T l l Γij (·, T ) − Γij (·, t) = (g −1 ∗ ∇Ric)(·, s)ds, ∇kg(·,T ) (Γ(·, T )

− Γ(·, t)) =

Z

t

T

t

k X l=0

∗···∗

∇k+1−l Ric ∗

ip ∇g(·,T ) (Γ(·, s)

X

i1 +1+···+ip +1=l

1 ∇ig(·,T ) (Γ(·, s) − Γ(·, T ))

− Γ(·, T )) ∗ g k ∗ g −(k+1) (·, s)ds.

(2.16)

By combining with the gradient estimates of Shi [20] and induction on k, we have Z T  1   √ ds, |Γ(·, T ) − Γ(·, t)| 6 C(n, k , T )  0  s t (2.17) |∇g(·,T ) (Γ(·, T ) − Γ(·, t))| 6 C(n, k0 , T )(1 + |logt|),    k−1  k |∇g(·,T ) (Γ(·, T ) − Γ(·, t))| 6 C(n, k0 , T, k)t− 2 , for k ≥ 2.

Since

∇kg(·,t) ϕa =

k−1 X l=0

a ∇k−l g(·,T ) ϕ ∗

X

i1 +1+···+ip +1=l

i

p 1 ∇ig(·,T ) (Γ(·, t)−Γ(·, T ))∗· · ·∗∇g(·,T ) (Γ(·, t)−Γ(·, T ))

for k > 1, the combination with (2.17) and (2.5) gives   |∇ ϕa | + |∇2g(·,t) ϕa | 6 C(n, k0 , T ),   g(·,t) |∇3g(·,t) ϕa | 6 C(n, k0 , T )(1 + |logt|),    |∇k ϕa | 6 C(n, k , T, k)t− k−3 2 , for k > 4. 0 g(·,t)

(2.18)

Then by combining (2.11) and (2.18), the curvature and the covariant derivatives of g a (·, t) can be estimated as follows aka

|∇ Rm |ga (·,t) 6 C(n, k0 , T, k)e−

k+2 a ϕ 2

Summing up, the above estimates give the following 8

k

t− 2 ,

for k > 0.

Lemma 2.4 There exists a sequence of constants k¯0 , k¯1 , · · · , with the following a property. For all a > 1, the metrics gija (·, t) = eϕ gij (·, t) on M n satisfy a a l

|∇ Rm |ga (·,t) 6 k¯l e−

k+2 a ϕ 2

l

t− 2 ,

for

(2.19)

l > 0.

on M n × [0, T ]. # We remark that the fact that the curvatures of and are uniformly bounded (independent of a) is essential in our argument. While the injectivity radius bound ia may depend on a. For the new family of metrics gija (·, t), we have the following lemma. haαβ

gija (·, t)

Lemma 2.5 ∂ a g ∂t ij ak

∂ Γ ∂t ij

a2

a

a

a

a

a

a

= eϕ (−2Rij + (∇ ϕa + ∇ϕa ∗ ∇ϕa ) ∗ g ∗ (g)−1 ), a

a

a

a

a

a

a3

a

a

= eϕ (g)−1 ∗ ∇Ric + eϕ (g)−2 ∗ g ∗ (Ric ∗ ∇ϕa + ∇ ϕa ) a

a

a3

a a

a

+eϕ (g)−3 ∗ (g)2 ∗ [(∇ϕ)3 + ∇ ϕa ], e

ϕa 2

a

a2

3

a

a3

k

a

ak

a

|∇ϕa |ga (·,t) + eϕ |∇ga (·,t) ϕa |ga (·,t) 6 C(n, k0 , T ), e 2 ϕ |∇ga (·,t) ϕa |ga (·,t) 6 C(n, k0 , T )(1 + | log t|), e 2 ϕ |∇ga (·,t) ϕa |ga (·,t) 6 C(n, k0 , T, k)

1

t

k−3 2

(2.20)

, for k > 4.

Proof. Note that a

Γ − Γ = g ∗ g −1 ∗ ∇ϕa a

a

∇2 ϕa = ∇2 ϕa + (Γ − Γ) ∗ ∇ϕa a X ∇k ϕa = g k−1 ∗ (g −1 )k−1 ∗ ∇i1 ϕa ∗ · · · ∗ ∇ip ϕa i1 +···+ip =k

where the summation is taken over all indices ij > 0. By combining this with (2.18), a

we get the desired estimates for |∇k ga (·,t) ϕa |ga (·,t) . One the other hand, since a

a

Rij = Rij + (∇2 ϕa + ∇ϕa ∗ ∇ϕa ) ∗ g ∗ g −1, it follows that a

a

a

∇i Rjl = ∇i Rjl + g ∗ g

a−1

a

a

∗ (Ric ∗ ∇ϕ + ∇3 ϕa ) a

a

a

a

+ (g a )2 ∗ (g a )−2 ∗ (∇2 ϕa ∗ ∇ϕa + (∇ϕa )3 ). 9

By combining this with ∂ ak ∂ ∂ Γij = Γkij + (g −1 ∗ g ∗ ∇ϕa ) ∂t ∂t ∂t kl = −g (∇i Rjl + ∇j Rli − ∇l Rij ) + g ∗ g −2 ∗ Ric ∗ ∇ϕa , we have proved the lemma. #

2.2

Modified harmonic map flow

The purpose of this subsection is to solve the equation (2.3)a . More precisely, we will prove the following theorem Theorem 2.6 There exists 0 < T1 < T , depending only on k0 , T and n such that for all a > 1 the modified harmonic map flow flow coupled with the Ricci flow  a a a   ∂ F (x, t) = △t F (x, t) ∂t (2.3)a a   F (·, 0) = identity has a solution on M n × [0, T0 ] satisfying the following estimates a a

|∇F | ≤ C(n, k0 , T ), aka

|∇ F | ≤ C(n, k0 , T, k)t−

k−2 2

(2.21) for all k ≥ 2,

,

for some constants C(n, k0 , T, k) depending only on n, k0 , T , and k but independent of a. a

Note that F is viewed as a map from (M n , gija (x, t)) and (N n , haαβ (y)), all the covariant derivatives and the norms in Theorem 2.6 are computed with respect gija (x, t) and haαβ (y). We begin with a easier short time existence of (2.3)a where the short time interval may depend on a. 2.2.1

Short time existence of the modified harmonic map flows

We consider (2.3)a with general initial data. Theorem 2.7 Let f be a smooth map from M n to N n with a

a

E0 = sup |∇f |gija (·,0),haαβ (x) + sup |∇2 f |gija (·,0),haαβ (x) < ∞. x∈M n

x∈M n

10

Then there exists a δ0 > 0 such that the initial problem  a a a   ∂ F (x, t) = △ F (x, t), t ∂t a   F (x, 0) = f (x),

(2.3)a ′

has a smooth solution on M n × [0, δ0 ] satisfying the following estimates sup (x,t)∈M n ×[0,δ0 ]

a a

|∇F |gija (·,0),haαβ (x, t)+

(x,t)∈M n ×[0,δ0 ]

a a k

sup (x,t)∈M n ×[0,δ0 ]

a a

sup

|∇2 F |gija (·,0),haαβ (x, t) 6 C(n, k0 , T, a, E0 ),

|∇ F |gija (·,0),haαβ (x, t) 6

C(n, k0 , T, k, a) t

k−2 2

(2.22)

,

for k ≥ 3. We will prove the theorem by solving the corresponding initial-boundary value problem on a sequence of exhausted bounded domains D1 ⊆ D2 ⊆ · · · with smooth boundaries and Dj ⊇ Bgaa (·,0) (P, j + 1) :     

a

   

a a ∂ aj F (x, t) = △t F j (x, t), for x ∈ Dj and t > 0, ∂t F j (x, 0) = f (x) for x ∈ Dj ,

(2.23)

a

F j (x, t) = f (x)

for x ∈ ∂Dj , a

and F will be obtained as the limit of a convergent subsequence of F j as j → ∞. Here P is a fixed point on M n and Bgaa (·,0) (P, j + 1) is the geodesic ball centered at P of radius j + 1 with respect to the metric gija (·, 0) a

The following lemma gives the zero-order estimate of F j . Lemma 2.8 There exist positive constants 0 < T2 < T and C > 0 such that for a any j, if (2.23) has a smooth solution F j on D¯j × [0, T3 ] with T3 ≤ T2 , then we have a √ d(N n ,ha ) (f (x), F j (x, t)) ≤ C t,

for any (x, t) ∈ Dj × [0, T3 ].

(2.24)

a

Proof. For simplicity, we drop the superscripts a and j of F j . Note that the distance function d(N n ,ha ) (y1 , y2 ) can be regarded as a function on N n × N n . Set ψ(y1 , y2 ) = 21 d2(N n ,ha ) (y1 , y2) and ρ(x, t) = ψ(f (x), F (x, t)). Then ψ(x, t) is smooth when ψ < 12 i2a . Now we compute the equation of ρ(x, t): (

a ∂dha a α ∂ − △t )ρ = −dha (f (x), F (x, t)) △t f − Hess(ψ)(Xi, Xj )(g a )ij ∂t ∂y1 α

11

(2.25)

where the vector fields Xi , i = 1, 2, · · · , n, in local coordinates (y1α , y2β ) on N n × N n are defined as follows ∂f α ∂ ∂F β ∂ Xi = . + ∂xi ∂y1α ∂xi ∂y2β To handle the first term on the right hand side of (2.25), we use a

a

Γkij (x, t)−Γkij (x, 0) = Γkij (x, t)−Γkij (x, 0)+g(·, t)∗g −1(·, t)∗∇ϕa +g(·, 0)∗g −1(·, 0)∗∇ϕa, to conclude that

a

|△t f |ga (·,t),ha 6 C(n, k0 , T )E0 .

Recall from Lemma 2.3 that the curvature of the metric haαβ is bounded by C¯0 . We claim that if dha (f (x), F (x, t)) ≤ min{ i4a , √π ¯ }, then 4

C0

1 a a Hess(ψ)(Xi, Xj )(g a )ij ≥ |∇F |2ga ,ha − C 2

(2.26)

where C = C(E0 , C¯0) depends only on E0 and C¯0 . Indeed, recall the computation of Hess(ψ) in [19]. For any (u, v) ∈ D = {(u, v) : (u, v) ∈ N n × N n , dN n (u, v) < min{ i2a , √π ¯ }}, let γuv be the minimal geodesic from 2

C0

u to v and e1 ∈ Tu N n be the tangent vector to γuv at u. Then e1 (u, v) defines a smooth vector field on D. Let {ei } be an orthonormal basis for Tu N n which depends u smoothly. By parallel translation of {ei } along γ, we define {¯ ei } an orthonormal basis for Tv N n . Thus {e1 , · · · en , e¯1 , · · · e¯n } is a local frame on D. Then For any X = X (1) + X (2) ∈ T(u,v) D, where X

(1)

=

n X

ξiei , and X

(2)

=

i=1

n X

ηi e¯i ,

i=1

by the formula (16) in [19], Hess(ψ)(X, X) =

n X i=1

−

2

(ξi − ηi ) +

Z

r 0

Z

r 0

th∇e1 V, ∇e1 V i +

thR(e1 , V )V, e1 i −

Z

r 0

Z

0

r

th∇e¯1 V, ∇e¯1 V i

thR(¯ e1 , V )V, e¯1 i

where V is a Jacobi field on geodesic σ (connecting (v, v) to (u, v)) and σ ¯ (connecting (u, u) to (u, v)) with X as the boundary values, where X is extended to be a local vector field by letting its coefficients with respect to {e1 , · · · en , e¯1 , · · · e¯n } be constant(see [19]). By the Jacobi equation, |V |, |∇e1 V | and |∇e¯1 V | are bounded. Thus we have |Hess(ψ)|ha ≤ C(ia , C¯0 ) 12

under the assumption of the claim. So the mixed term a a

∂2ψ f α F β (g a )ij ∂y1α ∂y2β i j

in Hess(ψ)(Xi , Xj )(g a )ij

can be bounded by C(E0 , C¯0 )E0 |∇F |ga ,ha . On the other hand, the Hessian comparison theorem for the points which are not in the cut locus gives a a ∂ψ ∂ψ π γ − ( Γ ◦ F ) γ ≥ haαβ , αβ α β ∂y2 ∂y2 ∂y2 4 a ∂ψ π a ∂ψ γ − ( Γ ◦ f ) ≥ h . γ αβ ∂y1 α ∂y1 β ∂y1 4 αβ

Thus the claim follows. Let π T2′ = max{t ≤ T : sup dha (f (x), F (x, t)) ≤ min{ia , p }}. D 4 C¯0 a

¯ × [0, T3 ] with T3 6 T2′ , by (2.25) and If F (x, t) is a smooth solution of (2.23) on D (2.26), we get (

a ∂ 1 a a √ − △t )ρ ≤ − |∇F |2ga ,ha + C ρ + C ∂t 2

(2.27)

on D ×[0, T3 ], for some constant C depending on E0 , ia and C¯0 . Note that the initial and boundary values of ρ are zero, so by the maximum principle, we get a √ dha (f (x), F (x, t)) ≤ C t.

This implies T2′ ≥ min{

min{ia , √π ¯ }2 4

C0

C2

, T3 }.

Hence the lemma holds with T2 = min{

min{ia , √π ¯ }2 4

C2

C0

, T }.

. # After we have the zero order estimate (2.24), we now apply the standard parabolic equation theory to get the following short time existence for (2.23). Lemma 2.9 There exists a positive constant T3 ≤ T2 depending only on the dimension n, a, T2 and C in Lemma 2.8 such that for each j, the initial-boundary value a problem (2.23) has a smooth solution F j on D¯j × [0, T3 ].

13

Proof. For an arbitrarily fixed point x0 in D¯j , choose normal coordinates {xi } and {y α } on (M n , g a (·, 0)) and (N n , ha ) around x0 and f (x0 ) respectively. The equation (2.23) can be written as a a ∂y α ∂2yα ∂y α ∂y β ∂y γ (x, t) = (g a )ij (x, t){ i j − Γkij (x, t) k + Γαβγ (y 1 (x, t), · · · , y n (x, t)) i j }. ∂t ∂x ∂x ∂x ∂x ∂x (2.28) a

Note that Γαβγ (f (x0 )) = 0. By applying (2.24) and a result of Hamilton (Corollary (4.12) in [11]), we know that the coefficients of the quadratic terms on the RHS of (2.28) can be as small as we like provided T3 > 0 sufficiently small (independent of x0 and j). Now for fixed j, we consider the corresponding parabolic system of the difference a

of the map F j and f (x). Clearly the coefficients of the quadratic terms of the gradients are also very small. Thus, whenever (2.23) has a solution on a time interval [0, T3′ ] with T3′ ≤ T3 , we can argue exactly as in the proof of Theorem 6.1 a a in Chapter VII of the book [14] to bound the norm of ∇F j over D¯j × [0, T ′ ] by 3

a

∞

a constant depending only on the L bound of F in (2.23), the map f (x), the domain Dj , and the metrics gija (·, t) and haαβ over the domain Dj+1. Hence by the same argument as in the proof of Theorem 7.1 in Chapter VII of the book [14], we a

deduce that the initial-boundary value problem (2.23) has a smooth solution F j on D¯j × [0, T3 ]. # a

Unfortunately, the gradient estimates of F j in the proof of the above lemma a

depend also on the domain Dj . In order to get a convergent subsequence of F j , we a

have to estimate the covariant derivatives of F j uniformly in each compact subsets. Before we proceed, we need some preliminary estimates. a

Lemma 2.10 The covariant derivatives of F j satisfy the following equations a a a a a a a a a ∂ a aj ∇F = △t ∇F j + Ric(M n ) ∗ ∇F j + RN ∗ (∇F j )3 , ∂t k−1 a a a a X a a a a a a a ∂ ak a j a a a k j ∇ F = △t ∇ F + ∇l [(RM + RN ∗ (∇F j )2 + eϕ RM + ∇2 eϕ ) ∗ ∇k−l F j ], ∂t l=0

(2.29)

a

a

a

a

where ∇l (A ∗ B) represents the linear combinations of ∇l A ∗ B,∇l−1 A ∗ ∇B, · · · , a

a

a

a

a

a

a

A ∗ ∇l B, and ∇2 eϕ = eϕ (∇2 ϕa + ∇ϕa ∗ ∇ϕa ).

Proof For k = 1, by direct computation and Ricci formula, we have a a a a a a aα a a a a a a ∂ a aα ∇i F = △t ∇i F α − Ril ∇l F α + Rβδγ ∇i F β ∇k F δ ∇l F γ (g a )kl . ∂t

14

For k > 2, by Ricci formula, it follows a a

a

a

a

a

a

a

a

a

a a

a

a

∇△∇k−1F j = △∇k F j + ∇[(RM + RN ∗ (∇F j )2 ) ∗ ∇k−1 F j ]. Recall from (2.20) that a a ∂ ai a a a Γjk = ∇(eϕ RM + ∇2 eϕ ). ∂t

Then we have a a a a a a a a a a a a ∂ ∂ ak aj k j k−1 j ϕa 2 ϕa k−1 j ∇ F − △t ∇ F =∇[( − △t )∇ F ] + ∇(e RM + ∇ e ) ∗ ∇ F ∂t ∂t a

a a

a

a

a

a

a

a

a

a a

a

a

+ RN ∗ ∇F j ∗ ∇2 F j ∗ ∇k−1 F j + ∇[(RM + RN ∗ (∇F j )2 ) ∗ ∇k−1 F j ] a a a a a a a a a a a a ∂ a a a =∇[( − △)∇k−1 F j ] + ∇{(RM + RN ∗ (∇F j )2 + (eϕ RM + ∇2 eϕ ) ∗ ∇k−1F j } ∂t k−1 a a a a X a a a a a a a = ∇l [(RM + RN ∗ (∇F j )2 + eϕ RM + ∇2 eϕ ) ∗ ∇k−l F j ]. l=0

This proves the lemma. # For each k > 0, let ξk be a smooth non-increasing function from (−∞, +∞) to 1 [0, 1] so that ξk (s) = 1 for s ∈ (−∞, 12 + 2k+1 ], and ξk (s) = 0 for s ∈ [ 21 + 21k ); moreover for any ǫ > 0 there exists a universal Ck,ǫ > 0 such that |ξk′ (s)| + |ξk′′ (s)| ≤ Ck,ǫ ξk (s)1−ǫ . Lemma 2.11 There exists a positive constant T4 , 0 < T4 ≤ T3 independent of j such that for any geodesic ball Bga (·,0) (x0 , δ) ⊂ Dj , there is a constant C = C(a, δ, E0 , C¯0 , k¯0) such that the smooth solution of (2.23) satisfies a a

|∇F j |ga (·,t),ha 6 C on Bga (·,0) (x0 , 3δ4 ) × [0, T4 ].

a a

Proof. We compute the equation of |∇F j |2ga (·,t),ha . For simplicity, we drop the superscript j. By (2.20), we have (

a a a a a a a a a a a a a ∂ − △t )|∇F |2ga (·,t),ha = hRic(M n ) ∗ ∇F + RN ∗ (∇F )3 , ∇F iga ,ha − 2|∇2 F |2ga (·,t),ha ∂t a

ϕa

a

n

a

a

a a

a a

+ e (Ric(M ) + ∇2 ϕa + ∇ϕa ∗ ∇ϕa ) ∗ ∇F ∗ ∇F a a

a a

a a

6 −2|∇2 F |2ga (·,t),ha + C(n, k0 , T )|∇F |2ga (·,t),ha + C(n)C¯0 |∇F |4ga (·,t),ha . (2.30) 15

Setting

a a

ρA (x, t) = (d2ha (f (x), F (x, t)) + A)|∇F |2ga (·,t),ha where A is determined later, and combining with (2.27) and (2.24), we have a a a a a a ∂ ρA 6 △ρA − 2|∇2 F |2ga ,ha (d2ha (f (x), F (x, t)) + A) − |∇F |4ga ,ha ∂t a a a a a +C(n)C¯0 (d2ha (f (x), F (x, t)) + A)|∇F |4ga ,ha + C|∇F |2ga ,ha + C(n, k0 , T )ρA a

a a

+2|∇d2ha (f (x), F (x, t))|ga |∇|∇F |2ga (·,t),ha |ga . Since a

a

a a

a

|∇d2ha (f (x), F (x, t))|ga 6 2dha (f (x), F (x, t))(|∇F |ga ,ha + |∇f |ga ,ha ) √ a a √ 6 C t + C t|∇F |ga ,ha , a a

a a

a a

|∇|∇F |2ga (·,t),ha |ga 6 2|∇2 F |ga ,ha |∇F |ga ,ha , 1 , and applying Cauchy-Schwartz by choosing T4 = min{T3 , 4C(n)1C¯0 C 2 }, A = 4C(n) C¯0 inequality, we have a ∂ ( − △)ρA 6 −(C(n)C¯0 )ρ2A + C. ∂t Here and in the following we denote by C various constants depending only on n, k0 , T , E0 and a. d a (x ,·) We compute the equation of u = ξ1 ( g (·,0)δ 0 )ρA at the smooth points of dga (·,0) (x0 , ·), a

△dga (·,0) (x0 , ·) nk0 T |ξ1′′ | ∂ a +e )ρA . ( −△)u 6 Cξ1 −(C(n)C¯0 )ρ2A ξ1 −2(g a )ij ∇i ξ1 ∇j ρA +(−ξ1′ ∂t δ δ2 By the Hessian comparison theorem and the fact that −ξ1′ > 0, we have a

a

a

a

a

a

∇i ∇j dga (·,0) 6 ∇0i ∇0j dga (·,0) + (Γ(·, 0) − Γ(·, t)) ∗ ∇dga (·,0) 1 + k¯0 dga (·,0) 6 ( + C)gija (·, 0), dga (·,0) a C|ξ1′ | . −ξ1′ △dga (·,0) 6 δ These two inequalities hold on the whole manifold in the sense of support functions. Thus for any x1 ∈ M n , there is a function hx1 which is smooth on a neighborhood of x1 with hx1 (·) > dga (·,0) (x0 , ·), hx1 (x1 ) = dga (·,0) (x0 , x1 ) and a

−ξ1′ △hx1 |x1 6 2 16

C|ξ1′ | . δ

Indeed, hx1 can be chosen to having the form dga (·,0) (q, ·) + dga (·,0) (q, x0 ) for some q, a

so we may require |∇hx1 |ga (·,0) 6 1. Let (x1 , t0 ) be the maximum point of u over M n × [0, T4 ]. If t0 = 0, then ξ1 ρA 6 E0 . Assume t0 > 0. At the point (x1 , t0 ), we ∂ have ∂t (ξ1 ρA )(x1 , t0 ) > 0. If x1 does not lie on the cut locus of x0 , then 1 nk0 T |ξ1′ |2 2 ¯ 0 6 −C(n)C0 ρA ξ1 + 2 (e + 2C(|ξ1′ | + |ξ1′′ |))ρA + Cξ1 δ ξ1 p C 6 −C(n)C¯0 ρ2A ξ1 + 2 ξ1 ρA + Cξ1 δ C 6 −C(n)C¯0 ρ2A ξ1 + 4 δ C 6 −C(n)C¯0 (ρA ξ1 )2 + 4 . δ

We get ξ1 ρA 6 max{E0 ,

s

C } C(n)C¯0 δ 4

for all (x, t) ∈ Bga (·,0) (x0 , δ) × [0, T4 ]. If x1 lies on the cut locus of x0 , then by applying the standard support function technique (see for example [18]), the above maximum principle argument still works. So by the definition of ξ1 and ρA , we have a a

C δ 3δ on Bga (·,0) (x0 , 4 ) × [0, T4 ]. The proof of the lemma is completed. |∇F j |ga (·,t),ha 6

# The next lemma estimates the higher derivatives in terms of the bound of

a a

|∇F j |ga (·,t),ha .

a

Lemma 2.12 Let F be a smooth solution of equation a a ∂ − △)F = 0 ∂t on Bga (·,0) (x0 , δ) × [0, T¯], with T¯ 6 T . Suppose

(

a a

sup (x,t)∈Bga (·,0) (x0 , 3δ )×[0,T¯] 4

and

|∇F |gija (·,0),haαβ (x, t) 6 E1 , (2.31)

a a

sup x∈Bga (·,0) (x0 , 3δ ) 4

|∇2 F |gija (·,0),haαβ (x, 0) 6 E1 .

Then for any k > 2, there exists a positive constant C = C(k, E1 , δ, k0 , T ) > 0 such that a a

on

Bga (·,0) (x0 , 2δ )

× [0, T¯ ].

|∇k F |ga (·,t),ha ≤ Ct− 17

k−2 2

(2.32)

Proof. The proof is using the Bernstein trick. We assume δ < 1 without loss of generality. For k = 2, from (2.15), (2.19), (2.20) and (2.29), we have 1

a a a a a a X a a a a a a a ∂ a a a ( − △t )|∇2 F |2ga (·,t),ha = h ∇l [(RM + RN ∗ (∇F j )2 + eϕ RM + ∇2 eϕ ) ∗ ∇2−l F ], ∇2 F iga ,ha ∂t l=0

a a

a

a

a

a

a

a a

− 2|∇3 F |2ga (·,t),ha + eϕ (Ric(M n ) + ∇2 ϕa + ∇ϕa ∗ ∇ϕa ) ∗ (∇2 F )2

a a a a C a a 6 −2|∇3 F |2ga (·,t),ha + C|∇2 F |2ga (·,t),ha + √ |∇2 F |ga (·,t),ha . t (2.33)

In this lemma, we use C to denote various constants depending only on E1 , k0 , T , k and δ. Note that by (2.30) and (2.33), we have a a a a a ∂ − △t )|∇F |2ga (·,t),ha 6 −2|∇2 F |2ga (·,t),ha + C, ∂t a a a a a ∂ C 2 ( − △t )|∇ F |ga (·,t),ha 6 C|∇2 F |ga (·,t),ha + √ . ∂t t

(

So by setting a a √ a a √ v = |∇2 F |ga (·,t),ha − 2C t + 2C T + |∇F |2ga (·,t),ha ,

we have (

a a a a a ∂ − △t )v 6 −2|∇2 F |2ga (·,t),ha + C|∇2 F |ga (·,t),ha + C ∂t 6 −v 2 + C.

Since at t = 0,

√ v 6 2C T + E1 + E12

), we apply the maximum principle as in Lemma 2.11 to get on Bga (·,0) (x0 , 3δ 4 ξ2 (

dga (·,0) (x0 , ·) )v 6 C δ

on Bga (·,0) (x0 , 3δ4 ) × [0, T¯]. This implies a a

|∇2 F |ga (·,t),ha ≤ C on Bga (·,0) (x0 , ( 21 +

1 )δ) 23

× [0, T¯].

18

Now we estimate the third-order derivatives. From Shi’s gradient estimate [20], a a

the estimate of |∇2 F |ga (·,t),ha ≤ C and (2.15), (2.19), (2.20) and (2.29), we have: (

2 a a a a a a a X a a a a a a ∂ a a a − △t )|∇3 F |2ga (·,t),ha = h ∇l [(RM + RN ∗ (∇F )2 + eϕ RM + ∇2 eϕ ) ∗ ∇3−l F ], ∇3 F iga ,ha ∂t l=0 a a

a

a

a

a

a a

a

− 2|∇4 F |2ga (·,t),ha + eϕ (Ric(M n ) + ∇2 ϕa + ∇ϕa ∗ ∇ϕa ) ∗ (∇3 F )2 a a a a C a a 6 −2|∇4 F |2ga (·,t),ha + C|∇3 F |2ga (·,t),ha + |∇3 F |ga (·,t),ha . t (2.34) on Bga (·,0) (x0 , ( 21 + 18 )δ) × [0, T¯]. a

a

a

a

a

|∇3 eϕ |ga 6 C(1 + | log t|), and |∇2 Rm | 6 By (2.33), it follows (

a

Here we used the estimates |∇4 eϕ |ga 6

C √ , t

C . t

a a a a a ∂ C − △t )|∇2 F |2ga (·,t),ha 6 −2|∇3 F |2ga (·,t),ha + √ ∂t t a a

(2.35)

a a

on Bga (·,0) (x0 , ( 12 + 81 )δ) × [0, T¯ ]. Let v = (|∇2 F |2ga (·,t),ha + A)|∇3 F |2ga (·,t),ha , where A = 100

a a

sup Bga (·,0) (x0 ,( 12 +

1 )δ)×[0,T¯] 23

|∇2 F |gija (·,t),haαβ (x, t) + C. By a direct computation, it

follows a a a a a a a ∂ C ( − △)v ≤ |∇3 F |2ga (·,t),ha (−2|∇3 F |2ga (·,t),ha + √ ) + (|∇3 F |2ga (·,t),ha + A) ∂t t a a a a C a a ×(−2|∇4 F |2ga (·,t),ha + C|∇3 F |2ga (·,t),ha + |∇3 F |ga (·,t),ha ) t a a

a a

a a

+8|∇2 F |ga (·,t),ha |∇3 F |2ga (·,t),ha |∇4 F |ga (·,t),ha . Since a a 2

8|∇ F |

g a (·,t),ha

a a a a 3 2 |∇ F |ga (·,t),ha |∇4 F |ga (·,t),ha

6

a a a a a a 3 4 4 2 −|∇ F |ga (·,t),ha +16|∇ F |ga (·,t),ha |∇2 F |2ga (·,t),ha ,

we deduce (

a a a C a a C a a ∂ − △)v 6 −|∇3 F |4ga (·,t),ha + |∇3 F |ga (·,t),ha + √ |∇3 F |2ga (·,t),ha ∂t t t

and a a a a a √ a a ∂ ( − △)(tv) 6 v − t|∇3 F |4ga (·,t),ha + C|∇3 F |ga (·,t),ha + C t|∇3 F |2ga (·,t),ha ∂t a a a a √ √ √ a a 1 6 − {t2 |∇3 F |4ga (·,t),ha − C t( t|∇3 F |ga (·,t),ha ) − tv − C t(t|∇3 F |2ga (·,t),ha )} t 1 (tv)2 6 − { 5 2 − C}. t 10 C 19

So at the maximum point of ξ3 ( in Lemma 2.11, we have

dga (·,0) (x0 ,·) )(tv), δ

applying the maximum principle as

|ξ ′ |2 1 ξ3 (tv)2 0 6 − { 5 2 − Cξ3 } + C( 3 + |ξ3′′|)(tv) t 10 C ξ3 2 p 1 ξ3 (tv) 6 − { 5 2 − Cξ3 − Ct ξ3 (tv)} t 10 C 1 ξ3 (tv)2 6 − { 6 2 − C 4 }, t 10 C which gives ξ3 (tv) 6

√ 106 C 6 .

Thus by the definition of v and ξ3 , we get a a

1

|∇3 F |ga (·,t),ha ≤ Ct− 2

on B0 (x0 , ( 12 + 214 )δ) × [0, T¯ ]. Now we estimate the higher derivatives by induction. Suppose we have proved that a a l

l−2

for l = 3, · · · , k − 1 |∇ F |ga (·,t),ha ≤ Ct− 2 on B0 (x0 , ( 21 + 21k )δ) × [0, T¯ ]. By (2.29), we have k−1

(

a a a a a a X a a a a a a a ∂ a a a ∇l [(RM + RN ∗ (∇F )2 + eϕ RM + ∇2 eϕ ) ∗ ∇k−l F ], ∇3 F iga ,ha − △t )|∇k F |2ga (·,t),ha = h ∂t l=0

−

a a k+1 2|∇ F |2ga (·,t),ha a

a

ϕa

n

a

a

6 −2|∇k+1 F |2ga (·,t),ha + C|∇k F |2ga (·,t),ha + C(n) ϕa

a

a

ϕa

a a k−l

+ e RM + ∇2 e ]|ga ,ha |∇

20

a

a

a

a a k 2

a

+ e (Ric(M ) + ∇ ϕ + ∇ϕ ∗ ∇ϕ ) ∗ (∇ F ) a a

a

2

a a k

k−1 X l=1

a

a

a

a a

|∇l [RM + RN ∗ (∇F )2

F |ga ,ha |∇ F |ga (·,t),ha .

By the induction hypothesis, the local derivative estimate of Shi, and (2.15), (2.19) and (2.20), it follows k−1 X l=0

k−1 X l=0

a

a

a

l

a

a a k−l

|∇ RM | |∇ ga

F |ga (·,t),ha 6

a

a a

l=0

k−1 X l=0

a

a

a

a

a

a

C t

k−1 2

t

k−2 2

t

k−1 2

a

|∇l+2 eϕ |ga |∇k−l F |ga (·,t),ha 6 a

t

a

|∇l [RN ∗ (∇F )2 ]|ga |∇k−l F |ga (·,t),ha 6 k−1 X

C

C

a

|∇l eϕ RM |ga |∇k−l F |ga (·,t),ha 6

k−1 2

C

, a a

+ C|∇k F |ga (·,t),ha , ,

.

This gives a a a a a a a ∂ C a a − △t )|∇k F |2ga (·,t),ha 6 −2|∇k+1 F |2ga (·,t),ha + C|∇k F |2ga (·,t),ha + k−1 |∇k F |ga (·,t),ha , ∂t t 2 a a a a a C ∂ ( − △t )|∇k F |ga (·,t),ha 6 C|∇k F |ga (·,t),ha + k−1 , ∂t t 2 a a a a a ∂ C ( − △t )|∇k−1F |2ga (·,t),ha 6 −2|∇k F |2ga (·,t),ha + + k− 5 . ∂t t 2

(

Let ε =

2(k−3) k−2

(

− 1, then 0 ≤ ε < 1 for k ≥ 4. It is clear that

a a a a a C ak a ε ∂ k 1+ε − △t )|∇k F |1+ε 6 C| ∇ F | + a a a a k−1 |∇ F |g a (·,t),ha , g (·,t),h g (·,t),h ∂t t 2

and a a a a a a a a a ∂ k−1 2 k 2 k ( − △t )(|∇k F |1+ε + | ∇ F | ) 6 −2| ∇ F | C| ∇ F |1+ε a a a a a a g (·,t),h g (·,t),h g (·,t),h g a (·,t),ha ∂t C C a a + k−1 |∇k F |εga (·,t),ha + k− 5 , t 2 t 2 on Bga (·,0) (x0 , ( 21 + 21k )δ) × [0, T¯ ]. Let a a a

a

k−1 v = tk−3 (|∇k F |1+ε F |2ga (·,t),ha ). g a (·,t),ha + |∇

Then we have a a a v C a a ∂ C ( − △)v 6 (k − 3) + tk−3 (−|∇k F |2ga (·,t),ha + k−1 |∇k F |εga (·,t),ha + k− 5 ∂t t t 2 t 2 √ ε √ 2 1 1+ε − C tv 1+ε − C t} 6 − {v t 2 1 6 − {v 1+ε − C} t 21

d a (x ,·) on Bga (·,0) (x0 , ( 21 + 21k )δ) × [0, T¯ ]. Similarly, at the maximum point of ξk ( g (·,0)δ 0 )v, we have 2 |ξ ′ |2 1 0 6 − {ξk v 1+ε − Cξk } + C( k + |ξk′′ |)v t ξk 1+ε 2 1 6 − {ξk v 1+ε − Cξk 2 v − C} t 2 1 1 6 − { ξk v 1+ε − C} t 2 2 1 1 6 − { (ξk v) 1+ε − C}. t 2

since

2 1+ε

> 1. So we proved the k-th order estimate a a k

|∇ F |ga (·,t),ha ≤ Ct− on Bga (·,0) (x0 , ( 21 +

1 )δ) 2k+1

k−2 2

× [0, T¯ ]. This completes the proof of the lemma.

#

Now we are ready to prove Theorem 2.7. Proof of Theorem 2.7. Since Dj ⊇ Bga (·,0) (P, j + 1), by choosing δ = 1 and T¯ = T4 in Lemma 2.11 and a

Lemma 2.12, we get a convergent subsequence of F j (as j → ∞) on Bga (·,0) (P, j) × a

a

[0, T4 ]. Denote the limit by F (as j → ∞). Then F is the desired solution of (2.3)′a with estimates (2.22). Finally we prove a uniqueness theorem for the solutions of (2.3)′a with estimates (2.22). a

a

Lemma 2.13 Let F and F¯ be two solutions of the intial problem (2.3)′a on [0, T¯], a a T¯ 6 T , with estimates (2.22). Then F = F¯ on [0, T¯]. a

a

Proof Set ψ(y1 , y2 ) = 12 d2(N n ,ha ) (y1 , y2 ) and ρ(x, t) = ψ(F (x, t), F¯ (x, t)). Then ψ(x, t) is smooth when ψ < 12 i2a . Now by the same calculation as in Lemma 2.8, we have: a ∂ ( − △t )ρ = −Hess(ψ)(Xi , Xj )(g a )ij ∂t where the vector fields Xi , i = 1, 2, · · · , n, in local coordinates (y1α , y2β ) on N n × N n are defined as follows a a ∂F α ∂ ∂ F¯ β ∂ Xi = + . ∂xi ∂y1α ∂xi ∂y2β 22

By the estimates (2.22), we know that there is a constant 0 < T¯′ 6 T¯ such that there holds i2 π 2 ρ < min{ a , ¯ }. 8 8C0 on M n × [0, T¯′ ]. Similarly as in the proof of Lemma 2.8. By using the computation of Hess(ψ) in [19] (the formula (16) in [19]), for any (u, v) ∈ D = {(u, v) : (u, v) ∈ N n × N n with dN n (u, v) < min{ i2a , √π ¯ }}, and any X ∈ T(u,v) D, 2

C0

Hess(ψ)(X, X) > −

Z

r

thR(e1 , V )V, e1 i −

0

Z

r 0

thR(¯ e1 , V )V, e¯1 i

where V is a Jacobi field on geodesic σ (connecting (v, v) to (u, v)) and σ ¯ (connecting a a (u, u) to (u, v)) with X as the boundary values. Since |∇F |ga ,ha and |∇F¯ |ga ,ha are bounded, we know from above formula that Hess(ψ)(Xi , Xj )(g a )ij > −Cρ

on M n × [0, T¯′ ]. Thus we have

(

a ∂ − △t )ρ 6 Cρ ∂t

on M n × [0, T¯′ ]. By the maximum principle, it follows that ρ = 0 on M n × [0, T¯′]. Then the lemma follows by continuity method. # 2.2.2

Proof of theorem 2.6 and Theorem 2.1

Proof of Theorem 2.6. Let us check the initial data. Now f = identity, so a

|∇f |2ga (·,0),ha = g ij (·, 0)gij (·, T ) 6 ne2nk0 T

a

a

(2.36)

a

|∇2 f |2ga (·,0),ha = |Γkij (·, 0) − Γkij (·, T )|ga (·,0),ha Z T a a a a a 6 C(n, k0 , T ) eϕ (|∇RM |ga (·,t) + |RM ∗ ∇ϕa |ga (·,t) ) 0

a

a

a

+ |∇ϕa |ga (·,t) |∇2 ϕa |ga (·,t) + |∇3 ϕa |ga (·,t) )dt Z T 1 √ + | log t|dt 6 C(n, k0 , T ) t 0 6 C(n, k0 , T ). 23

(2.37)

By applying Theorem 2.7, we know that there is δ0 > 0 such that (2.3)a has a a

smooth solution F on M n × [0, δ0 ] with estimates (2.22). In views of Lemma 2.12 a a

and Lemma 2.13, in order to prove Theorem 2.6, we only need to bound |∇F |2ga (·,t),ha uniformly on a uniformly interval [0, T1 ] with T1 independent of a. To this end, let T˜ = sup{T˜0 |T˜0 6 T, (2.3)a has a smooth solution on M n × [0, T˜0 ] with

a a

|∇F |2ga (·,t),ha < ∞},

sup M n ×[0,T˜0 ]

We will estimate T˜ from below. a a We come back to the equation (2.30) of |∇F |2ga (·,t),ha , where there holds (

a a a a a a a a a ∂ − △t )|∇F |2ga (·,t),ha 6 −2|∇2 F |2ga (·,t),ha + C1 (n, k0 , T )|∇F |2ga (·,t),ha + C2 (n, k0 , T )|∇F |4ga (·,t),ha ∂t a

on M n × [0, T˜ ]. We remark that F is defined on a complete manifold with bounded a a curvature and supM n ×[0,T˜0 ] |∇F |2ga (·,t),ha < ∞, for each T˜0 < T˜ . So by applying the maximum principle on complete manifolds, we have a a a a a a d+ (sup |∇F |2ga (·,t),ha ) 6 C1 (n, k0 , T ) sup |∇F |2ga (·,t),ha + C2 (n, k0 , T ) sup |∇F |4ga (·,t),ha dt M n Mn Mn

where

d+ dt

is the upper right derivative defined by d+ u(t + △t) − u(t) u = lim sup . dt △t △tց0

By combining with (2.36), we have sup M n ×[0,T˜0 ]

a a

|∇F |2ga (·,t),ha 6 2ne2nk0 T ,

log 2 provided T˜0 6 min{T, C1 (n,k0 ,T )+2ne 2nk0 T C (n,k ,T ) }. 2 0

a

By Lemma 2.12 and Lemma 2.13 and Theorem 2.7, the solution F exists smoothly a a log 2 ˜ until |∇F |2a }. By a blows up, so we know T > min{T, 2nk T g (·,t),h

C1 (n,k0 ,T )+2ne

0

C2 (n,k0 ,T )

log 2 choosing T1 = min{T, C1 (n,k0 ,T )+2ne 2nk0 T C (n,k ,T ) }, Theorem 2.6 follows. 2 0

#

Proof of Theorem 2.1. a

a

Note that ϕa = 0 on Bg(·,T ) (P, a), and gija (x, t) = eϕ gij (x, t), haαβ (y) = eϕ hαβ . It follows that gija (x, t) = gij (x, t) on Bg(·,T ) (P, a), 24

haαβ (y) = hαβ (y) on Bg(·,T ) (P, a). a

By Theorem 2.6 and estimates (2.21) and letting a → ∞, the solutions F of (2.3)a on M n × [0, T1 ] have a convergent subsequence so that the limit is a solution of (2.3) with the estimates (2.4). #

3 3.1

The uniqueness of the Ricci flow Preliminary estimates for the Ricci-De Turck flow

Let F (x, t) be a solution to (2.3) in Theorem 2.1 on M n × [0, T0 ]. Let g˜ij (x, t) = α ∂F β hαβ (F (x, t)) ∂F be the one-parameter family of pulled back metrics F ∗ h. We ∂xi ∂xj will estimate gij (x, t) in terms of g˜ij (x, t). Proposition 3.1 There exists a constant 0 < T5 ≤ T0 depending only on k0 and T such that for all (x, t) ∈ M n × [0, T5 ], we have 1 g˜ij (x, t) ≤ gij (x, t) ≤ C(n, k0 , T )˜ gij (x, t) C(n, k0 , T ) 0 , T, k) ˜ k g|g˜ ≤ C(n, kk−1 |∇ t 2

(3.1)

for k = 1, 2, · · · Proof. We first consider the zero-order estimate of gij (x, t). The estimate |∇F |2 = g˜ij g ij ≤ C in (2.4) implies g˜ij (x, t) ≤ Cgij (x, t). For the reverse inequality, we compute the equation of g˜ij (x, t): ∂ β kl α g˜ij = △˜ gij − 2Rik Flα Fjβ hαβ g kl + 2Rαβγδ Fiα Fkβ Fjγ Flδ g kl − 2hαβ Fk,i Fl,j g ∂t ≥ △˜ gij − 2Rik g˜jl g kl − 2k0 |∇F |2 gij − 2|∇2 F |2gij ≥ △˜ gij − 2Rik g˜jl g kl − C(n, k0 , T )gij , by (2.4). Combining this with the Ricci flow equation gives (

1 1 ∂ − △)(˜ gij + C(n, k0 , T )tgij − gij ) > −2Rik (˜ glj + C(n, k0 , T )tglj − glj )g kl . 2nk T 2nk 0 ∂t 2ne 2ne 0 T

Note that at t = 0, (˜ gij + C(n, k0 , T )tgij −

1 2ne2nk0 T

gij ) |t=0 = gij (·, T ) − 25

1 2ne2nk0 T

gij (·, 0) > 0.

By applying the maximum principle to above equation, we obtain g˜ij + C(n, k0 , T )tgij −

1 2ne2nk0 T

gij > 0

on M n × [0, T0 ]. Let T5 = min{T0 , 4ne2nk0 T 1C(n,k0 ,T ) }. Then we have g˜ij ≥

1 4ne2nk0 T

on M n × [0, T5 ].

gij ,

This gives the zero-order estimate of gij (x, t). For the first order derivative of gij , we compute ˜ k gij = (∇ ˜ k − ∇k )gij = (Γl − Γ ˜ l )glj + (Γl − Γ ˜ l )gli ∇ ki ki kj kj and 2 ˜ l )|2 = |(Γp − Γ ˜ p )˜ |(Γlki − Γ ki g˜ ki ki glp |g˜

= |∇k ∇i F α

∂F β hαβ |g˜ ∂xl

∂F β hαβ |g ∂xl 6 C(n, k0 , T )|∇2F |g,h |∇F |g,h 6 C(n, k0 , T ). 6 C(n, k0 , T )|∇k ∇i F α

This gives the first order estimate. For higher order estimates, we prove it by induction. Suppose we have showed ˜ l g|g˜ 6 C |∇ l−1 t 2

for l = 1, 2, · · · , k − 1,

˜ l (Γ − Γ)| ˜ g˜ 6 Cl |∇ t2

for l = 0, 1, · · · , k − 2.

Since by induction ˜ k−1 (Γ − Γ)| ˜ g˜ = |∇ ˜ k−1 [(Γ − Γ) ˜ ∗ g˜]|g˜ |∇ k−1 X ˜ ∗ g˜] ∗ = | ∇k−1−j [(Γ − Γ) j=0

6 C(n, k0 , T ) ×

X

k−1 X j=0

6

t

˜ i1 (Γ − Γ)| ˜ g˜ · · · |∇ ˜ iq (Γ − Γ)| ˜ g˜ |∇ 1

1

k−1−j 2

j−2 2

t

C(n, k0 , T, k) t

i1 +1+···+iq +1=j

˜ g˜ ˜ i1 (Γ − Γ) ˜ ∗···∗∇ ˜ iq (Γ − Γ)| ∇

|∇k−1−j (∇2 F ∗ ∇F )|g,h

i1 +1+···+iq +1=j

6 C(n, k0 , T, k)(

X

k−1 2

26

+

1 t

k−1 2

)

and ˜ kg = ∇ ˜ k−1 ((Γ − Γ) ˜ ∗ g) ∇ k−1 X ˜i (Γ − Γ) ˜ ∗∇ ˜ k−1−i g, = ∇ i=0

then we have ˜ k g|g˜ 6 |∇

C

. t This completes the induction argument and the proposition is proved. k−1 2

# Proposition 3.2 Let F (x, t) be the solution of (2.3) in Theorem 2.1. Then F (·, t) are diffeomorphisms for all t ∈ [0, T5 ]; moreover, there exists a constant C(n, k0 , T ) > 0 depending only on n, k0 and T such that dh (F (x1 , t), F (x2 , t)) > e−C(n,k0 ,T ) dh (x1 , x2 ) for all x1 , x2 ∈ M n , t ∈ [0, T5 ]. Proof. Note that

1 g˜ij (x, t) ≤ gij (x, t) ≤ C g˜ij (x, t) C implies that F are local diffeomorphisms. So we only need to prove that F (·, t) is injective. Suppose not. Then there exist two points x1 6= x2 , such that F (x1 , t) = F (x2 , t), for some t0 ∈ (0, T5 ]. Assume t0 > 0 be the first time so that F (x1 , t) = ˜ of F (x1 , t0 ) F (x2 , t). Choose small δ > 0, such that there exist a neighborhood O −1 ˜ to O for and a neighborhood O of x1 such that F (·, t) is a diffeomorphism from O all t ∈ [t0 − δ, t0 ], moreover, letting γ˜t be a shortest geodesic( parametrized by arc ˜ length) on the target (N n , hαβ ) connecting F (x1 , t) and F (x2 , t), we require γ˜ ∈ O for t ∈ [t0 − δ, t0 ]. We compute ∂ dh (F (x1 , t), F (x2 , t)) = hV, γ˜ ′ (l)ih − hV, γ˜ ′ (0)ih ∂t where γ˜ (0) = F (x1 , t) , γ˜ (l) = F (x2 , t), and V α = △F α . Now we pull back everything by F −1 to O, ∂ dh (F (x1 , t), F (x2 , t)) = hP−˜γ V − V, γ ′ (0)iF ∗ h ∂t ˜ |(x, t)dh (F (x1 , t), F (x2 , t)) ≥ − sup |∇V x∈F −1 γ ˜

27

where Pγ˜ is the parallel translation along F −1 γ˜ using the metric F ∗ h. By (2.4), ∂F α hαβ )|g˜ ∂xl ∂F α ˜ g˜|∇2 F |g,h |∇F |g,h 6 |∇k (△F α l hαβ )|g˜ + C|Γ − Γ| ∂x 6 C(n, k0 , T )(|∇3F |g,h |∇F |g,h + |∇2 F |2g,h + |∇2 F |2g,h|∇F |g,h ) C(n, k0 , T ) √ 6 . t

˜ k V l |g˜ = |∇ ˜ k (△F α |∇

It follows that we have √ √ t0 − t0 −δ)

dh (F (x1 , t), F (x2 , t)) 6 eC(

dh (F (x1 , t0 ), F (x2 , t0 )) = 0,

for t ∈ [t0 − δ, t0 ], which contradicts with the choice of t0 . So F (·, t) are diffeomorphisms. ˜ = N n , O = M n , the above computation also gives By choosing O dh (F (x1 , t), F (x2 , t)) > e−C(n,k0 ,T ) dh (x1 , x2 ). The proof of the proposition is completed. #

3.2

Ricci-De Turck flow

From the previous section, we know that the harmonic map flow coupled with Ricci flow (2.3)with identity as initial data has a short time solution F (x, t) on M n × [0, T5 ], which remains being a diffeomorphism with good estimates (2.4). Let ∗ (F −1 ) g be one-parameter family of pulled back metrics on the target (N n , hαβ ). Denote gαβ (y, t). Then gαβ (y, t) satisfies the so called Ricci-De Turck flow: ∂ gαβ (y, t) = −2Rαβ (y, t) + ∇α Vβ + ∇β Vα ∂t

(3.3)

where V α = g βγ (Γαβγ (g) − Γαβγ (h)), Γαβγ (g) and Γαβγ (h) are the Christoffel symbols of the metrics gαβ (y, t) and hαβ (y) respectively. By (3.1) of Proposition 3.1, we already have the following estimates for gαβ (y, t) 1 hαβ (y) ≦ gαβ (y, t) ≦ C(n, k0 , T )hαβ (y) C(n, k0 , T ) |∇kh g|h ≦

C(n, k0 , T, k) t

on N n × [0, T5 ]. 28

k−1 2

.

(3.4)

Let gij (x, t) and g¯ij (x, t) be two solutions to the Ricci flow with bounded curvature and with the same initial value as assumed in Theorem 1.1. We solve the corresponding harmonic map flow with same target (N n , hαβ ) = (M n , gij (·, T )) by   ∂ F (x, t) = △F (x, t), ∂t (3.5)  F (·, 0) = identity, and

  

∂ ¯ ¯ F¯ (x, t), F (x, t) = △ ∂t F¯ (·, 0) = identity,

(3.6)

respectively. Then we obtained two solutions F (x, t) and F¯ (x, t) on M n × [0, T5 ]. It is clear that F¯ (x, t) still satisfies (2.4), Proposition 3.1 and Proposition 3.2. Let ∗ g¯αβ (y, t) = (F¯ −1 ) g¯(y, t), then g¯αβ (y, t) still satisfies (3.4). Now we have two solutions gαβ (y, t) and g¯αβ (y, t) to the Ricci De-Turck flow with same initial data and with good estimates (3.4). Proposition 3.3 There holds gαβ (y, t) = g¯αβ (y, t) on N n × [0, T5 ]. Proof. We can write the Ricci-De Turck flow (3.3) by using the fixed metric hαβ (y) in the following form (see [20]): ∂ ˜ γ∇ ˜ δ gαβ − g γδ gαξ g˜ξη R ˜ βγηδ − g γδ gβξ g˜ξη R ˜ αγηδ + 1 g γδ g ξη (∇ ˜ α gξγ ∇ ˜ β gηδ gαβ =g γδ ∇ ∂t 2 ˜ γ gβξ ∇ ˜ η gαδ − 2∇ ˜ γ gβξ ∇ ˜ δ gαη − 2∇ ˜ β gξγ ∇ ˜ δ gαη − 2∇ ˜ α gξγ ∇ ˜ δ gβη ) + 2∇ (3.7) ˜ and R ˜ are the covariant derivative and the curvature of g˜αβ . where g˜αβ = hαβ , ∇ Note that g¯αβ also satisfies (3.7), then the difference gαβ − g¯αβ satisfies the following equation: ∂ ˜ γ∇ ˜ δ (g − g¯) + g −1 ∗ g¯−1 ∗ ∇ ˜ 2 g¯ ∗ (¯ (g − g¯) =g γδ ∇ g − g) ∂t ˜ ∗ (g − g¯) + g −1 ∗ g¯−1 ∗ g ∗ g˜−1 ∗ Rm ˜ ∗ (g − g¯) + g¯−1 ∗ g˜−1 ∗ Rm ˜ ∗ ∇g ˜ ∗ (g − g¯) + g −1 ∗ g¯−1 ∗ g¯−1 ∗ ∇g ˜ ∗ ∇g ˜ ∗ (g − g¯) + g −1 ∗ g −1 ∗ g¯−1 ∗ ∇g ˜ ∗ ∇(g ˜ − g¯) + g¯−1 ∗ g¯−1 ∗ ∇¯ ˜ g ∗ ∇(g ˜ − g¯) + g¯−1 ∗ g¯−1 ∗ ∇g

since g αβ − g¯αβ = g αξ g¯ηβ (¯ gηξ − gηξ ). Let |g − g¯|2 = g˜αγ g˜βδ (gαβ − g¯αβ )(gγδ − g¯γδ ). 29

(3.8)

It follows from (3.8) that: ∂ ˜ γ∇ ˜ δ )|g − g¯|2 6 −2g ξη g˜αγ g˜βδ (∇ ˜ ξ gαβ − ∇ ˜ ξ g¯αβ )(∇ ˜ η gγδ − ∇ ˜ η g¯γδ ) ( − g γδ ∇ ∂t ˜ ˜ 2 g¯||¯ +100[|Rm|(1 + |g||g −1|)|¯ g −1| + |∇ g −1||g −1| ˜ 2(|¯ +|∇g| g −1|2 |g −1 | + |¯ g −1 ||g −1k2 )]|g − g¯|2 ˜ + |∇¯ ˜ g |)|∇(g ˜ − g¯)||g − g¯| +100|¯ g −1|2 (|∇g|

where all the norms are computed with the metric g˜ = h. By Cauchy-Schwartz inequality and (3.4), we have ∂ ˜ γ∇ ˜ δ )|g − g¯|2 6 − 2g ξη g˜αγ g˜βδ (∇ ˜ ξ gαβ − ∇ ˜ ξ g¯αβ )(∇ ˜ η gγδ − ∇ ˜ η g¯γδ ) ( − g γδ ∇ ∂t C ˜ − g¯)||g − g¯| + √ |g − g¯|2 + C|∇(g (3.9) t C 6 √ |g − g¯|2 t n on N × [0, T5 ]. Let ϕ1 be the nonnegative function in Lemma 2.2 with a = 1, then 1 ˜ p)) 6 ϕ1 (y) 6 C0 d(y, ˜ p) on N n \B(P, 2), (1 + d(y, C ˜ 1 | + |∇ ˜ 2 ϕ1 | 6 C, on N n . |∇ϕ For any fixed t and any ε > 0, consider the maximum of |g − g¯|2 − εϕ. Clearly, the maximum is achieved at some point Pεt and there hold |g − g¯|2 (Pεt ) > |g − g¯|2 (y) − εϕ(y), ˜ − g¯|2 |(P t ) 6 Cε , ∇ ˜ α∇ ˜ β |g − g¯|2 (P t) 6 Cε˜ |∇|g gαβ (Pεt ) ε ε

for all y ∈ N n . This gives

lim sup |g − g¯|2 (Pεt ) = sup |g − g¯|2 ε⇀0

˜ α∇ ˜ β |g − g¯|2 (P t ) 6 Cε g ∇ ε αβ

(3.10)

by the equivalence of g and g˜. Define a function

|g − g¯|2max (t) = sup |g − g¯|2 (y, t). y∈N n

By (3.9), and (3.10), we have C d+ |g − g¯|2max (t) 6 √ |g − g¯|2max (t) dt t and then |g − g¯|2max (t) 6 eC

√

T

|g − g¯|2max (0) = 0

Therefore the proof of the Proposition 3.3 is completed.

# 30

3.3

Proof of the main theorem

Let gij (x, t) and g¯ij (x, t) be two solutions to the Ricci flow (1.1) with bounded curvature and with the same initial data. We solve the corresponding harmonic map flow (3.5) and (3.6) with the same target (N n , hαβ ) = (M n , gij (·, T )) respectively. We obtain two solutions F (x, t) and F¯ (x, t) which are diffeomorphisms for t ∈ [0, T5 ], ∗ ∗ where T5 > 0 depends only on n, k0 , T . Then (F −1 ) g and (F¯ −1 ) g¯ are two solutions to the Ricci-De Turck flow with the same initial value. It follows from Proposition 3.3 that ∗ ∗ (F −1 ) g = (F¯ −1 ) g¯, on N n × [0, T5 ]. So in order to prove gij (x, t) ≡ g¯ij (x, t), we only need to show F ≡ F¯ . Let ˜ α − Γα ) = −(△F ◦ F −1 )α V α (y, t) = g βγ (Γ βγ βγ α βγ ˜ α α ¯ F¯ ◦ F¯ −1 )α . ¯ ¯ V (y, t) = g¯ (Γβγ − Γβγ ) = −(△ ∗

be two one-parameter family of vector fields on N n , where gαβ (y, t) = ((F −1 ) g)αβ (y, t) ∗ and g¯αβ (y, t) = ((F¯ −1) g¯)αβ (y, t). By Proposition 3.3, we have gαβ (y, t) = g¯αβ (y, t), thus the vector fields V ≡ V¯ on the target N n . Therefore, F and F¯ satisfy the same ODE equation with the same initial value: ∂ F = V ◦ F, ∂t F (·, 0) = identity, and ∂ ¯ F = V ◦ F¯ , ∂t F¯ (·, 0) = identity, By the same calculation as in the proof of Proposition 3.2, we have ∂ dN n (F (x, t), F¯ (x, t)) 6 ∂t

˜ |(y, t)dN n (F (x, t), F˜ (x, t)) sup |∇V

y∈N n

C 6 √ dN n (F (x, t), F˜ (x, t)). t This gives dN n (F (x, t), F¯ (x, t)) 6 eC which concludes that

√ T

dN n (F (x, 0), F¯ (x, 0)) = 0,

¯ t). F (x, t) ≡ F (x,

Thus g(x, t) = g¯(x, t), for all x ∈ M n and t ∈ [0, T5 ] and for some T1 > 0. Clearly, we can extend the interval [0, T1 ] to the whole [0, T ] by continuity method. 31

Therefore we complete the proof of the Theorem 1.1. # Finally, Corollary 1.2 is a direct consequence of Theorem 1.1. Indeed, since G is the isometry group of gij (x, 0), then for any σ ∈ G, σ ∗ g(·, t) is still a solution to the Ricci flow with bounded curvature and σ ∗ g(·, t) |t=0 = σ ∗ g(·, 0) = g(·, 0). By applying Theorem 1.1, we have σ ∗ g(·, t) = g(·, t), ∀t ∈ [0, T ]. So the corollary follows. #

References [1] Cheeger, J., Gromov, M., and Taylor, M., Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom. 17 (1982), 15-53. [2] Chen, B. L. and Zhu, X. P., Ricci flow with surgery on four manifolds with positive isotropic curvature, arXiv:math.DG/0504478 v1 April 2005. Preprint. [3] Cheng, S. Y., Li, P. and Yau, S. T., On the upper estimate of the heat kernel of complete Riemannian manifold, Amer. J. Math. 103 (1981) no.5, 1021-1063. [4] De Turck, D., Deforming metrics in the direction of their Ricci tensors, J. Diff. Geom. 18 (1983), 157-162. [5] Ding, W. Y.and Lin, F. H., A generalization of Eells-Sampson’s theorem, J. Partial Differential Equations 5 (1992), no.4, 13-22. [6] Eells, J. and Sampson, J. Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. [7] Green, R. E. and Wu, H., C ∞ approximation of convex, subharmonic and plurisubharmonic functions, Ann. Sci.Ec. Norm. Sup., 12 (1979), 47-84. [8] Hamilton, R. S., Three manifolds with positive Ricci curvature , J. Diff. Geom. 17 (1982), 255-306. [9] Hamilton, R. S., Four manifolds with positive curvature operator, J. Diff. Geom. 24 (1986), 153-179. [10] Hamilton, R. S., The Ricci flow on surfaces, Contemporary Mathematics 71 (1988) 237-261. [11] Hamilton, R. S., A compactness property for solution of the Ricci flow, Amer. J. Math. 117 (1995), 545-572.

32

[12] Hamilton, R. S., The formation of singularities in the Ricci flow, Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136, International Press, Combridge, MA,1995. [13] Hamilton, R. S., Four manifolds with positive isotropic curvature, Comm. in Analysis and Geometry, 5(1997), 1-92. (or see, Collected Papers on Ricci Flow, Edited by H.D.Cao, B.Chow, S.C.Chu and S.T.Yau, International Press 2002). [14] Ladyzenskaja, O. A., Solonnikov, V. A. and Uralceva, N. N., Linear and quasilinear equations of parabolic type, Transl. Amer. Math. Soc. 23(1968). [15] Li, P. and Tam, L. F., The heat equation and harmonic maps of complete manifolds, Invent. Math.105 (1991) 1-46. [16] Perelmann, G., The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159 v1 November 11, 2002. Preprint. [17] Perelmann, G., Ricci flow with surgery on arXiv:math.DG/0303109 v1 March 10, 2003. Preprint.

three

manifolds

[18] Schoen, R. and Yau, S. T., Lectures on differential geometry, in conference proceedings and Lecture Notes in Geometry and Topology, Volume 1, International Press Publications, 1994. [19] Schoen, R. and Yau, S. T., Module space of harmonic maps, compact group actions and the topology of manifolds with non-positive curvature in Lectures on harmonic maps, International Press, 1997. [20] Shi, W. X., Deforming the metric on complete Riemannian manifold, J. Differential Geometry 30 (1989), 223-301.

33

PERSPECTIVES ON GEOMETRIC ANALYSIS

arXiv:math/0602363v2 [math.DG] 16 Feb 2006

SHING-TUNG YAU

This essay grew from a talk I gave on the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicate the lecture to the memory of my teacher S. S. Chern who had passed away half a year before (December 2004). During my graduate studies, I was rather free in picking research topics. I [727] worked on fundamental groups of manifolds with non-positive curvature. But in the second year of my studies, I started to look into differential equations on manifolds. However, at that time, Chern was very much interested in the work of Bott on holomorphic vector fields. Also he told me that I should work on Riemann hypothesis. (Weil had told him that it was time for the hypothesis to be settled.) While Chern did not express his opinions about my research on geometric analysis, he started to appreciate it a few years later. In fact, after Chern gave a course on Calabi’s works on affine geometry in 1972 at Berkeley, S. Y. Cheng told me about these inspiring lectures. By 1973, Cheng and I started to work on some problems mentioned in Chern’s lectures. We did not realize that the great geometers Pogorelov, Calabi and Nirenberg were also working on them. We were excited that we solved some of the conjectures of Calabi on improper affine spheres. But soon after we found out that Pogorelov [559] published his results right before us by different arguments. Nevertheless our ideas are useful in handling other problems in affine geometry, and my knowledge about MongeAmpère equations started to broaden in these years. Chern was very pleased by my work, especially after I [732] solved the problem of Calabi on Kähler Einstein metric in 1976. I had been at Stanford, and Chern proposed to the Berkeley Math Department that they hire me. I visited Berkeley in 1977 for a year and gave a course on geometric analysis with emphasis on isometric embedding. Chern nominated me to give a plenary talk at the International Congress in Helsinki. The talk [733] went well, but my decision not to stay at Berkeley did not quite please him. Nevertheless he recommended me for a position on the faculty at the Institute for Advanced Study. Before I accepted a faculty position at the Institute, I organized a special year on geometry in 1979 at the Institute at the invitation of Borel. That was an exciting year because most people in geometric analysis came. In 1979, I visited China at the invitation of Professor L. K. Hua. I gave a series of talks on the bubbling process of Sacks-Uhlenbeck [577]. I suggested to the Chinese mathematicians that they apply similar arguments for a Jordan curve bounding two surfaces with the same constant mean curvature. I thought it would be a good exercise for getting into this exciting field of geometric analysis. The problem was indeed picked up by a group of students of Professor G. Y. Wang [360]. But unfortunately it also initiated some ugly fights during the meeting of the sixtieth anniversary of the Chinese Mathematical Society. Professor Wang was forced to resign, and this event hampered development of this beautiful subject in China in the past ten years. In 1980, Chern decided to develop geometric analysis on a large scale. He initiated a series of international conferences on differential geometry and differential equations to be held each year in China. For the first year, a large group of the most distinguished mathematicians was gathered in Beijing to give lectures (see [147]). I lectured on open problems in geometry [735]. It took a much longer time than I expected for Chinese mathematicians to pick up some of these problems. To his disappointment, Chern’s enthusiasm about developing differential equations and differential geometry in China did not stimulate as much activity as he had hoped. Most Chinese To appear in Survey in Differential Geometry, Vol. X, 2005. This research is supported by NSF grants DMS-0244464, DMS-0354737 and DMS-0306600. 1

mathematicians were trained in analysis but were rather weak in geometry. The goal of geometric analysis for understanding geometry was not appreciated. The major research center on differential geometry came from students of Chern, Hua and B. C. Su. The works of J. Q. Zhong (see, e.g., [751, 523, 524]) were remarkable. Unfortunately he died about twenty years ago. Q. K. Lu studied the Bergman metric extensively. C. H. Gu [294] studied gauge theory and considered harmonic map where the domain is R1,1 . J. X. Hong (see, e.g., [343, 316]) did some interesting work on isometric embedding of surfaces into R3 . In the past five years, the research center at the Chinese University of Hong Kong, led by L. F. Tam and X. P. Zhu, has produced first class work related to Hamilton’s Ricci flow (see, e.g., [124, 125, 128, 129, 112]). In the hope that it will advance Chern’s ambition to build up geometric analysis, I will explain my personal view to my Chinese colleagues. I will consider this article to be successful if it conveys to my readers the excitement of developments in differential geometry which have been taking place during the period when it has been my good fortune to contribute. I do not claim this article covers all aspects of the subject. In fact, I have given priority to those works closest to my personal experience, and, consequently, I have given insufficient space to aspects of differential geometry in which I have not participated. In spite of these shortcomings, I hope that my readers, particularly those too young to know the origins of geometric analysis, will be interested to learn how the field looks to someone who was there. I would like to thank comments given by R. Bryant, H. D. Cao, J. Jost, H. Lawson, N. C. Leung, T. J. Li, Peter Li, J. Li, K. F. Liu, D. Phong , D. Stroock , X. W. Wang, S. Scott, S. Wolpert and S. W. Zhang. I am also grateful to J. X. Fu, especially for his help of tracking down references for the major part of this survey. When Fu went back to China, this task was taken up by P. Peng and X. F. Sun to whom I am grateful also. In this whole survey, I follow the following:

Basic Philosophy: Functions, tensors and subvarieties governed by natural differential equations provide deep insight into geometric structures. Information about these objects will give a way to construct a geometric structure. They also provide important information for physics, algebraic geometry and topology. Conversely it is vital to learn ideas from these fields. Behind such basic philosophy, there are basic invariants to understand how space is twisted. This is provided by Chern classes [148], which appear in every branch of mathematics and theoretical physics. So far we barely understand the analytic meaning of the first Chern class. It will take much more time for geometers to understand the analytic meaning of the higher Chern forms. The analytic expression of Chern classes by differential forms have opened up a new horizon for global geometry. Professor Chern’s influence on mathematics is forever.

2

An old Chinese poem says: The reeds and rushes are abundant, and the white dew has yet to dry. The man whom I admire is on the bank of the river. I go against the stream in quest of him, But the way is difficult and turns to the right. I go down the stream in quest of him, and Lo! He is on the island in the midst of the water.

May the charm and beauty be always the guiding principle of geometry!

3

Contents 1. History and contributors of the subject 5 1.1. Founding fathers of the subject 5 1.2. Modern Contributors 7 2. Construction of functions in geometry 8 2.1. Polynomials from ambient space. 8 2.2. Geometric construction of functions 10 2.3. Functions and tensors defined by linear differential equations 13 3. Mappings between manifolds and rigidity of geometric structures 26 3.1. Embedding 26 3.2. Rigidity of harmonic maps with negative curvature 28 3.3. Holomorphic maps 29 3.4. Harmonic maps from two dimensional surfaces and pseudoholomorphic curves 30 3.5. Morse theory for maps and topological applications 31 3.6. Wave maps 31 3.7. Integrable system 32 3.8. Regularity theory 32 4. Submanifolds defined by variational principles 33 4.1. Teichm¨ uller space 33 4.2. Classical minimal surfaces in Euclidean space 33 4.3. Douglas-Morrey solution, embeddedness and application to topology of three manifolds 34 4.4. Surfaces related to classical relativity 34 4.5. Higher dimensional minimal subvarieties 35 4.6. Geometric flows 37 5. Construction of geometric structures on bundles and manifolds 39 5.1. Geometric structures with noncompact holonomy group 39 5.2. Uniformization for three manifolds 40 5.3. Four manifolds 43 5.4. Special connections on bundles 43 5.5. Symplectic structures 45 5.6. Kähler structure 46 5.7. Manifolds with special holonomy group 50 5.8. Geometric structures by reduction 51 5.9. Obstruction for existence of Einstein metrics on general manifolds 51 5.10. Metric Cobordism 52 References 53

4

1. History and contributors of the subject 1.1. Founding fathers of the subject. Since the whole development of geometry depends heavily on the past, we start out with historical developments. The following are samples of work before 1970 which provided fruitful ideas and methods. • Fermat’s principle of calculus of variation (Shortest path in various media). • Calculus (Newton and Leibnitz): Path of bodies governed by law of nature. • Euler, Lagrange: Foundation for the variational principle and the study of partial differential equations. Derivations of equations for fluids and for minimal surfaces. • Fourier, Hilbert: Decomposition of functions into eigenfunctions, spectral analysis. • Gauss, Riemann: Concept of intrinsic geometry. • Riemann, Dirichlet, Hilbert: Solving Dirichlet boundary value problem for harmonic function using variational method. • Maxwell: Electromagnetism, gauge fields, unification of forces. • Christoffel, Levi-Civita, Bianchi, Ricci: Calculus on manifolds. • Riemann, Poincar´ e, Koebe, Teichm¨ uller: Riemann surface uniformization theory, conformal deformation. • Frobenius, Cartan, Poincar´ e: Exterior differentiation and Poincaré lemma. • Cartan: Exterior differential system, connections on fiber bundle. • Einstein, Hilbert: Einstein equation and Hilbert action. • Dirac: Spinors, Dirac equation, quantum field theory. • Riemann, Hilbert, Poincar´ e, Klein, Picard, Ahlfors, Beurling, Carlsson: Application of complex analysis to geometry. • K¨ ahler, Hodge: K¨ ahler metric and Hodge theory. • Hilbert, Cohn-Vossen, Lewy, Weyl, Hopf, Pogorelov, Efimov, Nirenberg: Global surface theory in three space based on analysis. • Weierstrass, Riemann, Lebesgue, Courant, Douglas, Rad´ o, Morrey: Minimal surface theory. • Gauss, Green, Poincar´ e, Schauder, Morrey: Potential theory, regularity theory for elliptic equations. • Weyl, Hodge, Kodaira, de Rham, Milgram-Rosenbloom, Atiyah-Singer: de RhamHodge theory, integral operators, heat equation, spectral theory of elliptic self-adjoint operators. • Riemann, Roch, Hirzebruch, Atiyah-Singer: Riemann-Roch formula and index theory. • Pontrjagin, Chern, Allendoerfer-Weil: Global topological invariants defined by curvature forms. • Todd, Pontrjagin, Chern, Hirzebruch Grothendieck, Atiyah: Characteristic classes and K-theory in topology and algebraic geometry. • Leray, Serre: Sheaf theory. • Bochner-Kodaira: Vanishing of cohomology groups based on the curvature consideration. • Birkhoff, Morse, Bott, Smale: Critical point theory, global topology, homotopy groups of classical groups. • De Giorgi-Nash-Moser: Regularity theory for the higher dimensional elliptic equation and the parabolic equation of divergence type. • Kodaira, Morrey, Grauert, Hua, H¨ ormander, Bergman, Kohn, Andreotti-Vesentini: ¯ Embedding of complex manifolds, ∂-Neumann problem, L2 method, kernel functions. • Kodaira-Spencer, Newlander-Nirenberg: Deformation of geometric structures. • Federer-Fleming, Almgren, Allard, Bombieri, De Giorgi, Giusti: Varifolds and minimal varieties in higher dimensions. 5

• Eells-Sampson, Al’ber: Existence of harmonic maps into manifolds with non-positive curvature. • Calabi: Affine geometry and conjectures on Kähler Einstein metric.

6

1.2. Modern Contributors. The major contributors can be roughly mentioned in the following periods: I. 1972 to 1982: M. Atiyah, R. Bott, I. Singer, E. Calabi, L. Nirenberg, A. Pogorelov, R. Schoen, L. Simon, K. Uhlenbeck, S. Donaldson, R. Hamilton, C. Taubes, W. Thurston, E. Stein, C. Fefferman, Y. T. Siu, L. Caffarelli, J. Kohn, S. Y. Cheng, M. Kuranishi, J. Cheeger, D. Gromoll, R. Harvey, H. Lawson, M. Gromov, T. Aubin, V. Patodi, N. Hitchin, V. Guillemin, R. Melrose, Colin de Verdière, M. Taylor, R. Bryant, H. Wu, R. Greene, Peter Li, D. Phong, S. Wolpert, J. Pitts, N. Trudinger, T. Hildebrandt, S. Kobayashi, R. Hardt, J. Spruck, C. Gerhardt, B. White, R. Gulliver, F. Warner, J. Kazdan. Highlights of the works in this period include a deep understanding of the spectrum of elliptic operators, introduction of self-dual connections for four manifolds, introduction of a geometrization program for three manifolds, an understanding of minimal surface theory, Monge-Ampére equations and the application of the theory to algebraic geometry and general relativity. II. 1983 to 1992: In 1983, Schoen and I started to give lectures on geometric analysis at the Institute for Advanced Study. J. Q. Zhong took notes on the majority of our lectures. The lectures were continued in 1985 in San Diego. During the period of 1985 and 1986, K. C. Chang and W. Y. Ding came to take notes of some part of our lectures. The book Lectures on Differential Geometry was published in Chinese around 1989 [602]. It did have great influence for a generation of Chinese mathematicians to become interested in this subject. At the same time, a large group of my students made contributions to the subject. This includes A. Treibergs, T. Parker, R. Bartnik, S. Bando, L. Saper, M. Stern, H. D. Cao, B. Chow, W. X. Shi, F. Y. Zheng and G. Tian. At the same time, J. Bismut, C. S. Lin, N. Mok, J. Q. Zhong, J. Jost, G. Huisken, D. Jerison, P. Sarnak, K. Fukaya, T. Mabuchi, T. Ilmanen, C. Croke, D. Stroock, Price, F. H. Lin, S. Zelditch, D. Christodoulou, S. Klainerman, V. Moncrief, C. L. Terng, Michael Wolf, M. Anderson, C. LeBrun, M. Micallef, J. Moore, John Lee, A. Chang, N. Korevaar were making contributions in various directions. One should also mention that in this period important work was done by the authors in the first group. For example, Donaldson, Taubes [651] and Uhlenbeck [684, 685] did spectacular work on Yang-Mills theory of general manifolds which led Donaldson [194] to solve the outstanding question on four manifold topology. Donaldson [195], Uhlenbeck-Yau [687] proved the existence of Hermitian Yang-Mills connection on stable bundles. Schoen [586] solved the Yamabe problem. III. 1993 to now: Many mathematicians joined the subject. This includes P. Kronheimer, B. Mrowka, J. Demailly, T. Colding, W. Minicozzi, T. Tao, R. Thomas, Zworski, Y. Eliashberg, Toth, Andrews, L. F. Tam, N. C. Leung, Y. B. Ruan, W. D. Ruan, R. Wentworth, A. Grigor’yan, L. Saloff-Coste, J. X. Hong, X. P. Zhu, M. T. Wang, A. K. Liu, K. F. Liu, X. F. Sun, T. J. Li, X. J. Wang, J. Loftin, H. Bray, J. P. Wang, L. Ni, P. F. Guan, N. Kapouleas, P. Ozsváth, Z. Szab´ o and Y. I. Li. The most important event is of course the first major breakthrough of Hamilton [313] in 1995 on the Ricci flow. I did propose to him in 1982 to use his flow to solve Thurston’s conjecture. But after this paper by Hamilton, it is finally realized that it is feasible to solve the full geometrization program by geometric analysis. (A key step was the estimates on parabolic equations initiated by Li-Yau [443] and accomplished by Hamilton for Ricci flow [310, 311].) In 2002, Perelman [547, 548] brought in fresh new ideas to solve important steps that are remained in the program. Many contributors, including Colding-Minicozzi [172], Shioya-Yamaguchi [612] and Chen-Zhu [128], [129] have helped in filling important gaps in the arguments of HamiltonPerelman. Cao-Zhu has just finished a long manuscript which gives a complete detail account of the program. The monograph will be published by International Press. In the other direction, we see the important development of Seiberg-Witten theory [717]. Taubes [657, 658, 659, 660] was able to prove the remarkable theorem for counting pseudo-holomorphic curves in terms of his invariants. Kronheimer-Mrowka [400] were able to solve the Thom conjecture that holomorphic curves provide the lowest genus surfaces in representing homology in algebraic surfaces. (Ozsváth-Szabó had a symplectic version [544].) 7

2. Construction of functions in geometry The following is the basic principle [733]: Linear or non-linear analysis is developed to understand the underlying geometric or combinatorial structure. In the process, geometry will provide deeper insight of analysis. An important guideline is that space of special functions defined by the structure of the space can be used to define the structure of this space itself. Algebraic geometers have defined the Zariski topology of an algebraic variety using ring of rational functions. In differential geometry, one should extract information about the metric and topology of the manifolds from functions defined over it. Naturally, these functions should be defined either by geometric construction or by differential equations given by the underlying structure of the geometry. (Integral equations have not been used extensively as the idea of linking local geometry to global geometry is more compatible with the ideas of differential equations.) A natural generalization of functions consists of the following: differential forms, spinors, and sections of vector bundles. The dual concepts of differential forms or sections of vector bundles are submanifolds or foliations. From the differential equations that arise from the variational principle, we have minimal submanifolds or holomorphic cycles. Naturally the properties of such objects or the moduli space of such objects govern the geometry of the underlying manifold. A very good example is Morse theory on the space of loops on a manifold (see [514]). I shall now discuss various methods for constructing functions or tensors of geometric interest. 2.1. Polynomials from ambient space. If the manifold is isometrically embedded into Euclidean space, a natural class of functions are the restrictions of polynomials from Euclidean space. However, isometric embedding in general is not rigid, and so functions constructed in such a way are usually not too useful. On the other hand, if a manifold is embedded into Euclidean space in a canonical manner and the geometry of this submanifold is defined by some group of linear transformations of the Euclidean space, the polynomials restricted to the submanifold do play important roles. 2.1.1. Linear functions being the harmonic function or eigenfunction of the submanifold. For minimal submanifolds in Euclidean space, the restrictions of linear functions are harmonic functions. Since the sum of the norm square of the gradient of the coordinate functions is equal to one, it is fruitful to construct classical potentials using coordinate functions. This principle was used by Cheng-Li-Yau [139] in 1982 to give a comparison theorem for a heat kernel for minimal submanifolds in Euclidean space, sphere and hyperbolic space. Li-Tian [437] also considered a similar estimate for complex submanifolds of CP n . But this follows from [139] as such submanifolds can be lifted to a minimal submanifold in S 2n+1 . Another very important property of a linear function is that when it is restricted to a minimal hypersurface in a sphere S n+1 , it is automatically an eigenfunction. When the hypersurface is embedded, I conjectured that the first eigenfunction is linear and the first eigenvalue of the hypersurface is equal to n (see [735]). While this conjecture is not completely solved, the work of Choi-Wang [154] gives strong support. They proved that the first eigenvalue has a lower bound depending only on n. Such a result was good enough for Choi-Schoen [152] to prove a compactness result for embedded minimal surfaces in S 3 . 2.1.2. Support functions. An important class of functions that are constructed from the ambient space are the support functions of a hypersurface. These are functions defined on the sphere and are related to the Gauss map of the hypersurface. The famous Minkowski problem reduces to solving some Monge-Ampère equation for such support functions. This was done by Nirenberg [536], Pogorelov [556], Cheng-Yau [143]. The question of prescribed symmetric functions of principal curvatures has been studied by many people: Pogorelov [560], Caffarelli-Nirenberg-Spruck [91], P. 8

F. Guan and his coauthors (see [296, 295]), Gerhardt [248], etc. It is not clear whether one can formulate a useful Minkowski problem for higher codimensional submanifolds. The question of isometric embedding of surfaces into three space can also be written in terms of the Darboux equation for the support function. The major global result is the Weyl embedding theorem for convex surfaces, which was proved by Pogorelov [557, 558] and Nirenberg [536]. The rigidity part was due to Cohn-Vossen and an important estimate was due to Weyl himself. For local isometric embeddings, there is work by C. S. Lin [453, 454], which are followed by Han-Hong-Lin [316]. The global problem for surfaces with negative curvature was studied by Hong [343]. In all these problems, infinitesimal rigidity plays an important role. Unfortunately they are only well understood for a convex hypersurface. It is intuitively clear that generically, every closed surface is infinitesimally rigid. However, significant works only appeared for very special surfaces. Rado studied the set of surfaces that are obtained by rotating a curve around an axis. The surfaces constructed depend on the height of the curve. it turns out that such surfaces are infinitesimally rigid except on a set of heights which form part of a spectrum of some Sturm-Liouville operator. 2.1.3. Gradient estimates of natural functions induced from ambient space. A priori estimates are the basic tools for nonlinear analysis. In general the first step is to control the ellipticity of the problem. In the case of the Minkowski problem, we need to control the Hessian of the support function. For minimal submanifolds and other submanifold problem, we need gradient estimates which we shall discuss in Chapter 4. In 1974 and 1975, S. Y. Cheng and I [142, 146] developed several gradient estimates for linear or quadratic polynomials in order to control metrics of submanifolds in Minkowski spacetime or affine space. This kind of idea can be used to deal with many different metric problems in geometry. The first theorem concerns a spacelike hypersurface M in P the Minkowski space Rn,1 . The foln,1 lowing important question arose: Since the metric on R is (dxi )2 − dt2 , the restriction of this metric on M need not be complete even though it may be true for the induced Euclidean metric. In order to prove the equivalence of these two concepts for hypersurfaces whose mean curvatures are controlled, Cheng and I proved the gradient estimate of the function X hX, Xi = (xi )2 − t2 i

restricted on the hypersurface. By choosing a coordinate system, the function hX, Xi can be assumed to be positive and proper on M . For any positive proper function f defined on M , if we prove the following gradient estimate | ▽f | ≤C f where C is independent of f , then we can prove the induced metric on M is complete. This is obtained by integrating the inequality to get | log f (x) − log f (y) |≤ Cd(x, y) so that when f (y) → ∞, d(x, y) → ∞. Once we knew the metric was complete, we proved the Bernstein theorem which says that maximal spacelike hypersurface must be linear. Such work was then generalized by Treibergs [681], C. Gerhardt [247] and R. Bartnik [40] for hypersurfaces in more general spacetime. (It is still an important problem to understand the behavior of a maximal spacelike hypersurface foliation for general spacetime when we assume the spacetime is evolved by Einstein equation from a nonsingular data set.) Another important example is the study of affine hypersurfaces M n in an affine space An+1 . These are the improper affine spheres det(uij ) = 1 9

where u is a convex function or the hyperbolic affine spheres 1 n+2 det(uij ) = − u

where u is convex and zero on ∂Ω and Ω is a convex domain. Note that these equations describe hypersurfaces where the affine normals are either parallel or converge to a point. For affine geometry, there is an affine invariant metric defined on M which is 1 X (det hij )− n+2 hij dxi dxj

where hij is the second fundamental form of M . A fundamental question is whether this metric is complete or not. A coordinate system in An+1 is chosen so that the height function is a proper positive function defined on M . The gradient estimate of the height function gives a way to prove completeness of the affine metric. Cheng and I [146] did find such an estimate which is similar to the one given above. Once completeness of the affine metric is known, it is straight forward to obtain important properties of the affine spheres, some of which were conjectured by Calabi. For example we proved that an improper affine sphere is a paraboloid and that every proper convex cone admits a foliation of hyperbolic affine spheres. The statement about improper affine sphere was first proved by Jörgens [362], Calabi [93] and Pogorelov [559]. Conversely, we also proved that every hyperbolic affine sphere is asymptotic to a convex cone. (The estimate of Cheng-Yau was reproduced again by a Chinese mathematician who claimed to prove the result ten years afterwards.) Much more recently, Trudinger and X. J. Wang [683] solved the Bernstein problem for an affine minimal surface, thereby settling a conjecture by Chern. They found a counterexample for dim≥ 10. These results are solid contributions to fourth order elliptic equations. The argument of using gradient estimates for some naturally defined function was also used by me to prove that the K¨ ahler Einstein metric constructed by Cheng and myself is complete for any bounded pseudo-convex domain [144]. (It appeared in my paper with Mok [522] who proved the converse statement which says that if the Kähler Einstein metric is complete, the domain is pseudo-convex.) It should be noted that in most cases, gradient estimates amount to control of ellipticity of the nonlinear elliptic equation. Comment: To control a metric, find functions that are capable of describing the geometry and give gradient or higher order estimates for these functions. 2.2. Geometric construction of functions. 2.2.1. Distance function and Busemann function. When manifolds cannot be embedded into the linear space, we construct functions adapted to the metric structure. Obviously the distance function is the first major function to be constructed. A very important property of the distance function is that when the Ricci curvature of the manifold is greater than the Ricci curvature of a model manifold which is spherical symmetric at one point, the Laplacian of the distance function is not greater than the Laplacian of the distance function of the model manifold in the sense of distribution. This fact was used by Cheeger-Yau [123] to give a sharp lower estimate of the heat kernel of such manifolds. An argument of this type was also used by Perelman in his recent work. Gromov [282] developed a remarkable Morse theory for the distance function (a preliminary version was developed by K. Grove and K. Shiohama [293]) to compare the topology of a geodesic ball to that of a large ball, thereby obtaining a bound on the Betti numbers of compact manifolds with nonnegative sectional curvature. (He can also allow the manifolds to have negative curvature. But in this case the diameter and the lower bound of the curvature will enter into the estimate.) We can also take the distance function from a hypersurface and compute the Hessian of the distance function. In general, one can prove comparison theorems, and the principle curvatures of 10

the hypersurface will come into the estimates. However the upper bound of the Laplacian of the function depends only on the Ricci curvature of the ambient manifold and the mean curvature of the hypersurface. This kind of calculation was used in the sixties by Penrose and Hawking to study the focal locus of a closed surface under the assumption that the surface is ”trapped” which means the mean curvatures are negative in both the ingoing and the outgoing null directions. This information allowed them to prove the first singularity theorem in general relativity (which demonstrates that the black hole singularity is stable under perturbation). The distance to hypersurfaces can be used as barrier functions to prove the existence of a minimal surface as was shown by Meeks-Yau [503], [504]. T. Frankel used the idea of minimizing the distance between two submanifolds to detect the topology of minimal surfaces. In particular, two maximal spacelike hypersurfaces in spacetime which satisfy the energy condition must be disjoint if they are parallel at infinity. Out of the distance function, we can construct the Busemann function in the following way: Given a geodesic ray γ : [0, ∞) → M so that where k

dγ dt

distance(γ(t1 ), γ(t2 )) = t2 − t1 ,

k= 1, one defines

Bγ (x) = lim (d(x, γ(t)) − t). t→∞

This function generalizes the notion of a linear function. For a hyperbolic space form, its level set defines horospheres. For manifolds with positive curvature, it is concave. Cohn-Vossen (for surface) and Gromoll-Meyer [277] used it to prove that a complete noncompact manifold with positive curvature is diffeomorphic to Rn . A very important property of the Busemann function is that it is superharmonic on complete manifolds with nonnegative Ricci curvature in the sense of distribution. This is the key to prove the splitting principle of Cheeger-Gromoll [118]. Various versions of this splitting principle have been important for applications to the structure of manifolds. When I [732] proved the Calabi conjecture, the splitting principle was used by me and others to prove the structure theorem for Kähler manifolds with a nonnegative first Chern Class. (The argument for the structure theorem is due to Calabi [92] who knew how to handle the first Betti number. Kobayashi [385] and Michelsohn [512] wrote up the formal argument and Beauville [45] had a survey article on this development.) In 1974, I was able to use the Busemann function to estimate the volume of complete manifolds with nonnegative Ricci curvature [730]. After long discussions with me, Gromov [283] realized that my argument of Busemann function amounts to compare volumes of geodesic balls. The comparison theorem of Bishop-Gromov had been used extensively in metric geometry. If we consider inf γ Bγ , where γ ranges from all geodesic rays from a point on the manifold, we may be able to obtain a proper exhaustion of the manifold. When M is a complete manifold with finite volume and its curvature is pinched by two negative constants, Siu and I [630] did prove that such a function gives a concave exhaustion of the manifold. If the manifold is also Kähler, we were able to prove that one can compactify the manifold by adding a point to each end to form a compact complex variety. In the other direction, Schoen-Yau [599] was able to use the Busemann function to construct a barrier for the existence of minimal surfaces to prove that any complete three dimensional manifold with positive Ricci curvature is diffeomorphic to Euclidean space. The Busemann function also gives a way to detect the angular structure at infinity of the manifold. It can be used to construct the Poisson kernel of hyperbolic space form. For a simply connected complete manifold with bounded and strongly negative curvature, it is used as a barrier to solve the Dirichlet problem for bounded harmonic functions, after they are mollified at infinity. This was achieved by Sullivan [643] and Anderson [8]. Schoen and Anderson [9] obtained the Harnack inequality for a bounded harmonic function and identified the Martin boundary of such manifolds. W. Ballmann [27] then studied the Dirichlet problem for manifolds of non-positive curvature. Schoen and I [602] conjectured that nontrivial bounded harmonic function exists if the manifold has bounded geometry and a positive first eigenvalue. Many important cases were settled 11

in [602]. Lyons-Sullivan [483] proved the existence of nontrivial bounded harmonic functions using the non-amenability of groups acting on the manifold. The abundance of bounded harmonic functions on the universal cover of a compact manifold should mean that the manifold is “hyperbolic”. Hence if the Dirichlet problem is solvable on the universal cover, one expects the Gromov volume of the manifold to be greater than zero. The Martin boundary was studied by L. Ji and MacPherson (see [301, 359]) for the compactification of various symmetric spaces. For product of manifolds with negative curvature, it was determined by A. Freire [231]. For rank one complete manifolds with non-positive curvature, work has been done by Ballman-Ledrappier [28] and Cao-Fan-Ledrappier [104]. It should be nice to generalize the work of L. K. Hua on symmetric spaces with higher rank to general manifolds with non-positive curvature. Hua found that bounded harmonic functions satisfy extra equations (see [346]). 2.2.2. Length function defined on loop space. If we look at the space of loops in a manifold, we can take the length of each loop and thereby define a natural function on the space of loops. This is a function for which Morse theory found rich application. Bott [70] made use of it to prove his periodicity theorem. Bott [67, 69] and Morse also developed formula for computing index a geodesic. Bott showed that the index of a closed geodesic and its linearized Poincaré map determine the indexes of iterates of this geodesic. Starting from the famous works of Poincaré, Birkhoff, Morse and Ljusternik-Shnirel’man, there has been extensive work on proving the existence of a closed geodesic using Morse theory on the space of loops. Klingenberg and his students developed powerful tools (see [384]). Gromoll-Meyer [278] did important work in which they proved the existence of infinitely many closed geodesics assuming the Betti number of the free loop space of the manifold grows unboundedly. They used the results of Bott [69], Serre and some version of degenerate Morse theory. There was also later work by Ballmann, Ziller, G. Thorbergsson, Hingston and Kramer (see, e.g., [30, 326, 399]), who improved the Gromoll-Meyer theorem to give a low estimate of the growth of the number of geometrically distinct closed geodesics of length ≤ t. In most cases, they grow at least as fast as the prime numbers. The classical important question that every metric on S 2 supports an infinite number of closed geodesics was also solved affirmatively by Franks [227], Bangert [35] and Hingston [327]. An important achievement was made by Vigué-Poirrier and D. Sullivan [691] who proved that the Gromoll-Meyer condition for the existence of infinite numbers of closed geodesics is satisfied if and only if the rational cohomology algebra of the manifold has at least two generators. They made use of Sullivan’s theory of the rational homotopic type. When the metric is Finsler, the most recent work of Victor Bangert and Yiming Long [36] showed the existence of two closed geodesics on the two dimensional sphere. (Katok [375] produced an example which shows that two is optimal.) Length function is a natural concept in Finsler geometry. In the last fifty years, Finsler geometry has not been popular in western world. But under the leadership of Chern, David Bao, Z. Shen, X. H. Mo and M. Ji did develop Finsler geometry much further (see, e.g., [37]). A special class of manifolds, all of whose geodesics are closed, have occupied quite a lot of interest of distinguished geometers. It started from the work of Zoll (1903) for surfaces where Guillemin did important contributions. Bott [68] has determined the cohomology ring of these manifolds. The well known Blaschke conjecture was proved by L. Green [268] for two dimension and by M. Berger and J. Kazdan (see [51]) for higher dimensional spheres. Weinstein [709] and C. T. Yang [723, 724, 725] made important contributions to the conjecture for other homotopic types. 2.2.3. Displacement functions. When the manifold has negative curvature, the length function of curves is related to the displacement function defined in the following way: If γ is an element of the fundamental group acting on the universal cover of a complete manifold with non-positive curvature, we consider the function d(x, γ(x)): The study of such a function gives rise to properties of compact manifolds with non-positive curvature. For example, in my thesis, I generalized the Preissmann theorem to the effect that every solvable subgroup of the 12

fundamental group must be a finite extension of an Abelian group which is the fundamental group of a totally geodesic flat sub-torus [727]. Gromoll-Wolf [279] and Lawson-Yau [410] also proved that if the fundamental group of such a manifold has no center and splits as a product, then the manifold splits as a metric product. Strong rigidity result for discrete group acting on product of manifolds irreducibly was obtained by Jost-Yau [370] where they proved that these manifolds are homogeneous if the discrete group also appears as fundamental group of compact manifolds with nonpositive curvature. When the manifold has bounded curvature, Margulis studied those points where d(x, γ(x)) is small and proved the famous Margulis lemma which was used extensively by Gromov [280] to study the structure of manifolds with non-positive curvature. Comment: The lower bound of sectional curvature (or Ricci curvature) of a manifold gives upper estimate of the Hessian (or the Laplacian) of the distance functions. Since most functions constructed in geometry come from distance functions, we have partial control of the Hessian of these functions. The information provides us with basic tools to construct barrier functions for harmonic analysis or to produce convex functions. The Hessian of distance functions come from computations of second variation of geodesics. If we consider the second variation of closed geodesic loops, we get information about the Morse index of the loop, which enable us to link global topology to the existence of many closed geodesics or curvatures of the manifold. We always look for canonical objects through geometric constructions and deform them to find their global properties. 2.3. Functions and tensors defined by linear differential equations. Direct construction of functions or tensors based on geometric intuitions alone is not rich enough to handle the very complicated geometric world. One should produce global geometric objects based on global differential equations. Often the construction depends on the maximal principle, integration by part, or the method of contradictions, and they are not necessarily geometric intuitive. On the other hand, basic principle of global differential equations does fit well with modern geometry in relating local data to global behavior. In order for the theory to be effective, the global differential operator has to be constructed from a geometric structure naturally. The key to understanding any self-adjoint linear elliptic differential operator is to understand its spectral resolution and the detail of the structure of objects in the process of the resolution: eigenvalues or eigenfunctions are particularly important for their relation to geometry. Low eigenvalues and low eigenfunctions give deep information about global geometry such as topology or isoperimetric inequalities. High eigenvalues and high eigenfunctions are related to local geometry such as curvature forms or characteristic forms. Semiclassical analysis in quantum physics give a way to relate these two ends. This results in using either the heat equation or the hyperbolic equation. ¯ Dirac operator. All these There are many important first order differential operators: d, δ, ∂, operators have contributed to a deeper understanding of geometry. They form systems of equations. Our understanding of them is not as deep as the Laplacian acting on functions. The future of geometry will rest on an understanding of global systems of equations and their relation to global topology. The index theorem is the most important contribution. It provides information about the kernel (or cokernel). We still need to have a deeper understanding of the spectrum of these operators. 2.3.1. Laplacian. (a). Harmonic functions. The spectral resolution of the Laplacian gives rise to eigenfunctions. Harmonic functions are therefore the simplest functions that play important roles in geometry. 13

If the manifold is compact, the maximum principle shows that harmonic functions are constant. However, when we try to understand the singularities of compact manifolds, we may create noncompact manifolds by scaling and blowing up processes, at which point harmonic functions can play an important role. The first important question about harmonic functions on a complete manifold is the Liouville theorem. I started my research on analysis by understanding the right formulation of the Liouville theorem. In 1971, I thought that it is natural to prove that for complete manifolds with a nonnegative Ricci curvature, there is no nontrivial harmonic function [728]. I also thought that in the opposite case, when a complete manifold has strongly negative curvature and is simply connected, one should be able to solve Dirichlet problem for bounded harmonic functions. The gradient estimates [728] that I derived for a positive harmonic function come from a suitable interpretation of the Schwarz lemma in complex analysis. In fact, I generalized the Ahlfors Schwarz lemma before I understood how to work out the gradient estimates for harmonic functions. The generalized Schwarz lemma [734] says that holomorphic maps, from a complete Kähler manifold with Ricci curvature bounded from below to a Hermitian manifold with holomorphic bisectional curvature bounded from above by a negative constant, are distance decreasing with constants depending only on the bound on the curvature. This generalization has since found many applications such as the study of the geometry of moduli spaces by Liu-Sun-Yau [467, 468]. They used it to prove the equivalence of the Bergman metric with the Kähler-Einstein metric on the moduli space. They also proved that these metrics are equivalent to the Teichm¨ uller metric and the McMullen metric. The classical Liouville theorem has a natural generalization: Polynomial growth harmonic functions are in fact polynomials. Motivated by this fact and several complex variables, I asked whether the space of polynomial growth harmonic functions with a fixed growth rate is finite dimension with the upper bound depending only on the growth rate [737]. This was proved by Colding-Minicozzi [167] and generalized by Peter Li [435]. (Functions can be replaced by sections of bundles). In a beautiful series of papers (see, e.g., [438, 439]), P. Li and J. P. Wang studied the space of harmonic functions in relation to the geometry of manifolds. In the case when harmonic functions are holomorphic, they form a ring. I am curious about the structure of this ring. In particular, is it finitely generated when the manifold is complete and has a nonnegative Ricci curvature? A natural generalization of such a question is to consider holomorphic sections of line bundles, especially powers of canonical line bundles. This is part of Mori’s minimal model program. (b). Eigenvalues and eigenfunctions. Eigenvalues reflect the geometry of manifolds very precisely. For domains, estimates of them date back to Lord Rayleigh. Hermann Weyl [707] solved a problem of Hilbert’s on the asymptotic behavior of eigenvalues in relation to the volume of the domain and hence initiated a new subject of spectral geometry. Pólya-Szegö, Faber, Krahn and Levy gave estimates of eigenvalues of various geometric problems. On a general manifold, Cheeger [113] was the first person to relate a lower estimate of the first eigenvalue with the isoperimetric constant (now called the Cheeger constant). One may note that many questions on the eigenvalue for domains are still unsolved. The most noted one is the Pólya conjecture which gave a sharp lower estimate of the Dirichlet problem in terms of volume. Li-Yau [442] did settle the average version of the P´ olya conjecture. The gradient estimate that I found for harmonic functions can be generalized to cover eigenfunctions and Peter Li [434] was the first one to apply it to finding estimates for eigenvalues for manifolds with positive Ricci curvature. (If the Ricci curvature has a positive lower bound, this is due to Lichnerowicz.) Li-Yau [440] then solved the well-known problem of estimating eigenvalues of manifolds in terms of their diameter and the lower bound on their Ricci curvature. Li-Yau conjectured the sharp constant for their estimates, and Zhong-Yang [751] were able to prove this conjecture by sharpening Li and Yau’s arguments. A probabilistic argument was developed by Chen and Wang [132] to derive these inequalities. The precise upper bound for the eigenvalue was 14

first obtained by S. Y. Cheng [136] also in terms of diameter and lower bound of the Ricci curvature. Cheng’s theorem gives a very good demonstration of how the analysis of functions provides information about geometry. As a corollary of his theorem, he proved that if a compact manifold M n has a Ricci curvature ≥ n − 1 and the diameter is equal to π, then the manifold is isometric to the sphere. He used a lower estimate for eigenvalues based on the work of Lichnerowicz and Obata. Colding [166] was able to use functions with properties close to those of the first eigenfunction to prove a pinching theorem which states that: When the Ricci curvature is bounded below by n − 1 and the volume is close to that of the unit sphere, the manifold is diffeomorphic to the sphere. There is extensive work by Colding-Cheeger [115, 116, 117] and Perelman (see, e.g., [87]) devoted to the understanding of Gromov’s theory of Hausdorff convergence for manifolds. The tools they used include the comparison theorem, the splitting theorem of Cheeger and Gromoll, and the ideas introduced earlier by Colding. A very precise estimate of eigenvalues of the Laplacian has been important in many areas of mathematics. For example, the idea of Szegö [647]-Hersch [325] on the upper bound of the first eigenvalue in terms of the area alone was generalized by me to the higher genus in joint works with P. Yang [726] and P. Li [441]. For genus one, this was Berger’s conjecture, as I was informed by Cheng. After Cheng showed me the paper of Hersch, I realized how to create trial functions by taking the branched conformal cover of S 2 . While the constant in the paper of Yang-Yau [726] for torus is not the best possible, the recent work of Jakobson, Levitin, Nadirashvili, Nigam and Polterovich [356] demonstrated that the constant for a genus two surface may be the best possible and may be achieved by Bolza’s surface. Shortly afterwards, I applied the argument of [726] to prove that a Riemann surface defined by an arithmetic group must have a relative high degree when it is branched over the sphere. This observation of using Selberg’s estimate coupled with Li-Yau [441] was made in 1985 when I was in San Diego, where I also used similar idea to estimate genus of mini-max surface in a three dimensional manifolds and also to prove positivity of Hawking mass. After I arrived in Harvard, I discussed the idea with my colleague N. Elkies and B. Mazur. The paper was finally written up and published in 1995 [741]. In the mean while, ideas of using my work on eigenvalue coupled with Selberg’s work to study congruence subgroup was generalized by D. Abramovich [1] (my idea was conveyed by Elkies to him) and by P. Zograf [754] to the case where the curve has cusps. Most recently Ian Agol [2] also used similar idea to study arithmetic Kleinian reflection groups. In a beautiful article, N. Korevaar [395] gave an upper bound, depending only on genus and n, for the n-th eigenvalue λn of a Riemann surface. His result answered a challenge of mine (see [735]) when I met him in Utah in 1989. Grigor’yan, Netrusov and I [276] were able to give a simplified proof and apply the estimate to bound the index of minimal surfaces. There are also works by P. Sarnak (see, e.g., [582, 355]) on understanding eigenfunctions for such Riemann surfaces. IwaniecSarnack [355] showed that the estimate of the maximum norm of the n-th eigenfunction on an arithmetic surface has significant interest in number theory. Wolpert [721] analyzes perturbation stability of embedded eigenvalues and applies asymptotic perturbation theory and harmonic map theory to show that stability is equivalent to the non-vanishing of certain standard quantities in number theory. There was also the work of Schoen-Wolpert-Yau [591] on the behavior of eigenvalues λ1 , · · · , λ2g−3 for a compact Riemann surface of genus g. These are eigenvalues that may tend to zero for metrics with curvature −1. However, λ2g−2 , λ2g−1 , · · · , λ4g−1 always appear in [cg , 41 ] where cg > 0 depends only on g. It will be nice to find the optimal cg . In this regard, one may mention the very deep problem of Selberg on lower estimate of λ1 for 3 surfaces defined by an arithmetic group. Selberg proved that it is grater than 16 and it was later improved by Luo-Rudnick-Sarnak [478]. For a higher dimensional locally symmetric space, there is a similar question of Selberg and results similar to Selberg’s were found by J. S. Li [423] and Cogdell-Li-Piatetski-Shapiro-Sarnak [165]. Many researchers attempt to use Kazdhan’s property T for discrete groups to study Selberg’s problem. 15

There are many important properties of eigenfunctions that were studied in the seventies. For example, Cheng [137] found a beautiful estimate of multiplicities of eigenvalues of Riemann surfaces based only on genus. The idea was used by Colin de Verdière [174] to embedded graphes into R3 when they satisfy nice combinatorial properties. The connectivity and the topology of nodal domains are very interesting questions. Melas [506] did prove that for a convex planar domain, the nodal line of second eigenfunctions must intersect the boundary in exactly two points. Very little is known about the number of nodal domains except the famous theorem of Courant that the number of nodal domains of the m-th eigenfunction is less than m. There are several important questions related to the size of nodal sets and the number of critical points of eigenfunctions. I made a conjecture (see [735]) about the area of nodal sets, and significant progress toward its resolution was made by Donnelly-Fefferman [206], Dong [205] and F. H. Lin [456]. The number of critical points of an eigenfunction is difficult to determine. I [742] managed to prove the existence of a critical point near the nodal set. Jakobson and Nadirashvili [357] gave a counterexample to my conjecture that the number of critical points of the n-th eigenfunction is unbounded when n tends to infinity. I believe the conjecture is true for generic metrics and deserves to be studied extensively. Nadirashvili and his coauthors [342, 317] were also the first to show that the critical sets of eigenfunctions in n-dimensional manifold have a finite H n−2 -Hausdorff measure. Afterwards, Han-Hardt-Lin [315] gave an explicit estimate. When there is potential V , the eigenvalues of − △ +V are also important. When V is convex, with Singer, Wong and Stephen Yau, I applied the argument that I had with Peter Li to estimate the gap λ2 − λ1 [620]. When V is arbitrary, I [743] observed how this gap depends on the lower eigenvalue of the Hessian of − log ψ, where ψ is the ground state. The method of continuity was used by me in 1980 to reprove the work of Brascamp-Lieb [78] on the convexity of − log ψ when V is convex (This work appeared in the appendix of [620]). When V is the scalar curvature, this was studied by Schoen and myself extensively. In fact, in [600], we found an upper estimate of the first 2 Dirichlet eigenvalue of the operator −△ + 21 R in terms of 3π where r is a certain concept of radius 2r 2 related to loops in a three dimensional manifold. (If we replace loops with higher dimensional spheres, one can define a similar concept of radius. It will be nice if such a concept can be related to eigenvalues of differential forms.) This operator is naturally related to conformal deformation, stability of minimal surfaces, etc. (The works of D. Fischer-Colbrie and Schoen [222], Micallef [508], Schoen-Yau [593, 599] on stable minimal surfaces all depend on an understanding of spectrum of this operator.) The parabolic version appears in the recent work of Perelman. If there is a closed non-degenerate elliptic geodesic in the manifold, Babiˇc [25], Guillemin and Weinstein [300] found a sequence of eigenvalues of the Laplacian which can be expressed in terms of the length, the rotation angles and the Morse index of the geodesic. Comment: It is important to understand how harmonic functions or eigenfunctions oscillate. Gradient estimate is a good tool to achieve this. Gradient estimate for the log of the eigenfunction can be used to prove the Louville theorem or give a good estimate of eigenvalues. For higher eigenfunctions, it is important to understand its zero set and its growth. By controlling this information, one can estimate the dimension of these functions. A good upper estimate for eigenvalues depends on geometric intuition which may lead to construction of trial functions that are more adaptive to geometry. It should be emphasized that a clean and sharp estimate for the linear operator is key to obtaining good estimates for the nonlinear operator. (c). Heat kernel. Most of the work on the heat kernel over Euclidean space can be generalized to those manifolds where Sobolev and Poincaré inequalities hold. (It should be noted that Aubin [22, 24] and Talenti [648] did find best constant for various Sobolev inequalities on Euclidean space.) These inequalities are all related to isoperimetric inequalities. C. Croke [177] was able to follow my work [729] on Poincaré inequalities to prove the Sobolev inequality depending only on volume, diameter and the lower bound of Ricci curvature. Arguments of John Nash were then used by 16

Cheng-Li-Yau [138] to give estimates of the heat kernel and its higher derivatives. In this paper, an estimate of the injectivity radius was derived and this estimate turns out to play a role in Hamilton’s theory of Ricci flow. A year later, Cheeger-Gromov-Taylor [121] made use of the wave kernel to reprove this estimate. D. Stroock (see [534]) was able to use his methods from Malliavin’s calculus to give remarkable estimates for the heat kernel at a pair of points where one point is at the cut locus of another point. The estimate of the heat kernel was later generalized by Davies [184, 185], Saloff-Coste [578] and Grigor’yan [274, 275] to complete manifolds with polynomial volume growth and volume doubling property. Since these are quasi-isometric invariants, their analysis can be applied to graphs or discrete groups. See Grigor’yan’s survey [275] and Saloff-Coste’s survey [579]. On the other hand, the original gradient estimate that I derived is a pointwise inequality that is much more adaptable to nonlinear theory. Peter Li and I [443] were able to find a parabolic version of it in 1984. We observed its significance for estimates on the heat kernel and its relation to the variational principle for paths in spacetime. Coupled with the work of Cheeger-Yau [123], this gives a very precise estimate of the heat kernel. Such ideas turn out to provide fundamental estimates which play a crucial role in the analysis of Hamilton’s Ricci flow [310, 311]. Not much is known about the heat kernel on differential forms or differential forms with twisted coefficients. The fundamental idea of using the heat equation to prove the Hodge theory came from Milgram-Rosenbloom. The heat kernel for differential forms with twisted coefficients does play an important role in the analytic proof of the index theorem, as was demonstrated by AtiyahBott-Patodi [13]. It is the alternating sum that exhibits cancellations and gives rise to index of elliptic operators. When t is small, the alternating sum reduces to a calculation of curvature forms. When t is large, it gives global information on harmonic forms. Since the index of the operator is independent of t, we can relate the index to characteristic forms. If a compact manifold is the quotient of a non-compact manifold by a discrete group and if the heat kernel of the non-compact manifold can be computed explicitly, it can be averaged to give the heat kernel of the quotient manifold. Since the integral of the later kernel on the diagonal can be P −tλ computed by the spectrum to be e i , one can relate the displacement function of the discrete group to the spectrum. This is the Selberg trace formula relating length of closed geodesics to the spectrum of the Laplacian. Comment: Understanding the heat kernel is almost the same as understanding the heat equation. However, heat kernel satisfies semi-group properties, which enables one to give a good estimate of the maximum norm or higher order derivative norms as long as the Sobolev inequality can be proved. It is useful to look at the heat equation in spacetime where the Li-Yau gradient estimate is naturally defined. The estimate provides special pathes in spacetime for the estimate of the kernel. However, the effects of closed geodesics have not been found in the heat equation approach. A sharp improvement of the Li-Yau estimate may lead to such information. (d). Isoperimetric inequalities. Isoperimetric inequality is a beautiful subject. It has a long history. Besides its relation to eigenvalues, it reviews the deep structure of manifolds. The best constant for the inequality is important. P´ olya-Szegö [561], G. Faber (1923), E. Krahn (1925) and P. Lévy (1951) made fundamental contributions. Gromov generalized the idea of Lévy to obtain a good estimate of Cheeger’s constant (see [287]). C. Croke [178] and Cao-Escobar [103] have worked on domains in a simply connected manifold with non-positive curvature. It is assumed that the inequality holds for any minimal subvariety in Euclidean space. But it is not known to be true for the best constant. Li-Schoen-Yau [436] did prove it in the case of a minimal surface with a connected boundary, and E. Lutwak, Deane Yang and G. Y. Zhang did some beautiful work in the affine geometry case (see, e.g., [480, 481]). In Hamilton’s proof of Ricci flow convergent to the round metric on S 2 , he demonstrated that the isoperimetric constant of the metric is improving. One sees how the constant controls the geometry of the manifold. 17

Comment: The variational principle has been the most important method in geometry since the Greek mathematicians. Fixing the area of the domain and minimizing the length of the boundary is the most classical form of isoperimetric inequality. This principle has been generalized to much more general situations in geometry and mathematical physics. In most cases, one tries to prove existence of the extremal object and establish isoperimetric inequalities by calculating corresponding quantities for the extremal object. There is also the idea of rearrangement or symmetrization to prove isoperimetric inequalities. In the other direction, there is the duality principle in the alculus of variation: instead of minimizing the length of the boundary, one can fix it and maximize the area it encloses. The principle can be effective in complicated variational problems. (e). Harmonic analysis on discrete geometry. There are many other ideas in geometric analysis that can be discretized and applied to graph theory. This is especially true for the theory of spectrum of graphs. Some of these were carried out by F.Chung, Grigor’yan and myself (see the reference of Chung’s survey [163]). But the results in [163] are far away from being exhaustive. On the other hand, Margulis [487] and Lubotzky-Phillips-Sarnak [476] were able to make use of discrete group and number theory to construct expanding graphes. Methods to construct and classify these expanding graphs are important for application in computer science. It should be noted that Kazhdan’s property (T) [378] did play an important role in such discussions. It is also important to see how to give a good decomposition of any graph using the spectral method. The most important work for the geometry of a finitely presented group was done by Gromov [282]. He proved the fundamental structure theorem of groups where volume grows at most polynomially. These groups must be virtually nilpotent. Geometric ideas were developed by Varopoulos and his coauthors [689, 38] on the precise behaviors of the heat kernel in terms of volume growth. As an application of the theory of amenable groups, R. Brooks [80] was able to prove that if a manifold covers a compact set by a discrete group Γ, then it has positive eigenvalue if and only if Γ is non-amenable. Gromov [281] also developed a rich theory of hyperbolic groups using concepts of isoperimetric type inequalities. It would be nice to characterize these groups that are fundamental groups of compact manifolds with non-positive curvature or locally symmetric spaces. Comment: The geometry of a graph or complex can be used as a good testing ground for geometric ideas. They can be important in understanding smooth geometric structures. Many rough geometric concepts such as isoperimetric inequalities, can be found on graphs and in fact they play some roles in computer network theory. On the other hand, many natural geometric concepts should be generalized to graphs. For example, the concept of the fiber bundle, bundle theory over graphs and harmonic forms. It is likely one needs to have a good way to define the concept of equivalence between such objects. When we approximate a smooth manifold by a graph or complex, we only care about the limiting object and therefore some equivalence relations should be allowed. In the case of Cayley graph of a finitely generated group, it depends on the choice of the generating set, and properties independent of this generating set are preferable if we are only interested in the group itself. In the other direction, computer networks and other practical subjects have independent interest in graph theory. A close collaboration between geometer and computer scientists would be fruitful. (f ). Harmonic analysis via hyperbolic operators. There are important works of Fefferman, Phong, Lieb, Duistermaat, Guillemin, Melrose, Colin de Verdier, Taylor, Toth, Zelditch and Sarnak on understanding the spectrum of the Laplacian from the point of view of semi-classical analysis (see, e.g. [220, 209, 323, 583]). Some of their ideas can be traced back to the geometric optics analysis of J. Keller. The fundamental work of Duistermaat and Hörmander [208] on propagation 18

of singularities was also used extensively. There has been a lot of progress on the very difficult question of determining when one ”Can hear the shape of a drum” by, among others, Melrose (see [507]), Guillemin [297] and Zelditch [749]. (Priori to this, Guillemin and Kazhdan [298] proved that no negatively curved closed surface can be isospectrally deformed.) The first counterexample for closed manifolds was given by J. Milnor [515] on a 16 dimensional torus. The idea was generalized by Sunada [645], Gordon-Wilson [261]. For domains in Euclidean spaces, there were examples by Urakawa in three dimensions. Two dimensional counterexamples were given by Gordon-WebbWolpert [260], Wilson and Szab´ o [646]. Most of the ideas for counterexamples are related to the Selberg trace formula discussed in the section of heat kernel. The semi-classical analysis based on the hyperbolic operator also gives a very precise estimate or relation between the geodesic and the P √−1t√λi spectrum. The support of the singularities of the trace of the wave kernel e is a subset of the set of the lengths of closed geodesics. It is difficult to achieve such results by elliptic theory. However, most results are asymptotic in nature. It will be remarkable if both methods can be combined. Comment: Fourier expansion has been a very powerful tool in analysis and geometry. Practically any general theorem in classical Fourier analysis should have a counterpart in analysis of the spectrum of the Laplacian. The theory of geometric optics and the propagation of a singularity gives deep understanding of the singularity of a wave kernel. Geodesic and closed geodesic becomes an important means to understand eigenvalues. However, the theory has not been fruitful for the Laplacian acting on differential forms. Should areas of minimal submanifolds play a role? In the case of K¨ ahler manifolds, holomorphic cycles or the volume of special Lagrangian cycles should be important, as the length of close geodesics appear in the exponential decay term of the heat kernel. It would be useful to sharpen the heat equation method to capture this lower order information. (g). Harmonic forms. Natural generalizations of harmonic or holomorphic functions are harmonic or holomorphic sections of bundles with connections. The most important bundles are the exterior power of cotangent bundles. Using the Levi-Civita connection, harmonic sections are harmonic forms which, by the theory of de Rham and Hodge, give canonical representation of cohomology classes. The major research on harmonic forms comes from Bochner’s vanishing theorem [58]. But our understanding is still poor except for 1-forms or when the curvature operator is positive, in which case, the Bochner argument proved the manifold to be a homology sphere. If there is any nontrivial operator which commutes with the Laplacian, the eigenforms split accordingly. Making use of special structures of such splitting, the Bochner method can be more effective. For example, when the manifold is K¨ ahler, differential forms can be decomposed further to (p, q)-forms and the Kodaira vanishing theorem [388] yields much more powerful information, when the (p, q) forms are twisted with a line bundle or vector bundles. Similar arguments can be applied to manifolds with a special holonomy group depending on the representation theory of the holonomy group. When the complex structure moves holomorphically, the subbundles of (p, q) forms in the bundle of (p+q) forms do not necessarily deform holomorphically. The concept of Hodge filtration is therefore introduced. When we deform the complex structure around a point where the complex structure degenerates, there is a monodromy group acting on the Hodge filtration. The works of GriffithsSchmid [273] and Schmid’s SL2 (R) theorem [584] give powerful control on the degeneration of the Hodge structure. Deligne’s theory of mixed Hodge structure [186] plays a fundamental role for studying singular algebraic varieties. The theory of variation of Hodge structures is closely related to the study of period of the differential forms. This theory also appears in the subject of mirror symmetry. It is desirable to give a precise generalization of these works to higher dimensional moduli spaces where Kaplan-Cattani-Schmid made important contributions. Harmonic forms give canonical representation to de Rham cohomology. However the wedge product of harmonic forms need not be harmonic. The obstruction comes from secondary cohomology 19

cooperation. K. T. Chen [131] studied the case of 1-forms and Sullivan [642] studied the general ¯ case and gave a minimal model theory for a rational homotopic type of a manifold. Using ∂ ∂-lemma of Kähler manifolds, Deligne-Griffiths-Morgan-Sullivan [187] showed that the rational homotopic type is formal for K¨ ahler manifolds. The importance of harmonic forms is that they give canonical representation to the de Rham cohomology which is isomorphic to singular cohomology over real numbers. It gives a powerful tool to relate local geometry to global topology. In fact the vanishing theorem of Bochner-KodairaLichnerowicz allows one to deduce from sign of curvature to vanishing of cohomology. This has been one of the most powerful tools in geometry in the past fifty years. The idea of harmonic forms came from fluid dynamics and Maxwell equations. The non-Abelian version is the Yang-Mills theory. Most of the works on Yang-Mills theory have been focused on these gauge fields where the absolute minimum is achieved by some (topological) characteristic number. (These are called BPS state in physics literature.) When the dimension of the manifold is four, the star operator maps two form to two form and it makes sense to require the curvature form to be self-dual or anti-self-dual depending whether the curvature form is invariant or anti-invariant under the star operator. These curvature forms can be interpreted as non-Abelian harmonic forms. The remarkable fact is that when the metric is Kähler, the anti-self-dual connections give rise to holomorphic bundles. The moduli space of such bundles can often be computed using tools from algebraic geometry. If we take the product space M × M where M is the four dimensional manifold and M is the moduli space of anti-self-dual connections, there is a universal bundle V over M × M. By studying the slant product and the Chern classes of V , we can construct polynomials on the cohomology of M that are invariants of the differentiable structure of M . These are Donaldson polynomials (see [203]). In general M is not compact and Donaldson has to construct cycles in M for such operations. Donaldson invariants are believed to be equivalent to Seiberg-Witten invariants, where the vanishing theorem can apply and powerful geometric consequences can be found. Kronheimer and Mrowka [400] built an important concept of simple type for Donaldson invariants. It is believed that Donaldson invariants of algebraic surfaces of general type are of simple type. If the manifold is symplectic, we can look at the moduli space of pseudo-holomorphic curves. (These are J-invariant maps from Riemann surfaces to the manifold. J is an almost complex structure that is tame to the symplectic form.) Symplectic invariants can be created and they are called Gromov-Witten invariants. Y. Ruan [575] has observed that they need not be diffeomorphic invariants. It may still be interesting to know whether Gromov-Witten invariants are invariants of differentiable structures for Calabi-Yau manifolds. De Rham cohomology can only capture the non-torsion part of the singular cohomology. Weil [706] and Allendoerfer-Eells [5] attempted to use differential forms with poles to compute cohomology with integer coefficients. Perhaps one should study Chern forms of a complex bundle with a connection that satisfies the Yang-Mills equation and whose curvature is square integrable. The singular set of the connection may be allowed to be minimal submanifolds. The moduli space of such objects may give information about integral cohomology. It should be noted that CheegerSimons [122] did develop a rich theory of differential character with values in R/Z. It depends on the connections of the bundle. Witten managed to integrate the Chern-Simons forms [151] on the space of connections to obtain the knot invariants of Jones [361]. When we look for different operators acting on different forms, we may have to look into different kinds of harmonic forms. For example, if we are interesting in ∂ ∂¯ cohomology, we may look for the ∗ ¯ ¯ operator ∂ ∂ ∂ ∂ + ∂∂ ∗ + ∂¯∂¯∗ . It would be interesting to see how super-symmetry may generalize the concept of harmonic forms. Comment: The theory of harmonic form is tremendously powerful because it provides a natural link between global topology, analysis, geometry, algebraic geometry and arithmetic geometry. However, our analytic understanding of high degree forms is poor. For one forms, we can integrate along paths. For two forms, we can take an 20

interior product with a vector field to create a moment map. For closed (1, 1)-forms ¯ . However, we do not have in a K¨ ahler manifold, we can express them locally as ∂ ∂f good ways to reduce a high degree form to functions which are easier to understand. Good estimates of higher degree forms will be very important. ¯ 2.3.2. ∂-operator. Construction of holomorphic functions or holomorphic sections of vector bundles and holomorphic curves are keys to understanding complex manifolds. In order to demonstrate the idea behind the philosophy of determining the structure of manifolds by function theory, I was motivated to generalize the uniformization theory of a Riemann surface to higher dimensions when I was a graduate student. During this period, I was influenced by the works of Greene-Wu [271] in formulating these conjectures. Greene and Wu were interested in knowing whether the manifolds are Stein or not. When the complete K¨ ahler manifold is compact with positive bisectional curvature, this is the Frankel conjecture, as was proved independently by Mori [526] and Siu-Yau [629]. Both arguments depend on the construction of rational curves of low degree. Mori’s argument is stronger, and it will be good to capture his result by the analytic method. When the manifold has nonnegative bisectional curvature and positive Ricci curvature, Mok-Zhong [523] and Mok [519], using ideas of Bando [31] in his thesis on Hamilton’s Ricci flow, proved that the manifold is Hermitian symmetric. When the complete K¨ ahler manifold is noncompact with positive bisectional curvature, I conjectured that it must be biholomorphic to Cn (see [735]). Siu-Yau [628] made the first attempt to prove such a conjecture by using the L2 -method of Hörmander [345] to construct holomorphic functions with slow growth. (Note that Hörmander’s method goes back to Kodaira, which was also generalized by Calabi-Vesentini [94].) Singular weight functions were used in this paper and later much more refined arguments were developed by Nadel [531] and Siu [626] using what is called the multiplier ideal sheaf method. Siu found important applications of this method in algebraic geometry and also related the idea to the powerful work of J. Kohn on weakly pseudo-convex domain. This work of Siu-Yau was followed by Siu-Mok-Yau [520] and Mok [517, 518] under assumptions about the decay of curvature and volume growth. Shi [607, 608, 609] introduced Hamilton’s Ricci flow to study my conjecture, and his work is fundamental. This was followed by beautiful works of Cao [100, 101], Chen-Zhu [126, 127] and Chen-Tang-Zhu [124]. Assuming the manifold has maximal Euclidean volume growth and bounded curvature, Chen-Tang-Zhu [124] (for complex dimension two) and then Ni [535] (for all higher dimension) were able to prove the manifold can be compactified as a complex variety. Last year, Albert Chau and Tam [112] were finally able to settle the conjecture assuming maximal Euclidean volume growth and bounded curvature. An important lemma of L. Ni [535] was used, where a conjecture of mine (see [738] or the introduction of [535]) was proved. The conjecture says that maximal volume growth implies scalar curvature decays quadratically in the average sense. While we see great accomplishments for Kähler manifolds with positive curvature, very little is known for K¨ ahler manifolds, which are complete simply connected with strongly negative curvature. It is conjectured to be a bounded domain in Cn . (Some people told me that Kodaira considered a similar problem. But I cannot find the appropriate reference.) The major problem is to construct bounded holomorphic functions. The difficulty of construction of bounded holomorphic functions is that the basic principle of the 2 L -method of H¨ ormander comes from Kodaira’s vanishing theorem. It is difficult to obtain elegant results by going from weighted L2 space to bounded functions. In this connection, I was able to show that non-trivial bounded holomorphic functions do not exist on a complete manifold with non-negative Ricci curvature [734]. If the manifold is the universal cover of a compact Kähler manifold M which has a homotopically nontrivial map to a compact Riemann surface with genus > 1, then one can construct a bounded holomorphic function, using arguments of Jost-Yau [368]. In particular, if M has a map to a 21

product of Riemann surfaces with genus > 1 with nontrivial topological degree, the universal cover should have a good chance to be a bounded domain. Of course, this kind of construction is based on the fact that holomorphic functions are harmonic. Certain rigidity based on curvature forced the converse to be true. For functions, the target space has no topology and rigidity is not expected. Bounded holomorphic functions can not be constructed by solving the Dirichlet problem unless some boundary condition is assumed. This would make good sense if the boundary has a nice CR structure. Indeed, for odd dimensional real submanifold in Cn which has maximal complex linear subspace on each tangent plane, Harvey-Lawson [319, 320] proved the remarkable theorem that they bound complex submanifolds. Unfortunately the boundary of a complete simply connected manifold with bounded negative curvature does not have a smooth boundary. It will be nice to define a CR structure on such a singular boundary. One may mention the remarkable work of Kuranishi [403, 404, 405] on embedding of an abstract CR structure. Historically a motivation for the development of the ∂¯ operator came from the Levi problem, which was solved by Morrey, Grauert and greatly improved by Kohn and Hörmander. Their methods are powerful in studying pseudoconvex manifolds. In this regard, one may mention the conjecture of Shafarevich that the universal cover of an algebraic manifold is pseudoconvex. Many years ago, I conjectured that if the second homotopy group of the manifold is trivial, its universal cover can be embedded into a domain of some algebraic manifold where the covering transformations act on the domain by birational transformations. One may also mention the work of S. Frankel [226] on proving that an algebraic manifold is Hermitian symmetric if the universal cover is a convex domain in complex Euclidean space. Comment: The ∂¯ operator is the fundamental operator in complex geometry. Classically it was used to solve the uniformization theorem, the Levi problem and the Corona problems. We have seen much progress on the higher dimensional generalizations of the first two problems. However, due to poor understanding of the construction of bounded holomorphic functions, we are far away from understanding the Corona problem in a higher dimension and many related geometric questions. 2.3.3. Dirac operator. A very important bundle is the bundle of spinors. The Dirac operator acting on spinors is the most mysterious but major geometric operator. Atiyah-Singer were the first mathematicians to study it in geometry and by thoroughly understanding the Dirac operator, they were able to prove their celebrated index theorem [20]. On a Kähler manifold, the Dirac operator can be considered as a ∂¯ + ∂¯∗ operator acting on differential forms with coefficients on the square root of the canonical line bundle. Atiyah-Singer’s original proof can be traced back to the celebrated Riemann-Roch-Hirzebruch formula and the Hirzebruch index formula. The formulas of Gauss-Bonnet-Chern and Atiyah-Singer-Hirzebruch should certainly be considered as the most fundamental identities in geometry. The vanishing theorem of Lichnerowicz [451] on harmonic spinors over spin manifolds with positive scalar curvature gives strong information. Through the ˆ Atiyah-Singer index theorem, it gives the vanishing theorem for the A-genus and the α invariants for spin manifolds with positive scalar curvature. The method was later sharpened by Hitchin [329] to prove that every Einstein metric over K3−surfaces must be Kähler and Ricci flat. An effective use of Lichnerowicz formula for a spinC structure for a four dimensional manifold is important for Seiberg-Witten theory, which couples the Dirac operator with a complex line bundle. Lawson-Yau [411] were able to use Lichnerowicz’s work coupled with Hitchin’s work to prove a large class of smooth manifolds have no smooth non-Abelian group action and, by using modular forms, K. F. Liu proved a loop space analogue of the Lawson-Yau’s theorem for the vanishing of the Witten genus in [465]. On the basis of the surgery result of Schoen-Yau [593, 596] and Gromov-Lawson [288, 289], one expects that a suitable converse to Lichnerowicz’s theorem exists. The chief result is that surgery on spheres with codimension ≥ 3 preserves a class of metrics with positive scalar curvature. Once 22

geometric surgery is proved, standard works on cobordism theory allow one to deduce existence results for simply connected manifolds with positive scalar curvature. The best work in this direction is due to Stolz [637] who gave a complete answer in the case of simply connected manifolds with dimension greater then 4. I also suggested the possibility of performing surgery on an asymptotic hyperbolic manifold with conformal boundary whose scalar curvature is positive. This is related to the recent work of Witten-Yau [718] on the connectedness of the conformal boundary. The study of metrics with positive scalar curvature is the first important step in understanding the positive mass conjecture in general relativity. Schoen-Yau [594, 598] gave the first proof using ideas of minimal surfaces. Three years later, Witten [712] gave a proof using harmonic spinors. Both approaches have been fundamental to questions related to mass and other conserved quantities in general relativity. In the other direction, Schoen-Yau [596] generalized their argument in 1979 to find topological obstructions for higher dimensional manifolds with positive scalar curvature. Subsequently Gromov-Lawson [288, 289] observed that the Lichnerowicz theorem can be coupled with a fundamental group and give topological obstructions for a metric with positive scalar curvature. This work was related to the Novikov conjecture where many authors, including Lusztig [479], Rosenberg [567], Weinberger [708] and G. L. Yu [747] made contributions. Besides its importance in demonstrating the stability of Minkowski spacetime, the positive mass conjecture was used by Schoen [586] in a remarkable manner to finish the proof of the Yamabe problem where Trudinger [682] and Aubin [21] made substantial contributions. Comment: The Dirac operator is perhaps one of the most mysterious operators in geometry. When it is twisted with other bundles, it gives the symbol of all first order elliptic operators. When it couples with a complex line bundle it gives the Seiberg-Witten theory which provides powerful information for four manifolds. On the other hand, there were two different methods to study metrics with positive scalar curvature. It should be fruitful to combine both methods: the method of Dirac operator and the method of minimal submanifolds.

2.3.4. First order operator twisted by vector fields or endomorphisms of bundles. Given a vector field X on a manifold, we can consider the complex of differential forms ω so that LX ω = 0. On such complex, d + ιX defines a differential and the resulting cohomology is called equivariant cohomology. During the seventies, Bott [71] and Atiyah-Bott [12] developed the localization formula for equivariant cohomology. Both the concepts of a moment map and equivariant cohomology have become very important tools for computations of various geometric quantities, especially Chern numbers of natural bundles. The famous work of Atiyah, Guillemin-Sternberg on the convexity of the image of the moment map gives a strong application of equivariant cohomology to toric geometry. The formula of Duistermaat-Heckman [207] played an important role in motivation for evaluation of path integrals. These works have been used by Jeffrey and Kirwan [358] and by K. F. Liu and his coauthors on several topics: the mirror principle (Lian-Liu-Yau [447, 448, 449, 450]), topological vertex (Li-Liu-Liu-Zhou [428]), etc. The idea of applying localization to enumerative geometry was initiated by Kontsevich [391] and later by Givental [255] and Lian-Liu-Yau [447] independently. (Lian-Liu-Yau [447] formulated a functorial localization formula which has been fundamental for various calculations in mirror geometry.) These works solve the identities conjectured by Candelas et al [98] based on mirror symmetry, and provide good examples of the ways in which conformal field theory can be a source of inspiration when looking at classical problems in mathematics. If we twist the ∂¯ operator with an endomorphism valued holomorphic one form s so that s◦s = 0, it gives rise to a complex ∂¯ + s. This was the Higgs theory initiated by Hitchin [330] and studied extensively by Simpson [617]. There is extensive work of Zuo Kang and Jost-Zuo (see [755]) on Higgs theory and representation of fundamental groups of algebraic manifold. 23

In string theory, there is a three form H and the cohomology of dc + H has not been well understood. It would be interesting to develop a deeper understanding of such twisted cohomology and its localization. Comment: The idea of deforming a de Rham operator by twisting with some other zero order operators has given powerful information to geometry. Witten’s idea of the analytic proof of Morse theory is an example. Equivariant cohomology is another example. We expect to see more works in such directions. 2.3.5. Spectrum and global geometry. Weyl made a famous address in the early fifties. The title of his talk was The Eigenvalue Problem Old and New. He was excited P by the work of Minakshisundaram and Pleijel which asserts that the zeta function ζ(s) = λ λ−s , where λ are eigenvalues of the Laplacian, not only makes sense for Re’s large, but also has meromorphic extension to the whole complex s-plane, the position of whose poles could be described explicitly. In particular, it is analytic near s = 0. Formally dζ(s) ds |s=0 can be viewed as − log det(∆). This gives a definition of determinant of Laplacian which entered into the fundamental work of Ray-Singer relating Reidemeister’s combinational invariant of a manifold with analytic torsion defined by determinants of the Laplacians acting on differential forms of various degrees. Other application of zeta function expressed in terms of kernel is the calculation of the asymptotic growth of eigenvalues in terms of volume of the manifold. Tauberian type theorem is needed. This initiated the subject of finding formula to relate spectrum of manifolds with their global geometry. Atiyah and Singer [20] were the most important contributors to this beautiful subject. Atiyah-Bott-Patodi [13] applied the heat kernel expansion to a proof of the local index theorem. Atiyah-Patodi-Singer [17, 18, 19] initiated the study of spectrum flow and gave important global spectral invariants on odd dimensional manifolds. These global invariants become boundary terms for the L2 -index theorem developed by Atiyah-Donnelly-Singer [14] and Mark Stern [636]. (A method of Callias [95] has been used for such calculations.) Witten [713, 714] has introduced supersymmetry and analytic deformation of the de Rham complex to Morse theory, and thereby revealed a new aspect of the connection between global geometry and theoretical physics. Witten’s work has been generalized by Demailly [189] and Bismut-Zhang [55, 56] to study the holomorphic Morse inequality and analytic torsion. Novikov [537] also studied Morse theory for one forms. Witten’s work on Morse theory inspired the work of Floer (see, e.g., [223, 224, 225]) who used his ideas in Floer cohomology to prove Arnold’s conjecture in case where the manifold has vanishing higher homotopic group. Floer’s theory is related to knot theory (through Chern-Simon’s theory [151]) on three manifolds. Atiyah, Donaldson, Taubes , Dan Freed, P. Braam, and others (see, e.g., [10, 654, 76, 228]) all contributed to this subject. Fukaya-Ono [240], Oh [540], Kontsevich [392], Hofer-Wysocki-Zehnder [339], G. Liu-Tian [464], all studied such a theory in symplectic geometry. Some part of Arnold’s conjecture on fixed points of groups acting on symplectic manifolds was claimed to be proven. But a completely satisfactory proof has not been forthcoming. One should also mention here the very important work of Cheeger [114] and M¨ uller [529] in which they verify the conjecture of Ray-Singer equating analytic torsion with the combinational torsion of the manifold. The fundamental idea of Ray-Singer [563] on holomorphic torsion is still being vigorously developed. It appeared in the beautiful work of Vafa et al [50]. Many more works on analytic torsion were advanced by Quillen, Todorov, Kontsevich, Borcherds , Bismut, Lott, Zhang and Z. Q. Lu (see [54] and it’s reference, [363], [63, 64]). The local version of the index theorem by Atiyah-Bott-Patodi [13] was later extended in an sophisticated way by Bismut [53] to an index theorem for a family of elliptic operators.(The local index argument dates back to the foundational work of McKean-Singer [495] where methods were developed to calculate coefficients of heat kernel expansion.) The study of elliptic genus by Witten [715], Bott-Taubes [72], Taubes [653], K. F. Liu [466] and M. Hopkins [344] has built a bridge between topology and modular form. Comment: The subject of relating the spectrum to global topology is extremely rich. It is likely that we have only touched part of this rich subject. The deformation 24

of spectrum associated with the deformation of geometric structure is always a fascinating subject. Global invariants are created by spectral flows. Determinants of elliptic operators are introduced to understand measures of infinite dimensional space. Geometric invariants that are created by asymptotic expansion of heat or wave kernels are in general not well understood. It will be a long time before we have a much better understanding of the global behavior of spectrum.

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3. Mappings between manifolds and rigidity of geometric structures There is a need to exhibit a geometric structure in a simpler space: hence we embed algebraic manifolds into complex projective space, we isometrically embed a Riemannian manifold into Euclidean space and we classify structures such as bundles by studying maps into Grassmannian. We are also interested in probing the structure of a manifold by mapping Riemann surfaces inside the manifold, an important example being holomorphic curves in algebraic manifolds. Of course, we are also interested in maps that can be used to compare the geometric structures of different manifolds. 3.1. Embedding. 3.1.1. Embedding theorems. Holomorphic sections of holomorphic line bundles have always been important in algebraic geometry. The Riemann-Roch formula coupled with vanishing theorems gave very powerful existence results for sections of line bundles. The Kodaira embedding theorem [389] which said that every Hodge manifold is projective has initiated the theory of holomorphic embedding of K¨ ahler manifolds. For example, Hirzebruch-Kodaira [328] proved that every odd (complex) dimensional K¨ ahler manifold diffeomorphic to projective space is biholomorphic to projective space. (I proved the same statement for even dimensional Kähler manifolds based on K¨ ahler Einstein metric.) Given an orthonormal basis of holomorphic sections of a very ample line bundle, we can embed the manifold into projective space. The induced metric is the Bergman metric associated with the line bundle. Note that the original definition of the Bergman metric used the canonical line bundle and L2 -holomorphic sections. In the process of understanding the relation between stability of a manifold and the existence of the Kähler Einstein metric, I [737] proposed that every Hodge metric can be approximated by the Bergman metric as long as we allow the power of the line bundle to be large. Following the ideas of the paper of Siu-Yau [628], Tian [672] proved the C 2 convergence in his thesis under my guidance. My other student W. D. Ruan [571] then proved C ∞ convergence in his thesis. This work was followed by Lu [475], Zelditch [748] and Catlin [109] who observed that the asymptotic expansion of the kernel function follows from some rather standard expressions of the Szegö kernel, going back to Fefferman [219] and Boutet de Monvel-Sjöstrand [74] on the circle bundle associated with the holomorphic line bundle over the K¨ ahler manifold. Recently, Dai, Liu, Ma and Marinescu [181] [484] obtained the asymptotic expansion of the kernel function by using the heat kernel method, and gave a general way to compute the coefficients, thus also extended it to symplectic and orbifold cases. Kodaira’s proof of embedding Hodge manifolds by the sufficiently high power of a positive line bundle is not effective. Matsusaka [489] and later Kollár [390], Siu [624] were able to provide effective estimate of the power. Demailly [191, 192] and Siu [624, 626] made a remarkable contribution toward the solution of the famous Fujita conjecture [236] (see also Ein and Lazarsfeld [212]). Siu’s powerful method also leads to a proof of the deformation invariance of plurigenera of algebraic manifolds [625]. It should be noticed that the extension theorem of Ohsawa-Takegoshi played an important role in this last work of Siu. Comment: The idea of embedding a geometric structure is clearly important as once they are put in the same space, we can compare them and study the moduli space of the geometric structure much better. For example, one can define Chow coordinate of a projective manifold and we can study various concepts of geometric stability of these structures. At this moment, there is no natural universal space of Kähler manifolds or complex manifolds as we may not have a positive holomorphic line bundle over such manifolds to embed into complex projective space. In a similar vein, it will be nice to find a universal space for symplectic manifolds. 26

3.1.2. Compactification. Problem of compactification of the manifold dates back to Siegel, Satake, Baily-Borel [26] and Borel-Serre [66]. They are important for representation theory, for algebraic geometry and for number theory. For geometry of non-compact manifolds, we like to control behavior of differentiable forms at infinity. A good exhaustion function is needed. Construction of a proper exhaustion function with a bounded Hessian on a complete manifold with a bounded curvature was achieved by Schoen-Yau [602] in 1983 in our lectures in Princeton. Based on this exhaustion function, M. Dafermos [179] was able to reprove a theorem of CheegerGromov [120] that such manifolds admit an exhaustion by compact hypersurfaces with bounded second fundamental form. Such exhaustions are useful to understand characteristic forms on noncompact manifolds as the boundary term can be controlled by the second fundamental form of the hypersurfaces. After my work with Siu [630] on compactification of a strongly, negatively curved Kähler manifold with finite volume, I proposed that every complete Kähler manifold with bounded curvature, finite volume and finite topology should be compactifiable to be a compact complex variety. I suggested this problem to Mok and Zhong in 1982 who did significant work [524] in this direction. (The compactification by Mok-Zhong is not canonical and it is desirable to find an algebraic geometric analogue of Borel-Baily compactification [26] so that we can study the L2 -cohomology in terms of the intersection cohomology of the compactification.) Recall that the important conjecture of Zucker on identifying L2 -cohomology with the intersection cohomology of the Borel-Baily compactification was settled by Saper-Stern [581] and Looijenga [472]. (Intersection cohomology was introduced by Goresky-MacPherson [263, 264]. It is a topological concept and hence the Zucker conjecture gives a topological meaning of the L2 -cohomology.) It would be nice to find compactification for algebraic varieties so that suitable form of intersection cohomology can be used to understand L2 cohomology. Goresky-Harder-MacPherson [262] and Saper [580] have contributed a lot toward this kind of question. For moduli space of bundles, or polarized projective structures, compactification means degeneration of these structures in a suitable canonical manner. For algebraic curves, there is Deligne-Mumford compactification [188] which has played a fundamental role in understanding algebraic curves. Geometric invariant theory (see [530]) gives a powerful method to introduce the concept of stable structures. Semi-stable structures can give points at infinity. The compactification based on the geometric invariant theory for moduli space of surfaces of the general type was done by Gieseker [253]. For a higher dimension, this was done by Viehweg [690]. Detailed analysis of the divisors at infinity is still missing. Comment: Compactification of a manifold is very much related to the embedding problem. One needs to construct functions or sections of bundles near infinity. For the moduli space of geometric structures, it amounts to degenerately the structures canonically. It will be important to study the degeneration of Hermitian Yang-Mills connections and K¨ ahler Einstein metrics. 3.1.3. Isometric embedding. Given a metric tensor on a manifold, the problem of isometric embedding is equivalent to find enough functions f1 , · · · , fN so that the metric can be written as P 2 (dfi ) . Much work was accomplished for two dimensional surfaces as was mentioned in section 2.1.2. Isometric embedding for the general dimension was solved in the famous work of J. Nash [532, 533]. Nash used his famous implicit function theorem which depends on various smoothing operators to gain derivatives. In a remarkable work, G¨ unther [305] was able to avoid the Nash procedure. He used only the standard Hölder regularity estimate for the Laplacian to reproduce the Nash isometric embedding with the same regularity result. In his book [285], Gromov was able to lower the codimension of the work of Nash. He called his method the h-principle. When the dimension of the manifold is n, the expected dimension of the Euclidean space for the manifold to be isometrically embedded is n(n+1) . It is important to understand manifolds 2 isometrically embedded into Euclidean space with this optimal dimension. Only in such a dimension 27

does it make sense to talk about rigidity questions. It remains a major open problem whether one can find a nontrivial smooth family of isometric embeddings of a closed manifold into Euclidean space with an optimal dimension. Such a nontrivial family was found for a polyhedron in Euclidean three space by Connelly [175]. For a general manifold, it is desirable to find a canonical isometric embedding into a given Euclidean space by minimizing the L2 norm of its mean curvature within the space of isometric embeddings. Chern told me that he and H. Lewy studied local isometric embedding of a three manifold into six dimensional Euclidean space. But they didn’t wrote any manuscript on it. The major work in this subject was done by E. Berger, Bryant, Griffiths and Yang [84] [47]. They showed that a generic three dimensional embedding system is strictly hyperbolic, and the generic four dimensional system is a real principal type. Local existence is true for a generic metric using a hyperbolic operator and the Nash-Moser implicit function theorem. If the target space of isometric embedding is a linear space with indefinite metric, it is possible that the problem is easier. For example, by a theorem of Pogorelov [557, 558], any metric on the two dimensional sphere can be isometrically embedded into a three dimensional hyperbolic spaceform (where the sectional curvature may be a large negative constant). Hence it can always be embedded into the hyperboloid of the Minkowski spacetime. This statement may also be true for surfaces with higher genus. The fundamental group may cause obstruction, hence the first step should be an attempt to canonically embed any complete metric (with bounded curvature) on a simply connected surface into a three dimensional hyperbolic space form. It should be also very interesting to study the rigidity problem of a space-like surface in Minkowski spacetime. Besides requesting the metric to be the induced metric, we shall need one more equation. Such an equation should be related to the second fundamental form. A candidate appeared in the work of M. Liu-Yau [462, 463] on the quasi-local mass in general relativity. In the other direction, Calabi found the condition for a Kähler metric to be isometrically and holomorphically embedded into Hilbert space with an indefinite signature. In the course of his investigation, he introduced some kind of distance function that can be defined by the K¨ ahler potential and enjoys many interesting properties. Calabi’s work in this direction which should be relevant to the flat coordinate appeared in the recent works of Vafa et al [50]. Comment: The theory of isometric embedding is a classical subject. But our knowledge is still rather limited, especially in dimension greater than three. Many difficult problems are related to nonlinear mixed type equation or hyperbolic differential systems over a closed manifold. 3.2. Rigidity of harmonic maps with negative curvature. One can define the energy of maps between manifolds and the critical maps are called harmonic maps. In 1964, Eells-Sampson [211] and Al’ber [3] independently proved the existence of such maps in their homotopy class if the image manifold has a non-positive curvature. When I was working on manifolds with non-positive curvature, I realized that it is possible to use harmonic map to reprove some of the theorems in my thesis. I was convinced that it is possible to use harmonic maps to study rigidity questions in geometry such as Mostow’s theorem [527]. In 1976, I proved the Calabi conjecture and applied the newly proved existence of the K¨ ahler Einstein metric and the Mostow rigidity theorem to prove uniqueness of a complex structure on the quotient of the ball [731]. Motivated by this theorem, I proposed to use the harmonic map to prove the rigidity of a complex structure for Kähler manifolds with strongly negative sectional curvature. I proposed this to Siu who carried out the idea when the image manifold satisfies a stronger negative curvature condition [621]. Jost-Yau [366] proved that for harmonic maps into manifolds with non-positive curvature, the fibers give rise to holomorphic foliations even when the map is not holomorphic. Such a work was found to be used in the work of Corlette, Simpson et al. A further result was obtained by Jost-Yau [369] and Mok-Siu-Yeung [521] on the proof of the superrigidity theorem of Margulis [486], improving an earlier result of Corlette [176] who proved 28

superrigidity for a certain rank one locally symmetric space. Complete understanding of superrigidity for the quotient of a complex ball is not yet available. One needs to find more structures for harmonic maps which reflect the underlying structure of the manifold. The analytic proof of super-rigidity was based on an argument of Matsushima [491] as was suggested by Calabi. (This was a topic discussed by Calabi in the special year on geometry in the Institute for Advanced Study.) The discrete analogue of harmonic maps is also important. When the image manifold is a metric space, there are works by Gromov-Schoen [290], Korevaar-Schoen [397] and Jost [365]. Margulis knew that the super-rigidity for both the continuous and the discrete case is enough to prove Selberg’s conjecture for the arithmeticity of lattices in groups with rank ≥ 2. Unfortunately, the analytic argument mentioned above only works if the lattices are cocompact as it is difficult to find a degree one smooth map with finite energy for non-cocompact lattices. Harmonic maps into a tree have given interesting applications to group theory. When the domain manifold is a simplicial ´ complex, there are articles by Ballmann-Swwi¸ atkowski [29] and M. T. Wang [699, 700], where they introduce maps from complices which are generalizations of buildings. They also generalized the work of H. Garland [244] on the vanishing of the cohomology group for p-adic buildings. Using the concept of the center of gravity, Besson-Courtois-Gallot [52] give a metric rigidity theorem for rank one locally symmetric space. They also proved a rigidity theorem for manifolds with negative curvature: if the fundamental group can be split as a nontrivial free product over some other group C, the manifold can be split along a totally geodesic submanifold with the fundamental group C. Comment: The harmonic map gives the first step in matching geometric structures of different manifolds. Eells-Sampson derived it from the variational principle. One can also use different elliptic operators to define maps which satisfy elliptic equations. Higher dimensional applications are mostly based on the assumption that the image manifold has a metric with non-positive curvature. In such a case, existence is easier and uniqueness (as shown by Hartman) is also true. Up to now, significant results on higher dimensional harmonic maps are based on such assumptions. Generalization to kähler manifold should be reasonable. The second homotopic group should play a role as one may look at it as a generalization of the work of Sacks-Uhlenbeck. It may be possible to use harmonic maps to study the moduli of geometric structure on a fixed manifold as was done by Michael Wolf for Riemann surfaces. It will also be nice to see how a harmonic map can be used to compare graphs. 3.3. Holomorphic maps. The works of Liouville, Picard, Schwarz-Pick and Ahlfors show the importance of hyperbolic complex analysis. Grauèrt-Reckziegel [266] generalized this kind of analysis to higher dimensional complex manifolds. Kobayashi [386] and H. Wu [722] put this theory in an elegant setting. Kobayashi introduced the concept of hyperbolic complex manifolds. Its elegant formulation has been influential. An important application of the negative curvature metric is the extension theorem for holomorphic maps, as was achieved by the work of Griffiths-Schmid [273] on maps to a period domain and by the extension theorem of Borel [65] on compactification of Hermitian symmetric space. A major question was Lang’s conjecture: on an algebraic manifold of a general type, there exists a proper subvariety such that the image of any holomorphic map from C must be a subset of this subvariety. It has deep arithmetic geometric meaning. In terms of the Kobayashi metric, it says that the Kobayashi metric is nonzero on a Zariski open set. Many works were done towards subvarieties of Abelian variety by Bloch, Green-Griffiths, Kobayashi-Ochiai, Voitag and Faltings. For generic hypersurfaces in CP n , there is work by Siu [627]. They developed the tool of jet differentials and meromorphic connections. For algebraic surfaces with C12 > 2C2 , Lu-Yau [473] proved Lang’s conjecture, based on the ideas of Bogomolov. Comment: Holomorphic maps have been studied for a long time. There is no general method to construct such maps based on the knowledge of topology alone, 29

except the harmonic map approach proposed by me and carried out by Siu, JostYau and others. But the approach is effective only for manifolds with negative curvature. For rigidity questions, the most interesting manifolds are Kähler manifolds with non-positive Ricci curvature, which give the major chunk of algebraic manifolds of a general type. The Kähler-Einstein metric should provide tools to study such problems. Is there any intrinsic way, based on the metric, to find the largest subvariety where the image of all holomorphic maps from the complex line lie? Deformation theory of such a subvariety should be interesting. There is also the question of when the holomorphic image of the complex line will intersect a divisor. Cheng and I did find good conditions for the complement of a divisor to admit the complete K¨ ahler-Einstein metric. For such a geometry, the holomorphic line should either intersect the divisor or a subset of some subvarieties. This kind of questions are very much related to arithmetic questions if the manifolds are defined over number fields. 3.4. Harmonic maps from two dimensional surfaces and pseudoholomorphic curves. Harmonic maps behave especially well for Riemann surface. Morrey was the first one who solved the Dirichlet problem for energy minimizing harmonic map into any Riemannian manifold. Another major breakthrough was made by Sacks-Uhlenbeck [577] in 1978 where they constructed minimal spheres in Riemannian manifolds representing elements in the second homotopy group using a beautiful extension theorem of a harmonic map at an isolated point. By pushing their method further, Siu-Yau [629] studied the bubbling process for the harmonic map and made use of it to prove a stable harmonic map must be holomorphic under curvature assumptions. As a consequence, they proved the famous conjecture of Frankel that a Kähler manifold with positive bisectional curvature is CP n , as was discussed in section [2.3.2]. Gromov [284] then realized that a pseudoholomorphic curve for an almost complex structure can be used in a similar way to prove rigidity of a symplectic structure on CP n . The bubbling process mentioned above was sharpened further to give compactification of the moduli space of pseudoholomorphic maps by Ye [745] and Parker-Wolfson [545]. Based on these ideas, Kontsevich [391] introduced the concept of stable maps and the compactification of their moduli spaces. The formal definitions of Gromov-Witten invariants and quantum cohomology were based on these developments and the ideas of physicists. For example, quantum cohomology was initiated by Vafa (see, e.g., [688]) and his coauthors (the name was suggested by Greene and me). Associativity in quantum cohomology was due to four physicists WDVV [716, 193]. The mathematical treatment (done by Ruan [575] and subsequently by Ruan-Tian [576]) followed the gluing ideas of the physicists. Ruan-Tian made use of the ideas of Taubes [652]. But important points were overlooked. A. Zinger [752, 753] has recently completed these arguments. In close analogy with Donaldson’s theory, one needs to introduce the concept of virtual cycle in the moduli space of stable maps. The algebraic setting of such a concept is deeper than the symplectic case and is more relevant to the development for algebraic geometry. The major idea was conceived by Jun Li who also did the algebraic geometric counterpart of Donaldson’s theory (see [424, 429]). (The same comment applies to the concept of the relative Gromov-Witten invariant, where Jun Li made the vital contribution in the algebraic setting [426, 427].) The symplectic version of Li-Tian [430] ignores difficulties, many of which were completed recently by A. Zinger [752, 753]. Sacks-Uhlenbeck studied harmonic maps from higher genus Riemann surfaces. Independently, Schoen-Yau [597] studied the concept of the action of an L21 map on the fundamental group of a manifold. It was used to prove the existence of a harmonic map with prescribed action on the fundamental group. Jost-Yau [367] generalized such action on fundamental group to a more general setting which allows the domain manifold to be higher dimensional. Recently F. H. Lin developed this idea further [458]. He studied extensively geometric measure theory on the space of maps (see, 30

e.g., [455, 457]). The action on the second homotopy group is much more difficult to understand. I think there should exist a harmonic map with nontrivial action on the second homotopic group if such a continuous map exists. Such an existence theorem will give interesting applications to Kähler geometry. There is a supersymmetric version of harmonic maps studied by string theorists. This is obtained by coupling the map with Dirac spinors in different ways (which corresponds to different string theories). While this kind of world sheet theory is fundamental for the development of string theory, geometers have not paid much attention to the supersymmetric harmonic map. Interesting applications may be found. The most recent paper of Chen, Jost, Li and Wang [133] does address to a related problem where they studied the regularity and energy identities for Dirac-Harmonic maps. Comment: Maps from circle or Riemann surfaces into a Riemannian manifold give a good deal of information about the manifold. The capability to construct holomorphic or pseudo-holomorphic maps from spheres with low degree was the major reason that Mori, Siu-Yau and Taubes were able to prove the rigidity of algebraic or symplectic structures on the complex projective space. It will be desirable to find more ways to construct such maps from low genus curves to manifolds that are not of a rational type. Their moduli space can be used to produce various invariants. An outstanding problem is to understand the invariants on counting curves of a higher genus which appeared in the fundamental paper of Vafa et al [50]. 3.5. Morse theory for maps and topological applications. The energy functional for maps from S 2 into a manifold does not quite give rise to Morse theory. But the perturbation method of Sacks-Uhlenbeck did provide enough information for Micallef-Moore [509] to prove some structure theorem for manifolds with positive isotropic curvature. (Micallef and Wang [510] then proved the vanishing of second Betti number in the even dimensional case. If the manifold is irreducible, has non-negative isotropic curvature and non-vanishing second Betti number, then they proved that its second Betti number equals to one and it is Kähler with positive first Chern class.) If the image manifold has negative curvature, the theorem of Eells-Sampson [211] says that any map can be canonically deformed by the heat flow to a unique harmonic map. Hence the topology of the space of maps is given by the space of homomorphism between the fundamental groups of the manifolds. This gives some information of the topology of manifolds with negative curvature. Farrell and Jones [217] have done much deeper analysis on the differentiable structure of manifolds with negative curvature. Schoen-Yau [597] exploited the uniqueness theorem for harmonic maps to demonstrate that only finite groups can act smoothly on a manifold which admits a non-zero degree map onto a compact manifold with negative curvature. The size of the finite group can also be controlled. If the image manifold has non-positive curvature, then the only compact continuous group actions are given by the torus. The topology of the space of maps into Calabi-Yau manifolds should be very interesting for string theory. Sullivan [644] has developed an equivariant homology theory for loop space. It will be interesting to link such a theory with quantum cohomology when the manifold has a symplectic structure. Comment: Morse theory has been one of the most powerful tools in geometry and topology as it connects local to global information. One does not expect full Morse theory for harmonic maps as we have difficulty even proving their existence. However, if their existence can be proven, the perturbation technique may be used and powerful conclusions may be drawn. 3.6. Wave maps. In early eighties, C. H. Gu [294] studied harmonic maps when the domain manifold is the two dimensional Minkowski spacetime. They are called wave maps. Unfortunately, good global theory took much longer to develop as there were not many good a priori estimates. 31

This subject was studied extensively by Christodoulou , Klainerman, Tao, Tataru and M. Struwe (see, e.g., [160, 383, 649, 650, 606]). It is hoped that such theory may shed some light on Einstein equations. Comment: The geometric or physical meaning of wave maps should be studied. The problem of vibrating membrane gives a good motivation to study time-like minimal hypersurface in a Minkowski spacetime. One can study the vibration of submanifold by looking into the minimal time-like hypersurface with the boundary given by the submanifold. It is a mystery how such vibrations can be related to the eigenvalues of the submanifold. 3.7. Integrable system. Classically, Bäcklund (1875) was able to find a nonlinear transformation to create a surface with constant curvature in R3 from another one. The nonlinear equation behind it is the Sine-Gordon equation. Then in 1965, Kruskal and Zabusky (see [401]) discovered solitons and subsequently in 1967, Gardner, Greene, Kruskal and Miura [243] discovered the inverse scatting method to solve the KdV equations. The subject of a completely integrable system became popular. Uhlenbeck [686] used techniques from integrable systems to construct harmonic maps from S 2 to U (n), Bryant [82] and Hitchin [332] also contributed to related constructions using twistor theory and spectral curves. These inspired Burstall, Ferus, Pedit and Pinkall [88] to construct harmonic maps from a torus to any compact symmetric space. In a series of papers, Terng and Uhlenbeck [664, 665] used loop group factorizations to solve the inverse scattering problem and to construct B¨ acklund transformations for soliton equations, including Schrödinger maps from R1,1 to a Hermitian symmetric space. There are recent attempts by Martin Schmidt [585] to use an integrable system to study the Willmore surface. The integrable system also appeared naturally in several geometric questions such as the Schottky problem (see Mulase [528]) and the Witten conjecture on Chern numbers of bundles over moduli space of curves. Geroch found the Backlund transformation for axially symmetric stationary solutions of Einstein equations. It will be nice to find such nonlinear transformations for more general geometric structures. Comment: It is always important to find an explicit solution to nonlinear problem. Hopefully an integrable system can help us to understand general structures of geometry. 3.8. Regularity theory. The major work on regularity theory of harmonic maps in higher dimensions was done by Schoen-Uhlenbeck [588, 589]. (There is a weaker version due to Giaquinta-Giusti [251] and also the earlier work of Ladyzhenskaya-Ural’ceva and Hildebrandt-Kaul-Widman where the image manifolds for the maps are more restrictive.) Leon Simon (see [616]) made a deep contribution to the structure of harmonic maps or minimal subvarieties near their singularity. This was followed by F. H. Lin [457]. The following is still a fundamental problem: Are singularities of harmonic maps or minimal submanifolds stable when we perturb the metric of the manifolds? Presumably some of them are. Can we characterize them? How big is the codimension of generic singularities? In the other direction Schoen-Yau [592] also proved that degree one harmonic maps are one to one if the image surface has a non-positive curvature. Results of this type work only for two dimensional surfaces. It will be nice to study the set where the Jacobian vanishes. Comment: There is a very rich theory of stable singularity for smooth maps. However, in most problems, we can only afford to deform certain background geometric structures, while the extremal objects are still constrained by the elliptic variational problem. Understanding this kind of stable singularity should play fundamental roles in geometry.

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4. Submanifolds defined by variational principles 4.1. Teichm¨ uller space. The totality of the pair of polarized Kähler manifolds with a homotopic equivalence to a fixed manifold gives rise to the Teichm¨ uller space. For an Algebraic curve, this is the classical Teichm¨ uller space. This space is important for the construction of the mapping problem for minimal surfaces of a higher genus. In fact, given a conformal structure on a Riemann surface Σ, a harmonic map from Σ to a fixed Riemannian manifold may minimize energy within a certain homotopy class. However, it may not be conformal and may not be a minimal surface. In order to obtain a minimal surface, we need to vary the conformal structure on Σ also. Since the space of conformal structures on a surface is not compact, one needs to make sure the minimum can be achieved. If the map f induces an injection on the fundamental group of the domain surface, Schoen-Yau [593] proved the energy of the harmonic map is proper on the moduli space of conformal structure on this surface by making use of a theorem of Linda Keen [379]. Based on a theory of topology of the L21 map, they proved the existence of incompressible minimal surfaces. As a product of this argument, it is possible to find a nice exhaustion function for the Teichm¨ uller space. Michael Wolf [719] was able to use harmonic maps to give a compactification of Teichm¨ uller space which he proved to be equivalent to the Thurston compactification. S. Wolpert studied extensively the behavior of the Weil-Petersson metric (see Wolpert’s survey [720]). A remarkable theorem of Royden [570] says that the Teichm¨ uller metric is the same as the Kobayashi metric. C. McMullen [498] introduced a new Kähler metric on the moduli space which can be used to demonstrate that the moduli space is hyperbolic in the sense of Gromov [286]. The great detail of comparison of various intrinsic metrics on the Teichm¨ uller space had been a major problem [737]. It was accomplished recently in the works of Liu-Sun-Yau [467, 468]. Actually Liu-Sun-Yau introduced new metrics with bounded negative curvature and geometry and found the stability of the logarithmic cotangent bundle of the moduli spaces. Recently L. Habermann and J. Jost [306, 307] also study the geometry of the Weil-Petersson metric associated to the Bergmann metric on the Riemann surface instead of the Poincaré metric. Comment: For a conformally invariant variational problem, Teichm¨ uller space plays a fundamental role. It covers the moduli space of curves and in many ways behaves like a Hermitian symmetric space of noncompact type. Unfortunately, there is no good canonical realization of it as a pseudo-convex domain in Euclidean space. For example, we do not know whether it can be realized as a smooth domain or not. There is also Teichm¨ uller space for other algebraic manifolds, such as Calabi-Yau manifolds. It is an important question in understanding their global behavior. 4.2. Classical minimal surfaces in Euclidean space. There is a long and rich history of minimal surfaces in Euclidean space. Recent contributions include works by Meeks, Osserman, Lawson, Gulliver, White, Hildebrandt, Rosenberg, Collin, Hoffman, Karcher, Ros, Colding, Minicozzi, Rodríguez, Nadirashvili and others (see the reference in Colding and Minicozzi’s survey [173]) on embedded minimal surfaces in Euclidean space. They come close to classifying complete embedded minimal surfaces and a good understanding of complete minimal surface in a bounded domain. For example, Meeks-Rosenberg [499] proved that the plane and helicoid are the only properly embedded simply connected minimal surfaces in R3 . Calabi also initiated the study of isometric embedding of Riemann surfaces into S N as minimal surfaces. The geometry of minimal spheres and minimal torus was then pursued by many geometers [106], [149], [82], [332], [408]. Comment: This is one of the most beautiful subjects in geometry where Riemann made important contributions. Classification of complete minimal surface is nearly accomplished. However a similar problem for compact minimal surfaces in S 3 is far from being solved. It is also difficult to detect which set of disjoint Jordan curves can 33

bound a connected minimal surface. The classification of moduli space of complete minimal surfaces with finite total curvature should be studied in detail. 4.3. Douglas-Morrey solution, embeddedness and application to topology of three manifolds. In a series of papers started in 1978, Meeks-Yau [501, 502, 503, 504] settled a classical conjecture that the Douglas solution for the Plateau problem is embedded if the boundary curve is a subset of a mean convex boundary. (One should note that Osserman [542] had already settled the old problem of non-existence of branched points for the Douglas solution while Gulliver [304] proved non-existence of false branched points.) We made use of the area minimizing property of minimal surfaces to prove these surfaces are equivariant with respect to the group action. Embedded surfaces which are equivariant play important roles for finite group actions on manifolds. Coupling with a theorem of Thurston, we can then prove the Smith conjecture [744] for cyclic groups acting on the spheres: that the set of fixed points is not a knotted curve. The Douglas-Morrey solution of the Plateau problem is obtained by fixing the genus of the surfaces. However, it is difficult to minimize the area when the genus is allowed to be arbitrary large. This was settled by Hardt-Simon [318] by proving the boundary regularity of the varifold solution of the Plateau problem. In the other direction, Almgren-Simon [7] succeeded in minimizing the area among embedded disks with a given boundary in Euclidean space. The technique was used by Meeks-Simon-Yau [500] to prove the existence of embedded minimal spheres enclosing a fake ball. This theorem has been important to prove that the universal covering of an irreducible three manifold is irreducible. They also gave conditions for the existence of embedding minimal surfaces of a higher genus. This work was followed by topologists Freedman-Hass-Scott [230]. Pitts [553] used the mini-max argument for varifolds to prove the existence of an embedded minimal surfaces. Simon-Smith (unpublished) managed to prove the existence of an embedded minimax sphere for any metric on the three sphere. J. Jost [364] then extended it to find four mini-max spheres. Pitts-Rubinstein (see, e.g., [554]) continued to study such mini-max surfaces. Since such minimax surfaces have Morse index one, I was interested in representing such minimal surface as a Heegard splitting of the three manifolds. I estimated its genus based on the fact that the second eigenvalue of the stability operator is nonnegative. This argument (dates back to Szego-Hersch) is to map the surface conformally to S 2 . Hence we can use three coordinate functions, orthogonal to the first eigenfunction, to be trial functions. The estimate gave an upper bound of the genus for mini-max surfaces in compact manifolds with a positive scalar curvature. About twenty years ago, I was hoping to use such an estimate to control a Heegard genus as a way to prove Poincaré conjecture. While the program has not materialized, three manifold topologists did adapt the ideas of Meeks-Yau to handle combinational type minimal surfaces and gave applications in three manifold topology. The most recent works of Colding and Minicozzi [168, 169, 170, 171] on lamination by minimal surfaces and estimates of minimal surfaces without the area bound are quite remarkable. They [172] made contributions to Hamilton’s Ricci flow by bounding the total time for evolution. Part of the idea came from the above mentioned inequality. Comment: The application of minimal surface theory to three manifold topology is a very rich subject. However, one needs to have a deep understanding of the construction of minimal surfaces. For example, if minimal surfaces are constructed by the method of mini-max, one needs to know the relation of their Morse index to the dimension of the family of surfaces that we use to perform the procedure of mini-max. A detail understanding may lead to a new proof of the Smale conjecture, as we may construct a minimal surface by a homotopic group of embeddings of surfaces. Conversely, topological methods should help us to classify closed minimal surfaces. 4.4. Surfaces related to classical relativity. Besides minimal surfaces, another important class of surfaces are surfaces with constant mean curvature and also surfaces that minimize the L2 -norm 34

of the mean curvature. It is important to know the existence of such surfaces in a three dimensional manifold with nonnegative scalar curvature, as they are relevant to question in general relativity. The existence of minimal spheres is related to the existence of black holes. The most effective method was developed by Schoen-Yau [600] where they [595] proved the existence theorem for the equation of Jang. It should be nice to find new methods to prove existence of stable minimal spheres. The extremum of the Hawking mass is related to minimization of the L2 norm of mean curvature. Their existence and behavior have not been understood. For surfaces with constant mean curvature, we have the concept of stability. (Fixing the volume it encloses, the second variation of area is non-negative.) Making use of my work on eigenvalues with Peter Li, I proved with Christodoulou [161] that the Hawking mass of such a surface is positive. (This was part of my contribution to the proposed joint project with Christodoulou-Klainerman which did not materialize.) This fact was used by Huisken and me [352] to prove uniqueness and the existence of foliation by constant mean curvature spheres for a three dimensional asymptotically flat manifold with positive mass. (We initiated this research in 1986. Ye studied our work and proved existence of similar foliations under various conditions, see [746].) This foliation was used by Huisken and Yau [352] to give a canonical coordinate system at infinity. It defines the concept of center of gravity where important properties for general relativity are found. The most notable is that total linear momentum is equal to the total mass multiple with the velocity of the center of the mass. One expects to find good asymptotic properties of the tensors in general relativity along these canonical surfaces. We hope to find a good definition of angular momentum based on this concept of center of gravity so that global inequality like total mass can dominate the square norm of angular momentum. The idea of using the foliation of surfaces satisfying various properties (constant Gauss curvature, for example) to study three manifolds in general relativity is first developed by R. Bartnik [41]. His idea of quasi-spherical foliation gives a good parametrization of a large class of metrics with positive scalar curvature. Some of these ideas were used by Shi-Tam [610] to study quantities associated to spheres which bound three manifolds with positive scalar curvature. Such a quantity is realized to be the quasilocal mass of Brown-York [81]. At the same time, Melissa Liu and Yau [462, 463] were able to define a new quasi-local mass for general spacetimes in general relativity, where some of the ideas of Shi-Tam were used. Further works by M. T. Wang and myself generalized Liu-Yau’s work by studying surfaces in hyperbolic space-form. My interest in quasi-local mass dates back to the paper that I wrote with Schoen [600] on the existence of a black hole due to the condensation of matter. It is desirable to find a quasi-local mass which includes the effect of matter and the nonlinear effect of gravity. Hopefully one can prove that when such a mass is larger than a constant multiple of the square root of the area, a black hole forms. This has not been achieved. Comment: When surfaces theory appears in general relativity, we gain intuitions from both geometry and physics together. This is a fascinating subject. 4.5. Higher dimensional minimal subvarieties. Higher dimensional minimal subvarieties are very important for geometry. There are works by Federer-Fleming [218], Almgren [6] and Allard [4]. The attempt to prove the Bernstein conjecture, that minimal graphs are linear, was a strong drive for its development. Bombieri, De Giorgi and Giusti [62] found the famous counterexample to the Bernstein problem. It initiated a great deal of interest in the area minimizing cone (as a graph must be area minimizing). Schoen-Simon-Yau [587] found a completely different approach to the proof of Bernstein problem in low dimensions. This paper on stable minimal hypersurfaces initiated many developments on curvature estimates for the codimension one stable hypersurfaces in higher dimension. There are also works by L. Simon with Caffarelli and Hardt [90] on constructing minimal hypersurfaces by deforming stable minimal cones. Recently N. Wickramasekera [710, 711] did some deep work on stable minimal (branched) hypersurfaces which generalizes Schoen-Simon-Yau. 35

Michael-Simon [511] proved the Sobolev inequality and mean valued inequalities for such manifolds. This enables one to apply the classical argument of harmonic analysis to minimal submanifolds. For a minimal graph, Bombieri-Giusti [61] used ideas of De Giorgi-Nash to prove gradient estimates of the graph. N. Korevaar [394] was able to reprove this gradient estimate based on the maximal principle. The best regularity result for higher codimension was done by F. Almgren [6] when he proved that for any area minimizing variety, the singular set has the codimension of at least two. How such a result can be used for geometry remains to be seen. It was observed by Schoen-Yau [593] that for a closed stable minimal hypersurface in a manifold with positive scalar curvature, the first eigenfunction of the second variational operator can be used to conformally deform the metric so that the scalar curvature is positive. This provides an induction process to study manifolds with a positive scalar curvature. For example, if the manifold admits a nonzero degree map to the torus, one can then construct stable minimal hypersurfaces inductively until we find a two dimensional surface with higher genus which cannot support a metric with positive scalar curvature. At this moment, the argument encounters difficulty for dimensions greater than seven as we may have problems of singularity. In any case, we did apply the argument to prove the positive action conjecture in general relativity. The question of which type of singularities for minimal subvariety are generic under metric perturbation remains a major question for the theory of minimal submanifolds. Perhaps the most important possible application of the theory of minimal submanifolds is the Hodge conjecture: whether a multiple of a (p, p) type integral cohomology class in a projective manifold can be represented by an algebraic cycle. Lawson made an attempt by combining a result of Lawson-Simons [409] and work of J. King [381] and Harvey-Shiffman [322]. (Lawson-Simons proved that currents in CP n which are minimizing with respect to the projective group action are complex subvarieties.) The problem of how to use the hypothesis of (p, p) type has been difficult. In general, the algebraic cycles are not effective. This creates difficulties for analytic methods. The work of King [381] and Shiffman [611] on complex currents may be relevant. Perhaps one should generalize the Hodge conjecture to include general (p, q) classes, as it is L possible that every integral cycle in ki=−k H p−i,p+i is rationally homologous to an algebraic sum of minimal varieties such that there is a p − k dimensional complex space in the tangent space for almost every point of the variety: it may be important to assume the metric to be canonical, e.g. the Kähler Einstein metric. A dual question is how to represent a homology class by Lagrangian cycles which are minimal submanifolds also. When the manifold is Calabi-Yau, these are special Lagrangian cycles. Since they are supposed to be dual to holomorphic cycles, there should be an L analogue of the Hodge i,j should be conjecture. For example if dimC M = n is odd, any integral element in i+j=n H representable by special Lagrangian cycles up to a rational multiple provided the cup product of it with the Kähler class is zero. A very much related question is: if the Chern classes of a complex vector bundle are of (p, p) type, does the vector bundle, after adding a holomorphic vector bundle, admit a holomorphic structure? If the above generalization of the Hodge conjecture holds, there should be a similar generalization for the vector bundle. It should also be noted that Voisin [692] observed that Chern classes of all holomorphic bundles do not necessarily generate all rational (p, p) classes. On the other hand, the Kähler manifold that she constructed is not projective. These questions had a lot more success for four dimensional symplectic manifolds by the work of Taubes both on the existence of pseudoholomorphic curves [661] and on the existence of antiself-dual connections [651, 652]. On a Kähler surface, anti-self-dual connections are Hermitian connections for a holomorphic vector bundle. In particular, Taubes gave a method to construct holomorphic vector bundles over K¨ ahler surfaces. Unfortunately this theorem does not provide much information on the Hodge conjecture as it follows from Lefschetz theorem in this dimension. 36

Another important class of minimal varieties is the class of special Lagrangian cycles in CalabiYau manifolds. Such cycles were first developed by Harvey-Lawson [321] in connection to calibrated geometry. Major works were done by Schoen-Wolfson [590], Yng-Ing Lee [415] and Butscher [89]. One expects Lagrangian cycles to be mirror to holomorphic bundles and special Lagrangian cycles to be mirror to Hermitian-Yang-Mills connections. Hence by the Donaldson-Uhlenbeck-Yau theorem, it is related to stability. The concept of stability for Lagrangian cycles was discussed by Joyce and Thomas. Since the Yang-Mills flow for Hermitian connection exists for all time, ThomasYau [667] suggested an analogy with the mean curvature flow for Lagrangian cycles. For stable Lagrangian cycles, mean curvature flow should converge to special Lagrangian cycles. See M. T. Wang [701, 702], Smoczyk [632] and Smoczyk-Wang [633]. The geometry of mirror symmetry was explained by Strominger-Yau-Zaslow in [639] using a family of special Lagrangian tori. There are other manifolds with special holonomy group. They have similar calibrated submanifolds. Conan Leung has contributed to studies of such manifolds and their mirrors (see, e.g. [418, 419]). Submanifolds of space forms are called isoparametric if the normal bundle is flat and the principal curvatures are constants along parallel normal fields. These were studied by E. Cartan [107]. Minimal submanifolds, with constant scalar curvature are believed to be isoparametric surfaces. There is work done by Lawson [407], Chern-de Carmo-Kobayashi [150] and Peng-Terng [546]. Recently there has been extensive work by Terng and Thorbergsson (see Terng’s survey [663] and Thorbergsson [668]). Terng [662] related isometric embedded hyperbolic spaces in Euclidean space to soliton theory. A nice theory of Lax pair and loop groups related to geometry has been developed. Comment: The theory of higher dimensional minimal submanifolds is one of the deepest subjects in geometry. Unfortunately our knowledge of the subject is not mature enough to give applications to solve outstanding problems in geometry, such as the Hodge conjecture. But the future is bright. 4.6. Geometric flows. The major geometric flows are flows of submanifold driven by mean curvature, gauss curvature, inverse mean curvature. Flows that change geometric structures are Ricci flows and Einstein flow. Mean curvature flow for varifolds was initiated by Brakke [77]. The level set approach was studied by many people: S. Osher, L. Evans, Giga, etc (see [541, 216, 135]). Huisken [347, 348] did the first important work when the initial surface is convex. His recent work with Sinestrari [350, 351] on mean convex surfaces is remarkable and gives a good understanding of the structure of singularities of mean curvature flow. Mean curvature flow has many geometric applications. For example, the work of Huisken-Yau mentioned in 4.4 was achieved by mean curvature flow. Mean curvature flow for spacelike hypersurfaces in Lorentzian manifolds should be very interesting. Ecker [210] did interesting work in this direction. It will be nice to find the Li-Yau type estimate for such flows. The inverse mean curvature was proposed by Geroch [249] to understand the Penrose conjecture relating the mass with the area of the black hole. Such a procedure was finally carried out by Huisken-Ilmanen [349] when the scalar curvature is non-negative. There was a different proof by H. Bray [79] subsequently. Ricci flow has had spectacular successes in recent years. However, not much progress has been made on the Calabi flow (see Chang’s survey [110]) for Kähler metrics. They are higher order problems where the maximal principle has not been effective. An important contribution was made by Chr´ usciel [162] for Riemann surface. Inspired by the concept of the Bondi mass in general relativity, Chru´sciel was able to give a new estimate for the Calabi flow. Unfortunately, a higher dimensional analogue had not been found. Natural higher order elliptic problems are difficult to handle. Affine minimal surfaces and Willmore surfaces are such examples. L. Simon [615] made an important contribution to the regularity of the Willmore surfaces. The corresponding flow problem should be interesting. 37

The dynamics of Einstein equations for general relativity is a very difficult subject. The Cauchy problem was considered by many people: A. Lichnerowicz, Y. Choquet-Bruhat, J. York, V. Moncrief, H. Friedrich, D. Christodoulou, S. Klainerman, H. Lindblad, M. Dafermos (see, e.g., [452], [157], [156], [233], [158], [382], [459], [180]). But the global behavior is still far from being understood. The major unsolved problem is to formulate and prove the fundamental question of Penrose on Cosmic censorship. I suggested to Klainerman and Christodoulou to consider small initial data for the Einstein system. The treatment of stability of Minkowski spacetime was accomplished by Christodoulou-Klainerman [159] under small perturbation of flat spacetime and fast fall off conditions. Recently Lindblad and Rodnianski [459] gave a simpler proof. A few years ago, N. Zipser (Harvard thesis) added Maxwell equation to gravity and still proved stability of Minkowski spacetime. There is remarkable progress on the problem of Cosmic censorship by M. Dafermos [180]. He made an important contribution for the spherical case. Stability for Schwarzschild or Kerr solutions is far from being known. Finster-Kamran-Smoller-Yau [221] had studied decay properties of Dirac particles with such background. The work does indicate the stability of these classical spacetimes. The no hair theorem for stationary black holes is a major theorem in general relativity. it was proved by W. Israël [353], B. Carter [108], D. Robinson [566] and S. Hawking [324]. But the proof is not completely rigorous for the Kerr metric. In any case, the existing uniqueness theorem does assume regularity of the horizon of the black hole. It is not clear to me whether a nontrivial asymptotically flat solitary solution of a vacuum Einstein equation has to be the Schwarzschild solution. There is a possibility that the Killing field is spacelike. In that case, there may be a new interesting vacuum solution. There is extensive literature on spacelike hypersurfaces with constant mean curvature. The foliation defined by them gives interesting dynamics of Einstein equation. These surfaces are interesting even for Rn,1 . A. Treiberges studied it extensively [681]. Li, Choi-Treibergs [153] and T. Wan [695] observed that the Gauss maps of such surfaces give very nice examples of harmonic maps mapping into the disk. Recently Fisher and Moncrief used them to study the evolution equation of Einstein in 2 + 1 dimension. Comment: The dynamics of submanifolds and geometric structures reveal the true nature of these geometric objects deeply. In the process of arriving at a stationary object or a solitary solution, it encounters singularities. Understanding the structures of such singularity will solve many outstanding conjectures in topology such as Shoenflies conjecture.

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5. Construction of geometric structures on bundles and manifolds A fundamental question is how to build geometric structures over a given manifold. In general, the group of topological equivalences that leaves this geometric structure invariant should be a special group. With the exception of symplectic structures, these groups are usually finite dimensional. When the geometric structure is unique (up to equivalence), it can be used to produce key information about the topological structure. The study of special geometric structures dates back to Sophus Lie, Klein and Cartan. In most cases, we like to be able to parallel transport vectors along paths so that we can define the concept of holonomy group. 5.1. Geometric structures with noncompact holonomy group. When the holonomy group is not compact, there are examples of projective flat structure, affine flat structure and conformally flat structure. It is not a trivial matter to determine which topological manifolds admit such structures. Since the structure is flat, there is a unique continuation property and hence one can construct a developing map from a suitable cover of the manifold to the real projective space, the affine space and the sphere respectively. The map gives rise to a representation of the fundamental group of the manifold to the real projective group, the special linear group and the Möbius group respectively. This holonomy representation gives a great deal of information for the geometric structure. Unfortunately, the map is not injective in general. In the case where it is injective, the manifold can be obtained as a quotient of a domain by a discrete subgroup of the corresponding Lie group. In this case, a lot more can be said about the manifold as the theories of partial differential equations and discrete groups can play important roles. 5.1.1. Projective flat structure. If a projective flat manifold can be projectively embedded as a bounded domain, Cheng-Yau [144] were able to construct a canonical metric from the real MongeAmpère equation which generalizes the Hilbert metric. When the manifold is two dimensional, there are works of C. P. Wang [696] and J. Loftin [469] on how to associate such metrics to a conformal structure and a holomorphic section of the cubic power of a canonical bundle. This is a beautiful theory related to the hyperbolic affine sphere mentioned in chapter one. There are fundamental works by S. Y. Choi, W. Goldman (see the reference of Choi-Goldman [155]), N. Hitchin [333] and others on the geometric decomposition and the moduli of projective structures on Riemann surfaces. It should be interesting to extend them to three or four dimensional manifolds. 5.1.2. Affine flat structure. It is a difficult question to determine which manifolds admit flat affine structures. For example, it is still open whether the Euler number of such spaces is zero, although great progress was made by D. Sullivan [640]. W. Goldman [257] has also found topological constraints on three manifolds in terms of fundamental groups. The difficulty arises as there is no useful metric that is compatible with the underlining affine structure. This motivated Cheng-Yau [145] to define the concept of affine K¨ ahler metric. When Cheng and I considered the concept of affine Kähler metric, we thought that it was a natural analogue of K¨ ahler metrics. However, compact nonsingular examples are not bountiful. Strominger-Yau-Zaslow [639] proposed the construction of mirror manifolds by constructing the quotient space of a Calabi-Yau manifold by a special Lagrangian torus. At the limit of the large Kähler class, it was pointed out by Hitchin [334] that the quotient space admits a natural affine structure with a compatible affine K¨ ahler structure. But in general, we do expect singularities of such structure. It now becomes a deep question to understand what kind of singularity is allowed and how we build the Calabi-Yau manifold from such structures. Loftin-Yau-Zaslow [470] have initiated the study of the structure of a ”Y” type singularity. Hopefully one can find an existence theorem for affine structures over compact manifolds with prescribed singularities along codimension two stratified submanifolds. 39

5.1.3. Conformally flat structure. Construction of conformally flat manifolds is also a very interesting topic. Similar to projective flat or affine flat manifolds, there are simple constraints from curvature representation for the Pontrjagin classes. The deeper problem is to understand the fundamental group and the developing map. When the structure admits a conformal metric with positive scalar curvature, Schoen-Yau [601] proved the rather remarkable theorem that the developing map is injective. Hence such a manifold must be the quotient of a domain in S n by a discrete subgroup of Möbius transformations. It would be interesting to classify such manifolds. In this regard, the Yamabe problem as was solved by Schoen [586] did provide a conformal metric with constant scalar curvature. One hopes to be able to use such metrics to control the conformal structure. Unfortunately the metric is not unique and a deep understanding of the moduli space of conformal metrics with constant scalar curvature should be important. Kazdan-Warner [377] and Korevaar-Mazzeo-Pacard-Schoen [396] developed a conformal method to understand Nirenberg’s problem on prescribed scalar curvature. It was followed by Chen-Lin [130], Chang-Gursky-Yang [111]. Chen-Lin has related this problem to mean field theory. Their computations in the relevant degree theory involves deep analysis. One should generalize their works to functions which are sections of a flat line bundle because it is related to the previously mentioned work of Loftin-Yau-Zaslow [470]. In any case, the integrability condition of Kazdan and Warner is still not fully understood. It is curious that while bundle theory was used extensively in Riemannian geometry, it has not been used in the study of these geometries. One can construct real projective space bundles, affine bundles or sphere bundles by mapping coordinate charts projectively, affinely or conformally to the corresponding model spaces (possible with dimensions different from the original manifold) and gluing the target model spaces together to form natural bundles. Perhaps one may study their associated Chern-Simons forms [151]. Many years ago, H. C. Wang [698] proved the theorem that if a compact complex manifold has trivial holomorphic tangent bundle, it is covered by a complex Lie group. It will be nice to generalize and interpret such a theorem in terms of Hermitian connections on the manifold with a special holonomy group and torsion. This program was discussed in my paper [740] on algebraic characterization of locally Hermitian symmetric spaces. For a holomorphic stable vector bundle V , we can form a stable vector bundles from V by taking N ∗ representation of GL(n, C) from decomposition of the tensor product N irreducible V representation q V . By twisting with powers of canonical line bundle, we can form irrep ducible stable bundles with trivial determinant line bundle. In general, such bundles may not have holomorphic sections. If they do, the section must be parallel with respect to the Hermitian-YangMills connection on the bundle, and the structure group of V can be reduced to a smaller group. Hermitian-Yang-Mills connections with reduced holonomy group have good geometric properties. We may formulate a principle: For stable holomorphic bundles, existence of nontrivial holomorphic invariants implies the existence of parallel tensors and therefore the reduction of structure group. If the holonomy group is reduced to discrete group, the bundle will provide representations of the fundamental group into unitary group. This should compare with Wang’s theorem when the bundle is the tangent bundle. Comment: Geometric structures with a noncompact holonomy group is less intuitive than Riemannian geometry. Perhaps we need to deepen our intuitions by relating them to other geometric structures, especially those structures that may carry physical meaning. 5.2. Uniformization for three manifolds. An important goal of geometry is to build a canonical metric associated to a given topology. Besides the uniformization theorem in two dimensions, the only (spectacular) work in higher dimensions is the geometrization program of Thurston (see [670]). W. Thurston made use of ideas from Riemann surface theory, W. Haken’s work [308] on threemanifolds, G. Mostow’s rigidity [527] to build his manifolds. Many mathematicians have contributed 40

to the understanding of this program of Thurston’s. (e.g., J. Morgan [525], C. McMullen [496, 497], J. Otal [543], J. Porti [562].) Thurston’s orbifold program was finally proved by M. Boileau, B. Leeb and J. Porti [60]. However, all this work has to assume that an incompressible surface (or corresponding surface in case of orbifold) exists. When R. Hamilton [309] had his initial success on his Ricci flow, I suggested (around 1981) to him to use his flow to break up the manifold and prove Thurston’s conjecture. His generalization of the theory of Li-Yau [443] to Ricci flow [310, 311] and his seminal paper in 1996 [313] on breaking up the manifold mark cornerstone of the remarkable program. Perelman’s recent idea [547, 548] built on these two works and has gone deeply into the problem. Detailed discussions have been pursued by Hamilton, Colding-Minicozzi, Shioya-Yamaguchi, Zhu, Cao and Huisken in the past two years. Hopefully it may lead to the final settlement of the geometrization program. This theory of Hamilton and Perelman should be considered as a crowning achievement of geometric analysis in the past thirty years. Most ideas developed in this period by geometric analysts are used. Let me now explain briefly the work of Hamilton and Perelman. In early 90’s, Hamilton [311, 312, 313] developed methods and theorems to understand the structure of singularities of the Ricci flow. Taking up my suggestions, he proved a fundamental Li-Yau type differential inequality (now called the Li-Yau-Hamilton estimate) for the Ricci flow with nonnegative curvature in all dimensions. He gave a beautiful interpretation of the work of Li-Yau and observed the associated inequalities should be equalities for solitary solutions. He then established a compactness theorem for (smooth) solutions to the Ricci flow, and observed (also independently by T. Ivey[354]) a pinching estimate for the curvature for three-manifolds. By imposing an injectivity radius condition, he rescaled the metric to show that each singularity is asymptotic to one of the three singularity models. For type I singularities in dimension three, Hamilton established an isoperimetric ratio estimate to verify the injectivity radius condition and obtained spherical or necklike structures. Based on the Li-Yau-Hamilton estimate, Hamilton showed that any type II model is either a Ricci soliton with a neck-like structure or the product of the cigar soliton with the real line. Similar characterization for type III model was obtained by Chen-Zhu [125]. Hence Hamilton had already obtained the canonical neighborhood structures (consisting of spherical, neck-like and cap-like regions) for the singularities of three-dimensional Ricci flow. But two obstacles remained: one is the injectivity radius condition and the other is the possibility of forming a singularity modelled on the product of the cigar soliton with a real line which could not be removed by surgery. Recently, Perelman [547] removed these two stumbling blocks in Hamilton’s program by establishing a local injectivity radius estimate (also called “Little Loop Lemma” by Hamilton in [312]). Perelman proved the Little Loop Lemma in two ways, one with an entropy functional he introduced in [547], the other with a reduced distance function based on the same idea as Li-Yau’s path integral in obtaining their inequality [547]. This reduced distance question gives rise to a Gaussian type integral which he called reduced volume. The reduced volume satisfies monotonicity property. Furthermore, Perelman [548] developed a refined rescaling argument (by considering local limits and weak limits in Alexandrov spaces) for singularities of the Ricci flow on three-manifolds to obtain a uniform and global version of the canonical neighborhood structure theorem. After obtaining the canonical neighborhoods for the singularities, one performs geometric surgery by cutting off the singularities and continue the Ricci flow. In [313], Hamilton initiated such a surgery procedure for four-manifolds with a positive isotropic curvature. Perelman [548] adapted Hamilton’s geometric surgery procedure to three-manifolds. The most important question is how to prevent the surgery times from accumulations and make sure there are only a finite number of surgeries on each finite time interval. When one performs the surgeries with a given accuracy at each surgery time, it is possible that the errors may add up which causes the surgery times to accumulate. Hence at each step of surgery one is required to perform the surgery more accurate than the former one. In [549], Perelman presented a clever idea on how to find “fine” necks, how to glue “fine” caps and how to use rescaling arguments to justify the discreteness of the surgery 41

times. In the process of rescaling for surgically modified solutions, one encounters the difficulty of how to use Hamilton’s compactness theorem, which works only for smooth solutions. The idea to overcome such difficulty consists of two parts. The first part, due to Perelman [548], is to choose cutoff radius (in neck-like regions) small enough to push the surgical regions far away in space. The second part, due to Chen-Zhu [129] and Cao-Zhu [102], is to show that the solutions are smooth on some uniform small time intervals (on compact subsets) so that Hamilton’s compactness theorem can be used. Once surgeries are known to be discrete in time, one can complete Schoen-Yau’s classification [599] for three-manifolds with positive scalar curvature. For simply connected three manifolds, if one can show solution to the Ricci flow with surgery extincts in finite time, Poincaré conjecture will be proved. Recently, such a finite extinction time result was proposed by Perelman [549] and a proof appeared in Colding-Minicozzi [172]. For the full geometrization program, one still needs to find the long time behavior of surgically modified solutions. In [314], Hamilton studied the long time behavior of the Ricci flow on a compact three-manifold for a special class of (smooth) solutions called “nonsingular solutions”. Hamilton proved that any (three-dimensional) nonsingular solution either collapses or subsequently converges to a metric with constant curvature on the compact manifold, or at large time it admits a thickthin decomposition where the thick part consists of a finite number of hyperbolic pieces and the thin part collapses. Moreover, by adapting Schoen-Yau’s minimal surface arguments to a parabolic version, Hamilton showed that the boundary of hyperbolic pieces are incompressible tori. Then by combining with the collapsing results of Cheeger-Gromov [119], any nonsingular solution to the Ricci flow is geometrizable. In [547, 548], Perelman modified Hamilton’s arguments to analyze the long-time behavior of arbitrary solutions to the Ricci flow and solutions with surgery in dimension three. Perelman also argued by showing a thick-thin decomposition, except that he can only show the thin part has (local) lower bound on sectional curvature. For the thick part, based on Li-Yau-Hamilton estimate, Perelman established a crucial elliptic type Harnack estimate to conclude the thick part consisting of hyperbolic pieces. For the thin part, he announced a new collapsing result which states that if a three-manifold collapses with a (local) lower bound on the sectional curvature, it is then a graph manifold. However, the proof of the new collapsing result has not been published. Shioya and Yamaguchi [612, 613] offered a proof for compact manifolds. Very recently, Cao-Zhu claimed to have a complete proof for compact manifolds based only on the Shioya-Yamaguchi’s collapsing result. Hopefully all these arguments can be checked thoroughly in the near future. It should also be interesting to see whether other famous problems in three manifold can be settled by analysis: Does every three dimensional hyperbolic manifold admit a finite cover with nontrivial first Betti number? Hyperbolic metrics have been used by topologists to give invariants for three dimensional manifolds. Thurston [669] observed that the volume of a hyperbolic metric is an important topological invariant. The associated Chern-Simons [151] invariant, which is defined by mod integers, can be looked upon as a phase for such manifolds. These invariants appeared later in Witten’s theory of 2+1 dimensional gravity [715] and S. Gukov [302] was able to relate them to fundamental questions in knot theory.

Comment: This is the most spectacular development in the last thirty years. Once the three manifold is hyperbolic, Ricci flow does not give much more information. Happily one may obtain further information by performing reduction from four dimension Ricci flow to three dimension by circle action. Is there any effective way to understand the totality of all hyperbolic manifolds with finite volume by constructing flows that may break up topology? 42

5.3. Four manifolds. The major accomplishment of Thurston, Hamilton, Perelman et al is the ability to create a canonical structure on three manifolds. Such a structure has not even been conjectured for four manifolds despite the great success of Donaldson invariants and Seiberg-Witten invariants. Taubes [655] did prove a remarkable existence theorem for self-dual metrics on a rather general class of four dimensional manifolds. Unfortunately their moduli space is not understood and their topological implication is not clear at this moment. Since the twistor space of Taubes metric admits integrable complex structure, ideas from complex geometry may be helpful. Prior to the construction of Taubes, Donaldson-Friedman [204] and LeBrun [412] have used ideas from twistor theory to construct self-dual metrics on the connected sum of CP 2 . The problem of four manifold is the lack of good diffeomorphic invariants. Donaldson or SeibergWitten provide such invariants. But they are not powerful enough to control the full structure of the manifold. A true understanding of four manifolds probably should come from understanding the question of existence of the integrable complex structures. The Riemann-Roch-Hirzebruch formula has been the basic tool to find the integrability condition. In the last twenty-five years, there are nonlinear methods from K¨ aher-Einstein metrics, harmonic maps, anti-self-dual connections and Seiberg-Witten invariants. However, one needs an existence theorem to find a canonical way to deform an almost complex structure to an integrable complex structure. What kind of obstructions do we expect? The work of Donaldson [198, 200] and Gompf [259] gave a good characterization of symplectic manifolds in terms of Lefschetz fibration. It may be useful to know under which condition such fibration will give rise to complex structures. I did ask several of my students to work on it. But no definite answer is known. J Jost and I [368] studied the rigidity part: if a K¨ ahler surface has a topological map to a Riemann surface with higher genus, it can be deformed to be a holomorphic map by changing the complex structure of the Riemann surface. One can derive from the work of Griffiths [272] that every algebraic surface has a Zariski open set which admits a complete Kähler-Einstein metric with finite volume and is covered by a contractible pseudo-convex domain. Perhaps one can classify these manifolds by topological means. While the Donaldson invariant gave the first counterexample to the h-cobordism theorem and irreducibility (nontrivial connected sum with manifolds not homotopic to CP 2 ) of four manifolds, the Seiberg-Witten invariant gave the remarkable result that an algebraic surface of general type can not be diffeomorphic to rational or elliptic surfaces. It also solves the famous Thom conjecture that holomorphic curves realize the lowest genus for embedded surface in a Kähler surface (KronheimerMrowka [400] and Ozsv´ ath-Szab´ o [544]). One wonders whether one can construct a diffeomorphic invariant based on metrics which are a generalization of Kähler-Einstein metrics. Comment: A good conjectural statement need to be made on the topology of four manifolds that may admit an integrable complex structure. Pseudo-holomorphic curve and fibration by Riemann surfaces should provide important information. Geometric flows may still be the major tool. 5.4. Special connections on bundles. In the seventies, theoretic physicists were very much interested in the theory of instantons: self-dual connections on four manifolds. Singer was able to communicate the favor of this excitement to the mathematical community which soon led to his paper with Atiyah and Hitchin [16] and also the complete solution of the problem over the four sphere by Atiyah-Hitchin-Drinfel’d-Manin [15] using twistor technique of Penrose. While the paper of Atiyah-Hitchin-Singer [16] laid the algebraic and geometric foundation for selfdual connections, the analytic foundation was laid by Uhlenbeck [684, 685] where she established the removable singularity theorem and compactness theorem for Yang-Mills connections. This eventually led to the fundamental works of Taubes [652] and Donaldson [194] which revolutionized four manifold topology. In the other direction, Atiyah-Bott [11] applied Morse theory to the space of connections over Riemann surface. They solved important questions on the moduli space of holomorphic bundles which was studied by Narasimhan, Seshadri, Ramanathan, Newstead and Harder. In the paper of 43

Atiyah-Bott, Morse theory, moment map and localization of equivariant cohomology were introduced on the subject of vector bundle. It laid the foundation of works in last twenty years. The analogue of anti-self dual connections over Kähler manifolds are Hermitian Yang-Mills connections, which was shown by Donaldson [195] for Kähler surfaces and Uhlenbeck-Yau [687] for general Kähler manifolds to be equivalent to the polystability of bundles. (That polystability of bundle is a consequence of existence of Hermitian Yang-Mills connection was first observed by L¨ uber [482]. Donaldson [196] was able to make use of the theorem of Mehta-Ramanathan [505] and ideas of above two papers to prove the theorem for projective manifold). It was generalized by C. Simpson [617], using ideas of Hitchin [331], to bundles with Higgs fields. It has important applications to the theory of variation of a Hodge structure [618, 619]. G. Daskalopoulos and R. Wentworth [183] studied such a theory for moduli space of vector bundles over curves. Li-Yau [431] generalized the existence of Hermitian Yang-Mills connections to non-Kähler manifolds. (Buchdahl [86] subsequently did the same for complex surfaces.) Li-Yau-Zheng [433] then used the result to give a complete proof of Bogomolov’s theorem for class VII0 surfaces. The only missing parts of the classification of non-K¨ ahler surfaces are those complex surfaces with a finite number of holomorphic curves. It is possible that the argument of Li-Yau-Zheng can be used. One may want to use Hermitian Yang-Mills connections with poles along such curves. I expect more applications of Donaldson-Uhlenbeck-Yau theory to algebraic geometry. It should be noted that the construction of Taubes [655] on the anti-self-dual connection is achieved by singular perturbation after gluing instantons from S 4 . The method is rather different from Donaldson-Uhlenbeck-Yau. While it applies to arbitrary four manifolds, it does require some careful choice of Chern classes for the bundle. It will be nice to find a concept of stability for a general complex bundle so that a similar procedure of Donaldson-Uhlenbeck-Yau can be applied. The method of singular perturbation has an algebraic geometric counterpart as was found by Gieseker-Li [254] and O’Grady [538] who proved that moduli spaces of algebraic bundles with fixed Chern classes over algebraic surfaces are irreducible. Li [425] also obtained information about Betti number of such moduli space. Not many general theorems are known for bundles over algebraic manifolds of a higher dimension. It will be especially useful for bundles over Calabi-Yau manifolds. D. Gieseker [252] developed the geometric invariant theory for the moduli space of bundles and introduced the Gieseker stability of bundles. Conan Leung [417] introduced the analytic counterpart of such bundles in his thesis under my guidance. While it is a natural concept, there is still an analytic problem to be resolved. (He assumed the curvature of the bundles to be uniformly bounded.) There were attempts by de Bartolomeis-Tian [43] to generalize Yang-Mills theory to symplectic manifolds and also by Tian [675] to manifolds with a special holonomy group, as was initiated by the work of Donaldson and Thomas. However, the arguments for both papers are not complete and still need to be finished. For a given natural structure on a manifold, we can often fix a structure and linearize the equation to obtain a natural connection on the tangent bundle. Usually we obtain Yang-Mills connections with the extra structure given by the holonomy group of the original structure. It is interesting to speculate whether an iterated procedure can be constructed to find an interesting metric or not. In any case, we can draw analogous properties between bundle theory and metric theory. The concept of stability for bundles is reasonably well understood for the holomorphic category. I believed that for each natural geometric structure, there should be a concept of stability. Donaldson [196] was able to explain stability in terms of moment map, generalizing the work of Atiyah-Bott [11] for bundles over Riemann surfaces. It will be nice to find moment maps for other geometric structures. Comment: Bundles with anti-self-dual connections or Hermitian Yang-Mills connections have been important for geometry. However, we do not have good estimates of their curvature of such connections. Such an estimate would be useful to handle important problems such as the Hartshorne question (see, e.g., [39]) on the splitting of rank two bundle over high dimension complex projective space. 44

5.5. Symplectic structures. Symplectic geometry had many important breakthroughs in the past twenty years. A moment map was developed by Atiyah-Bott [12], Guillemin-Sternberg [299] who proved the image of the map is a convex polytope. Kirwan and Donaldson had developed such a theory to be a powerful tool. The Marsden-Weinstein [488] reduction has become a useful method in many branches of geometry. At around the same time, other parts of symplectic topology were developed by Donaldson [199], Taubes [656], Gompf [258], Kronheimer-Mrowka [400] and others. The phenomenon of symplectic rigidity is manifested by the existence of symplectic invariants measuring the 2–dimensional size of a symplectic manifold. The first such invariant was discovered by Gromov [284] via pseudo-holomorphic curves. Hofer [338] then developed several symplectic invariants based on variational methods and successfully applied them to Weinstein conjectures. Ekeland-Hofer [213] introduced a concept of symplectic capacity and used it to provide a characterization of a symplectomorphism not involving any derivatives. The C 0 -closed property of the symplectomorphism group as a subgroup of the diffeomorphism group then follows, which was independently established by Y. Eliashberg [214] via wave front methods. Hofer-Zehnder [341] introduced another capacity and discovered the displacement-energy on R2n . By relating the two invariants with the energy-capacity inequality, Hofer [337] found a bi-invariant norm on the infinite dimensional group of Hamiltonian symplectomorphisms of R2n . The existence of such a norm has now been established for general symplectic manifolds by Lalonde-McDuff [406] via pseudoholomorphic curves and symplectic embedding techniques. The generalized Weinstein conjecture on the existence of a periodic orbits of Reeb flows for many 3-manifolds including the 3-sphere was also established in Hofer [338] by studying the finite energy pseudo-holomorphic plane in the symplectization of contact 3-manifolds. Eliashberg-Givental-Hofer [215] recently introduced the concept of symplectic field theory, which is about invariants of punctured pseudo-holomorphic curves in a symplectic manifold with cylindrical ends. Though it has not been rigorously established, some applications in contact and symplectic topology have been found. By analyzing the singularities of pseudo-holomorphic curves in a symplectic 4–manifold, D. McDuff [494] established rigorously the positivity of intersections of two distinct curves and the adjunction formula of an irreducible curve. Applying these basic properties to symplectic 4-manifolds containing embedded pseudo-holomorphic spheres with self-intersections at least −1, she was able to construct minimal models of general symplectic 4–manifolds, and classify those containing embedded symplectic spheres with non-negative self-intersections. A fundamental question in symplectic geometry is to decide which topological manifold admits a symplectic structure and how, as was pointed out by Smith-Thomas-Yau [631], mirrors of certain non-Kähler complex manifolds should be symplectic manifolds. Based on this point of view, they construct a large class of symplectic manifolds with trivial first Chern class by reversing the procedure of Clemens-Friedman on non-Kähler Calabi-Yau manifolds [164, 232]. In dimension four, the Betti numbers of such manifolds are determined by T. J. Li [444]. In the last ten years, there has been extensive work on symplectic manifolds, initiated by Gromov [284], Taubes [657, 658, 659, 660], Donaldson [197, 198, 200] and Gompf [259]. These works are based on the understanding of pseudo-holomorphic curves and Lefschetz fibrations. They are most successful for four dimensional manifolds. The major tools are Seiberg-Witten theory [603, 604, 717] and analysis. The work of Taubes on the existence of pseudo-holomorphic curves and the topological meaning of its counting is one of the deepest works in geometry. Based on this work, Taubes [657] was able to prove the old conjecture that there is only one symplectic structure on the standard CP 2 . However, the following question of mine is still unanswered: If M is a symplectic 4-manifold homotopic to CP 2 , is M symplectomorphic to the standard CP 2 ? (The corresponding question for complex geometry was solved by me in [731].) On the other hand, based on the work of Taubes [656], T. J. Li and A. K. Liu [445] did find a wall crossing formula for four dimensional manifolds that admit metrics with a positive scalar curvature. Subsequently A. Liu [460] gave the classification of such manifolds. (The surgery result by Stolz [637] based on Schoen-Yau-Gromov-Lawson 45

for manifolds with positive scalar curvature is not effective for four dimensional manifolds.) As another application of the general wall crossing formula in [445], it was proved by T. J. Li and A. Liu in [446] that there is a unique symplectic structure on S 2 -bundles over any Riemann surface. A main result of D. McDuff in [493] is used here. McDuff [492] also used a refined bordism type Gromov-Witten invariant to distinguish two cohomologous and deformation equivalent symplectic forms on S 2 × S 2 × T 2 , showing that they are not isotopic. Notice that there are also cohomologous but non-deformation equivalent symplectic forms on K3 × S 2 as shown by Y. Ruan [574]. In contrast, it is not known whether examples of this kind exist in dimension 4 or not. This phenomenon might be related to the special features of pseudo-holomorphic curves in a 4-manifold. Fukaya and Oh [238] have developed an elaborate theory for symplectic manifolds with Lagrangian cycles. Pseudo-holomorphic disks appeared as trace of motions of curves according to Floer theory. Due to boundary bubbles, the Lagrangian Floer homology is not always defined. Oh [539] developed some works on pseudo-holomorphic curves with Lagrangian boundary conditions and extended the Lagrangian Floer homology to all monotone symplectic manifolds. In order to understand open string theory, Katz-Liu [376] and Melissa Liu [461] developed the theory in analogue of the Gromov-Witten invariant for a holomorphic curve with boundaries on a given Lagrangian submanifold. Fukaya [237] discovered the underlying A∞ structure of the Lagrangian Floer homology on the chain level, leading to the Fukaya category. By carefully analyzing this A∞ structure, Fukaya, Oh, Ohta and Ono in [239] have constructed a sequence of obstruction classes which elucidate the rather difficult Lagrangian Floer homology theory to a great extent. Seidel-Thomas [605] and W. D. Ruan [573] discussed Fukaya’s category in relation to Kontsevich’s homological mirror conjecture [393]. One wonders whether Fukaya’s theory can help to construct canonical metrics for symplectic structures. For symplectic manifolds that admitan almost complex structure with zero first Chern class, it would be nice to construct Hermitian metrics with torsion that admit parallel spinor. Such structures may be considered as a mirror to the system constructed by Strominger on non-Kähler complex manifolds. Perhaps one can also gain some knowledge by reduction of G2 or Spin(7) structures to six dimensions. Comment: Geometry from the symplectic point of view has seen powerful development in the past twenty years. Its relation to Seiberg-Witten theory and mirror geometry is fruitful. More interesting development is expected. 5.6. K¨ ahler structure. The most interesting geometric structure is the Kähler structure. There are two interesting pre-K¨ ahler structures. One is the complex structure and the other is the symplectic structure. The complex structure is rather rigid for complex two dimensional manifolds. However it is much more flexible in dimension greater than two. For example, the twistor space of anti-self-dual four manifolds admit complex structures. Taubes [655] constructed a large class of such manifolds and hence a large class of complex three manifolds. There is also the construction of Clemens-Friedman for non-K¨ ahler Calabi-Yau manifolds which will be explained later. For quite a long time, it was believed that every compact Kähler manifold can be deformed to a projective manifold until C. Voisin [693, 694] found many counterexamples. We still need to digest the distinction between these two categories. Besides some obvious topological obstruction from Hodge theory and the rational homotopic type theory of Deligne-Griffiths-Morgan-Sullivan [187], it has been difficult to decide which complex manifolds admit K¨ ahler structure. Harmonic map argument does give some information. But it requires the fundamental group to be large. Many years ago, Sullivan [641] proposed to use the Hahn-Banach theorem to construct K¨ ahler metrics. This involves the concept of duality and hence closed currents. P. Gauduchon [245] has ¯ n−1 = 0. Siu [623] was able to use these ideas to proposed those Hermitian metrics ω which is ∂ ∂ω prove that every K3 surface is K¨ ahler. Demailly [190] did some remarkable work on regularization of closed positive currents. Singular K¨ ahler metrics have been studied and used by many researchers. 46

In fact, in my paper on proving the Calabi conjecture, I proved the existence of the Kähler metrics singular along subvarieties with control on volume element. They can be used to handle problems in algebraic geometry, including Chern number inequalities. Comment: The K¨ ahler structure is one of the richest structures in geometry. Deeper understanding may require some more generalized structure such as a singular K¨ ahler metric or balanced metrics. 5.6.1. Calabi-Yau manifolds. The construction of Calabi-Yau manifolds was based on the existence of a complex structure which can support a Kähler structure and a pluriharmonic volume form. A fundamental question is whether an almost complex manifold admits an integrable complex structure when complex dimension is greater than two. The condition that the first Chern class is zero is equivalent to the existence of pluriharmonic volume for Kähler manifolds. Such a condition is no more true for non-K¨ ahler manifolds. It would be nice to find a topological method to construct an integrable complex structure with pluriharmonic volume form. Once we have an integrable complex structure, we can start to search for Hermitian metrics with special properties. As was mentioned earlier, if we would like the geometry to have supersymmetries, then a Kähler metric is the only choice if the connection is torsion free. Further supersymmetry would then imply the manifold to be Calabi-Yau. However if we do not require the connection to be torsion free, Strominger [638] did derive a set of equations that exhibit supersymmetries without requiring the manifold to be K¨ ahler. It is a coupled system of Hermitian Yang-Mills connections with Hermitian metrics. Twenty years ago, I tried to develop such a coupled system. The attempt was unsuccessful as I restricted myself to Kähler geometry. My student Bartnik with Mckinnon [42] did succeed in doing so in the Lorentzian case. They found non-singular solutions for such a coupled system. (The mathematically rigorous proof was provided by Smoller-Wasserman-Yau-Mcleod [635] and [634]). The Strominger’s system was shown to be solvable in a neighborhood of a Calabi-Yau structure by Jun Li and myself [432]. Fu and I [234] were also able to solve it for certain complex manifolds which admit no K¨ ahler structure. These manifolds are balanced manifolds and were studied by M. Michelsohn [513]. These manifolds can be used to explain some questions of flux in string theory (see, e.g., [46, 105]). Since Strominger has shown such manifolds admits parallel spinors, I have directed my student C.C. Wu to decompose cohomology group of such manifolds correspondingly. It is expected that many theorems in Kähler geometry may have counterparts in such geometry. Such a structure may help to understand a proposal of Reid [564] in connecting Calabi-Yau manifolds with a different topology. This was initiated by a construction of Clemens [164] who proposed to perform complex surgery by blowing down rational curves with negative normal bundles in a Calabi-Yau manifold to rational double points. Friedman [232] found the condition to smooth out such singularities. Based on this Clemens-Friedman procedure, one can construct a complex structure on connected sums of S 3 × S 3 . It would be nice to construct Strominger’s system on these manifolds. The Calabi-Yau structure was used by me and others to solve important problems in algebraic geometry before it appeared in string theory. For example, the proof of the Torelli theorem (by Piatetskii-Shapiro and Shafarevich [555]) for a K3 surface by Todorov [679]-Siu [623] and the surjectivity of the period map of a K3 surface (by Kulikov [402]) by Siu [622]-Todorov [679] are important works for algebraic surfaces. The proof of the Bogomolov [59]-Tian [671]-Todorov [680] theorem also requires the metric. The last theorem helps us to understand the moduli space of Calabi-Yau manifolds. It is important to understand the global behavior of the Weil-Petersson geometry for Calabi-Yau manifolds. C. L. Wang [697] was able to characterize these points which have finite distance in terms of the metric. In my talk [733] in the Congress in 1978, I outlined the program and the results of classifying noncompact Calabi-Yau manifolds. Some of this work was written up in Tian-Yau [677, 678] and Bando-Kobayashi [32, 33]. During the period of 1984, there was an urgent request by string 47

theorists to construct Calabi-Yau threefolds with a Euler number equal to ±6. During the Argonne Lab conference, I [736] constructed such a manifold with a Z3 fundamental group by taking the quotient of a bi-degree (1, 1) hypersurface in the product of two cubics. Soon afterwards, more examples were constructed by Tian and myself [676]. However, it was pointed out by Brian Greene that all the manifolds constructed by Tian-Yau can be deformed to my original manifold. The idea of producing Calabi-Yau manifolds by the complete intersection of hypersurfaces in products of weighted projective space was soon picked up by Candelas et al [96]. By now, on the order of ten thousand examples of different homotopic types had been constructed. The idea of using toric geometry for construction was first performed by S. Roan and myself [565]. A few years later, the systematic study by Batyrev [44] on toric geometry allowed one to construct mirror pairs for a large class of Calabi-Yau manifolds, generalizing the construction of Greene-Plesser [269] and Candelas et al [96]. Tian and I [676] were also the first one to apply flop construction to change topology of Calabi-Yau manifolds. Greene-Morrison-Plesser [270] then made the remarkable discovery of isomorphic quantum field theory on two topological distinct Calabi-Yau manifolds. Most CalabiYau threefolds are a complete intersection of some toric varieties and they admit a large set of rational curves. It will be important to understand the reason behind it. Up to now all the Calabi-Yau manifolds that have a Euler number ±6 and a nontrivial fundamental group can be deformed from the birational model of the manifold (or their mirrors) that I constructed. It would be important if one could give a proof of this statement. The most spectacular advancement on Calabi-Yau manifolds come from the work of GreenePlesser, Candelas et al on construction of pairs of mirror manifolds with isomorphic conformal field theories attached to them. It allows one to calculate Gromov-Witten invariants. Existence of such mirror pairs was conjectured by Lerche-Vafa-Warner [416] and rigorous proof of mirror conjecture was due to Givental [256] and Lian-Liu-Yau [447] independently. The deep meaning of the symmetry is still being pursued. In [639], Strominger, Yau and Zaslow proposed a mathematical explanation for the mirror symmetry conjecture for Calabi-Yau manifolds. Roughly speaking, mirror Calabi-Yau manifolds should admit special Lagrangian tori fibrations and the mirror transformation is a nonlinear analog of the Fourier transformation along these tori. This proposal has opened up several new directions in geometric analysis. The first direction is the geometry of special Lagrangian submanifolds in Calabi-Yau manifolds. This includes constructions of special Lagrangian submanifolds ([415] and others) and (special) Lagrangian fibrations by Gross [291, 292] and W.D. Ruan [572], mean curvature of Lagrangian submanifolds in CalabiYau manifolds by Thomas and Yau [666] [667], structures of singularities on such submanifolds by Joyce [374] and Fourier transformations along special Lagrangian fibration by Leung-Yau-Zaslow [422] and Leung [420]. The second direction is affine geometry with singularities. As explained in [639], the mirror transformation at the large structure limit corresponds to a Legendre transformation of the base of the special lagrangian fibration which carries a natural special affine structure with singularities. Solving these affine problem is not trivial in geometric analysis [469] [470] and much work is still needed to be done here. The third direction is the geometry of special holonomy and duality and triality transformation in M-theory. In [303], Gukov, Yau and Zaslow proposed a similar picture to explain the duality in M-theory. The corresponding differential geometric structures are fibrations on G2 manifolds by coassociative submanifolds. These structures are studied by Kovalev [398], Leung and others [414] [421]. Comment: Although the first demonstration of the existence of Kähler Ricci flat metric was shown by me in 1976, it was not until the first revolution of string theory in 1984 that a large group of researchers did extensive calculations and changed the face of the subject. It is a subject that provides a good testing ground for analysis, geometry, physics, algebraic geometry, automorphic forms and number theory. 48

5.6.2. K¨ ahler metric with harmonic Ricci form and stability. The existence of a Kähler Einstein metric with negative scalar curvature was proved by Aubin [23] and me [732] independently. I [731] did find its important applications to solve classical problems in algebraic geometry, e.g., the uniqueness of complex structure over CP 2 [731], the Chern number inequality of Miyaoka [516]-Yau [731] and the rigidity of algebraic manifolds biholomorphic to Shimura varieties. The problem of existence of K¨ ahler Einstein metrics with positive scalar curvature in the general case is not solved. However, my proof of the Calabi conjecture already provided all the necessary estimates except some integral estimate on the unknown. This of course can be turned into hypothesis. I conjectured that an integral estimate of this sort is related to the stability of manifolds. Tian [674] called it K-stability. Mabuchi’s functional [485] made the integral estimate more intrinsic and it gave rise to a natural variational formulation of the problem. Siu has pointed out that the work of Tian [673] on two dimensional surfaces is not complete. The work of Nadel [531] on the multiplier ideal sheaf did give useful methods for the subject of the Kähler-Einstein metric. For Kähler Einstein manifolds with positive scalar curvature, it is possible that they admit a continuous group of automorphisms. Matsushima [490] was the first one to observe that such a group must be reductive. Futaki [241] introduced a remarkable invariant for general K¨ ahler manifolds and proved that it must vanish for such manifolds. In my seminars in the eighties, I proposed that Futaki’s theorem should be generalized to understand the projective group acting on the embedding of the manifold by a high power of anti-canonical embedding and that Futaki’s invariant should be relevant to my conjecture [739] relating the Kähler Einstein manifold to stability. Tian asked what happens when manifolds have no group actions. I explained that the shadow of the group action is there once it is inside the projective space and one should deform the manifold to a possibly singular variety to obtain more information. The connection of Futaki invariant to stability of manifolds has finally appeared in the recent work of Donaldson [201, 202]. Donaldson introduced a remarkable concept of stability based purely on concept of algebraic geometry. It is not clear that Donaldson’s algebraic definition has anything to do with Tian’s analytic definition of stability. Donaldson proved that the existence of Kähler-Einstein did imply his K-stability which in turn implies Hilbert stability and asymptotic Chow stability of the manifold. This theorem of Donaldson already gives nontrivial information for manifolds with negative first Chern class and Calabi-Yau manifolds, where existence of Kähler-Einstein metrics was proved. Some part of the deep work of Gieseker [252] and Viehweg [690] can be recovered by these theorems. One should also mention the recent interesting work of Ross-Thomas [568, 569] on the stability of manifolds. Phong-Strum [551] also studied solutions of certain degenerate Monge-Ampére equations and [552] the convergence of the K¨ ahler-Ricci flow. A Kähler metric with constant scalar curvature is equivalent to the harmonicity of the first Chern form. The uniqueness theorem for harmonic Kähler metric was due to X. Chen [134], Donaldson [201] and Mabuchi for various cases. (Note that the most important case of the uniqueness of the Kähler Einstein metric with positive scalar curvature was due to the remarkable argument of Bando-Mabuchi [34].) My general conjecture for existence of harmonic Kähler manifolds based on stability of such manifolds is still largely unknown. In my seminar in the mid-eighties, this problem was discussed extensively. Several students of mine, including Tian [672], Luo [477] and Wang [705] had written thesis related to this topic. Prior to them, my former students Bando [31] and Cao [99] had made attempts to study the problem of constructing Kähler-Einstein metrics by Ricci flow. The fundamental curvature estimate was due to Cao [100]. The Kähler Ricci flow may either converge to K¨ ahler Einstein metric or Kähler solitons. Hence in order for the approach, based on Ricci flow, to be successful, stability of the projective manifold should be related to such Kähler solitons. The study of harmonic Kähler metrics with constant scalar curvature on toric variety was initiated by S. Donaldson [202], who proposed to study the existence problem via the real Monge-Ampère equation. This problem of Donaldson in the Kähler-Einstein case was solved by Wang-Zhu [704]. LeBrun and his coauthors [380] also have found special constructions, based on twistor theory, for harmonic K¨ ahler surfaces. Bando was also interesting in Kähler manifolds 49

with harmonic i-th Chern form. (There should be an analogue of stability of algebraic manifolds associated to manifolds with harmonic i-th Chern form.) In the early 90’s, S.W. Zhang [750] studied heights of manifolds. By comparing metrics on Deligne pairings, he found that a projective variety is Chow semistable if and only if it can be mapped by an element of a special linear group to a balanced subvariety. (Note that a subvariety in CPN is called balanced if the integral of the moment map with respect to SU (N + 1) is zero, where the measure for the integral is induced from the Fubini-Study metric.) Zhang communicated his results to me. It is clearly related to Kähler-Einstein metric and I urged my students, including Tian, to study this connection. Zhang’s work has a nontrivial consequence on the previous mentioned development of Donaldson [201, 202]. Assume the projective manifold is embedded by an ample line bundle L into projective space. If the manifold has a finite automorphism group and admits a harmonic Kähler metric in c1 (L), then Donaldson showed that for k large, Lk gives rise to an embedding which is balanced. Furthermore, the induced Fubini-Study form divided by k will converge to the harmonic K¨ ahler form. Combining the work of Zhang and Luo, he then proved that the manifold is given by the embedding of Lk is stable in the sense of geometric invariant theory. Recently, Mabuchi generalized Donaldson’s theorem to certain case which allow nontrivial projective automorphism. Donaldson considered the problem from the point of view of symplectic geometry (Kähler form is a natural symplectic form). The Hamiltonian group then acts on the Hilbert space H of square integrable sections of the line bundle L where the first Chern class is the Kähler form. For each integrable complex structure on the manifold compatible with the symplectic form, the finite dimensional space of holomorphic sections gives a subspace of H. The Hamiltonian group acts on the Grassmannian of such subspaces. The moment map can be computed to be related to the Bergman P kernel α sα (x) ⊗ s∗α (y) where sα form an orthonormal basis of the holomorphic sections. On the other hand, Fujiki [235] and Donaldson [199] computed the moment map for the Hamiltonian group action on the space of integrable complex structure, which turns out to be the scalar curvature of the Kähler metric. These two moment maps may not match, but for the line bundle Lk with large k, one can show that they converge to each other after normalization. Lu [475] has shown the first term of the expansion (in terms of 1/k) of the Bergman kernel gives rise to scalar curvature. Hence we see the relevance of constant scalar curvature for a Kähler metric to these witha constant Bergman kernel function. S.W. Zhang’s result says that the manifold is Chow semistable if and only if it is balanced. The balanced condition implies that there is a Kähler metric where the Bergman kernel is constant. With the work of Zhang and Donaldson, what remains to settle my conjecture is the convergence of the balanced metric when k is large. In general, we should not expect this to be true. However, for toric manifolds, this might be the case. It may be noted that in my paper with Bourguignon and P. Li [73] on giving an upper estimate of the first eigenvalue of an algebraic manifold, this balanced condition also appeared. Perhaps first eigenfunction may play a role for questions of stability. Comment: K¨ ahler metrics with constant scalar curvature is a beautiful subject as it is related to structure questions of algebraic varieties including the concept of stability of manifolds. The most effective application of such metrics to algebraic geometry are still restricted to the Kähler-Einstein metric. The singular KählerEinstein metric as was initiated by my paper on Calabi conjecture should be studied further in application to algebraic geometry. 5.7. Manifolds with special holonomy group. Besides Kähler manifolds, there are manifolds with special holonomy groups. Holonomy groups of Riemannian manifolds were classified by Berger [48]. The most important ones are O(n), U (n), SU (n), G2 and Spin(7). The first two groups correspond to Riemannian and K¨ ahler geometry respectively. SU (n) corresponds to Calabi-Yau manifolds. A G2 manifold is seven dimensional and a Spin(7) is eight dimensional (assuming they are irreducible manifolds). These last three classes of manifolds have zero Ricci curvature. It may 50

be noted that before I [732] proved the Calabi conjecture in 1976, there was no known nontrivial compact Ricci flat manifold. Manifolds with a special holonomy group admit nontrivial parallel spinors and they correspond to supersymmetries in the language of physics. The input of ideas from string theory did give a lot of help to understand these manifolds. However, the very basic question of constructing these structures on a given topological space is still not well understood. In the case of G2 and Spin(7), it was initiated by Bryant (see [83, 85]). The first set of compact examples was given by Joyce [371, 372, 373]. Recently Dai-Wang-Wei [182] proved the stability of manifolds with parallel spinors. The nice construction of Joyce was based on a singular perturbation which is similar to the construction of Taubes [651] on anti-self-dual connections. However, it is not global enough to give a good parametrization of G2 or Spin(7) structures. A great deal more work is needed. The beautiful theory of Hitchin [335, 336] on three forms and four forms may lead to a resolution of these important problems. Comment: Recent interest in M-theory has stimulated a lot of activities on manifolds with special holonomy group. We hope a complete structure theorem for such manifolds can be found. 5.8. Geometric structures by reduction. One can also obtain new geometric structures by imposing some singular structures on a manifold with a special holonomy group. For example, if we require a metric cone to admit a G2 , Spin(7) or Calabi-Yau structure, the link of the cone will be a compact manifold with special structures. They give interesting Einstein metrics. When the cone is Calabi-Yau, the structure on the odd dimensional manifold is called Sasakian Einstein metric. There is a natural Killing field called the Reeb vector field defined on a Sasakian Einstein manifold. If it generates a circle action, the orbit space gives rise to a Kähler Einstein manifold with positive scalar curvature. However, it need not generate a circle action and J. Sparks, Gauntlelt, Martelli and Waldram [246] gave many interesting explicit examples of non-regular Sasakian Einstein structures. They have interesting properties related to conformal field theory. For quasiregular examples, there was work by Boyer, Galicki and Koll´ ar [75]. The procedure gave many interesting examples of Einstein metrics on odd dimensional manifolds. Sparks, Matelli and I have been pursuing general theory of Sasakian Einstein manifolds. I would like to consider them as a natural generalization of Kähler manifolds. Comment: The recent development of the Sasakian Einstein metric show that it gives a natural generalization of the Kähler-Einstein metric. Its relation with the recent activities on ADS/CFT theory is exciting. 5.9. Obstruction for existence of Einstein metrics on general manifolds. The existence of Einstein metrics on a fixed topological manifold is clearly one of the most important questions in geometry. Any metrics with a compact special holonomy group are Einstein. Besides K¨ ahler geometry, we do not know much of their moduli space. For an Einstein metric with no special structures, we know only some topological constraints on four dimensional manifolds. There is work by Berger [49], Gray [267] and Hitchin [330] in terms of inequalities linking a Euler number and the signature of the manifold. (This is of course based on Chern’s work [148] on the representation of characteristic classes by curvature forms.) Gromov [283] made use of his concept of Gromov volume to give further constraint. LeBrun [413] then introduced the ideas from Seiberg-Witten invariants to enlarge such classes and gave beautiful rigidity theorems on Einstein four manifolds. Unfortunately it is very difficult to understand moduli space of Einstein metrics when they admit no special structures. For example, it is still an open question of whether there is only one Einstein metric on the four dimensional sphere. M. Wang and Ziller [703] and C. Boehm [57] did use symmetric reductions to give many examples of Einstein metrics for higher dimensional manifolds. There may be much more examples of Einstein manifolds with negative Ricci curvature than we expected. This is certainly true for compact manifolds, with negative Ricci curvature. Gao-Yau 51

[242] was the first one to demonstrate that such a metric exists on the three sphere. A few years later, Lokhamp [471] used the h-principle of Gromov to prove such a metric exists on any manifold with a dimension greater than three. It would be nice to prove that every manifold with a dimension greater than 4 admits an Einstein metric with negative Ricci curvature. Comment: The Einstein manifold without extra special structures is a difficult subject. Do we expect a general classification for such an important geometric structure? 5.10. Metric Cobordism. In the last five years, a great deal of attention was addressed by physicists on the holographic principle: boundary geometry should determine the geometry in the interior. The ADS/CFT correspondence studies the conformal boundary of the Einstein manifold which is asymptotically hyperbolic. Gauge theory on the boundary is supposed to be dual to the theory of gravity in the bulk. Much fascinating work was done in this direction. Manifolds with positive scalar curvature appeared as conformal boundary are important for physics. Graham-Lee [265] have studied a perturbation problem near the standard sphere which bounds the hyperbolic manifold. Witten-Yau [718] proved that for a manifold with positive scalar curvature to be a conformal boundary, it must be connected. It is not known whether there are further obstructions. Cobordism theory had been a powerful tool to classify the topology of manifolds. The first fundamental work was done by Thom who determined the cobordism group. Characteristic numbers play important roles. When two manifolds are cobordant to each other, the theory of surgery helps us to deform one manifold to another. It is clear that any construction of surgery that may preserve geometric structures would play a fundamental role in the future of geometry. There are many geometric structures that are preserved under a connected sum construction. This includes the category of conformally flat structures, metrics with positive isotropic curvature and metrics with positive scalar curvature. For the last category, there was work by Schoen-YauGromov-Lawson where they perform surgery on spheres with a codimension greater than or equal to 3. A key part of the work of Hamilton-Perelman is to find a canonical neighborhood to perform surgery. If we can deform the spheres in the above SYGL construction to a more canonically defined position, one may be able able to create an extra geometric structure for the result of SYGL. In fact, the construction of Schoen-Yau did provide some information about the conformal structures of the manifold. In complex geometry, there are two important canonical neighborhoods given by the log transform of Kodaira and the operation of flop. There should be similar constructions for other geometric structures. The theory of quasi-local mass mentioned in section 4.4 is another example of how boundary geometry can be controlled by the geometry in the bulk. The work of Choi-Wang [154] on the first eigenvalue is also based on the manifold that it bounds. There can be interesting theory of metric cobordism. In the other direction, there are also beautiful rigidity of inverse problems for metric geometry by Gerver-Nadirashvili [250] and Pestov-Uhlmann [550] on recovering a Riemannian metric when one knows the distance functions between pair of points on the boundary, if the Riemannian manifold is reasonably convex. Comment: There should be a mathematical foundation of the holographic principle of physicists. Good understanding of metric cobordism may be useful.

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Geometrization of 3-Manifolds via the Ricci Flow Michael T. Anderson

Introduction The classification of closed surfaces is a milestone in the development of topology, so much so that it is now taught to most mathematics undergraduates as an introduction to topology. Since the solution of the uniformization problem for surfaces by Poincaré and Koebe, this topological classification is now best understood in terms of the geometrization of 2-manifolds: every closed surface Σ admits a metric of constant Gauss curvature +1 , 0 , or −1 and so is uniformized by one of the standard space-form geometries S2 , R2 , H2 . Hence any surface Σ is a quotient of either the 2-sphere, the Euclidean plane, or the hyperbolic disc by a discrete group Γ acting freely and isometrically. The classification of higher-dimensional manifolds is of course much more difficult. In fact, due to the complexity of the fundamental group, a complete classification as in the case of surfaces is not possible in dimensions ≥ 4 . In dimension 3 this argument does not apply, and the full classification of 3-manifolds has long been a dream of topologists. As a very special case, this problem includes the Poincaré Conjecture. In this article we report on remarkable recent work of Grisha Perelman [15]-[17], which may well have solved the classification problem for 3manifolds (in a natural sense). Perelman’s work is currently under intense investigation and scrutiny by many groups around the world. At this time, much of his work has been validated by experts in the area. Although at the moment it is still too soon to declare a definitive solution to the problem, Perelman’s ideas are highly original and of deep insight. Morever, his Michael T. Anderson is professor of mathematics at the State University of New York, Stony Brook. His email address is [email protected]. Partially supported by NSF Grant DMS 0305865.

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results are already being used by others in research on related topics. These circumstances serve to justify the writing of an article at this time, which otherwise might be considered premature. The work of Perelman builds on prior work of Thurston and Hamilton. In the next two sections we discuss the Thurston picture of 3-manifolds and the Ricci flow introduced and analyzed by Hamilton. For additional background, in particular on the Poincaré Conjecture, see Milnor’s Notices survey [14] and references therein. For much more detailed commentary and discussion on Perelman’s work, see [13].

The Geometrization Conjecture While the Poincaré Conjecture has existed for about one hundred years, the remarkable insights of Thurston in the late 1970s led to the realistic possibility of understanding and classifying all closed 3-manifolds in a manner similar to the classification of surfaces via the uniformization theorem. To explain this, we first need to consider what are the corresponding geometries in 3-dimensions. In terms of Riemannian geometry, a geometric structure on a manifold M is a complete, locally homogeneous Riemannian metric g . Thus, M may be described as the quotient Γ \ G/H , where G is the , g) and isometry group of the universal cover (M Γ, H are discrete and compact subgroups of the Lie group G respectively. Thurston showed that there are eight such simply connected geometries G/H in dimension 3 which admit compact quotients.1 As in two dimensions, the most important geometries are those of constant curvature: hyperbolic 1 The Thurston classification is essentially a special case of the much older Bianchi classification of homogeneous space-time metrics arising in general relativity; cf. [3] for further remarks on the dictionary relating these classifications.

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M

geometry H3 of curvature −1 , Euclidean geometry R3 of curvature 0 , and spherical geometry S3 of curvature +1 . The remaining five geometries are products or twisted products with the 2 -dimensional geometries. Trivial S 1 bundles over a surface of genus g > 1 have H2 × R geometry, while nontrivR) geometry; nontrivial S 1 ial bundles have SL(2, bundles over T 2 have Nil geometry, while nontrivial T 2 bundles over S 1 have Sol geometry (or Nil or R3 geometry); finally, S 1 bundles over S 2 have S2 × R (or S3 ) geometry. For example, any Seifert fibered 3 -manifold, a 3 -manifold admitting a locally free S 1 action, has such a geometric structure. Geometric 3 -manifolds, that is 3 -manifolds admitting a geometric structure, are the building blocks of more complicated 3 -manifolds. For simplicity, we assume throughout the article that all manifolds M are orientable. The building blocks are then assembled along 2-spheres S 2 , via connected sum, and along tori T 2. As a simple example of such an assembly, let {Mi } be a finite collection of Seifert fibered 3 -manifolds over surfaces Σi with nonempty boundary, so that ∂Mi consists of tori. These tori may then be glued together pairwise by diffeomorphisms to obtain a closed 3 -manifold or a 3 -manifold with toral boundary. A 3 -manifold assembled in this way is called a graph manifold. (One assigns a vertex to each Seifert fibered space and an edge to each torus connecting two such Seifert spaces). A torus bundle over S 1 is a graph manifold, since it is the union of two Seifert fibered spaces over S 1 × I . Graph manifolds were introduced, and their structure completely analyzed, by Waldhausen. Conversely, let M be an arbitrary closed 3 -manifold, as above always orientable. One then decomposes or splits it into pieces according to the structure of the simplest surfaces embedded in M , namely spheres and tori. Topologically, this is accomplished by the following classical results in 3 -manifold topology. Sphere (or Prime) Decomposition (Kneser, Milnor) Let M be a closed 3 -manifold. Then M admits a finite connected sum decomposition

(1)

M = (K1 #...#Kp )#(L1 #...#Lq )#(#1r S 2 × S 1 ).

The K and L factors here are closed irreducible 3 -manifolds; i.e. every embedded 2 -sphere S 2 bounds a 3 -ball. The K factors have infinite fundamental group and are aspherical 3 -manifolds (K(π , 1) ’s), while the L factors have finite fundamental group and have universal cover a homotopy 3 -sphere. Since M#S 3 = M , we assume no L factor is S 3 unless M = L = S 3 . The factors in (1) are then unique up to permutation and are obtained from M by performing surgery on a collection of essential, i.e. topologically nontrivial, 2 -spheres in M FEBRUARY 2004

S2 S2

S2 S2

L1

K2 K1 S 2×S 1

Figure 1. Sphere decomposition.

K S2

S1 H1

T1

H2 T3

T2 T4

S3

Figure 2. Torus decomposition, (Si Siefert fibered, Hj torusirreducible). (replacing regions S 2 × I by two copies of B 3); see Figure 1 for a schematic representation. The K factors in (1) may also contain topologically essential tori. A torus T 2 embedded in M is called incompressible if the inclusion map induces an injection on π1 . A 3 -manifold N is called torusirreducible if every embedded incompressible torus may be deformed to a torus in ∂N . Hence, if ∂N = ∅ , then N has no incompressible tori. Torus Decomposition (Jaco-Shalen, Johannsen) Let M be a closed, irreducible 3 -manifold. Then there is a finite collection, possibly empty, of disjoint incompressible tori in M that separate M into a finite collection of compact 3 -manifolds (with toral boundary), each of which is torus-irreducible or Seifert fibered. A coarser, but essentially equivalent, decomposition is given by tori separating M into torusirreducible and graph manifold components; see Figure 2. With the simple exceptions of S 2 × S 1 and its oriented Z2 quotient S 2 ×Z2 S 1 RP3 #RP3 , essential 2 -spheres are obstructions to the existence of a geometric structure on a 3 -manifold. The same is true

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for essential tori, unless M happens to be a Seifert fibered or a Sol 3 -manifold. Thus, the sphere and torus decompositions divide M topologically into pieces where these known obstructions are removed. Geometrization Conjecture (Thurston). Let M be a closed, oriented 3 -manifold. Then each component of the sphere and torus decomposition admits a geometric structure. The geometrization conjecture gives a complete and effective classification of all closed 3 -manifolds, closely resembling in many respects the classification of surfaces. More precisely, it reduces the classification to that of geometric 3 -manifolds. The classification of geometric 3 -manifolds is rather simple and completely understood, except for the case of hyperbolic 3 -manifolds, which remains an active area of research. As an illustration of the power of the Thurston Conjecture, let us see how it implies the Poincaré Conjecture. If M is a simply connected 3-manifold, then the sphere decomposition (1) implies that M must be an L factor. The geometrization conjecture implies that L is geometric, and so L = S 3 /Γ . Hence, M = L = S3. Thurston’s formulation and work on the geometrization conjecture revolutionized the field of 3-manifold topology; see [18], [19] and further references therein. He recognized that in the class of all (irreducible) 3 -manifolds, hyperbolic 3 manifolds are overwhelmingly the most prevalent, as is the case with surfaces, and developed a vast array of new ideas and methods to understand the structure of 3 -manifolds. Thurston and a number of other researchers proved the geometrization conjecture in several important cases, the most celebrated being the Haken manifold theorem: if M is an irreducible Haken 3 -manifold, i.e. M contains an incompressible surface of genus ≥ 1 , then the geometrization conjecture is true for M . An important ingredient in the Thurston approach is the deformation and degeneration of hyperbolic structures on noncompact manifolds (or the deformation of singular hyperbolic structures on compact manifolds). The eight geometric structures are rigid in that there are no geometries which interpolate continuously between them. Hence, on a composite 3 -manifold M , the geometric structure on each piece must degenerate in passing from one piece to the next; there is no single structure or metric giving the geometrization of all of M . For example, in Figure 2 the H pieces may be hyperbolic 3-manifolds separated by tori from Seifert fibered pieces S . Although this splitting is topologically well defined, the geometries do not match in the glueing region, and metrically there is no natural region in which to perform the glueing. 186

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Independently and around the same time as Thurston, Gromov [6], [7] also studied the deformation and degeneration of more general Riemannian metrics with merely bounded curvature in place of constant curvature. The idea is that one can control the behavior of a metric, or of a family of metrics, given a uniform bound on the Riemann curvature tensor Riem of the metric.2 This leads to the important Gromov compactness theorem, the structure theory of almost flat manifolds, and the theory of collapsing Riemannian manifolds, worked out in detail with Cheeger and Fukaya. One version of these results is especially relevant for our purposes. Let (M, g) be a closed Riemannian manifold, normalized to unit volume, and suppose

(2)

|Riem| ≤ Λ,

for some arbitrary constant Λ < ∞ . The metric g provides a natural decomposition of M into thick and thin parts, M = M ν ∪ Mν , where

(3)

M ν = {x ∈ M : volBx (1) ≥ ν}, Mν = {x ∈ M : volBx (1) < ν};

here Bx (1) is the geodesic ball about x of radius 1 and ν > 0 is an arbitrary but fixed small number. Now consider the class of all Riemannian n-manifolds of unit volume satisfying (2), and consider the corresponding decompositions (3). Then the geometry and topology of M ν is a priori controlled. For any given ν > 0, there are only a finite number, (depending on Λ and ν ), of possible topological types for M ν. Moreover, the space of metrics on M ν is compact in a natural sense; any sequence has a subsequence converging in the C 1,α topology, α < 1 (modulo diffeomorphisms). For ν sufficiently small, the complementary thin part Mν admits an F-structure in the sense of Cheeger-Gromov; in dimension 3 this just means that Mν is a graph manifold with toral (or empty) boundary. In particular, the topology of Mν is strongly restricted. A metric on Mν is highly collapsed in the sense that the circles in the Seifert fibered pieces of Mν and the tori glueing these pieces together have very small diameter, depending on ν ; see Figure 3 for a schematic picture. Moreover, for any fixed ν > 0, the distance between M ν and the arbitrarily thin part Mν becomes arbitrarily large as ν /ν → 0 . We point out that similar results hold locally and for complete noncompact manifolds; thus the unit volume normalization above is not essential. 2 The curvature tensor is a complicated (3,1) tensor expressed in terms of the second derivatives of the metric; in a local geodesic normal coordinate system at a given point, the components of Riem are given by l Rijk = − 12 (∂i ∂k gjl + ∂j ∂l gik − ∂j ∂k gil − ∂i ∂l gjk ).

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The Thurston approach to geometrization has made a great deal of progress on the “hyperbolic part” of the conjecture. In comparison with this, relatively little progress has been made on the “positive curvature part” of the conjecture, for example the Poincaré Conjecture. It is worth pointing out that among the eight geometries, the constant curvature geometries H3 and S3 are by far the most important to understand (in terms of characterizing which manifolds are geometric); the other (mixed) geometries are much simpler in comparison. From the point of view of Riemannian geometry, the Thurston conjecture essentially asserts the existence of a “best possible” metric on an arbitrary closed 3-manifold. In the case that M is not itself geometric, one must allow the optimal metric to have degenerate regions. The discussion and figures above suggest that the degeneration should be via the pinching off of 2-spheres (sphere decomposition) and collapse of graph manifolds along circles and tori (torus decomposition).

The Ricci Flow One method to find a best metric on a manifold is to find a natural evolution equation, described by a vector field on the space of metrics, and try to prove that the flow lines exist for all time and converge to a geometric limit. In case a flow line does not converge, the corresponding metrics degenerate, and one then needs to relate the degeneration with the topology of M . There is essentially only one simple and natural vector field (or more precisely family of vector fields) on the space of metrics. It is given by

(4)

d g(t) = −2Ricg(t) + λ(t) · g(t). dt

Here Ric is the Ricci curvature, in local co given k ordinates by Rij = (Ric)ij = k Rikj , so that Ric is a trace of the Riemann curvature. The constant 2 is just for convenience and could be changed by rescaling the time parameter; λ(t) is a constant depending on time t . The Ricci flow, introduced by Hamilton [11], is obtained by setting λ = 0 , i.e.

(5)

d g(t) = −2Ricg(t) . dt

The reason (4) is the only natural flow equation is essentially the same as that leading to the Einstein field equations in general relativity. The Ricci curvature is a symmetric bilinear form, as is the metric. Besides multiples of the metric itself, it is the only such form depending on at most the second derivatives of the metric, and invariant under coordinate changes, i.e. a (2, 0) tensor formed from the metric. By rescaling the metric and time variable t , one may transform (5) into (4). For FEBRUARY 2004

Mν

Mν

Mν Mν

Figure 3. Thick-thin decomposition. example, rescaling the Ricci flow (5) so that the volume of (M, g(t)) is preserved leads to the flow equation (4) with λ = 2 R , twice the mean value of the scalar curvature R . In a suitable local coordinate system, equation (5) has a very natural form. Thus, at time t , choose local harmonic coordinates so that the coordinate functions are locally defined harmonic functions in the metric g(t) . Then (5) takes the form

(6)

d gij = ∆gij + Qij (g, ∂g), dt

where ∆ is the Laplace-Beltrami operator on functions with respect to the metric g = g(t) and Q is a lower-order term quadratic in g and its firstorder partial derivatives. This is a nonlinear heat-type equation for gij . From the analysis of such PDE, one obtains existence and uniqueness of solutions to the Ricci flow on some time interval, starting at any smooth initial metric. This is the reason for the minus sign in (5); a plus sign leads to a backwards heat-type equation, which has no solutions in general. Here are a few simple examples of explicit solutions to the Ricci flow. If the initial metric g(0) is of constant Ricci curvature, Ric = a · g , then the evolution g(t) is just a rescaling of g(0) : g(t) = (1 − 2at)g(0) . Note that if a > 0 , then the flow contracts the metric, while if a < 0 , the flow expands the metric, uniformly in all directions. Hence, if one rescales g(t) to have constant volume, the resulting curve is constant. The stationary points of the volume-normalized Ricci flow are exactly the class of Einstein metrics, i.e. metrics of constant Ricci curvature. In dimension 3 , Einstein metrics are of constant curvature and so give the H3 , R3 and S3 geometries. More generally, if Ric(x, t) > 0 , then the flow contracts the metric g(t) near x , to the future, while if Ric(x, t) < 0, then the flow expands g(t) near x . At a general point, there will be directions of positive and negative Ricci curvature along which the metric will locally contract or expand. Suppose g(0) is a product metric on S 1 × Σ , where Σ is a surface with constant curvature metric. Then g(t) remains a product metric, where the length of the S 1 factor stays constant while the surface factor expands or contracts according to the sign of its curvature.

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Finally, the Ricci flow commutes with the action of the diffeomorphism group and so preserves all isometries of an initial metric. Thus, geometric 3 manifolds remain geometric. For the “nonpositive” R) , Nil , and Sol , mixed geometries H2 × R , SL(2, the volume-normalized Ricci flow contracts the S 1 or T 2 fibers and expands the base surface factor, while for the positive mixed geometry S2 × R , the volume-normalized flow contracts the S 2 and expands in the R -factor. Now consider the Ricci flow equation (5) in general. From its form it is clear that the flow g(t) will continue to exist if and only if the Ricci curvature remains bounded. This suggests one should consider evolution equations for the curvature, induced by the flow for the metric. The simplest of these is the evolution equation for the scalar curvature R = trg Ric = g ij Rij :

(7)

d R = ∆R + 2|Ric|2 . dt

Evaluating (7) at a point realizing the minimum Rmin of R on M gives the important fact that Rmin is monotone nondecreasing along the flow. In particular, the Ricci flow preserves positive scalar curvature (in all dimensions). Moreover, if Rmin (0) > 0, d 2 2 , then the same argument gives dt Rmin ≥ n Rmin n = dimM , by the Cauchy-Schwarz inequality |Ric|2 ≥ n1 R2 . A simple integration then implies

(8)

t≤

n . 2Rmin (0)

Thus, the Ricci flow exists only up to a maximal time T ≤ n/2Rmin (0) when Rmin (0) > 0 . In contrast, in regions where the Ricci curvature stays negative definite, the flow exists for infinite time. The evolution of the Ricci curvature has the same general form as (7):

(9)

d ij . Rij = ∆Rij + Q dt

is much more complicated The expression for Q than the Ricci curvature term in (7) but involves only quadratic expressions in the curvature. However, involves the full Riemann curvature tensor Riem Q of g and not just the Ricci curvature (as (7) involves Ricci curvature and not just scalar curvature). An elementary but important feature of dimension 3 is that the full Riemann curvature Riem is determined algebraically by the Ricci curvature. This implies that, in general, Ricci flow has a much better chance of “working” in dimension 3 . For ex shows that the Ricci flow ample, an analysis of Q preserves positive Ricci curvature in dimension 3 : if Ricg(0) > 0 , then Ricg(t) > 0 , for t > 0 . This is not the case in higher dimensions. On the other hand, in any dimension > 2 , the Ricci flow does not preserve negative Ricci curvature, nor does it preserve 188

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a general lower bound Ric ≥ −λ , for λ > 0 . For the remainder of the paper, we assume then that dimM = 3. In the Gromov compactness result and thick/thin decomposition (3), the hypothesis of a bound on |Riem| can now also be replaced by a bound on |Ric| (since we are in dimension 3). Further, on time intervals [0, t] where |Ric| is bounded, the metrics g(t) are all quasi-isometric to each other: cg(0) ≤ g(t) ≤ Cg(0) as bilinear forms, where c, C depend on t . Hence, the arbitrarily thin region Mν, ν > 1 only where R(x, t) >> 1 . Hence, to control the size |Riem| of the full curvature, it suffices to obtain just an upper bound on the scalar curvature R . This is remarkable, since the scalar curvature is a much weaker invariant of the metric than the full curvature. Moreover, at points where the curvature is sufficiently large, (10) shows that Riem(x, t)/R(x, t) ≥ −δ , for δ small. Thus,

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if one scales the metric to make R(x, t) = 1 , then Riem(x, t) ≥ −δ . In such a scale, the metric then has almost nonnegative curvature near (x, t) . • Harnack estimate [9]. Let (N, g(t)) be a solution to the Ricci flow with bounded and nonnegative curvature Riem ≥ 0 , and suppose g(t) is a complete Riemannian metric on N . Then for 0 < t1 ≤ t2 , dt21 (x1 , x2 ) t1 R(x1 , t1 ), (11) R(x2 , t2 ) ≥ exp − t2 2(t2 − t1 ) where dt1 is the distance function on (M, gt1 ) . This estimate allows one to relate or control the geometry of the solution at different space-time points. An estimate analogous to (11) in general, i.e. without the assumption Riem ≥ 0 , has been one of the major obstacles to further progress in the Ricci flow. The analysis above shows that the Ricci flow tends to favor positive curvature. The flow tends to evolve to make the curvature more positive, and the strongest results have been proved in the case of positive curvature, somewhat in contrast to the Thurston approach.

Singularity Formation The deeper analysis of the Ricci flow is concerned with the singularities that arise in finite time. As (8) already shows, the Ricci flow will not exist for an arbitrarily long time in general. In the case of initial metrics with positive Ricci curvature, this is resolved by rescaling the Ricci flow to constant volume. Hamilton’s space-form theorem shows that the volume-normalized flow exists for all time and converges smoothly to a round metric. However, the situation is necessarily much more complicated outside the class of positive Ricci curvature metrics. Consider for instance initial metrics of positive scalar curvature. Any manifold which is a connected sum of S 3 /Γ and S 2 × S 1 factors has metrics of positive scalar curvature (compare with the sphere decomposition (1)). Hence, for obvious topological reasons, the volume-normalized Ricci flow could not converge nicely to a round metric; even the renormalized flow must develop singularities. Singularities occur frequently in numerous classes of nonlinear PDEs and have been extensively studied for many decades. Especially in geometric contexts, the usual method to understand the structure of singularities is to rescale or renormalize the solution on a sequence converging to the singularity to make the solution bounded and try to pass to a limit of the renormalization. Such a limit solution serves as a model for the singularity, and one hopes (or expects) that the singularity models have special features making them much simpler than an arbitrary solution of the equation. FEBRUARY 2004

A singularity can form for the Ricci flow only where the curvature becomes unbounded. Suppose then that one has λ2i = |Riem|(xi , ti ) → ∞ , on a sequence of points xi ∈ M , and times ti → T < ∞ . It is then natural to consider the rescaled metrics and times

(12)

¯i (¯ g ti ) = λ2i g(t), ¯ ti = λ2i (t − ti ).

¯i are also solutions of the Ricci flow The metrics g and have bounded curvature at (xi , 0) . For suitable choices of xi and ti , the curvature will be bounded near xi , and for nearby times to the past, ¯ ti ≤ 0 ; for example, one might choose points where the curvature is maximal on (M, g(t)), 0 ≤ t ≤ ti . The rescaling (12) expands all distances by the factor λi and time by the factor λ2i . Thus, in effect one is studying very small regions, of spatial size on the order of ri = λ−1 about (xi , ti ), and “using a i microscope” to examine the small-scale features in this region on a scale of size about 1 . Implicit in this analysis is a change of coordinates near xi , i.e. use of local diffeomorphisms in conjunction with the metric rescaling. A local version of the Gromov compactness theorem will then allow one to pass to a limit solution of the Ricci flow, at least locally defined in space and time, provided that the local volumes of the rescalings are bounded below; more explicgi (¯ ti )) , for some fixed ν > 0; itly, one needs xi ∈ M ν (¯ see (3). In terms of the original unscaled flow, this means that the metric g(t) should not be locally collapsed, on the scale of its curvature, i.e. volBxi (ri , ti ) ≥ νri3 . ¯(¯ A maximal connected limit (N, g t ), x) containing the base point x = lim xi is then called a singularity model. Observe that the topology of the limit N may well be distinct from the original manifold M , most of which may have been blown off to infinity in the rescaling. To describe the potential usefulness of this process, suppose one does have local noncollapse on the scale of the curvature and that we have chosen points of maximal curvature in space and time 0 ≤ t ≤ ti. One then obtains, at least in a subse¯(¯ t ), x) , quence, a limit solution to the Ricci flow (N, g based at x , defined at least for times (−∞, 0]; more¯(¯ t ) is a complete Riemannian metric on N . over, g These are called ancient solutions of the Ricci flow in Hamilton’s terminology. The estimates in (10) and (11) can now be used to show that such singularity models do in fact have important features making them much simpler than general solutions of the Ricci flow. As discussed following (10), the pinching estimate implies that the limit has nonnegative curvature. Moreover, the topology of complete manifolds N of nonnegative curvature is completely understood in dimension 3. If N is

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noncompact, then N is diffeomorphic to R3, S 2 × R , or a quotient of these spaces. If N is compact, then a slightly stronger form of Hamilton’s theorem above implies N is diffeomorphic to S 3 /Γ , S 2 × S 1 or S 2 ×Z2 S 1 . Moreover, the Harnack estimate (11) holds on the limit. These general features of singularity models are certainly encouraging. Nevertheless, there are many problems to overcome to obtain any real benefit from this picture. I. One needs to prove noncollapse at the scale of the curvature to obtain a singularity model. II. In general, the curvature may blow up at many different rates or scales, and it is not nearly sufficient to understand just the structure of the singularity models at points of (space-time) maximal curvature. Somewhat analogous phenomena (usually called bubbling) arise in many other geometrical variational problems, for instance harmonic maps, Yang-Mills fields, Einstein metrics, and others. (In such elliptic contexts, these problems of multiple scales have been effectively resolved.) III. Even if one can solve the two previous issues, this leaves the main issue. One needs to relate the structure of the singularities with the topology of the underlying manifold. The study of the formation of singularities in the Ricci flow was initiated by Hamilton in [10]; cf. also [4] for a recent survey. Although there has been further technical progress over the last decade, the essential problems on the existence and structure of singularity models and their relation with topology remained unresolved until the appearance of Perelman’s work in 2002 and 2003.

Perelman’s Work Perelman’s recent work [15]-[17] (together with a less crucial paper still to appear) implies a complete solution of the Geometrization Conjecture. This is accomplished by introducing numerous highly original geometric ideas and techniques to understand the Ricci flow. In particular, Perelman’s work completely resolves issues I–III above. We proceed by describing, necessarily very briefly, some of the highlights.

I. Noncollapse Consider the Einstein-Hilbert action

R(g) =

M

This is a functional on the larger space M × C ∞ (M, R) , or equivalently a family of functionals on M , parametrized by C ∞ (M, R) .4 Fix any smooth measure dm on M and define the Perelman coupling by requiring that (g, f ) satisfy

(15)

e−f dVg = dm.

The resulting functional (16) F m (g, f ) = (R + |∇f |2 )dm M

becomes a functional on M . At first sight this may appear much more complicated than (13); however, for any g ∈ M there exists a large class of functions f (or measures dm) such that the L2 gradient flow of F m exists at g and is given simply by

(17)

d g = −2(Ricg + D2 f ), dt

where D 2 f is the Hessian of f with respect to g. The evolution equation (17) for g is just the Ricci flow (5) modified by an infinitesimal diffeomorphism: , where (d/dt)φt = ∇f . Thus, D 2 f = (d/dt)(φ∗ t g) the gradient flow of F m is the Ricci flow, up to diffeomorphisms. (Different choices of dm correspond to different choices of diffeomorphism.) In particular, the functional F m increases along the Ricci flow. What can one do with this more complicated functional? It turns out that, given any initial metric g(0) and t > 0 , the function f (and hence the measure dm) can be freely specified at g(t) , where g(t) evolves by the Ricci flow (5). Perelman then uses 3

The action (13) leads to the vacuum Einstein field equations in general relativity for Lorentz metrics on a 4-manifold. The term λ(t) in (4) is of course analogous to the cosmological constant.

(13)

metrics of constant Ricci curvature.3 It is natural to try to relate the Ricci flow with R; for instance, is the Ricci flow the gradient flow of R (with respect to a natural L2 metric on the space M )? However, while rather close to being true, it has long been recognized that this is not the case. In fact, the gradient flow of R does not even exist, since it implies a backwards heat-type equation for the scalar curvature R (similar to (7) but with a minus sign before ∆). Consider now the following functional enhancing R: (14) F (g, f ) = (R + |∇f |2 )e−f dVg .

R(g)dVg M

4

as a functional on the space of Riemannian metrics M on a manifold M . Critical points of R are Ricciflat metrics (Ric = 0). The action may be adjusted, for instance by adding a cosmological constant −2Λ , to give an action whose critical points are Einstein 190

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The functional (14) arises in string theory as the lowenergy effective action [5, §6]; the function or scalar field f is called the dilaton. It is interesting to note in this context that the gravitational field and the dilaton field arise simultaneously from the low-energy quantization of the string world sheet (σ-model) [5, p. 837].

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this freedom to probe the geometry of g(t) with suitable choices of f . For instance, he shows by a very simple study of the form of F m that the collapse or noncollapse of the metric g(t) near a point x ∈ M can be detected from the size of F m (g(t)) by choosing e−f to be an approximation to a delta function centered at x . The more collapsed g(t) is near x , the more negative the value of F m (g(t)) . The collapse of the metric g(t) on any scale in finite time is then ruled out by combining this with the fact that the functional F m is increasing along the Ricci flow. In fact, this argument is carried out with respect to a somewhat more complicated scale-invariant functional than F ; motivated by certain analogies in statistical physics, Perelman calls this the entropy functional.

II. Singularity Models A second highlight of [15] is essentially a classification of all complete singularity models (N, g(t)) that arise in finite time. Complete here means the metric g(0) is a complete Riemannian metric on N ; we also drop the overbar from the notation from now on. If N is smooth and compact, then it follows from Hamilton’s space-form theorem that N is diffeomorphic to S 3 /Γ, S 2 × S 1 or S 2 ×Z2 S 1. In the more important and difficult case where N is complete and noncompact, Perelman proves that the geometry of N near infinity is as simple and natural as possible. At time 0 and at points x with r (x) = dist(x, x0 ) 1 , for a fixed base point x0, a large neighborhood of x in the scale where R(x) = 1 is ε -close to a large neighborhood in the standard round product metric on S 2 × R . Here ε may be made arbitrarily small by choosing r (x) sufficiently large. Such a region is called an ε -neck. Thus the geometry near infinity in N is that of a union of ε necks, where the slowly varying radius of S 2 may either be uniformly bounded or diverge to infinity, but only at a rate much smaller than r (x) . Moreover, this structure also holds on a long time interval to the past of 0 , so that on such regions the solution is close to the (backwards) evolving Ricci flow on S 2 × R . Topologically, N is diffeomorphic to R3 or (N, g) is isometric to S 2 × R . Perelman shows that this structural result for the singularity models themselves also holds for the solution g(t) very near any singularity time T . Thus, at any base point (x, t) where the curvature is sufficiently large, the rescaling as in (12) of the space-time by the curvature is smoothly close, on large compact domains, to corresponding large domains in a complete singularity model. The “ideal” complete singularity models do actually describe the geometry and topology near any singularity. Consequently, one has a detailed understanding of the small-scale geometry and topology everywhere on (M, g(t)) , for t near T . In particular, this basically proves a general version of the FEBRUARY 2004

Ω

Ω Ω

Figure 4. Horns on singular limit. Harnack inequality (11). These results are of course rather technical, and the proofs are not simple. However, they are not exceptionally difficult and mainly rely on new insights and tools to understand the Ricci flow. A key idea is the use of the noncollapse result above on all relevant scales.

III. Relation with Topology The basic point now is the appearance of 2 -spheres S 2 near the singularities. Recall from (1) that one first needs to perform the sphere decomposition on M before it can be geometrized. There is no geometry corresponding to the sphere decomposition.5 While the sphere decomposition is the simplest operation to carry out topologically, geometrically and analytically it is by far the hardest to understand. How does one detect 2-spheres in M on which to perform surgery from the geometry of a metric? We now see that such 2 -spheres, embedded in the ε -necks above, arise naturally near the singularities of the Ricci flow. The idea then is to surger the 3 -manifold M along the 2 -spheres just before the first singularity time T . Figure 4 gives a schematic picture of the partially singular metric g(T ) on M . The metric g(T ) is smooth on a maximal domain Ω ⊂ M , where the curvature is locally bounded but is singular, i.e. ill-defined, on the complement where the curvature blows up as t → T . Suppose first that Ω = ∅ , so that the curvature of g(t) blows up everywhere on M as t → T . One says that the solution to the Ricci flow becomes extinct at time T . Note that R(x, t) 1 for all x ∈ M and t near T (by the pinching estimate (10)). Given the understanding of the singularity models above, it is not difficult to see that M is then diffeomorphic to 5

One might think that the S 2 × R geometry corresponds to sphere decomposition, but this is not really correct; at best, this can be made sense of only in an idealized or limiting context.

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S 3 /Γ , S 2 × S 1 , or S 2 ×Z2 S 1 . In this situation we are done, since M is then geometric. If Ω ≠ ∅ , then the main point is that small neighborhoods of the boundary ∂Ω consist of horns. A horn is a metric on S 2 × [0, δ] where the S 2 factor is approximately round of radius ρ(r ) , with ρ(r ) small and ρ(r )/r → 0 as r → 0 . Thus, a horn is a union of ε-necks assembled on smaller and smaller scales. The boxed figure in Figure 4 represents a partially singular metric on the smooth manifold S 2 × I , consisting of a pair of horns joined by a degenerate metric. At time T there may be infinitely many components of Ω , of arbitrarily small size, containing such horns. However, all but finitely many of these components are doubled horns, each topologically again of the form S 2 × I . In quantitative terms, there is a small constant ρ0 > 0 such that if Ω contains no horns with sphere S 2 × {δ} of radius ≥ ρ0 , then, as above when Ω = ∅ , M is diffeomorphic to S 3 /Γ , S 2 × S 1 , or S 2 ×Z2 S 1 , and we are done. If there are horns containing a sphere S 2 × δ of a definite size ρ0 in Ω , one then performs a surgery on each such horn by truncating it along the S 2 of radius ρ0 and glueing in a smooth 3 -ball, giving then a disjoint collection of 3 -manifolds. Having now disconnected M by surgery on 2 -spheres into a finite number of components, one then continues with the Ricci flow separately on each component. A conceptually simple, but technically hard, argument based on the decrease of volume associated with each surgery shows that the surgery times are locally finite: on any finite time interval there are only finitely many times at which singularities form. As a concrete example, suppose the initial metric g(0) on M has positive scalar curvature. Then the estimate (8) shows that Ricci flow completely terminates, i.e. becomes extinct, in finite time. Hence only finitely many surgeries are applied to M during the Ricci flow and it follows from the work above that M is diffeomorphic to a finite connected sum of S 3 /Γ and S 2 × S 1 factors. The upshot of this procedure is that if one successively throws away or ignores such components which become extinct in finite times (and which have already been identified topologically), the Ricci flow with surgery then exists for infinite time [0, ∞). What then does the geometry of the remaining ˆ i } of M look like at a sufficiently components {M large time T0 ? Here the thick-thin decomposition ˆ ∈ {M î } of Gromov-Thurston appears. Fix any {M} ˆ(t) = t −1 g(t) , for and consider the rescaled metric g t = T0 ; it is easy to see from the Ricci flow equation ˆ g ˆ(t)) is uniformly bounded. For ν sufthat vol(M, ficiently small, Perelman proves that there is sufˆ ν , as defined ficient control on the ν -thick part M ν ˆ in (3), to see that M is diffeomorphic to a complete hyperbolic 3-manifold H (with finitely many 192

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ˆν components). The smooth Ricci flow exists on M −1 for infinite time, and the rescalings t g(t) converge to the hyperbolic metric of curvature − 14 as t → ∞. (Since the Ricci flow exists for all time, it is reasonable to expect that the volume-normalized flow converges to an Einstein metric, necessarily a hyperbolic metric in our situation.) While there is ˆ ν, there is enough less control on the ν -thin part M ˆ ν is diffeomorphic to a graph to conclude that M manifold G (with finitely many components). Although there may still be infinitely many surgeries required to continue the Ricci flow for all ˆ ν = G .6 time, all further surgeries take place in M Thus, the original 3-manifold M has been decomposed (at large finite time) topologically as (18)

q

M = (K1 #...#Kp )#(#1 S 3 /Γi )#((#1r S 2 × S 1 ).

Perelman has recently shown [17] that the S 3 /Γ and S 2 × S 1 factors necessarily become extinct in bounded time (with bound depending on the initial metric), so that only the K factors exist after a sufficiently long time. (This result is not needed, however, for the geometrization conjecture.) Moreover, each K = Ki decomposes via the thick/thin decomposition as a union

(19)

K = H ∪ G,

where H is a complete hyperbolic manifold of finite volume (possibly disconnected) and G is a graph manifold (possibly disconnected). The union of H and G is along a collection of embedded tori. Perelman uses the proofs in [11] or [1], [2] to conclude that each such torus is incompressible in K. This process gives then both the sphere and torus decomposition of the manifold M . Although it is not asserted that the Ricci flow detects the further decomposition of G into Seifert fibered components, this is comparatively elementary from a topological standpoint. The torus-irreducible components of K have been identified as hyperbolic manifolds. This completes our brief survey of the geometrization conjecture. Perelman’s work has created a great deal of excitement in the mathematical research community, as well as in the scientifically interested public at large. While at the moment further evaluation of the details of his work are still being carried out, the beauty and depth of these new contributions are clear. I am very grateful to Bruce Kleiner, John Lott, and Jack Milnor for their many suggestions and comments, leading to significant improvements in the paper. 6

ˆ ν for all t It is not asserted that the bound (2) holds on M large, for some Λ < ∞ .

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References [1] M. ANDERSON, Scalar curvature and geometrization conjectures for 3-manifolds, in Comparison Geometry (Berkeley 1993–94), MSRI Publications, vol. 30, 1997, pp. 49–82. [2] ——— , Scalar curvature and the existence of geometric structures on 3-manifolds. I, J. Reine Angew. Math. 553 (2002), 125–182. [3] L. ANDERSSON, The global existence problem in general relativity, preprint, 1999, gr-qc/9911032; to appear in 50 Years of the Cauchy Problem in General Relativity, P. T. Chrusćiel and H. Friedrich, eds. [4] H.-D. CAO and B. CHOW, Recent developments on the Ricci flow, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 59–74. [5] E. D’H OKER, String theory, in Quantum Fields and Strings: A Course for Mathematicians, Vol. 2, Amer. Math. Soc., Providence, RI, 1999. [6] M. GROMOV, Structures Métriques pour les Variétés Riemanniennes, Cedic/Fernand Nathan, Paris, 1981. [7] ——— , Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 50 (1982), 5–100. [8] R. HAMILTON, Three manifolds of positive Ricci curvature, J. Differential Geom. 17 (1982), 255–306. [9] ——— , The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993), 225–243. [10] ——— , Formation of singularities in the Ricci flow, Surveys in Differential Geometry, Vol. 2, International Press, 1995, pp. 7–136. [11] ——— , Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. 7 (1999), 695–729. [12] T. IVEY, Ricci solitons on compact three-manifolds, Differential Geom. Appl. 3 (1993), 301–307. [13] B. KLEINER and J. LOTT, Ricci flow website: http:// www.math.lsa.umich.edu/research/ricciflow/ perelman.html. [14] J. MILNOR, Towards the Poincaré Conjecture and the classification of 3-manifolds, preprint, 2003; Notices Amer. Math. Soc. 50 (2003), 1226–1233. [15] G. PERELMAN, The entropy formula for the Ricci flow and its geometric applications, preprint, 2002, math.DG/0211159. [16] ——— , Ricci flow with surgery on three-manifolds, preprint, 2003, math.DG/0303109. [17] ——— , Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint, 2003, math.DG/0307245. [18] W. THURSTON, The Geometry and Topology of ThreeManifolds, preprint, 1978, Princeton University, available online at http://www.msri.org/publications/ books/gt3m; and Three-Dimensional Geometry and Topology, Vol. 1, Princeton University Press, 1997. [19] ——— , Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N. S.) 6 (1982), 357–381.

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arXiv:math/0607821v2 [math.DG] 21 Aug 2006

Structure of Three-Manifolds – Poincaré and geometrization conjectures Shing-Tung Yau1,2 Ladies and gentlemen, today I am going to tell you the story of how a chapter of mathematics has been closed and a new chapter is beginning. Let me begin with some elementary observations. A major purpose of Geometry is to describe and classify geometric structures of interest. We see many such interesting structures in our day-to-day life. Let us begin with topological structures of a two dimensional surface. These are spaces where locally we have two degrees of freedom. Here are some examples:

genus 0

genus 1

genus 2

genus 3

Genus of a surface is the number of handles of the surface. An abstract and major way to construct surfaces is by connecting along some deleted disk of each surface. The connected sum of two surfaces S1 and S2 is denoted by S1 #S2 . It is formed by deleting the interior of disks Di from each Si and attaching the resulting punctured surfaces Si −Di to each other by a one-to-one continuous 1

This was a talk given at the Morningside Center of Mathematics on June 20, 2006. All the computer graphics are provided by David Gu, based on the joint paper of David Gu, Yalin Wang and S.-T. Yau. 2

1

map h : ∂D1 → ∂D2 , so that S1 #S2 = (S1 − D1 ) ∪h (S2 − D2 ). D1 S1

D2 +

S2

Example:

A genus 8 surface, constructed by connected sum. The major theorem for the two dimensional surfaces is the following: Theorem (Classification Theorem for Surfaces). Any closed, connected orientable surface is exactly one of the following surfaces: a sphere, a torus, or a finite number of connected sum of tori. Note that a surface is called orientable if each closed curve on it has a well-defined continuous normal field.

1

Conformal geometry

In order to understand surfaces in a deep manner, Riemann, Poincaré and others proposed to study conformal structure on these two dimensional ob2

jects. Such structures allow us to measure angles in the neighborhood of each point on the surface. For example, if we take a standard atlas of the globe, we have longitude and latitude. They are orthogonal to each other. When we map the atlas, which is a square, onto the globe; distances are badly distorted. For example, the region around the north pole is shown to be a large region on the square. However, the fact that longitude and latitude is orthogonal to each other is preserved under the map. Hence if a ship moves in the ocean, we can use the atlas to determine its direction accurately, but not the distance travelled. Globe

Poincaré found that at any point, we can draw a longitude (blue curve) and latitude (red curve) on any surface of genus zero in three space. These curves are orthogonal to each other and they converge to two distinct points, on the surface, just like north pole and south pole on the sphere. This theorem of Poincaré also works for arbitrary abstract surface with genus zero. It is a remarkable theorem that for any two closed surfaces with genus zero, we can always find a one-to-one continuous map mapping longitude and latitude of one surface to the corresponding longitude and latitude of the other surface. This map preserve angles defined by the charts. In such a situation, we say that these two surfaces are conformal to each other. And there is only one conformal structure for a surface with genus zero. For genus equal to one, the surface looks like a donut, and we can draw longitude and latitude with no north or south poles. However, there can be distinct surfaces with genus one that are not conformal to each other. In fact, there are two parameters of conformal structures on a genus one surface. For genus g greater than one, one can still draw longitude and latitude (the definition of such curves needs to be made precise). But they have many poles, the number of which depends on the genus. The number 3

of parameters of conformal structures over a surface with genus g is 6g − 6. In order to find a global atlas of the surface, we can cut along some special curves of a surface and then spread the surface on the plane or the disk. In this procedure, the longitude and the latitude will be preserved.

A fundamental theorem for surfaces with metric structure is the following theorem. Theorem (Poincar´ e’s Uniformization Theorem). Any closed two-dimensional space is conformal to another space with constant Gauss curvature. • If curvature > 0, the surface has genus = 0; • If curvature = 0, the surface has genus = 1; • If curvature < 0, the surface has genus > 1. The generalization of this theorem plays a very important role in the field of geometric analysis. In particular, it motivates the works of Thurston and Hamilton. This will be discussed later in this talk.

2

Hamilton’s equation on Surfaces

Poincaré’s theorem can also be proved by the equation of Hamilton. We can deform any metric on a surface by the negative of its curvature. After 4

Spherical

Euclidean

Hyperbolic

normalization, the final state of such deformation will be a metric with constant curvature. This is a method created by Hamilton to deform metrics on spaces of arbitrary dimensions. In higher dimension, the typical final state of spaces for the Hamilton equation is a space that satisfies Einstein’s equation. As a consequence of the works by Richard Hamilton and B. Chow, one knows that in two dimension, the deformation encounters no obstruction and will always converge to one with constant curvature. This theorem was used by David Gu, Yalin Wang, and myself for computer graphics. The following sequence of pictures is obtained by numerical simulation of the Ricci flow in two dimension.

5

3

Three-Manifolds

So far, we have focused on spaces where there are only two degrees of freedom. Instead of being a flat bug moving with two degrees of freedom on a surface, we experience three degrees of freedom in space. While it seems that our three dimensional space is flat, there are many natural three dimensional spaces, which are not flat. Important natural example of higher dimensional spaces are phase spaces in mechanics. In the early twentieth century, Poincaré studied the topology of phase space of dynamics of particles. The phase space consists of (x; v), the position and the velocity of the particles. For example if a particle is moving freely with unit speed on a two dimensional surface Σ, there are three degrees of freedom in the phase space of the particle. This gives rise to a three dimensional space M. Such a phase space is a good example for the concept of fiber bundle. If we associate to each point (x; v) in M the point x ∈ Σ, we have a map from M onto Σ. When we fix the point x, v can be any vector with unit length. The totality of v forms a circle. Therefore, M is a fiber bundle over Σ with fiber equal to a circle.

4

The Poincar´ e Conjecture

The subject of higher dimensional topology started with Poincaré’s question: Is a closed three dimensional space topologically a sphere if every closed curve in this space can be shrunk continuously to a point? This is not only a famous difficult problem, but also the central problem for three dimensional topology. Its understanding leads to the full structure theorem for three dimensional spaces. I shall describe its development chronologically.

5

Topological Surgery

Topologists have been working on this problem for over a century. The major tool is application of cut and paste, or surgery, to simplify the topology of a space:

6

Two major ingredients were invented. One is called Dehn’s lemma which provides a tool to simplify any surface which cross itself to one which does not.

Theorem (Dehn’s lemma) If there exists a map of a disk into a three dimensional space, which does not cross itself on the boundary of the disk, then there exists another map of the disc into the space which does not cross itself and is identical to the original map on the boundary of the disc. This is a very subtle theorem, as it took almost fifty years until Papakyriakopoulos came up with a correct proof after its discovery. The second tool is the construction of incompressible surfaces introduced by Haken. It was used to cut three manifolds into pieces. Walhausen proved important theorems by this procedure. (Incompressible surfaces are embedded surfaces which have the property whereby if a loop cannot be shrunk to a point on the surface, then it cannot be shrunk to a point in the three dimensional space, either.)

7

6

Special Surfaces

There are several important one dimensional and two dimensional spaces that play important roles in understanding three dimensional spaces. 1. Circle Seifert constructed many three dimensional spaces that can be described as continuous family of circles. The above mentioned phase space is an example of a Seifert space. 2. Two dimensional spheres We can build three dimensional spaces by removing balls from two distinguished ones and gluing them along the boundary spheres. Conversely

S2

Kneser and Milnor proved that each three dimensional space can be uniquely decomposed into irreducible components along spheres. (A space is called irreducible if each embedded sphere is the boundary of a three dimensional ball in this space.) 3. Torus A theorem of Jaco-Shalen, Johannson says that one can go one step further by cutting a space along tori.

T2

7

Structure of Three Dimensional Spaces

A very important breakthrough was made in the late 1970s by W. Thurston. He make the following conjecture. 8

Geometrization Conjecture (Thurston): The structure of three dimensional spaces is built on the following atomic spaces: (1) The Poincaré conjecture: three dimensional space where every closed loop can be shrunk to a point; this space is conjectured to be the three-sphere. (2) The space-form problem: spaces obtained by identifying points on the three-sphere. The identification is dictated by a finite group of linear isometries which is similar to the symmetries of crystals. (3) Seifert spaces mentioned above and their quotients similar to (2). (4) Hyperbolic spaces according to the conjecture of Thurston: threespace whose boundaries may consist of tori such that every two-sphere in the space is the boundary of a ball in the space and each incompressible torus can be deformed to a boundary component; it was conjectured to support a canonical metric with constant negative curvature and it is obtained by identifying points on the hyperbolic ball. The identification is dictated by a group of symmetries of the ball similar to the symmetries of crystals. An example of a space obtained by identifying points on the three dimensional hyperbolic space

Hyperbolic Space Tiled with Dodecahedra, by Charlie Gunn (Geometry Center). from the book ”Three-dimensional geometry and topology” by Thurston, Princeton University press Thurston’s conjecture effectively reduced the classification of three dimensional spaces to group theory, where many tools were available. He and his followers proved the conjecture when the three space is sufficiently large in 9

the sense of Haken and Walhausen. (A space is said to be sufficiently large if there is a nontrivial incompressible surface embedded inside the space. Haken and Walhausen proved substantial theorem for this class of manifolds.) This theorem of Thurston covers a large class of three dimensional hyperbolic manifolds. However, as nontrivial incompressible surface is difficult to find on a general space, the argument of Thurston is difficult to use to prove the Poincaré conjecture.

8

Geometric Analysis

On the other hand, starting in the seventies, a group of geometers applied nonlinear partial differential equations to build geometric structures over a space. Yamabe considered the equation to conformally deform metrics to metrics with constant scalar curvature. However, in three dimension, metrics with negative scalar curvature cannot detect the topology of spaces. A noted advance was the construction of Kähler-Einstein metrics on Kähler manifolds in 1976. In fact, I used such a metric to prove the complex version of the Poincaré conjecture. It is called the Severi conjecture in complex geometry. It says that every complex surface that can be deformed to the complex projective plane is itself the complex projective plane. The subject of combining ideas from geometry and analysis to understand geometry and topology is called geometric analysis. While the subject can be traced back to 1950s, it has been studied much more extensively in the last thirty years. Geometric analysis is built on two pillars: nonlinear analysis and geometry. Both of them became mature in the seventies based on the efforts of many mathematicians. (See my survey paper “Perspectives on geometric analysis” in Survey in Differential Geometry, Vol. X. 2006.)

9

Einstein metrics

I shall now describe how ideas of geometric analysis are used to solve the Poincaré conjecture and the geometrization conjecture of Thurston. In the case of a three dimensional space, we need to construct an Einstein metric, a metric inspired by the Einstein equation of gravity. Starting from

10

an arbitrary metric on three space, we would like to find a method to deform it to the one that satisfies Einstein equation. Such a deformation has to depend on the curvature of the metric. Einstein’s theory of relativity tells us that under the influence of gravity, space-time must have curvature. Space moves dynamically. The global topology of space changes according to the distribution of curvature (gravity). Conversely, understanding of global topology is extremely important and it provides constraints on the distribution of gravity. In fact, the topology of space may be considered as a source term for gravity. From now on, we shall assume that our three dimensional space is compact and has no boundary (i.e., closed). In a three dimensional space, curvature of a space can be different when measured from different directions. Such a measurement is dictated by a quantity Rij , called the Ricci tensor. In general relativity, this gives rise to the matter tensor of space. An important quantity that is independent of directions is the scalar curvature R. It is the trace of Rij and can be considered as a way to measure the expansion or shrinking of the volume of geodesic balls: 4π 3 1 (r − R(p)r 5 ), 3 30 where B(p, r) is the ball of radius r centered at a point p, and R(p) is the scalar curvature at p. This can be illustrated by a dumbbell surface where, near the neck, curvature is negative and where, on the two ends which are convex, curvature is positive. Volume(B(p, r)) ∼

Two-dimensional dumbbell surface Two-dimensional surfaces with negative curvature look like saddles. Hence a two dimensional neck has negatives curvature. However, in three dimension, the slice of a neck can be a two dimensional sphere with very large 11

positive curvature. Since scalar curvature is the sum of curvatures in all direction, the scalar curvature at the three dimensional neck can be positive. This is an important difference between a two-dimensional neck and a three-dimensional neck.

Three dimensional neck.

10

The dynamics of Einstein equation

In general relativity, matter density consists of scalar curvature plus the momentum density of space. The Dynamics of Einstein equation drives space to form black holes which splits space into two parts: the part where scalar curvature is positive and the other part, where the space may have a black hole singularity and is enclosed by the apparent horizon of the black hole, the topology tends to support metrics with negative curvature. There are two quantities in gravity that dictate the dynamics of space: metric and momentum. Momentum is difficult to control. Hence at this 12

time, it is rather difficult to use the Einstein equation of general relativity to study the topology of spaces.

11

Hamilton’s Equation

In 1979, Hamilton developed a new equation to study the dynamics of space metric. The Hamilton equation is given by ∂gij = −2Rij . ∂t Instead of driving space metric by gravity, he drives it by its Ricci curvature which is analogous to the heat diffusion. Hamilton’s equation therefore can be considered as a nonlinear heat equation. Heat flows have a regularizing effect because they disperse irregularity in a smooth manner. Hamilton’s equation was also considered by physicists. (It first appeared in Friedan’s thesis.) However, this point of view was completely different. Physicists considered it as beta function for deformations of the sigma model to conformal field theory.

12

Singularity

Despite the fact that Hamilton’s equation tend to smooth out metric structure, global topology and nonlinear terms in the equation coming from curvature drive the space metric to points where the space topology collapses. We call such points singularity of space. In 1982, Hamilton published his first paper on the equation. Starting with a space with positive Ricci curvature, he proved that under his equation, space, after dilating to keep constant volume, never encounters any singularity and settles down to a space where curvature is constant in every direction. Such a space must be either a 3-sphere or a space obtained by identifying the sphere by some finite group of isometries. After seeing the theorem of Hamilton, I was convinced that Hamilton’s equation is the right equation to carry out the geometrization program. (This paper of Hamilton is immediately followed by the paper of Huisken on deformation of convex surfaces by mean curvature. The equation of mean 13

curvature flow has been a good model for understanding Hamilton’s equation.) We propose to deform any metric on a three dimensional space which shall break up the space eventually. It should lead to the topological decomposition according to Kneser, Milnor, Jacob-Shalen and Johannson. The asymptotic state of Hamilton’s equation is expected to be broken up into pieces which will either collapse or produce metrics which satisfy the Einstein equation. In three dimensional spaces, Einstein metrics are metrics with constant curvature. However, along the way, the deformation will encounter singularities. The major question is how to find a way to describe all possible singularities. We shall describe these spectacular developments.

13

Hamilton’s Program

Hamilton’s idea is to perform surgery to cut off the singularities and continue his flow after the surgery. If the flow develops singularities again, one repeats the process of performing surgery and continuing the flow. If one can prove there are only a finite number of surgeries in any finite time interval, and if the long-time behavior of solutions of the Hamilton’s flow with surgery is well understood, then one would be able to recognize the topological structure of the initial manifold. Thus Hamilton’s program, when carried out successfully, will lead to a proof of the Poincaré conjecture and Thurston’s geometrization conjecture. The importance and originality of Hamilton’s contribution can hardly be exaggerated. In fact, Perelman said: “The implementation of Hamilton’s program would imply the geometrization conjecture for closed three-manifolds.” “In this paper we carry out some details of Hamilton’s program”. We shall now describe the chronological development of Hamilton’s program. There were several stages: I. A Priori Estimates In the early 1990s, Hamilton systematically developed methods to understand the structure of singularities. Based on my suggestion, he proved the

14

fundamental estimate (the Li-Yau-Hamilton estimate) for his flow when curvature is nonnegative. The estimate provides a priori control of the behavior of the flow. An a prior estimate is the key to proving any existence theorem for nonlinear partial differential equations. An intuitive example can be explained as follows: when a missile engineer designs trajectory of a missile, he needs to know what is the most likely position and velocity of the missile after ten seconds of its launch. Yet a change in the wind will cause reality to differ from his estimate. But as long as the estimate is within a range of accuracy, he will know how to design the missile. How to estimate this range of accuracy is called a prior estimate. The Li-Yau-Hamilton Estimate In proving existence of a nonlinear differential equation, we need to find an a priori estimate for some quantity which governs the equation. In the case of Hamilton’s equation, the important quantity is the scalar curvature R. An absolute bound on the curvature gives control over the nonsingularity of the space. On the other hand, the relative strength of the scalar curvature holds the key to understand the singularity of the flow. This is provided by the Li-Yau-Hamilton estimate: For any one-form Va we have ∂R R + + 2∇a R · Va + 2Rab Va Vb ≥ 0. ∂t t In particular, tR(x, t) is pointwise nondecreasing in time. In the process of applying such an estimate to study the structure of singularities, Hamilton discovered (also independently by Ivey) a curvature pinching estimate for his equation on three-dimensional spaces. It allows him to conclude that a neighborhood of the singularity looks like space with nonnegative curvature. For such a neighborhood, the Li-Yau-Hamilton estimate can be applied. Then, under an additional non-collapsing condition, Hamilton described the structure of all possible singularities. However, he was not able to show that all these possibilities actually occur. Of particular concern to him was a singularity which he called the cigar. II. Hamilton’s works on Geometrization

15

In 1995, Hamilton developed the procedure of geometric surgery using a foliation by surfaces of constant mean curvature, to study the topology of four-manifolds of positive isotropic curvature. In 1996, he went ahead to analyze the global structure of the space time structure of his flow under suitable regularity assumptions (he called them nonsingular solutions). In particular, he showed how three-dimensional spaces admitting a nonsingular solution of his equation can be broken into pieces according to the geometrization conjecture. These spectacular works are based on deep analysis of geometry and nonlinear differential equations. Hamilton’s two papers provided convincing evidence that the geometrization program could be carried out using his approach. Main Ingredients of these works of Hamilton In this deep analysis he needed several important ingredients: (1) a compactness theorem on the convergence of metrics developed by him, based on the injectivity radius estimate proved by Cheng-Li-Yau in 1981. (The injectivity radius at a point is the radius of the largest ball centered at that point that the ball would not collapse topologically.) (2) a beautiful quantitative generalization of Mostow’s rigidity theorem which says that there is at most one metric with constant negative curvature on a three-dimensional space with finite volume. This rigidity theorem of Mostow is not true for two dimensional surfaces. (3) In the process of breaking up the space along the tori, he needs to prove that the tori are incompressible. The ingredients of his proof depend on the theory of minimal surfaces as was developed by Meeks-Yau and SchoenYau. At this stage, it seems clear to me that Hamilton’s program for the Poincaré and geometrization conjectures could be carried out. The major remaining obstacle was to obtain certain injectivity radius control, in terms of local curvature bound, in order to understand the structure of the singularity and the process of surgery to remove the singularity. Hamilton and I worked together on removing this obstacle for some time. III. Perelman’s Breakthrough In November of 2002, Perelman put out a preprint, “The entropy formula for Hamilton’s equation and its geometric applications”, wherein major ideas were introduced to implement Hamilton’s program. 16

Parallel to what Li-Yau did in 1986, Perelman introduced a space-time distance function obtained by path integral and used it to verify the noncollapsing condition in general. In particular, he demonstrated that cigar type singularity does not exist in Hamilton’s equation. His distance function can be described as follows. Let σ be any space-time path joining p to q, we define the action to be Z τ √ 2 s(R + |σ(s)| ˙ )ds. 0

By minimizing among all such paths joining p to q, we obtain L(q, τ ). Then Perelman defined his reduced volume to be Z 1 −n 2 (4πτ ) exp − √ L(q, τ ) 2 τ and observed that under the Hamilton’s equation it is nonincreasing in τ . In this proof Perelman used the idea in the second part of Li-Yau’s paper in 1986. As recognized by Perelman: “in Li-Yau, where they use ‘length’, associated to a linear parabolic equation, is pretty much the same as in our case.” Rescaling Argument Furthermore, Perelman developed an important refined rescaling argument to complete the classification of Hamilton on the structure of singularities of Hamilton’s equation and obtained a uniform and global version of the structure theorem of singularities. Hamilton’s Geometric Surgery Now we need to find a way to perform geometric surgery. In 1995, Hamilton had already initiated a surgery procedure for his equation on fourdimensional spaces and presented a concrete method for performing such surgery. One can see that Hamilton’s geometric surgery method also works for Hamilton’s equation on three-dimensional spaces. However, in order for surgeries to be done successfully, a more refined technique is needed. Discreteness of Surgery Times The challenge is to prove that there are only a finite number of surgeries on each finite time interval. The problem is that, when one performs the surgeries with a given accuracy at each surgery time, it is possible that the error may add up to so fast that they force the surgery times to accumulate. 17

Ωρ

R @

Ωρ

R @

I @

ε-tube

I @

ε-horn

6

double ε-horn

6

capped ε-horn

The Structure of Singularity

Rescaling Arguments In March of 2003, Perelman put out another preprint, titled “Ricci flow with surgery on three manifolds”, where he designed an improved version of Hamilton’s geometric surgery procedure so that, as time goes on, successive surgeries are performed with increasing accuracy. Perelman introduced a rescaling argument to prevent the surgery time from accumulating. When using the rescaling argument for surgically modified solutions of Hamilton’s equation, one encounters the difficulty of applying Hamilton’s compactness theorem, which works only for smooth solutions. The idea of overcoming this difficulty consists of two parts: 1. (Perelman): choose the cutoff radius in the neck-like regions small enough to push the surgical regions far away in space. 2. (Cao-Zhu): establish results for the surgically modified solutions so that Hamilton’s compactness theorem is still applicable. To do so, they need a deep understanding of the prolongation of the surgical regions, which in turn relies on the uniqueness theorem of Chen-Zhu for solutions of Hamilton’s equation on noncompact manifolds. 18

I @

I @

ε-horn

neck

I @

the gluing cap geometric surgery

Conclusion of the proof of the Poincar´ e Conjecture One can now prove Poincaré conjecture for simply connected three dimensional space, by combining the discreteness of surgeries with finite time extinction result of Colding-Minicozzi (2005). IV. Proof of the geometrization conjecture: Thick-thin Decomposition To approach the structure theorem for general spaces, one still needs to analyze the long-time behavior of surgically modified solutions to Hamilton’s equation. As mentioned in II, Hamilton studied the long time behavior of his equation for a special class of (smooth) solutions – nonsingular solutions. In 1996, Hamilton proved that any three-dimensional nonsingular solution admits of a thick-thin decomposition where the thick part consists of a finite number of hyperbolic pieces and the thin part collapses. Moreover, by adapting Schoen-Yau’s minimal surface arguments, Hamilton showed that the boundary of hyperbolic pieces are incompressible tori. Consequently, any nonsingular solution is geometrizable. 19

thin part R @

6

thick part (hyperbolic piece)

thick part (hyperbolic piece)

thick part (hyperbolic piece) ?

thin part

Thick-thin decomposition

Even though the nonsingularity assumption seems restrictive, the ideas and arguments of Hamilton are used in an essential way by Perelman to analyze the long-time behavior for general surgical solutions. In particular, he also studied the thick-thin decomposition. For the thick part, based on the Li-Yau-Hamilton estimate, Perelman established a crucial elliptic type estimate, which allowed him to conclude that the thick part consists of hyperbolic pieces. For the thin part, since he could only obtain a lower bound on the sectional curvature, he needs a new collapsing result. Assuming this new collapsing result, Perelman claimed that the solutions to Hamilton’s equation with surgery have the same long-time behavior as nonsingular solutions in Hamilton’s work, a conclusion which would imply the validity of Thurston’s geometrization conjecture. Although the proof of this new collapsing result was promised by Perelman, it still has yet to appear. (Shioya-Yamaguchi has published a proof of the collapsing result in the special case when the space is closed.) Nonethe-

20

less, based on the previous results, Cao-Zhu gave a complete proof of Thurston’s geometrization conjecture.

Conclusion The success of Hamilton’s program is the culmination of efforts by geometric analysts in the past thirty years. It should be considered as the crowning achievement of the subject of geometric analysis, a subject that is capable of proving hard and difficult topological theorems by geometry and analysis solely. Hamilton’s equation is a complicated nonlinear system of partial differential equations. This is the first time that mathematicians have been able to understand the structure of singularity and development of such a complicated system. Similar systems appear throughout the natural world. The methods developed in the study of Hamilton equation should shed light on many natural systems such as the Navier-Stokes equation and the Einstein equation. In addition, the numerical implementation of the Hamilton flow should be useful in computer graphics, as was demonstrated by Gu-Wang-Yau for two dimensional figures.

Impact on the future of geometry Poincar´ e: “Thought is only a flash in the middle of a long night, but the flash that means everything.” The Flash of Poincaré in 1904 has illuminated a major portion of the topological developments in the last century. Poincaré also initiated development of the theory of Riemann surfaces. It has been one of the major pillars of all mathematics development in the twentieth century. I believe that the full understanding of the three dimensional manifolds will play a similar role in the twenty-first century. Remark In Perelman’s work, many key ideas of the proofs are sketched or outlined, but complete details of the proofs are often missing. The recent paper of 21

Cao-Zhu, submitted to The Asian Journal of Mathematics in 2005, gives the first complete and detailed account of the proof of the Poincaré conjecture and the geometrization conjecture. They substituted several arguments of Perelman with new approaches based on their own studies. The materials were presented by Zhu in a Harvard seminar from September 2005 to March 2006, where faculties and postdoctoral fellows of Harvard University and MIT attended regularly. Some of the key arguments, that has been important for the completion of the Poincare conjecture, has already appeared in the paper of Chen-Zhu [2]. In the last three years, many mathematicians have attempted to see whether the ideas of Hamilton and Perelman can hold together. Kleiner and Lott (in 2004) posted on their web page some notes on several parts of Perelman’s work. However, these notes were far from complete. After the work of Cao-Zhu was accepted and announced by the journal in April, 2006 (it was distributed on June 1, 2006). On May 24, 2006, Kleiner and Lott put up another, more complete, version of their notes. Their approach is different from Cao-Zhu’s. It will take some time to understand their notes which seem to be sketchy at several important points. Most recently, a manuscript of Morgan-Tian appeared in the web. In a letter to the author, Jim Carlson of the Clay institute stated that the first version of this manuscript was submitted to the Clay institute on May 19, 2006, and the revised version was submitted on July 23, 2006.

References [1] H.-D. Cao and X.-P. Zhu, A complete proof of the Poincar and geometrization conjectures - application of the Hamilton-Perelman theory of the Ricci flow, Asian Journal of Mathematics, 10 no.2 (2006), 165-492 [2] Bing-Long Chen and Xi-Ping Zhu, Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature, arxiv.org: math.DG/0504478. [3] Siu-Yuen Cheng, Peter Li, Shing-Tung Yau, On the upper estimate of the heat kernel of a complete Riemannian manifold. Amer. J. Math. 103 (1981), no. 5, 1021–1063.

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[4] Colding, Tobias H.; Minicozzi, William P., II, Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman. J. Amer. Math. Soc. 18 (2005), no. 3, 561–569. [5] Freedman, Michael Hartley, The topology of four-dimensional manifolds. J. Differential Geom. 17 (1982), no. 3, 357–453. [6] R. Hamilton, Four-Manifolds with Positive Curvature Operator, J. Diff. Geom. 24(1986), 153–179. [7] R. Hamilton, The Ricci Flow on Surfaces, in Mathematics and General Relativity, Contemp. Math. 71, AMS, Providence, RI, p. 237–262 (1988). [8] R. Hamilton, The Harnack Estimate for the Ricci Flow, J. Diff. Geom. 37(1993), 225–243. [9] R. Hamilton, A Compactness Property for Solutions of the Ricci Flow Amer. J. Math. 117(1995), 545–572. [10] R. Hamilton, The Formation of Singularities in the Ricci Flow, in Surveys in Differential Geometry, Vol. II, Internat. Press, Cambridge (1995) 7–136. [11] R. Hamilton, Nonsingular Solutions of the Ricci Flow on ThreeManifolds, Comm. Anal. Geom. 7(1999) 695–729. [12] R. Hamilton, Three-Manifolds with Positive Ricci Curvature, J. Diff. Geom. 17(1982), 255–306. [13] R. Hamilton, Four-Manifolds with Positive Isotropic Curvature, Comm. Anal. Geom. 5(1997), 1–92. [14] P. Li and S.-T. Yau, On the parabolic kernel of the Schr¨ odinger operator, Acta Math., 156(1986), 153–201. [15] Milnor, John, The work of M. H. Freedman. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 13–15, Amer. Math. Soc., Providence, RI, 1987. [16] Milnor, John, Towards the Poincar conjecture and the classification of 3-manifolds. Notices Amer. Math. Soc. 50 (2003), no. 10, 1226–1233. 23

[17] G. Perelman, The Entropy Formula for the Ricci Flow and its Geometric Applications, arXiv.org: math.DG/0211159. [18] G. Perelman, Ricci Flow with Surgery on Three-Manifolds, arxiv.org: math.DG/0303109. [19] G. Perelman, Finite Extinction Time for the Solutions to the Ricci Flow on Certain Three-Manifolds, arxiv.org: math.DG/0307245. [20] H. Poincaré, Cinquième complément ` a l’analysis situs, Rend. Circ. Mat. Palermo, 18 (1904), 45–110. (See Oeuvres, Tome VI, Paris, 1953, p. 498.) [21] W. X. Shi, Deforming the metric on complete Riemannian manifold, J. Differential Geometry, 30 (1989), 223–301. [22] Shioya, Takashi; Yamaguchi, Takao, Volume collapsed three-manifolds with a lower curvature bound. Math. Ann. 333(2005), no. 1, 131–155. [23] Smale, Stephen, The generalized Poincar conjecture in higher dimensions. Bull. Amer. Math. Soc. 66(1960) 373–375. [24] Stallings, John R. Polyhedral homotopy-spheres. Bull. Amer. Math. Soc. 66(1960) 485–488. [25] Thurston, William P. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. [26] Yau, Shing Tung, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. [27] Yau, Shing Tung, Some 3-manifold topological works related to the Poincare conjecture before 1990.

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Uniqueness of standard solutions in the work of Perelman Peng Lu and Gang Tian The short time existence of Ricci flow on complete noncompact Riemannian manifolds with bounded curvature is proven by Shi [Sh1]. The uniqueness of such solutions is a difficult problem. Hsu studies this problem in dimension two [Hs], otherwise there is not any result about this problem. In the fundamental paper [Pe2] Perelman discussed a special family of solutions of Ricci flow on R3 , the so-called standard solutions, the solutions are used to construct the geometric-topological surgeries and their uniqueness are used to construct the longtime existence and to study the properties of the Ricci flow with surgery. A particular nice feature about these solutions is that at space infinity these solutions are asymptotic to round infinity cylinder. In [Pe2] §2 Perelman gives a proof of the uniqueness of the standard solutions. The idea is to reduce the Ricci flow equation to ft = f 00 + a1 f 0 + b1 g 0 + c1 f + d1 g gt = a2 f 0 + b2 g 0 + c2 f + d2 g by using the rotational symmetry of the solutions, then prove the uniqueness of solutions of above equations for given initial data. However there is difficulty to bound the coefficients in above equations near the origin of R3 , the first named author thanks John Lott for discussions in understanding this difficulty. In this paper we give a proof of the uniqueness of standard solutions through the uniqueness of the DeTurk-Ricci flow. The general idea of using DeTurk-Ricci flow to prove the uniqueness of the Ricci flow is due to Hamilton [H95b] §6. Our proof uses the rotationally symmetry of the standard solutions which is used to prove the short time existence and the certain asymptotic behavior of harmonic map flow. We make two remarks. First through a private communication we have learned that Perelman has similar idea of using the DeTurk-Ricci flow to prove the uniqueness; Second properties of standard solutions are proven in [Pe2] §2, for completeness we include here proofs of those properties needed to show the uniqueness.

1

1 Standard solutions Let integer n ≥ 3. Denote S n−1 (r) the round (n − 1)-sphere of radius r and let dσ be the standard metric on S n−1 (1). Let Gn be the set of rotationally symmetric metric g0 on Rn which satisfies the following condition: (i) The curvature operator of g0 is nonnegative and at some point; ¯ is positive ¯ (ii) The curvature |Rmg0 | and its derivatives ¯∇i Rmg0 ¯ , i = 1, 2, 3, 4 are bounded; (iii) There of points yi → ∞ in Rn , (Rn , g0 , yi ) converges to ´ ³p is a sequence n−1 2 (n − 2) in pointed C 3 Cheeger-Gromov topology. R×S We ¡ construct a¢ rotationally symmetric metric to show that ¡Gn is nonempty.¢ Let θ1 , · · · , θn−1 be normal coordinates on S n−1 (1), then θ1 , · · · , θn−1 , r are local coordinates on Rn . Consider rotationally symmetric complete metric 2 dr2 + f (r) dσ on Rn , the curvatures are given by 2

Rijji = f 2 − f 2 (f 0 ) Rinnj = 0

Rijkn = 0 Rinni = −f f 00

(1)

where 1 ≤ i 6= j, k ≤ n − 1 (see, for example, [BW] §9). We choose a smooth convex function f0 (r) satisfies ½ π psin r if 0 ≤ r ≤ 100 √ . f0 (r) = 2 (n − 2) if r ≥ 3n 2

Then clearly metric g∗ + dr2 + f0 (r) dσ has nonnegative sectional curvature. Using (1) it is easy to check that g∗ satisfies (i), condition (ii) and (iii) also hold and hence Gn is not empty. Given g0 ∈ Gn , since g0 is complete and has bounded curvature tensor, by Shi’s existence theorem ([Sh1]) there is a solution g∗ (t), t ∈ [0, T∗ ], T∗ > 0 of the Ricci flow ∂g = −2Rij , g (0) = g0 (2) ∂t ¯ ¯ on Rn . g∗ (t) has uniform bounded curvature supRn ×[0,T∗ ] ¯Rmg∗ (t) (x)¯ < +∞ and g∗ (t), t > 0 has positive curvature operator ([Sh2] Theorem 4.14). ¯ We call ¯ any solution g(t), t ∈ [0, T ], T > 0 of (2) with supRn ×[0,T ] ¯Rmg(t) (x)¯ < +∞ a standard solution, again g(t), t > 0 has positive curvature operator ([Sh2] Theorem 4.14). The uniqueness problem of the standard solutions is to show that g(t) = g∗ (t) on t ∈ [0, min{T∗ , T }]. To prove the uniqueness, we need to establish a few properties of standard solutions.

1.1 Asymptotic behavior of standard solutions g(t) at space infinity Let yi → ∞ of points in Rn such that (Rn , g0 , yi ) converges to ³pbe the sequence ´ n−1 R×S 2 (n − 2) in pointed C 3 Cheeger-Gromov topology. 2

Lemma 1 There is a subsequence of (Rn , g (t) , yi ) , t ∈ [0, T ] which converges in pointed C 3 Cheeger-Gromov topology to the round cylinder solution dr2 + 2 (n − 2) (1 − t) dσ on R × S n−1 . In particular T < 1. Proof. We will apply Hamilton’s Cheeger-Gromov type compactness theorem [H95a], but we need to make some modification of the compactness theorem since time 0 is not an interior point of [0, T ]. In general if there is a sequence of complete solution of the Ricci flow (Mkn , gk (t), pk ), t ∈ [0, T ], k ∈ N satisfying ¯ ¯ sup sup ¯∇i Rmgk (0) (x)¯ < +∞, 0 ≤ i ≤ i0 + 1 for some i0 ∈ N k

Mkn

sup

sup

k

Mkn ×[0,T ]

¯ ¯Rmg

k (t)

¯ (x)¯ < +∞,

one can improve Shi’s derivative estimate and get ¯ ¯ sup sup ¯∇i Rmgk (t) (x)¯ < +∞ k

Mkn ×[0,T ]

i ≤ i0 + 1.

We will give a proof of this estimate in the appendix at the end. This estimate implies that in normal coordinates the C i0 +1 -norm of metric tensor gk (t) are bounded independent of k, t. Suppose we have injectivity radius bound igk (0) (pk ) ≥ δ > 0, then one can follow the proof of compactness theorem in [H95a] to conclude the following. There is a subsequence (Mknj , gkj (t), pkj ), t ∈ [0, T ] which converges in pointed C i0 Cheeger-Gromov topology to an complete n , g∞ (t), p∞ ), t ∈ [0, T ], with bounded curvature solution of the Ricci flow (M∞ tensor and ig∞ (0) (p∞ ) ≥ δ. Applying this compactness statement to (Rn , g(t), yi ), t ∈ [0, T ], we conclude that there is a subsequence yij (still denoted by yi ) such that (Rn , g(t), yi ) → (M∞ , g∞ (t), y∞ ),

t ∈ [0, T ]

in the C 3 -topology. Note³ that by assumption (iii) (M∞ , g∞ (0)) is isometric to ´ p n−1 2 (n − 2) . (M∞ , g∞ (t)) has nonnegative curvature round cylinder R × S operator because g(t) has nonnegative curvature operator. Since M∞ has two ends, (M∞ , h∞ (t)) has a line for each t ∈ [0, T ]. From the local version of Hamilton’s strong maximal principle ([?] §8), the line direction is preserved by the Ricci flow. By the Toponogov splitting theorem the metric g∞ (t), t ∈ [0, T ] splits and has the following form g∞ (t) = dr2 + gS n−1 (t) , where gS n−1 (t) is the solution of the Ricci flow on sphere S n−1 with initial metric 2 (n − 2) dσ. It follows from the uniqueness of Ricci flow solution on closed manifold that gS n−1 (t) = 2 (n − 2) (1 − t) dσ. 3

1.2 Standard solutions are rotationally symmetric Lemma 2 Let X be a vector field evolving by ∂ i X = ∆X i + Rki X k ∂t

(3)

∂ Vij = ∆Vij − 2Rikjl Vkl − Rik Vkj − Rjk Vik . ∂t

(4)

¢ ¡ and Vij + ∇i gjk X k = gjk ∇i X k , then V evolves by

Proof. The evolution equation of Xi + gik X k is ∂ ∂ Xi = −2Rik X k + gik X k ∂t ∂t ¡ ¢ = −2Rik X k + gik ∆X k + Rjk X j = ∆Xi − Rik Xk . We compute ∂ ∂ Vij = (∇i Xj ) = − ∂t ∂t

µ

¶ µ ¶ ∂ l ∂ Γ Xl + ∇i Xj ∂t ij ∂t

= (−∇l Rij + ∇i Rjl + ∇j Ril ) Xl + ∇i (∆Xj − Rjk Xk ) . From ∇i (∆Xj ) = ∇k ∇i ∇k Xj − Rikkl ∇l Xj − Rikjl ∇k Xl = ∇k (∇k ∇i Xj − Rikjl Xl ) − Ril ∇l Xj − Rikjl ∇k Xl = ∆Vij − Xl ∇k Rikjl − 2Rikjl Vkl − Ril Vlj , we get ∂ Vij = ∆Vij − 2Rikjl Vkl − Rik Vkj − Rjk Vik ∂t − Xl ∇l Rij + Xl ∇j Ril − Xl ∇k Rikjl . (4) follows from the second Bianchi identity 0 = ∇k Rikjl + ∇j Riklk + ∇l Rikkj = ∇k Rikjl − ∇j Ril + ∇l Rij . Let hij + Vij + Vji . It follows from (4) that ∂ hij = ∆L hij , (5) ∂t where ∆L hij + ∆hij +2Riklj hkl −Rik hkj −Rjk hki is the Lichnerowicz Laplacian. A simple calculation shows that there is a constant C > 0 such that µ ¶ ∂ 2 2 − ∆ |hij | = −2 |∇k hij | + 4Rijkl hjk hil (6) ∂t ∂ 2 2 2 2 |hij | ≤ ∆ |hij | − 2 |∇k hij | + C |hij | . (7) ∂t 4

Note X i (t) is Killing vector fields for g(t) if and only if hij (t) = 0. For any given Killing vector field X i (0) for metric g(0), (3) has a bounded solution 2 X i (t) for t ∈ [0, T ). Then |hij (t)| is a bounded function and satisfying (7) and 2 |hij | (0) = 0. Since the metric g(t) has bounded sectional curvature, we can 2 apply the maximum principle to |hij | on complete manifolds ([Sh2] Theorem 4.6) to conclude that hij (t) = 0 for all t ≥ 0. So X i (t) is Killing vector fields for g (t). The following is a very nice observation of Bennett Chow, we thank him for allowing us to use it here. From hij = 0 we have ∇j X i + ∇i X j = 0. Taking ∇j derivative and summing over j we get ∆X i + Rki X k = 0 for all t. Hence ∂ (3) gives ∂t X i = 0 and X i (t) = X i (0), i.e., the rotation group O(n) of Rn are contained in the isometry group of g(t). We conclude Lemma 3 The standard solution g(t), t ∈ [0, T ] are rotationally symmetric.

2 From Ricci flow to DeTurk-Ricci flow In this section we discuss the DeTurk-Ricci flow and the harmonic map flow.

2.1 Converting Ricci flow solutions to Deturk-Ricci flow solutions Let (M n , h(t)), t ∈ [0, T ] be a solution of the Ricci flow and let ψt : M → M, t ∈ [0, T1 ] be a solution of harmonic map flow ∂ψt = ∆h(t),h(0) ψt , ∂t

ψ0 = Id.

(8)

In local coordinates (xi ) on domain M and (y α ) on target M , the harmonic map flow (8) can be written as µ ¶ ∂ ∂ψ β (x, t) ∂ψ γ (x, t) − ∆h(t) ψ α (x, t) = hij (x, t) Γα (ψ (x, t)) (9) βγ ∂t ∂xi ∂xj where Γα βγ is the Christoffel symbols of h(0). Suppose ψ (x, t) is a solution with bounded |∇ψ|C 2 norm, then ψ (t) , t ∈ [0, T1 ] are when T1 > 0 is small. For 0 ≤ t ≤ T1 , define ¡ diffeomorphisms ¢ ˆ (t) + ψ −1 ∗ h (t), then h ˆ (t) is a solution of Ricci–DeTurck flow by (see h t [D], [H95b] or [CK] Chapter 3 for details) ∂ˆ ˆ ij + ∇ ˆ i Wj + ∇ ˆ j Wi hij = −2R ∂t

ˆ (0) = h(0), h

ˆ reˆ ij and ∇ ˆ i are the Ricci curvature and Levi-Civita connection of h where R ³ ´ ˆ is defined by spectively and the time-dependent 1-form W = W h 5

³

ˆ W (h)

´ j

³ ´ ˆ jk h ˆ pq Γ ˆ kpq − Γkpq (h(0)) . +h

In local coordinates the Ricci-DeTurk flow takes the following form ([Sh1] Lemma 2.1) ˆ ij ∂h ˆ kl ∇k ∇l h ˆ ij − h ˆ kl h(0)ip h ˆ pq Rjkql (h(0)) − h ˆ kl h(0)jp h ˆ pq Rikql (h(0)) =h ∂t · ¸ ˆ pk ∇j h ˆ ql + 2∇k h ˆ jp ∇q h ˆ il 1 ˆ kl ˆ pq ∇i h h (10) + h ˆ jp ∇l h ˆ iq − 2∇j h ˆ pk ∇l h ˆ iq − 2∇i h ˆ pk ∇l h ˆ jq . 2 −2∇k h where ∇ is the Levi-Civita connection of h(0). This is a parabolic system. ˆ of the Ricci-DeTurck Recall the following derivative estimate for solution h(t) flow from [Sh1] Lemmas 4.1 and 4.2, one can check easily that the γ dependence of constant C below is not necessary. ³ ´ ˆ (t) , t ∈ [0, T ), be a solution of the Ricci-DeTurck Proposition 4 Let M n , h flow. For a given m ∈ N, suppose sup δ x∈Bg˜ (x0 ,γ+ m+1 ),i≤k

Rmh(0) |h(0) ≤ Bk . |∇ih(0) ˆ ˆ ˆ

There exists a constant C = C (n, m, δ, T, Bk ) depending only on n, m, δ, T, Bk such that if µ ¶ µ ¶ 1 1 ˆ ˆ ˆ 1− h(0) ≤ h (t) ≤ 1 + h(0), 0≤t≤T 256000n10 256000n10 then

¯ ¯ ¯ ˜ mˆ¯ ¯∇ h ¯ ˆ

h(0)

≤C

µ in Bh(0) x0 , γ + ˆ

δ m+1

¶ × [0, T )

(11)

2.2 Solutions of harmonic map flow In this section we study the existence of harmonic flow (8) and its asymptotic behavior at the space infinity when h(t) = g(t) is a standard solutions. Here we use of the rotationally symmetric property and asymptotic property at infinity of g(t). Let θ = (θ1 , · · · , θn−1 ) be local coordinates on the round (n − 1)-sphere of radius 1, and let dσ the volume form on the sphere. Since g(t) is rotationally symmetric and n ≥ 3, we can write g(t) = dr2 + f (r, t)2 dσ

g0 = dr2 + f0 (r)2 dσ

where r be the radial coordinate on Rn depending on time t, i.e. clear that f (r, 0) = f0 (r). We want to solve (8) by maps of form φ(t) : Rn → Rn

(r, θ) → (ρ(r, t), θ). 6

(12) ∂r ∂t

6= 0. It is (13)

2.2.1 The harmonic map flow equation. Using (12) and (13) it is easy to calculate the energy functional

E(φ(t)) +

1 2

Z

|∇φ(t)|g(t),g0 dVg(t) # Z "µ ¶2 1 ∂ρ 2 −2 = + (n − 1)f0 (ρ)f (r, t) dVg(t) . 2 Rn ∂r Rn

If we have a compact-supported variation δρ = w, then ¸ Z · 1 ∂ρ ∂w ∂f0 −2 δE(φ(t))(w) = 2 + 2(n − 1)f0 (ρ) f (r, t)w dVg (t) 2 Rn ∂r ∂r ∂ρ ¸ Z Z +∞ · ∂f0 n−3 ∂ρ ∂w n−1 + (n − 1)f (ρ) f (r, t)w dr · dVdσ = f (r, t) 0 ∂r ∂r ∂ρ S n−1 0 µ ¶ ¸ Z · ∂ ∂ρ n−1 ∂f0 −2 = −f 1−n f f (r, t) wdVg (t). + (n − 1)f0 (ρ) ∂r ∂r ∂ρ Rn Hence for rotationally symmetric maps the harmonic map flow equation (8) has the following form µ ¶ 1 ∂ ∂ρ ∂f0 dρ n−1 = n−1 f (r, t) − (n − 1)f −2 (r, t)f0 (ρ) dt f (r, t) ∂r ∂r ∂ρ 2 ∂ρ ∂ ρ n − 1 ∂f ∂ρ n−1 ∂f0 ∂ρ dr = 2 + − 2 f0 (ρ) − . ∂t ∂r f (r, t) ∂r ∂r f (r, t) ∂ρ ∂r dt

(14)

Note that coordinate r on domain Rn depends on time t, this leads us to take ∂ρ ∂ρ dr full derivative of ρ(r, t) with respect to t in above formula, dρ dt = ∂t + ∂r dt . 2.2.2 An equation equivalent to the harmonic map flow. We want to make change of variables and turn (14) to an equation more easily to solve. Let ˜ 2 ˜ 2 f (r, t) = ref (r ,t) f0 (ρ) = ρef0 (ρ ) . Note that f˜(w, 0) = f˜0 (w). We claim that f˜(w, t) and f˜0 (w) are both smooth functions of w ≥ 0 and t. Let f (r, t) = rfˆ(r2 , t), to see this claim we only need to show that fˆ(w, t) is a smooth function of w, t and fˆ(0, t) 6= 0. Write metric g(t) = gij dxi dxj and let x1 = rˆ cos θ1 , x2 = rˆ sin θ1 cos θ2 , · · · , xn = rˆ sin θ1 · · · sin θn−1 , we compute f (r, t) by choosing rˆ = x1 and θ1 = · · · = θn−1 = 0, i.e., x2 = · · · = xn = 0. g(t) = g11 (ˆ r, 0, · · · , 0, t)dˆ r2 + g22 (ˆ r, 0, · · · , 0, t)ˆ r2 dσ. So

Z r=

rˆ p

Z g11 (ˆ s, 0, · · · , 0, t)dˆ s = rˆ

0

0

7

1

p g11 (ˆ rs, 0, · · · , 0, t)ds

and p p rˆ g22 (ˆ r, 0, · · · , 0, t) g22 (ˆ r, 0, · · · , 0, t) 2 ˆ f (r , t) = R rˆ p = R1p . g11 (ˆ rs, 0, · · · , 0, t)ds g11 (ˆ s, 0, · · · , 0, t)dˆ s 0 0 For any k > 0 suppose F (ˆ r, t) is an even function in rˆ and is differentiable up √ to order 2k, it is clear that function wk F ( w, t), w ≥ 0 is differentiable up to order k in w ≥ 0 and its (left-)derivatives up to order k − 1 at w = 0 are 0. Since by rotational symmetry g22 (ˆ r, 0, · · · , 0, t) is even in rˆ and by Taylor series we can write for any k > 0 g22 (ˆ r, 0, · · · , 0, t) = c0 (t) + c1 (t)ˆ r2 + · · · + ck−1 (t)ˆ r2k−2 + rˆ2k F (ˆ r, t) for some smooth even function F (ˆ r, t). Let gˆ22 (ˆ r2 , 0, · · · , 0, t) = g22 (ˆ r, 0, · · · , 0, t). 2k Since rˆ F (ˆ r, t) haspk-derivative, gˆ22 (w, 0, · · · , 0, t) is a smooth function of w ≥ 0, t. Similarly √ g11 (ˆ rs, 0, · · · , 0, t) is a smooth function of rˆ2 , t . Hence R1p g (ˆ r ,0,··· ,0,t) rs, 0, · · · , 0, t)ds is a and r2 = rˆ2 0 g11 (ˆ we conclude that R 1 √ 22 0

g11 (ˆ r s,0,··· ,0,t)ds

smooth function of rˆ2 , t. Furthermore r2 is a smooth invertible function of rˆ2 . We now conclude that fˆ(w, t) is a smooth function of w ≥ 0, t. Since R1 ∂ p g11 (ˆ rs, 0, · · · , 0, t)ds −1 dr , = 0R 1∂tp r dt g11 (ˆ rs, 0, · · · , 0, t)ds 0

a simple consequence of above arguments is that r˜(w, t) is a smooth function R 2 1 r of w ≥ 0, t for r˜(r2 , t) + r−1 dr ˜(w, t)dw. Clearly dr dt . Let B(w, t) + 2 0 r dt = ∂ 2 B(r , t) and B(w, t) is a smooth function of w ≥ 0, t. ∂r We will solve (14) for solutions of form ˜ ρ(r, t) = reρ(r,t) .

Then some straight forward calculation shows that (14) becomes µ ¶2 ∂ f˜ ¡ 2 ¢ ∂ ρ˜ ∂ ρ˜ 1 dr ∂ ρ˜ ∂ 2 ρ˜ n + 1 ∂ ρ˜ + = + + (n − 1) r ,t + r dt ∂t ∂r2 r ∂r ∂r ∂r ∂r h i n−1 ∂ f˜ ¡ 2 ¢ ˜ 2 ˜ 2 + 1 − e2f0 (ρ )−2f (r ,t) + 2 (n − 1) r ,t 2 r ∂w ˜ 1 dr dr ∂ ρ˜ ˜ 2 ˜ f˜(r 2 ,t) ∂ f0 (ρ2 ) − − . − 2 (n − 1) e2f0 (ρ )+2ρ−2 ∂w r dt dt ∂r Note that from the definition of f˜ (0, t) = 0 we can write f˜ (w, t) = wf˜∗ (w, t) and f˜0 (w) = wf˜0∗ (w) where both f˜∗ (w, t) and f˜0∗ (w) are smooth functions. So i n−1h i n−1h 2f˜0 (ρ2 )−2f˜(r 2 ,t) 2r 2 [e2ρ˜f˜0∗ (ρ2 )−f˜∗ (r 2 ,t)] 1 − e = 1 − e r2 r2

8

which is a smooth function of ρ˜, r2 , t. Let G1 (r2 , ρ˜, t) +

i n−1h ∂ f˜ ¡ 2 ¢ 2f˜0 (ρ2 )−2f˜(r 2 ,t) 1 − e + 2 (n − 1) r ,t 2 r ∂w ˜ 2 dr ˜ 2 ˜ f˜(r 2 ,t) ∂ f0 (ρ2 ) − . − 2 (n − 1) e2f0 (ρ )+2ρ−2 ∂w r dt

(15)

G(w, ρ˜, t) is a smooth function. ∂ 2 Recall dr dt = ∂r B(r , t). (14) can be written as " # µ ¶2 ∂ f˜ ∂B ¡ 2 ¢ ∂ ρ˜ ∂ ρ˜ ∂ ρ˜ ∂ 2 ρ˜ n + 1 ∂ ρ˜ = 2 + + (n − 1) − r ,t + + G1 (r2 , ρ˜, t). ∂t ∂r r ∂r ∂r ∂r ∂r ∂r Now we think ρ˜Pas a rotational symmetric function defined on Rn+2 and let n+2 G(x, ρ˜, t) + G1 ( i=1 (xi )2 , ρ˜, t). Then the equation above can be written as ∂ ρ˜ 2 = ∆˜ ρ + ∇[(n − 1)f˜ − B] · ∇˜ ρ + |∇˜ ρ| + G(x, ρ˜, t) ∂t

(16)

where ∇ and ∆ are the Levi-Civita connection and Laplacian defined by EuPn+2 clidean metric on Rn+2 respectively. Note that f˜ = f˜( i=1 (xi )2 , t), B = Pn+2 B( i=1 (xi )2 , t) are smooth functions on Rn+2 . From the nonnegativity of the curvature operator of g(t) and Lemma 1, we have the following properties of f˜, f˜0 for large r, ρ. ∂ f˜ 2 1 1 (r , t) ∼ 2 (1 − 2t)r ∂w r 1 1 ∂ f˜0 ˜ (ρ) ∼ 2 ef0 (ρ) ∼ ρ ∂w ρ 1 ∂B 2 1 −1 ∂r ∼ 2 (r , t) ∼ . r ∂t r ∂r r ˜

ef (r

2

,t)

∼

(17)

2.2.3 The short time existence. We will show that equation (16) with initial condition ρ˜(x, 0) = 0 has a solution on time interval [0, T ]. Let x, y be two points in Rn+1 and H(x, y, t) =

1 (4πt)(n+2)/2

e−

|x−y|2 4t

be the heat kernel of Rn+1 . We solve (16) by successive approximation [LT]. Define h i 2 F (x, ρ˜, ∇˜ ρ, t) + ∇ (n − 1)f˜ − B · ∇˜ ρ + |∇˜ ρ| + G(x, ρ˜, t). Let ρ˜0 (x, t) = 0. For i ≥ 1 we define ρ˜i inductively by Z tZ ρ˜i = H(x, y, t − s)F (y, ρ˜i−1 , ∇˜ ρi−1 , s)dyds 0

Rn+2

9

(18)

which solves ∂ ρ˜i = ∆˜ ρi + F (x, ρ˜i−1 , ∇˜ ρi−1 , t) ρ˜i (x, 0) = 0. (19) ∂t To show the existence of ρ˜i by induction, it suffices to prove the following statement: for any i ≥ 1 if |˜ ρi−1 |, |∇˜ ρi−1 | are bounded, then ρ˜i exists and |˜ ρi |, |∇˜ ρi | are bounded. Assume |˜ ρi−1 | ≤ C1 , |∇˜ ρi−1 | ≤ C2 are bounded on Rn+2 × [0, T ], then it follows from (17) that G(x, ρ˜i−1 , t) is bounded on Rn+2 × [0, T ] |G(x, ρ˜i−1 , t)| ≤ C3 where C3 depends on C1 , C2 . F (x, ρ˜i−1 , ∇˜ ρi−1 , t) is bounded h i |F (x, ρ˜i−1 , ∇˜ ρi−1 , t)| ≤ (n − 1) sup |∇f˜| + sup |∇B| C2 + C22 + C3 + C4 . Hence ρ˜i exists. The bounds of |˜ ρi | and |∇˜ ρi | follow from the following estimates Z tZ |˜ ρi | ≤ H(x, y, t − s)C4 dyds ≤ C4 t, 0

and

Rn+2

Z tZ |∇˜ ρi | = | [∇x H(x, y, t − s)]F (y, ρ˜i−1 , ∇˜ ρi−1 , s)dyds| 0 Rn+2 Z tZ ≤ |∇x H(x, y, t − s)|C4 dyds 0 Rn+2 Z tZ |x−y|2 1 − 4(t−s) |x − y| e C4 dyds = (n+2)/2 2(t − s) 0 Rn+2 (4π(t − s)) Z 1 2(n + 2)C4 √ (n + 2)C4 t √ √ √ t. ds = ≤ π π t−s 0 πC 2

2 1 If we define T1 + min{ C C4 , 4(n+2)2 C 2 }, then for 0 ≤ t ≤ T1 we have for all i 4

|˜ ρi | ≤ C 1

|∇˜ ρi | ≤ C 2 .

(20)

We prove the convergence of ρ˜i to a solution of (16) by showing that it is a Cauchy sequence in C 1 -norm. We assume 0 ≤ t ≤ T1 . Note that ρ˜i − ρ˜i−1 satisfies ∂(˜ ρi − ρ˜i−1 ) = ∆(˜ ρi − ρ˜i−1 ) + F (x, ρ˜i−1 , ∇˜ ρi−1 , t) − F (x, ρ˜i−2 , ∇˜ ρi−2 , t) ∂t (˜ ρi − ρ˜i−1 )(x, 0) = 0. (21) where F (x, ρ˜i−1 , ∇˜ ρi−1 , t) − F (x, ρ˜i−2 , ∇˜ ρi−2 , t) ˜ =[(n − 1)∇f − ∇B + ∇(˜ ρi−1 + ρ˜i−2 )] · ∇(˜ ρi−1 − ρ˜i−2 ) + G(x, ρ˜i−1 , , t) − G(x, ρ˜i−2 , t) 10

By lengthy but straight-forward calculations one can verify the Lipschitz property of G(x, ρ˜, t) |G(x, ρ˜i−1 , t) − G(x, ρ˜i−2 , t)| ≤ C5 · |˜ ρi−1 − ρ˜i−2 | where C5 depends on C1 . This and (20) implies |F (x, ρ˜i−1 , ∇˜ ρi−1 , t) − F (x, ρ˜i−2 , ∇˜ ρi−2 , t)| ≤C5 · |˜ ρi−1 − ρ˜i−2 | + C6 · |∇˜ ρi−1 − ∇˜ ρi−2 |

(22)

where C6 + [(n − 1) sup |∇f˜| + sup |∇B| + 2C2 ). Let Ai (t) = Bi (t) =

sup 0≤s≤t,x∈Rn+2

sup 0≤s≤t,x∈Rn+2

|˜ ρi − ρ˜i−1 |(x, s) |∇(˜ ρi − ρ˜i−1 )|(x, s).

From (21) and (22) we can estimate |˜ ρi − ρ˜i−1 | and |∇(˜ ρi − ρ˜i−1 )| in the same way as we estimate |˜ ρi | and |∇˜ ρi | above, we conclude Ai (t) ≤ [C5 Ai−1 (t) + C6 Bi−1 (t)] · t Bi (t) ≤

2(n + 2)[C5 Ai−1 (t) + C6 Bi−1 (t)] √ √ · t. π

Let C7 + max{C5 , C6 }, then we get √ ¶ µ 2(n + 2)C7 t √ · (Ai−1 (t) + Bi−1 (t)) . Ai (t) + Bi (t) ≤ C7 t + π If we choose T2 ∈ (0, T1 ] so that C7 T2 + Ai (t) + Bi (t) ≤

2(n+2)C7 √ π

√

T2

≤ 12 , then

1 (Ai−1 (t) + Bi−1 (t)) , 2

so ρ˜i is a Cauchy sequence in C 1 (Rn+2 ). Let limi→+∞ ρ˜i = ρ˜∞ . Then ∇˜ ρi → ∇˜ ρ∞ and F (x, ρ˜i−1 , ∇˜ ρi−1 , t) → F (x, ρ˜∞ , ∇˜ ρ∞ , t) uniformly. Hence we get from (18) Z tZ ρ˜∞ = H(x, y, t − s)F (y, ρ˜∞ , ∇˜ ρ∞ , s)dyds (23) 0

Rn+2

The argument below is similar to the argument in [LT] p.21. Since ρ˜i is a smooth solution of (19), ρ˜i (x, 0) = 0 and both ρ˜i and F (x, ρ˜i1 , ∇˜ ρi−1 , t) are uniformly bounded on Rn+2 × [0, T2 ], by Theorem 1.11 [LSU] p.211 and Theorem 12.1 [LSU] p.223, for any compact K ⊂ Rn+2 and any 0 < t∗ < T2 , there is C8 and α ∈ (0, 1) independent of i such that ´ ³ |∇˜ ρi (x, t) − ∇˜ ρi (y, s)| ≤ C8 · |x − y|α + |t − s|α/2 11

where x, y ∈ K and 0 ≤ t < s ≤ t∗ . Let i → ∞ we get ³ ´ |∇˜ ρ∞ (x, t) − ∇˜ ρ∞ (y, s)| ≤ C8 · |x − y|α + |t − s|α/2 .

(24)

Hence ∇˜ ρ∞ is Hölder continuous. From (23) we conclude that ρ˜∞ is a solution of (16) on Rn+2 × [0, T2 ] with ρ˜∞ (x, 0) = 0. 2.2.4 The asymptotic behavior of the solutions. In the rest of this subsection we study the asymptotic behavior of solution ρ˜(x, t) as x → ∞. First we prove inductively that there is a constant λ and T3 ∈ (0, T2 ] such that for x ∈ Rn+2 , t ∈ [0, T3 ] |˜ ρi (x, t)| ≤

λ (1 + |x|)2

|∇˜ ρi (x, t)| ≤

λ (1 + |x|)2

(25)

Clearly these estimates holds for i = 0. It follows from (20) and (17) that there is a constant C9 independent of i such that C9 (1 + |x|)2 h i (n − 1)|∇f˜| + |∇B| (x, t) ≤

|G(x, ρ˜i , t)| ≤

C9 . 1 + |x|

Now we assume the estimates hold for i, then for 0 ≤ t ≤ T2 · ¸ Z tZ C9 λ λ2 C9 |˜ ρi (x, t)| ≤ H(x, y, t − s) + + dyds (1 + |y|)2 (1 + |y|)2 (1 + |y|)2 0 Rn+2 · ¸ Z tZ 2 |x−y|2 1 − 4(t−s) C9 λ + λ + C9 = e dyds (n+2)/2 (1 + |y|)2 0 Rn+2 (4π(t − s)) C(n)t ≤ (C9 λ + λ2 + C9 ) · . (1 + |x|)2 Also we have

·

¸ C 9 λ + λ2 + C 9 dyds (1 + |y|)2 0 Rn+2 · ¸ Z tZ 2 |x−y|2 1 |x − y| − 4(t−s) C9 λ + λ + C9 dyds = e (n+2)/2 (1 + |y|)2 0 Rn+2 2(t − s) (4π(t − s)) √ C(n) t . ≤ (C9 λ + λ2 + C9 ) · (1 + |x|)2 Z tZ

|∇˜ ρi (x, t)| ≤

|∇x H(x, y, t − s)|

If we choose T3 ∈ (0, T2 ] such that (C9 λ + λ2 + C9 ) · C(n)T3 ≤ λ

and

p (C9 λ + λ2 + C9 ) · C(n) T3 ≤ λ,

then (25) hold for all i. From the definition of ρ˜∞ we conclude |˜ ρ∞ (x, t)| ≤

λ (1 + |x|)2

|∇˜ ρ∞ (x, t)| ≤

12

λ (1 + |x|)2

(26)

Now we consider ρ˜∞ as a solution of the following linear equation ∂υ = ∆υ + ∇[(n − 1)f˜ − B + ρ˜∞ ] · ∇υ + G(x, ρ˜∞ , t) ∂t υ(x, 0) = 0. From (24) and (17) we know that ∇[(n − 1)f˜− B + ρ˜∞ ] has C α,α/2 -Holder-norm bound. By some lengthy calculation we get |G(x, ρ˜∞ , t)|C α,α/2 ≤

C10 . (1 + |x|)2

By standard interior Schauder estimate for parabolic equation we conclude |˜ ρ∞ |C 2+α,1+α/2 ≤

C11 . (1 + |x|)2

Using this estimate one can further show by calculation |∇2 [(n − 1)f˜ − B + ρ˜∞ ]|C α,α/2 ≤ C12 C13 |∇G(x, ρ˜∞ , t)|C α,α/2 ≤ . (1 + |x|)2 By high order interior Schauder estimates for parabolic equation we conclude |∇˜ ρ∞ |C 2+α,1+α/2 ≤

C13 . (1 + |x|)2

We have proved the following Proposition 5 For standard solution (Rn , g(t)), there is a rotationally sym˜ metric solution φ(t)(x) = xeρ(x,t) to the harmonic map flow ∂φ(t) = ∆g(t),g(0) ∂t and |∇i ρ˜|(x, t) ≤

C14 (1+|x|)2

φ(0)(x) = x,

for 0 ≤ i ≤ 3.

3 The uniqueness of standard solutions Let φ(t) be the solution of harmonic map flow from §2.2, in this section we use the¡ asymptotic behaviors of g(t) and of φ(t) to prove the uniqueness of ¢∗ gˆ (t) = φ−1 g (t). Then the uniqueness of standard solutions follows easily. t

3.1 The uniqueness for the solutions of Deturk-Ricci flow We prove the following general uniqueness result for Deturk-Ricci flow on open manifolds. 13

Proposition 6 Let G (t) and H (t) , 0 ≤ t ≤ T be two bounded solution of the Deturk-Ricci flow on complete and noncompact manifold M n with initial metric G (0) = H (0) = g˜ and for some δ ∈ (0, 1) (1 − δ) g˜ ≤ G (t) ≤ (1 + δ) g˜ (1 − δ) g˜ ≤ H (t) ≤ (1 + δ) g˜ kG(t)kC 2 (Ωb ),˜g < +∞ kH(t)kC 2 (Ωb ),˜g < +∞. Suppose G (t) and H (t) has the same sequenctial asymptotical behavior at ∞ in the sense that there is a sequence of exhausting submanifolds of Ωk ⊂ M with Ωk ⊂ Ωk+1 and ∪Ωk = M . For any ² > 0, there is a k0 such that |G (t) − H (t)|C 1 (∂Ωk ),˜g ≤ ², 0 Then G (t) = H (t), i.e., the Deturk-Ricci flow has uniqueness in above allowable family of solutions. Proof. Using the orthonormal frame of g˜ and the Ricci-Deturk flow (10) ∂ for G and H we compute ∂t (Gij (t) − Hij (t)) and then estimate ¿ À ∂ ∂ 2 |G (t) − H (t)|g˜ = 2 (G (t) − H (t)) , G (t) − H (t) ∂t ∂t g ˜ ˜ α∇ ˜ β (Gij − Hij ) (Gij (t) − Hij (t)) ≤ 2Gαβ ∇ 2

2

+ C14 |G (t) − H (t)|g˜ + C14 |G (t) − H (t)|g˜ ³ ´ + C14 |G (t) − H (t)|g˜ + C14 |∇G (t) − ∇H (t)|g˜ |G (t) − H (t)|g˜ ˜ α (G(t) − H(t)) · ∇ ˜ β (G (t) − H (t)) ˜ α∇ ˜ β |G (t) − H (t)|2 − 4Gαβ ∇ ≤ 2Gαβ ∇ g ˜ 2

+ C14 |G(t) − H(t)|g˜ + C14 |∇ (G(t) − H(t))|g˜ |G(t) − H(t)|g˜ ˜ α∇ ˜ β |G (t) − H (t)|2 − 4 (1 − δ) |∇ (G(t) − H(t))|2 ≤ 2Gαβ ∇ g ˜ g ˜ 2

2

+ C14 |G(t) − H(t)|g˜ + (1 − δ) |∇ (G(t) − H(t))|g˜ +

2 C14 2 |G(t) − H(t)|g˜ . 4 (1 − δ)

We have proved ∂ 2 ˜ α∇ ˜ β |G (t) − H (t)|2 + C15 |G (t) − H (t)|2 |G (t) − H (t)|g˜ ≤ 2Gαβ ∇ g ˜ g ˜ ∂t

(27)

pointwise on Ωa with C15 depends only on n, δ, kG(t)kC 2 (Ωb ),˜g and kH(t)kC 2 (Ωb ),˜g . 2

If G (t) 6= H (t), then there is a point x0 such that |G (t0 ) − H (t0 )|g˜ (x0 ) > ²0 for some t0 > 0 and some ²0 > 0. We choose a k0 such that x0 ∈ Ωk0 and 2

sup |g (t) − h (t)|g˜ (x) ≤ ²

x∈∂Ωb

14

(28)

where ² > 0 is a constant to be chosen later. Recall we have initial condition 2 2 |G (0) − H (0)|g˜ = 0. Applying maximum principle to |G (t) − H (t)|g˜ in (27) on domain Ωk0 , we get 2

e−C15 t |G (t) − H (t)|g˜ (x) ≤ ². for all x ∈ Ωb . This is a contradiction if we choose ² ≤ ²0 e−C15 T . The proposition is proved. Let g(t) and g∗ (t) be two standard solutions with same initial condition. By Proposition 4 there are φ(t) for g(t) and φ∗ (t) for g∗ (t) which are two solutions ˆ ˆ of the harmonic map flow, 0 ≤ t ≤ T3 . Let G(t) + (φ−1 (t)) ∗ g(t) and H(t) + ˆ ˆ (t)) ∗ g (t). Then G(t) and H(t) are two solutions of Ricci-Deturk flow (φ−1 ∗ ∗ ˆ ˆ ˆ ˆ with G(0) = H(0). Choose T4 ∈ (0, T3 ] such that G(t) and H(t) is δ-close ˆ to G(0) as required in Proposition 5. It follows from Lemma 1 and the decay ˆ ˆ estimate in Proposition 4 that G(t) and H(t) are bounded solutions and they have same sequential asymptotic behavior at infinity. We can apply Proposition ˆ = H(t) ˆ 5 to conclude G(t) on 0 ≤ t ≤ T4 . We have proved ˆ and H(t) ˆ Lemma 7 The Ricci-Deturk solutions G(t) constructed from standard ˆ ˆ solutions g(t) and g∗ (t) with g(0) = g∗ (0) are the same, G(t) = H(t) for t ∈ [0, T4 ]. Remark 8 Another way to prove the uniqueness of Ricci-DeTurk flow is to use maximum principle on open manifolds, then we do not need using the asymptotic behavior. ˆ be a metric on complete manifold with injectivity raProposition 9 Let h(0) dius lower bound δ1 > 0 and curvature bound ˆ |∇i h(0)Rm |h(0) ≤ C for i = 0, 1, 2. ˆ ˆ h(0) ˆ 1 (t) and h ˆ 2 (t), 0 ≤ t ≤ T are two solutions of the Ricci-Deturk flow with Let h ˆ ˆ ˆ h1 (0) = h2 (0) = h(0). Suppose ¶ µ ¶ µ 1 1 ˆ ˆ (t) ≤ 1 + ˆ h(0) ≤ h h(0), 0≤t≤T 1− 256000n10 256000n10 Then h1 (t) = h2 (t), 0 ≤ t ≤ T . ¯ ¯2 ¯ˆ ¯ ˆ Proof. Let u(x, t) + ¯h (t) − h (t) ¯ . By (11) and the computation of 1 2 ˆ0 h

(27) we obtain that u is bounded and

∂ u ≤ 3∆h(0) u + C15 u ˆ ∂t

u(x, 0) = 0

If one check the proof of [Sh2] Theorem 4.6, it is clear the positive sectional curvature requirement in Assumption (A) can be replaced by lower bounded of sectional curvature. The requirement is used for constructing cut-off function on [Sh2] p310 which can be replaced by the injectivity radius assumption. We can apply [Sh2] Theorem 4.6 to conclude u(x, t) = 0 15

3.2 The uniqueness of standard solutions ˆ (t) of Deturk-Ricci Recall the following procedure of converting the solution h flow in §2.1 back to solution h (t) of the Ricci flow (see [CK], p.89-90). Given two metrics g and g∗ , we define 1-form ¡ ¢ W (g, g∗ )j + gjk g pq Γkpq (g) − Γkpq (g∗ ) . Define a family diffeomorphisms ϕt by solving the following ode ³ ´ d i ˆ ij (t)W h(t), ˆ (ϕt ) = h h(0) , dt j ϕ0 = Id. ˆ Then h(t) = ϕ∗ (t)h(t). For the rest of this subsection, we adopt the notation at the end of §3.1. Define diffeomorphisms ϕ1 (t) and ϕ2 (t) by ³ ´ d i ˆ ij (t)W G(t), ˆ ˆ ϕ1 (0) = Id (ϕ1 (t)) = G G(0) dt j ³ ´ d i ˆ ij (t)W H(t), ˆ ˆ ϕ2 (0) = Id. (ϕ2 (t)) = H H(0) dt j ³ ´ ³ ´ ˆ ˆ ˆ G(0) ˆ ˆ H(0) ˆ Since G(t) = H(t) by Lemma 6, W G, = W H, and hence ∗ ∗ ˆ ˆ ϕ1 (t) = ϕ2 (t). We have g(t) = ϕ1 (t)G(t) = ϕ2 (t)H(t) = g∗ (t) for 0 ≤ t ≤ T4 . To show g(t) = g∗ (t) for 0 ≤ t ≤ T , we repeat above argument using new initial time T4 . This proves the uniqueness of the standard solutions. Theorem 10 Let g(t) and g∗ (t), 0 ≤ t ≤ T be two standard solutions of the Ricci flow as defined in §1. Suppose g(0) = g∗ (0), then g(t) = g∗ (t) for 0 ≤ t ≤ T.

4 Appendix: Shi’s local derivative estimate when initial metrics have higher regularity If initial metric has better curvature bound, we can improve Shi’s local derivative estimates as following. Theorem 11 For any α, K, K1 , r, l ≥ 0, n and m ∈ N, there exists C = C (α, K, Kl , r, l, n, m) depending only on α, K, Kl , r, l, n and m such that if Mn is a manifold, p ∈ M, and g (t) , t ∈ [0, τ ] , 0 < τ ≤ α/K, is a solution to the ¯g(0) (p, r) as a compact Ricci flow on an open neighborhood U of p containing B subset, and if |Rm (x, t)| ≤ K for all x ∈ U and t ∈ [0, τ ], ¯ β ¯ ¯∇ Rm (x, 0)¯ ≤ Kl for all x ∈ U and β ≤ l 16

then |∇m Rm (y, t)| ≤

C tmax{m−l,0}/2

for all y ∈ Bg(0) (p, r/2) and t ∈ (0, τ ]. In particular if m ≤ l we have |∇m Rm (y, t)| ≤ C. Proof. Below the constant C may change from line to line and depends on some or all α, K, Kl , r, n, l and m. If l = 0, this is Shi’s local estimates for higher derivatives. Consider ³ ´ ¯ ¯2 2 Fm = C + tmax{m−l,0} |∇m Rm| tmax{m−l+1,0} ¯∇m+1 Rm¯ , where C is to be chosen. The main calculation is given by Lemma 12 ¶ µ c C ∂ 2 − ∆ Fm ≤ − sign{max{m−l+1,0}} (Fm ) + sign max{m−l+1,0} . ∂t t t We can easily obtain the ¡ theorem ¢from the lemma. Let η be a cutoff function with η = 1 on Bg(0) p, r/2m+1 and support in Bg(0) (p, r/2m ) . When sign {max {m − l + 1, 0}} ≤ 0, then we have ¶ µ ∂ 2 − ∆ Fm ≤ −c (Fm ) + C. ∂t We compute µ ¶ ³ ´ ∂ 2 − ∆ (ηFm ) ≤ η −c (Fm ) + C − ∆η · F − 2∇η · ∇F. ∂t Let (x0 , t0 ) be the point where ηFm attains its maximum in Bg(0) (p, r/2m ) . The maximum is finite by the assumption of the theorem. Then if t0 = 0, the estimate follows. If t0 > 0, then a simple maximum principle argument shows that ηFm is bounded. When sign {max {m − l + 1, 0}} > 0, again we use a maximum principle argument. We compute the evolution inequality for ηFm and conclude that ηFm is bounded. The theorem then follows from induction on m. Proof of the lemma. Given l, we argue by induction, assume that for j = 1, . . . , m there exist¡constants ¢ Cj depending only on α, K, Kl , r, n, l and m. such that for x ∈ Bg(0) p, r/2j and t ∈ [0, τ ], ¯ ¯ tmax{j−l,0}/2 ¯∇j Rm (x, t)¯ ≤ Cj . Recall that k ¯ ` ¯¯ ¯¯ ¯ ¯2 ¯ ¯2 ¯ ¯2 X ∂ ¯¯ k ¯∇ Rm¯ ¯∇k−` Rm¯ ¯∇k Rm¯ . ∇ Rm¯ ≤ ∆ ¯∇k Rm¯ − 2 ¯∇k+1 Rm¯ + ∂t `=0

(29) 17

Hence ³ ¯2 ´ ¯ ¯2 ´ ∂ ³ max{k−l,0} ¯¯ k t ∇ Rm¯ ≤ ∆ tmax{k−l,0} ¯∇k Rm¯ ∂t k X ¯ i ¯¯ ¯¯ ¯ ¯ ¯2 ¯∇ Rm¯ ¯∇k−i Rm¯ ¯∇k Rm¯ − 2tmax{k−l,0} ¯∇k+1 Rm¯ + tmax{k−l,0} i=0

+ max {k − l, 0} t

max{k−l,0}−1

¯ k ¯ ¯∇ Rm¯2 .

¯ ¯ In particular, using our induction hypothesis that tmax{j−l,0}/2 ¯∇j Rm (x, t)¯ ≤ Cj on Bg(0) (p, r/2m )×[0, τ ] for j = 1, . . . , m,, we have with ml + max {m − l + 1, 0} ³ ¯2 ´ ¯ ¯2 ´ ¯ ¯2 ∂ ³ ml ¯¯ m+1 t ∇ Rm¯ ≤ ∆ tml ¯∇m+1 Rm¯ − 2tml ¯∇m+2 Rm¯ ∂t m+1 X¯ ¯ ¯¯ ¯¯ ¯ ¯ ¯∇i Rm¯ ¯∇m−i Rm¯ ¯∇m+1 Rm¯ + ml tml −1 ¯∇m+1 Rm¯2 + t ml i=0

¯ ¯2 ¯ ¯2 ´ ≤ ∆ tml ¯∇m+1 Rm¯ − 2tml ¯∇m+2 Rm¯ ¯ ¯ ¯ ¯2 + Ctml /2 ¯∇m+1 Rm¯ + (2t |Rm| + ml ) tml −1 ¯∇m+1 Rm¯ ³ ¯ ¯2 ´ ¯ ¯2 ≤ ∆ tml ¯∇m+1 Rm¯ − 2tml ¯∇m+2 Rm¯ ³

¯ ¯2 + Ctml −1 ¯∇m+1 Rm¯ +

C tsign{ml }

.

Let m ˆ l + max {m − l, 0}. From (29) and the induction hypothesis, we have ´ ³ ´ ∂ ³m 2 2 ˆl t ˆ l |∇m Rm| ≤ ∆ tm |∇m Rm| ∂t ¯ ¯2 2 ˆ l ¯ m+1 ˆ l −1 − 2tm ∇ Rm¯ + m ˆ l tm |∇m Rm| + C.

18

2

Hence if C is chosen so that 4tmax{m−l,0} |∇m Rm| ≤ C, then µ ¶ h³ ´ ¯ ¯2 i ∂ 2 ˆl −∆ C + tm |∇m Rm| tml ¯∇m+1 Rm¯ ∂t ¶ ³ ´µ ¯ ¯2 ¯ ¯2 C 2 ˆl ≤ C + tm |∇m Rm| −2tml ¯∇m+2 Rm¯ + Ctml −1 ¯∇m+1 Rm¯ + sign{m } l t ³ ´ ¯ ¯ ¯ ¯ 2 2 2 ˆ l ¯ m+1 ˆ l −1 ˆ l tm + −2tm ∇ Rm¯ + m |∇m Rm| + C tml ¯∇m+1 Rm¯ ¯ ¯2 2 ˆ l +ml − 2tm ∇ |∇m Rm| · ∇ ¯∇m+1 Rm¯ ¯2 2¯ ˆ l +ml ≤ −10tm |∇m Rm| ¯∇m+2 Rm¯ ¯ ¯2 ¯ ¯ ¯ ¯4 ˆ l +ml − 8tm |∇m Rm| ¯∇m+1 Rm¯ ¯∇m+2 Rm¯ − 2tmax{m−l,0}+ml ¯∇m+1 Rm¯ ¶ ³ ´µ ¯ ¯2 C 2 ˆl ˆ l ¯ m+1 + C + tm |∇m Rm| Ctm ∇ Rm¯ + sign{m } l t ³ ´ ¯ ¯2 2 m ˆ l −1 m ml ¯ m+1 + max {m − l, 0} t |∇ Rm| + C t ∇ Rm¯ ³

¯ ¯2 ´ 2 ¯ ¯2 C ml ¯ m+1 m ˆ l ¯ m+1 ¯ ¯ + t ∇ Rm + C (1 + τ ) t ∇ Rm 5tsign{ml } tsign{ml } h³ ´ ¯ ¯2 i 2 C c 2 ˆl |∇m Rm| tml ¯∇m+1 Rm¯ + sign{m } . ≤ − sign{m } C + tm l l t t ≤−

2

The lemma follows. Peng Lu, Dept of Math, University of Oregon, Eugene, OR 97403 Gang Tian, Dept of Math, Princeton University, Princeton, NJ 08544

References [BW] I. Belegradek and G. Wei, Metrics of positive Ricci curvature on bundles, Int. Math. Res. Not. 57 (2004), 3079-3096. [CK] B. Chow and D. Knopf, The Ricci flow: An introduction, AMS, Providnece, RI, 2004. [D]

D. DeTurk, Deforming metrics in the direction of their Ricci tensors, improved version. In Collected Papers on Ricci Flow, ed. H.-D. Cao, B. Chow, S.-C. Chu and S.-T. Yau. Internat. Press, Somerville, MA, 2003.

[H95a] R. Hamilton, A compactness property for solutions of the Ricci flow, Amer. J. Math. 117 (1995), 545-572. [H95b] R. Hamilton, The formation of singularities in the Ricci flow, Survey in differential geometry, VOL II, 7-136, Internat Press, Cambidge, MA (1995)

19

[Hs] S.Y. Hsu, Global existence and uniqueness of solutions of the Ricci flow equation, Diff. Integral Equ. 14 (2001), 305–320. [LSU] . O. Ladyzhenskaja, V. Solonnikov and N. Ural’ceva, Linear and quasilinear eqautions of parabolic type, AMS, Providnece, RI, 1968. [LT] P. Li and L.-F. Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105 (1991), 1–46. [Pe2] G. Perelman, Ricci flow with surgery on three-manifolds, (2003) math.DG/0303109. [Sh1] W.X. Shi, Deforming the metric on complete Riemannian manifolds, J. Diff. Geom. 30 (1989), 223-301. [Sh2] W.X. Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Diff. Geom. 30 (1989), 303-394.

20

Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman) and some applications Natasa Sesum, Gang Tian

Abstract In this short note we present a result of Perelman with detailed proof. The result states that if g(t) is a K¨ ahler Ricci flow on a compact, K¨ ahler manifold M with c1 (M ) > 0, the scalar curvature and diameter of (M, g(t)) stay uniformly bounded along the flow, for t ∈ [0, ∞). We learned about this result and its proof from G.Perelman when he was visiting MIT in the spring of 2003. This may be helpful to people studying the K¨ ahler Ricci flow.

1

Introduction

We will consider a Kähler Ricci flow, d g ¯ = gi¯j − Ri¯j = ∂i ∂¯j u, dt ij

(1)

on a compact, Kähler manifold M , with c 1 (M ) > 0, of an arbitrary complex dimension n. Cao proved in ([1]) that (1) has a solution for all time t. One of the most important questions regarding the Kähler Ricci flow is whether it develops singularities at infinity, that is whether the curvature of g(t) blows up as t → ∞. This question was only answered in the case curvature operator or bisectional curvature is nonnegative (cf. [8], [7], [2]). In 2003, Perelman made a surprising claim that the scalar curvature of g(t) does not blow up as t → ∞. He also showed us a sketched proof.This result of Perelman strengthens the belief that the Kähler Ricci flow converges to a Kähler Ricci soliton as t tends to infinity, at least outside a subvariety of complex codimension 2. Partial progress was already made ([?]). The goal of this paper is to give a detailed proof of Perelman’s bound on a scalar curvature and a diameter. 1

Theorem 1 (Perelman). Let g(t) be a K¨ ahler Ricci flow (1) on a compact, K¨ ahler manifold M of complex dimension n, with c 1 (M ) > 0. There exists a uniform constant C so that • |R(g(t))| ≤ C, • diam(M, g(t)) ≤ C, • |u|C 1 ≤ C. The outline of the main steps of the proof of Theorem 1 is as follows: 1. Getting a uniform lower bound on Ricci potential u(t). 2. Bounding |∇u(t)| and a scalar curvature R(t) by C 0 norm of Ricci potential u(t). This can be achieved by considering the evolution equa|∇u|2 −∆u and u+2B , where B is a uniform constant such that tions for u+2B u + B > 0, whose existence is guaranteed by step 1. √ 3. Step 2 tells us that u + 2B is a Lipshitz function and that it is enough to bound diam(M, g(t)) in order to have uniform bounds on |u(t)| C 1 and scalar curvature R(t). 4. To show that the diamters are uniformly bounded along the flow, we will argue by contradiction. We will assume that the diameters are unbounded as we approach infinity. Using that, we will show that the integral of the scalar curvature over some large annulus is bounded by C · V , where C is a uniform constant and V is a volume of a slightly larger annulus than the one we started with. We can find such an annulus at every time t in the sequence of times for which diamters go to infinity. By choosing similar cut off functions as in the proof of Perelman’s noncollapsing theorem in [12] we will show that we get a contradiction if the diameters are unbounded as we approach infinity. The organization of the paper is as follows. In section 2 we will give the proof of Theorem 1. In section 3 we will discuss the convergence of the normalized Kähler Ricci flow, using Perelman’s results. Acknowledgements: We would like to thank Perelman for his generousity for telling us about his results on the Kähler Ricci flow. The first author would like to thank H.D.Cao for numerous useful discussions and helpful suggestions.

2

2

Ricci potential u(t)

In this section we will show that there is a uniform lower bound on u(x, t). We will also show that it is enough to bound diameters of (M, g(t)) in order to have Theorem 1. By taking the trace of (1) we get ∆u = n − R. Let φ(t) be a metric potential, that is, gi¯j (t) = gi¯j (0) + ∂i ∂¯j φ. Then we can take u(t) =

d dt φ(t).

Z

Normalize so that e−u = (2π)n .

M

Define µ(g, τ ) =

{f |

R

inf

M

e−f (4πτ )−n =1}

(4πτ )

−n

Z

M

e−f {2τ (R + |∇f |) + f − 2n}dV,

to be Perelman’s functional for g(t) as in [12]. Perelman has proved that µ(g, τ ) is achieved. Take f = u and τ = 1/2. Then by monotonicity of µ(g(t)) along the Kähler Ricci flow, A = µ(g(0), 1/2) ≤ µ(g(t), 1/2) Z ≤ (2π)−n e−u (R + |∇u|2 + u − n) M Z = (2π)−n e−u (−∆u + |∇u|2 − n + u) ZM Z = (2π)−n ∆e−u − n + (2π)−n e−u u M M Z −n −u = −n + (2π) e u.

(2)

M

We have just proved the following lemma. R Lemma 2. There is a uniform constant C 1 = C1 (A) so that e−u u ≥ C1 . R Define a = −(2π)−n M ue−u dV . In the following claim we will prove a lower bound on a. Claim 3. Moreover, there is a uniform constant C 2 > 0 so that a ≥ −C2 .

3

Proof. Let u− = min{u, 0} and u+ = max{0, u}. Then we have Z Z Z −n −u −n −u− −n a = −(2π) ue dV = −(2π) u− e dV − (2π) u+ e−u+ dV M M M Z −n ≥ −(2π) u+ e−u+ dV ≥ −C2 , M

for some constant C2 ≥ 0, since f (x) = xe−x is a bounded function for x ≥ 0. Remark 4. It is pretty well known that the scalar curvature is uniformly bounded from below along the flow. We may assume R > 0. Proof. Function u(t) satisfies, d ∂i ∂¯ ln det(gi¯j + ∂i ∂¯j φ)n dt j = ∂i ∂¯j (u + ∆u),

∂i ∂¯j ut = g¯j − Ri¯j + which implies

d u = ∆u + u + a, (3) dt R −u where we can choose a = − ue (2π)n ≤ C, unifomly bounded from above by the previous lemma. Lemma 5. Function u(t) is uniformly bounded from below. If the Ricci potential u is very negative for some time t 0 , say u(t0 ) ≤ −2(n + C1 ), from (3), by Lemma 2 and Remark 4 we have d u = n − R + u + a ≤ n + C1 + u < 0, dt

(4)

at t = t0 , which implies that u(t) stays very negative for t ≥ t 0 . In other words, if for some y0 we have u(y0 ) > u + 2B,

(17)

then a term on the right hand side of (16) becomes negative for large t and we have that Hmax is decreasing for big values of t, which gives that |∇u|2 ≤ C(u + 2B) for some constant C. This contradicts (17) and therefore we have (10). Our next goal is to prove that −∆u is bounded by C(u + C) which yields −∆u , where B is a uniform constant as (11), since ∆u = n − R. Let K = u+2B above. Similar computation as before gives that (−

¯ 2 (−∆u)(2B − a) ¯ · ∇K ∆u |∇∇u| ∇u )= + + 2 . u + 2B u + 2B (u + 2B)2 u + 2B

Take b > 1. Then (

−∆u + b|∇u|2 ) = u + 2B +

Let G =

−∆u+b|∇u|2 u+2B

¯ 2 (−∆u + b|∇u|2 )(2B − a) −b|∇∇u|2 − (b − 1)|∇∇u| + u + 2B (u + 2B)2 2 −∆u+b|∇u| ¯ ) 2∇u∇( u+2B . u + 2B

and by maximum principle,

¯ 2 (−∆u + b|∇u|2 )(2B − a) |∇∇u| d Gmax ≤ −(b − 1) + . dt u + 2B (u + 2B)2 In local coordinates, (∆u)2 = (

X i

ui¯i )2 ≤ n

X i

¯ 2, u2i¯i = n|∇∇u|

and therefore, (∆u)2 (−∆u + b|∇u|2 )(2B − a) d Gmax ≤ −(b − 1) + dt n(u + 2B) (u + 2B)2 1 (−∆u) b|∇u|2 (2B − a) (−∆u) 2B − a { − }+ . ≤ u + 2B u + 2B n u + 2B (u + 2B)2

(18)

2B−a By Lemmas 2 and 5 we may assume that u+2B is bounded from above by a uniform constant. We have also proved the estimate (10) on |∇u|. If

−∆u >> u + 2B, 7

(19)

d Gmax < 0 for big values of t. This would imply by (18) we would have that dt −∆u(t) ≤ C(u + 2B), for some uniform constant C and all big values of t, which contradicts (19). Therefore, there exists a uniform constant C such that (11) holds.

Proposition 7. There is a C = C(A), such that Vol(B(x, 1)) ≥ C, for any metric g satisfying |R| ≤ 1 on B(x, 1), where ∂B(x, 1) 6= ∅. Proof. Let g(t) be as before, a solution to a normalized Kähler Ricci flow d equation, and let g˜(s) be a solution to the equation ds g˜(s) = −2Ric(˜ g (s)). Reparametrization between these two flows is given by g˜(s) = (1−2s)g(t(s)), where t(s) = − ln(1 − 2s). The first flow has a solution for t ∈ [0, ∞) and the second one has a maximal solution for s ∈ [0, 1/2). The scalar curva1 ture rescales as R(˜ g (s)) = R(g(t(s))) ≤ 1−2s . The following improvement of 1−2s Perelman’s noncollapsing result (noticed by Perelman himself) that requires only a scalar curvature bound can be found in [9]. The result was communicated to Kleiner and Lott by Tian. It says that there is a universal constant d g˜(s) = −2Ric(˜ g (s)), κ = κ(˜ g (0)), so that for an unnormalized Ricci flow ds 1 2n if R(˜ g (s) ≥ − r2 in a ball Bg˜(s) (p, r), then Vol g˜(s) Bg˜(s) (p, r) ≥ κr . The detailed arguments of the proof can be found in [9] and [14], but for the convenience of a reader we will include it here as well. We argue by contradiction, that is, assume there are sequences p k ∈ M and tk → ∞ so that |R| ≤ rC2 , but Vol(Bk )rk−2n → 0 as k → ∞, where Bk = Btk (pk , rk ). Let k

τ = rk2 . Define

uk = eCk φ(rk−1 dist(x, pk ))

(20)

at tk , where φ is a smooth function on R, equal 1 on [0, 1/2], decreasing on [1/2, 1] and equal 0 on [1, ∞). Ck is a constant to make u satisfy the constraint Z (4π)n = e2Ck rk−2n φ(rk−1 dist(x, pk ))2 dV ≤

B(pk ,rk ) 2Ck −2n e rk Vol(Bk ).

8

Since rk−n VolBk → 0, this shows that Ck → +∞. We compute Z −n −2n 2Ck W(uk ) = (4π) rk e (4|φ0 (rk−1 dist(x, pk ))|2 − 2φ2 ln φ)dV B(pk ,rk ) Z + rk2 Ru2 (4π)−n rk−n dV − n − 2Ck B(pk ,rk ) Z ≤ (4π)−n rk−2n e2Ck (4|φ0 |2 − 2φ2 ln φ)dV B(pk ,rk )

+

rk2

max R − n − 2Ck . Bk

Let V (r) = Vol(B(pk , r)). The necessary ingredients of the argument are that (a) rk−2n Vol(B(pk , rk )) → 0. (b) rk2 R is uniformly bounded above. (c)

Vol(B(pk ,rk )) Vol(B(pk ,rk /2))

is uniformly bounded above.

Suppose that rk−2n Vol(B(pk , rk )) → 0 and rk2 R ≤ n(n − 1) on Bk . If Vol(B(pk ,rk )) n Vol(B(pk ,rk /2)) < 3 for all k, we are done. If not, for a given k we have that

Vol(B(pk ,rk )) Vol(B(pk ,rk /2))

≥ 3n . Let rk0 = rk /2. We have (rk0 )−2n Vol(B(pk , rk0 )) ≤

rk−2n Vol(B(pk , rk )) and (rk0 )2 R ≤ n(n − 1) on B(pk , rk0 ). Replace rk by rk0 . If Vol(B(pk ,rk )) n Vol(B(pk ,rk /2)) < 3 we stop. If not, then we repeat the process and replace Vol(B(pk ,rk )) rk by rk /2. At some point we will achieve that Vol(B(p < 3n . We k ,rk /2)) ∞ then take this new subsequence {pk , rk )}k=1 into further discussion. Hence V (rk ) − V (rk /2) ≤ C 0 V (rk /2). Therefore Z (4|φ0 |2 − 2φ2 ln φ)dV ≤ C (V (rk ) − V (rk /2)) B(pk ,rk )

≤ CV (rk /2) Z ≤ C φ2 dV. Bk

Plugging Rthis into the previous estimate for W and using the constraint (4πτk )−n M e−uk dVtk = 1, we get W(uk ) ≤ C 00 − 2Ck . 9

(21)

Since Ck → +∞ and µ(g(tk ), rk2 ) ≤ W(g(tk ), uk , rk2 ), we conclude that µ(g(tk ), rk2 ) → −∞. By the condition (a) we have A ≤ µ(g(t k ), rk2 ) → −∞ which is impossible. √ 1 The previous argument, since R(˜ g (s)) ≤ 1−2s implies Volg˜(s) Bg˜(s) (x, 1 − 2s) ≥ κ(1 − 2s)n , which by rescaling implies VolB(x, 1) ≥ κ at metric g(t), where κ is a constant depending only on an initial metric g(0). Claim 8. There is a uniform constant C such that u(y, t) ≤ Cdist2t (x, y) + C, R(y, t) ≤ Cdist2t (x, y) + C, |∇u| ≤ Cdistt (x, y) + C, where u(x, t) = miny∈M u(y, t). Proof. By Lemma 5 we can assume u ≥ δ > 0, since otherwise we can √ consider u + 2B + δ instead of u. From (10) it follows that u is a Lipshitz √ function since |∇( u)| ≤ C = C(δ) and therefore, √ √ | u(y, t) − u(z, t)| ≤

|∇u|(p, t) √ distt (y, z) 2 u ¯ ≤ Cdist t (y, z),

and therefore, ˜ u(y, t) ≤ (Cdist t (y, z) +

√

u(x, t))2

≤ C1 dist2t (x, y) + C1 u(x, t).

Assume u(x, t) ≥ K(t). Then u(y, t) ≥ K(t) for all y ∈ M and we would have, Z n (2π) = e−u dVt ≤ e−K(t) Vol(M ) → 0, M

if K(t) → ∞, which is not possible. Therefore, u(x, t) ≤ K, for a constant that does not depend on t and finally ˜ u(y, t) ≤ Cdist2t (y, x) + C,

(22)

˜ Other two estimates in the claim for some uniform constants C and C. follow from (22) and Proposition 6. By Claim 8 it follows that if we manage to estimate the diameter, we will get uniform bounds on the scalar curvature and the C 1 norm of u. 10

3

A uniform upper bound on diameters

In this section we want to prove the following proposition which will finish the proof of Theorem 1. Proposition 9. There is a uniform constant C such that diam(M, g(t)) ≤ C. The proof goes by contradiction argument. Assume that the diameters are unbounded in time. Denote by d t (z) = distt (x, z) where u(x, t) = miny∈M u(y, t). Let B(k1 , k2 ) = {z : 2k1 ≤ dt (z) ≤ 2k2 }. Consider an annulus B(k, k+1). By Claim 8 we have that R ≤ C22k on B(k, k + 1). Interval [2k , 2k+1 ] fits 22k balls of radii 21k . By Claim 8 and Proposition 7 we have that at time t X Vol(B(k, k + 1)) ≥ Vol(B(xi , 2−k )) ≥ 22k 2−2kn C. (23) i

Claim 10. For every > 0 we can find B(k 1 , k2 ), such that if diam(M, g(t)) is large enough, then (a) Vol(B(k1 , k2 )) < and (b) Vol(B(k1 , k2 )) ≤ 210n Vol(B(k1 + 2, k2 − 2)). Proof. Since Volt (M ) is constant along the flow and therefore uniformly bounded, if diameter is sufficiently big, there is k 0 such that for all k2 ≥ k1 ≥ k0 , we have that Vol(B(k1 , k2 )) < . If our estimate (b) did not hold, that is, if Vol(B(k1 , k2 )) ≥ 210n Vol(B(k1 + 2, k2 − 2)), we would consider B(k1 + 2, k2 − 2) instead and ask whether (b) holds for that ball. Assume that for every p, at the p-th step we are still not able to find our radii so that (a) and (b) are satisfied. In that case, at the p-th step we would have Vol(B(k1 , k2 )) ≥ 210np Vol(B(k1 + 2p, k2 − 2p)). In particular, assume we have the above estimate at the p-th step so that k1 + 2p + 1 ∼ k2 − 2p, which is for 2p ∼ k2 −k2 2 −1 (*). Take k1 = k/2 and k2 = 3k/2 for k >> 1. In that case (*) becomes p ∼ k/4, k 1 + 2p ∼ k and k2 − 2p ∼ k + 1. Combining this with (23) yields > Vol(B(k1 , k2 )) ≥ 210nk/4 Vol(B(k, k + 1)) ≥ 210nk/4 C22k 2−2nk . This leads to contradiction if we let k → ∞. This finishes the proof of our claim. 11

For every t for which diameter of (M, g(t)) becomes very big, find k 1 and k2 as in Claim 10. Then we have the following lemma. Lemma 11. There exist r1 , r2 and a uniform constant C such that 2k1 ≤ r1 ≤ 2k1 +1 , 2k2 −1 ≤ r2 ≤ 2k2 and Z R ≤ CV, B(r1 ,r2 )

where > 0 is the same as in Claim 10, B(r 1 , r2 ) = {z ∈ M : r1 ≤ distt (z) ≤ r2 } and V = Vol(B(k1 , k2 )).

Proof. We will first prove the existence of r 1 , such that 2k1 ≤ r1 ≤ 2k1 +1 and V VolS(r1 ) ≤ 2 k1 , (24) 2 where S(r) is a metric sphere of radius r. We have that, d Vol(B(r)) = VolS(r). dr

(25)

Assume that for all r ∈ [2k1 , 2k1 +1 ] we have Vol(S(r)) ≥ 2 2Vk1 . Integrate (25) in r. Then, Z 2 k2 Vol(B(k1 , k1 + 1)) = Vol(S(r))dr 2 k1

V k1 2 = 2V = 2Vol(B(k1 , k2 )), 2 k1 which is not possible, since k2 >> k1 by the proof of Claim 10. If for all r ∈ [2k2 −1 , 2k2 ] we have that Vol(S(r)) ≥ 2 2Vk2 , similarly as above we would get Vol(B(k2 − 1, k2 )) > V = B(k1 , k2 ), which is not possible. Therefore, there exists r2 ∈ [2k2 −1 , 2k2 ] such that > 2

V . (26) 2 k2 Estimates (24), (26), together with bounds on ∇u obtained in Claim 8 imply Z Z R = (R − n) + nVol(B(r1 , r2 )) B(r1 ,r2 ) B(r1 ,r2 ) Z = − ∆u + nVol(B(r1 , r2 )) B(r1 ,r2 ) Z Z V V ≤ |∇u| + |∇u| ≤ k1 C2k1 +1 + k2 C2k2 +1 2 2 S(r1 ) S(r2 ) ˜ < C. ˜ = CV VolS(r2 ) ≤ 2

12

We can now finish the proof of Proposition 9. Proof of Proposition 9. The proof of the proposition is similar to the proof of Perelman’s noncollapsing theorem from [12]. Assume diam(M, g(t)) is not uniformly bounded in t, that is, there exists a sequence t i → ∞ such that diam(M, g(ti )) → ∞. Let i → 0 be a sequence of positive numbers. By Claim 10 we can find sequences k1i and k2i such that Volti Bti (k1i , k2i ) < i ,

(27)

Vol(Bti (k1i , k2i )) ≤ 210n Vol(B(k1i + 2, k2i − 2)).

(28)

For each i, find r1i and r2i as in Lemma 11. Let φi be a sequence of cut off i i functions such that φ(z) = 1 for z ∈ [2k1 +2 , 2k2 −2 ] and equal zero Rfor z ∈ (−∞, r1i ] ∪ [r2i , ∞). Let ui (x) = eCi φi (distti (x, pi )) such that (2π)−n M u2i = 1. This implies Z n 2Ci (2π) = e φ2i M

≤ e2Ci Volti Bti (k1i , k2i + 1)

≤ e2Ci i .

Since i → 0, this is possible only if limi→∞ Ci = −∞. By Perelman’s monotonicity formula, A ≤ W(g(ti ), ui , 1/2) Z = (2π)−n e2Ci + (2π)−n

Z

Bti (r1i ,r2i )

Bti (r1i ,r2i )

(4|φ0i (distti (y))|2 − 2φ2i ln φi )dVti

Ru2i dVti − 2n − 2Ci .

First of all by Lemma 11 and (28) we have Z Z 2 2Ci Rui ≤ e Bti (r1i ,r2i )

R

Bti (r1i ,r2i )

˜ 2Ci Volt Bt (k i , k i ) ≤ Ce 1 2 i i 2Ci 10n ˜ ≤ Ce 2 Volti Bti (k1i + 2, k2i − 2) Z 10n ˜ ˜ 10n (2π)n . ≤ C2 u2i dVti = C2 M

13

(29)

By (28) we also have Z 2Ci e

Bti (r1i ,r2i )

(4|φ0i (distti (y))|2 − 2φ2i ln φi )dVti ≤

≤ Ce2Ci Volti Bti (k1i , k2i ) ≤ e2Ci C210n Volti Bti (k1i + 2, k2i − 2) Z 10n ≤ C2 u2i = C210n (2π)n . M

By (29) we get

A ≤ C¯ − 2Ci → −∞,

as i → ∞ and we get a contradiction. Therefore, there is a uniform bound on (M, g(t)), which gives us uniform bounds on scalar curvatures and |u(y, t)| C 1 .

4

Convergence of the K¨ ahler Ricci flow

In this section we will apply Perelman’s boundness results on the scalar curvature and the diameter of the Kähler Ricci flow to show its sequential convergence towards singular metrics that are smooth and satisfy the Kähler Ricci soliton equation outside a singular set. We would like to understand more closely a singular metric that we get as a limit in Theorem 12. Our final goal as a long term project is to prove that a singular set S is a variety, that is, that it has some structure than only being a closed set. Theorem 12. Let g(t) be a K¨ ahler Ricci flow on a compact, K¨ ahler manifold M , given by d g ¯ = gi¯j − Ri¯j = ∂i ∂¯j u. dt ij Assume the Ricci curvature is uniformly bounded along the flow. For every sequence ti → ∞ there is a subsequence so that (M, g(t i + t)) converges to (M∞ , g∞ (t)) in the following sense: (i) M∞ is smooth outside a singular set S which is of codimension at least 4 and the convergence is smooth off S. (ii) A singular metric g∞ (t) satisfies a K¨ ahler Ricci soliton equation g ∞ − ¯ Ric(g∞ ) = ∂ ∂f∞ , with (f∞ )ij = (f∞ )¯i¯j = 0 outside S. (iii) A potential function f∞ is smooth off S and there is a uniform constant C so that |f∞ (t)|C 1 (M∞ \S) ≤ C. 14

In [15] we have proved that if the Ricci curvatures are uniformly bounded, i.e. |Ric| ≤ C for all times t, then for every sequence t i → ∞ there exists a subsequence such that (M, g(ti + t)) → (M∞ , g∞ (t)) and the convergence is smooth outside a singular set S, which is at least of codimension four and M∞ is a smooth manifold off S. We also showed that a singular set S does not depend on time t. Moreover, g∞ (t) solves the Kähler-Ricci flow equation on M∞ \S. We want to show that g∞ (t) is actually a Kähler Ricci soliton off the singular set. Due to Perelman we have the following uniform estimates for the Kähler Ricci flow: there are uniform constants C and κ such that for all t, 1. |u(t)|C 1 ≤ C, 2. diam(M, g(t)) ≤ C, 3. |R(g(t))| ≤ C, 4. (M, g(t)) is κ-noncollapsed. This together with the uniform lower bound on the Ricci curvatures along the flow gives a uniform Sobolev constant, that is, there is a uniform constant CS so that for any v ∈ C01 (M ) we have that Z Z 2n 4n |∇v|2 dVg(t) , (30) ( v 2n−2 dVg(t) ) 2n−2 ≤ CS M

M

for all times t ≥ 0. This enables us to work with integral estimates. The proof of Theorem 12 will be completed after we prove Proposition 13 and Proposition 14. Proposition 13. A singular metric g ∞ that arises in Theorem 12 satisfies the K¨ ahler Ricci soliton equation on M ∞ \S. Proof. Notice that µ(g(t), 1/2) ≤ W(g(t), u(t), 1/2) ≤ C, for a uniform constant C, due to Perelman’s estimates mentioned above (recall that u(t) is the Ricci potential of metric g(t)). This yields a finite lim t→∞ µ(g(t), 1/2). Let ft be a minimizer of W with respect to metric g(t) and let f t (s) be a solution of the equation d ft (s) = −∆ft (s) + |∇ft (s)|2 − R(s) + 2n, ds

15

(31)

for s ∈ [0, t]. Fix A > 0. By monotonicity of Perelman’s functional W we also have Z AZ −n ¯ k ft +A (ti + s) − g ¯ |2 e−fti +A (ti +s) dVt +s ds + 0 ≤ (2π) |Rj k¯ + ∇j ∇ i i jk 0 M Z AZ ¯ j∇ ¯ k ft +A (ti + s)|2 )e−fti +A (ti +s) dVt +s ds + (2π)n (|∇j ∇k fti +A (ti + s)|2 + |∇ i i 0

M

= W(g(ti + A), fti +A , 1/2) − W(g(ti ), fti +A (ti ), 1/2) ≤ µ(g(ti + A), 1/2) − µ(g(ti ), 1/2),

where the right hand side of the previous estimate tends to zero as i → ∞, since there is a finite limt→∞ µ(g(t), 1/2). Moreover, for any compact set K ⊂ M∞ \S we have

Z AZ ¯ k ft +A (ti + s) − g ¯ (ti + s)|2 e−fti +A (ti +s) dVt +s ds + lim |Rj k¯ (ti + s) + ∇j ∇ i jk i i→∞ 0 K Z AZ ¯ j∇ ¯ k ft +A (ti + s)|2 )e−fti +A (ti +s) dVt +s ds = 0, + (|∇j ∇k fti +A (ti + s)|2 + |∇ i i 0

K

which implies for almost all s ∈ [0, A] and almost all x ∈ K, ¯ k ft +A (ti + s) − g ¯ (ti + s)|2 e−fti +A (ti +s) = 0. • limi→∞ |Rj k¯ (ti + s) + ∇j ∇ i jk • limi→∞ |∇j ∇k fti +A (ti + s)|2 e−fti +A (ti +s) = 0. ¯ l → S as Let Dl ⊂ M∞ be a sequence of open sets where S ⊂ Dl and D l → ∞. We know that g(ti + t) → g∞ (t) smoothly on M∞ \Dl . Function fti +A (ti + s) satisfies an evolution equation (31) and we have the geometries g(ti + s) are uniformly bounded on M∞ \Dl for all i and all s ∈ [0, A] (those bounds depend on l, that is, on the closeness to the singular set S). Standard parabolic estimates as in [15] and [16] give there are uniform con∂p q stants C(p, q, l) so that | ∂s p ∇ fti +A (ti + s)| ≤ C(p, q, l). Using the uniform derivative bounds from above, by Arzela Ascoli theorem and diagonalizing the sequence (by letting l bigger and bigger) we can find a subsequence so that fti +A (ti + s)

C k (M∞ \S)

→

f∞ (s). Moreover, this limit f∞ (s) satisfies

¯ k f∞ (s) − (g∞ ) ¯ (s) = 0. Rj k¯ (g∞ (s)) + ∇j ∇ jk

(32)

¯ i∇ ¯ j f∞ = 0. ∇i ∇j f∞ = ∇

(33)

16

Relations (32) and (33) give, sup |D 2 f∞ | ≤ C < ∞.

(34)

M∞ \S

The previous estimate helps us prove the following proposition. Proposition 14. There is a constant C so that |f ∞ |C 1 (M∞ \S) ≤ C.

Proof. We will first mention few results and some notation from [3] and [4], which our proof will rely upon. A point y ∈ M∞ is called regular, if for some k, every tangent cone at y is siometric to R2n . Let Rk denote the set of k-regular points and put R = ∪k Rk , the regular set. A point y ∈ M∞ is acalled singular, if it is not regular. Denote by S the set of singular points. In [3] it has been shown that under the assumption |Ric(g(t))| ≤ C, we have R = R n . Moreover, one of the results in [5] is that dim S ≤ n − 4. The -regular set, R , consists of those points y, such that every tangent cone, (Y y , y∞ ), satisfies dGH (B1 (y∞ , B1 (0)) < . In [3] it was shown that for the uniform bound on the Ricci curvatures, there is an 0 , so that for every < 0 , R = R and R ∩ S = ∅. That means we can write M∞ = R ∪ S, for < 0 . Fix x0 ∈ R. By Theorem 3.9 in [4], there exists C(x 0 ) ⊂ R, with ν(M∞ \C(x0 )) = 0 (ν is the unique limit measure, which in our noncollapsed case is exactly Hausdorff measure), such that for all y ∈ C(x 0 ) and > 0, there exists a minimal geodesic from x 0 to y, which is contained in R˙ . If we choose small enough, R = R and R is an open set. This means for almost all y ∈ R there is a minimal geodesic, call it γ, connecting x 0 and y, all contained in R. For such y, we have ˜ |Df∞ (x0 ) − Df∞ (y)| ≤ C sup |D 2 f∞ |length(γ) ≤ C, M∞ \S

since we have an estimate (34) and since length(γ) ≤ D, where D is a uniform bound on the diameters of (M, g(t)). Since |Df ∞ (x0 )| is a finite number, we get |Df∞ |(x) ≤ C˜1 for almost all x ∈ R. On the other hand, since f∞ is a smooth function on R = M∞ \S, we get sup |Df∞ | ≤ C˜1 . M∞ \S

By similar arguments, we also have sup |f∞ | ≤ C˜2 .

M∞ \S

17

References [1] H.D.Cao: Deformation of Kähler metrics to Käher-Einstein metrics on compact Kähler manifolds; Invent. math. 81 (1985) 359–372. [2] H.D.Cao, B.L.Chen, X.P.Zhu: Ricci flow on compact Kähler manifold of positive bisectional curvature; math.DG/0302087. [3] J.Cheeger, T.Colding: On the structure of spaces with Ricci curvature bounded below. I; J.Differential Geometry 45, (1997) 406–480. [4] J.Cheeger, T.Colding: On the structure of spaces with Ricci curvature bounded below. II; J.Differential Geometry 52, (1999) 13-35. [5] J.Cheeger, T.H.Colding, G.Tian: On the singularities of spaces with bounded Ricci curvatures; GAFA, Geom. funct. anal. 12, (2002) 873– 914. [6] Chen, X.X., G.Tian: Ricci flow on Kähler Einstein surfaces; Invent.Math., 147 (2002) 487–544. [7] Chen, X.X., G.Tian: Ricci flow on Kähler Einstein surfaces; Inventiones Math. 147 (2002) 487–544. [8] Chen X.X., G.Tian: Ricci flow on Kähler C.R.Acad.Sci.Paris Sr. I Math. 332 (2001) 245–248.

manifolds;

[9] B.Kleiner, J.Lott: Notes on Perelman’s papers; http://www.math.lsa.umich.edu/research/ricciflow/posting123004.pdf. [10] D.Gilbarg, N.S.Trudinger: Elliptic partial differential equations of second order; Springer-Verlag, 1983. [11] Q.Han, F.Lin: Elliptic partial differential equations; Courant Institute of Mathematical Sciences, 1997, ISBN 0-9658703-0-8. [12] G. Perelman: The entropy formula for the Ricci flow and its geometric applications; arXiv:math.DG/0211159. [13] G.Perelman: private correspondence. [14] N.Sesum, G.Tian, X.Wang: Notes on Perelman’s paper on the entropy fomula for the Ricci flow andits geometric applications.

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[15] N.Sesum: Convergence of a Kähler Ricci flow; preprint, arXiv:math.DG/0402238 (submitted to the Mathematical Research Letters). [16] N. Sesum: Convergence of a Kähler-Ricci flow, arXiv:math.DG/0402238, to appear in Mathematical Research Letters. [17] G.Tian, X.H.Zhu: Uniqueness of Kähler Ricci solitons; Acta Math. 184 (2000), 271–305. [18] G.Tian, X.H.Zhu: A new holomorphic invariant and uniqueness of Kähler Ricci solitons; Comm.Math.Helv. 77 (2002) 297-325. [19] Yau, S.T.: Harmonic functions on complete Riemannian manifolds; Comm. Pure Appl. Math. 28 (1975) 201-228.

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arXiv:math/0308090v3 [math.AP] 15 Aug 2006

ESTIMATES FOR THE EXTINCTION TIME FOR THE RICCI FLOW ON CERTAIN 3–MANIFOLDS AND A QUESTION OF PERELMAN TOBIAS H. COLDING AND WILLIAM P. MINICOZZI II Abstract. We show that the Ricci flow becomes extinct in finite time on any Riemannian 3–manifold without aspherical summands.

1. introduction In this note we prove some bounds for the extinction time for the Ricci flow on certain 3–manifolds. Our interest in this comes from a question that Grisha Perelman asked the first author at a dinner in New York City on April 25th of 2003. His question was “what happens to the Ricci flow on the 3–sphere when one starts with an arbitrary metric? In particular, does the flow become extinct in finite time?” He then went on to say that one of the difficulties in answering this is that he knew of no good way of constructing minimal surfaces for such a metric in general. However, there is a natural way of constructing such surfaces and that comes from the min–max argument where the minimal of all maximal slices of sweep–outs is a minimal surface; see, for instance, [2]. The idea is then to look at how the area of this min–max surface changes under the flow. Geometrically the area measures a kind of width of the 3–manifold and as we will see for certain 3–manifolds (those, like the 3–sphere, whose prime decomposition contains no aspherical factors) the area becomes zero in finite time corresponding to that the solution becomes extinct in finite time. Moreover, we will discuss a possible lower bound for how fast the area becomes zero. Very recently Perelman posted a paper (see [9]) answering his original question about finite extinction time. However, even after the appearance of his paper, then we still think that our slightly different approach may be of interest. In part because it is in some ways geometrically more natural, in part because it also indicates that lower bounds should hold, and in part because it avoids using the curve shortening flow that he simultaneously with the Ricci flow needed to invoke and thus our approach is in some respects technically easier. Let M 3 be a smooth closed orientable 3–manifold and let g(t) be a one–parameter family of metrics on M evolving by the Ricci flow, so (1.1)

∂t g = −2 RicMt .

Unless otherwise stated we will assume throughout that M is prime and non– aspherical (so πk (M ) 6= {0} for some k > 1). If M is prime but not irreducible, then M = S2 × S1 (proposition 1.4 in [5]) so π3 (M ) = Z. Otherwise, if M is irreducible, then the sphere theorem implies that π2 (M ) = 0 (corollary 3.9 in [5]). In the second 1991 Mathematics Subject Classification. Primary 53C44; Secondary 53C42, 57M50. Key words and phrases. Ricci flow, Finite extinction, 3-manifolds, Min-max surfaces. The authors were partially supported by NSF Grants DMS 0104453 and DMS 0104187. 1

2

TOBIAS H. COLDING AND WILLIAM P. MINICOZZI II

case, the Hurewicz isomorphism theorem then implies that π3 (M ) 6= {0} (since M is non–aspherical). Therefore, in either case, by suspension, as in lemma 3 of [8], the space of maps from S2 to M is not simply connected.

W

The min-max surface. Figure 1. The sweep–out, the min–max surface, and the width W. Fix a continuous map β : [0, 1] → C 0 ∩ L21 (S2 , M ) where β(0) and β(1) are constant maps so that β is in the nontrivial homotopy class [β]. We define the width W = W (g, [β]) by (1.2)

W (g) = min max E(γ(s)) . γ∈[β] s∈[0,1]

One could equivalently define the width using the area rather than the energy, but the energy is somewhat easier to work with. As for the Plateau problem, this equivalence follows using the uniformization theorem and the inequality Area(u) ≤ E(u) (with equality when u is a branched conformal map); cf. lemma 4.12 in [3].1 The next theorem gives an upper bound for the derivative of W (g(t)) under the Ricci flow which forces the solution g(t) to become extinct in finite time (see paragraph 4.4 of [11] for the precise definition of extinction time when surgery occurs). Theorem 1.1. Let M 3 be a closed orientable prime non–aspherical 3–manifold equipped with a Riemannian metric g = g(0). Under the Ricci flow, the width W (g(t)) satisfies (1.3)

3 d W (g(t)) ≤ −4π + W (g(t)) , dt 4(t + C)

in the sense of the limsup of forward difference quotients. Hence, g(t) must become extinct in finite time. 1It may be of interest to compare our notion of width, and the use of it, to a well–known approach to the Poincar´ e conjecture. This approach asks to show that for any metric on a homotopy 3–sphere a min–max type argument produces an embedded minimal 2–sphere. Note that in the definition of the width it plays no role whether the minimal 2–sphere is embedded or just immersed, and thus, the analysis involved in this was settled a long time ago. This well–known approach has been considered by many people, including Freedman, Meeks, Pitts, Rubinstein, Schoen, Simon, Smith, and Yau; see [2].

EXTINCTION TIME FOR THE RICCI FLOW AND A QUESTION OF PERELMAN

3

The 4π in (1.3) comes from the Gauss–Bonnet theorem and the 3/4 comes from the bound on the minimum of the scalar curvature that the evolution equation implies. Both of these constants matter whereas the constant C depends on the initial metric and the actual value is not important. To see that (1.3) implies finite extinction time rewrite (1.3) as d (1.4) W (g(t)) (t + C)−3/4 ≤ −4π (t + C)−3/4 dt and integrate to get h i (1.5) (T + C)−3/4 W (g(T )) ≤ C −3/4 W (g(0)) − 16 π (T + C)1/4 − C 1/4 .

Since W ≥ 0 by definition and the right hand side of (1.5) would become negative for T sufficiently large we get the claim. Arguing as in 1.5 of [9] (or alternatively using Section 4), we get as a corollary of this theorem finite extinction time for the Ricci flow on all 3–manifolds without aspherical summands. Corollary 1.2. Let M 3 be a closed orientable 3–manifold whose prime decomposition has only non–aspherical factors and is equipped with a Riemannian metric g = g(0). Under the Ricci flow with surgery, g(t) must become extinct in finite time. Part of Perelman’s interest in the question about finite time extinction comes from the following: If one is interested in geometrization of a homotopy three-sphere (or, more generally, a three-manifold without aspherical summands) and knew that the Ricci flow became extinct in finite time, then one would not need to analyze what happens to the flow as time goes to infinity. Thus, in particular, one would not need collapsing arguments. 2. Upper bounds for the rate of change of area of minimal 2–spheres

Suppose that Σ ⊂ M is a closed immersed surface (not necessarily minimal), then using (1.1) an easy calculation gives (cf. page 38–41 of [4]) Z d Areag(t) (Σ) = − [R − RicM (n, n)] . (2.1) dt t=0 Σ If Σ is also minimal, then Z Z d (2.2) Areag(t) (Σ) = −2 KΣ − [|A|2 + RicM (n, n)] dt t=0 ZΣ Z Σ 1 [|A|2 + R] . =− KΣ − 2 Σ Σ Here KΣ is the (intrinsic) curvature of Σ, n is a unit normal for Σ (our Σ’s below will be S2 ’s and hence have a well–defined unit normal), A is the second fundamental form of Σ so that |A|2 is the sum of the squares of the principal curvatures, RicM is the Ricci curvature of M , and R is the scalar curvature of M . (The curvature is normalized so that on the unit S3 the Ricci curvature is 2 and the scalar curvature is 6.) To get (2.2), we used that by the Gauss equations and minimality of Σ 1 (2.3) KΣ = KM − |A|2 , 2 where KM is the sectional curvature of M on the two–plane tangent to Σ.

4


Our first lemma gives an upper bound for the rate of change of area of minimal 2–spheres. Lemma 2.1. If Σ ⊂ M 3 is a branched minimal immersion of the 2–sphere, then (2.4)

Areag(0) (Σ) d min R(0) . Areag(t) (Σ) ≤ −4π − M dt t=0 2

Proof. Let {pi } be the set of branch points of Σ and bi > 0 the order of branching at pi . By (2.2) Z Z Z X d 1 1 (2.5) R = −4π − 2π bi − R, Areag(t) (Σ) ≤ − KΣ − dt t=0 2 Σ 2 Σ Σ where the equality used the Gauss–Bonnet theorem with branch points. Note that branch points only help in the inequality (2.4). For an immersed minimal surface Σ ⊂ M we set (2.6)

LΣ φ = ∆Σ φ + |A|2 φ + RicM (n, n) φ .

By (2.2) and the Gauss–Bonnet theorem Z Z Z d (2.7) Areag(t) (Σ) = −2 KΣ − 1 · LΣ 1 = −4πχ(Σ) − 1 · LΣ 1 . dt t=0 Σ Σ Σ (Note that by the second variational formula (see, for instance, section 1.7 of [3]), then Z ∂2 (2.8) Area(Σr ) = − φ LΣ φ , ∂r2 r=0 Σ where Σr = {x+ r φ(x) nΣ (x) | x ∈ Σ}.) Recall also that by definition the index of a minimal surface Σ is the number of negative eigenvalues (counted with multiplicity) of LΣ . (A function η is an eigenfunction of LΣ with eigenvalue λ if LΣ η + λ η = 0.) Thus in particular, since Σ is assumed to be closed, the index is always finite. 3. Extinction in finite time We begin by recalling a result on harmonic maps which gives the existence of minimal spheres realizing the width W (g). The results of Sacks and Uhlenbeck give the harmonic maps but potentially allow some loss of energy. This energy loss was ruled out by Siu and Yau (using also arguments of Meeks and Yau), see Chapter VIII in [13]. For our purposes, the most convenient statement of this is given in theorem 4.2.1 of [6]. (The bound for the index is not stated explicitly in [6] but follows immediately as in [8].) Proposition 3.1. Given a metric g on M and a nontrivial [β] ∈ π1 (C 0 ∩L21 (S2 , M )), there exists a sequence of sweep–outs γ j : [0, 1] → C 0 ∩ L21 (S2 , M ) with γ j ∈ [β] so that (3.1)

W (g) = lim max E(γsj ) . j→∞ s∈[0,1]

Furthermore, there exist sj ∈ [0, 1] and branched conformal minimal immersions u0 , . . . , um : S2 → M with index at most one so that, as j → ∞, the maps γsjj


5

converge to u0 weakly in L21 and uniformly on compact subsets of S2 \ {x1 , . . . , xk }, and m X (3.2) W (g) = E(ui ) = lim E(γsjj ) . i=0

j→∞

Finally, for each i > 0, there exists a point xki and a sequence of conformal dilations Di,j : S2 → S2 about xki so that the maps γsjj ◦ Di,j converge to ui .

Remark 3.2. It is implicit in Proposition 3.1 that W (g) > 0. This can, for instance, be seen directly using [6]. Namely, page 125 in [6] shows that if maxs E(γsj ) is sufficiently small (depending on g), then γ j is homotopically trivial. We will also need a standard additional property for the min–max sequence of sweep–outs γ j of Proposition 3.1 which can be achieved by modifying the sequence as in section 4 of [2] (cf. proposition 4.1 on page 85 in [2]). Loosely speaking this is the property that any subsequence γskk with energy converging to W (g) converges (after possibly going to a further subsequence) to the union of branched immersed minimal 2–spheres each with index at most one. Precisely this is that we can choose γ j so that: Given ǫ > 0, there exist J and δ > 0 (both depending on g and γ j ) so that if j > J and E(γsj ) > W (g) − δ ,

(3.3)

then there is a collection of branched minimal 2–spheres {Σi } each of index at most one and with dist (γsj , ∪i Σi ) < ǫ .

(3.4)

Here, the distance means varifold distance (see, for instance, section 4 of [2]). Below we will use that, as an immediate consequence of (3.4), if F is a quadratic form on M and Γ denotes γsj , then (3.5) Z XZ [Tr(F ) − F (nΣi , nΣi )] < C ǫ kF kC 1 Area(Γ) . [Tr(F ) − F (nΓ , nΓ )] − Γ Σi i

In the proof of the result about finite extinction time we will also need that the evolution equation for R = R(t), i.e. (see, for instance, page 16 of [4]), 2 (3.6) ∂t R = ∆R + 2|Ric|2 ≥ ∆R + R2 , 3 implies by a straightforward maximum principle argument that at time t > 0 1 3 (3.7) R(t) ≥ =− . 1/[min R(0)] − 2t/3 2(t + C)

In the derivation of (3.7) we implicitly assumed that min R(0) < 0. If this was not the case, then (3.7) trivially holds with C = 0, since, by (3.6), min R(t) is always non–decreasing. This last remark is also used when surgery occurs. This is because by construction any surgery region has large (positive) scalar curvature. Proof. (of Theorem 1.1) Fix a time τ . Below C˜ denotes a constant depending only on τ but will be allowed to change from line to line. Let γ j (τ ) be the sequence of sweep–outs for the metric g(τ ) given by Proposition 3.1. We will use the sweep–out at time τ as a comparison to get an upper bound for the width at times t > τ .

6


The key for this is the following claim (the inequality in (3.8) below): Given ǫ > 0, ¯ then there exist J and ¯ h > 0 so that if j > J and 0 < h < h, Areag(τ +h) (γsj (τ )) − max Eg(τ ) (γsj (τ )) s

(3.8)

≤ [−4π + C˜ ǫ +

3 max Eg(τ ) (γsj (τ ))] h + C˜ h2 . 4(τ + C) s

To see why (3.8) implies (1.3), we use the definition of the width to get (3.9)

W (g(τ + h)) ≤ max Areag(τ +h) (γsj (τ )) , s∈[0,1]

and then take the limit as j → ∞ (so that maxs Eg(τ ) (γsj (τ )) → W (g(τ ))) in (3.8) to get (3.10)

W (g(τ + h)) − W (g(τ )) 3 ≤ −4π + C˜ ǫ + W (g(τ )) + C˜ h . h 4(τ + C)

Taking ǫ → 0 in (3.10) gives (1.3). It remains to prove (3.8). First, let δ > 0 and J, depending on ǫ (and on τ ), be given by (3.3)–(3.5). If j > J and Eg(τ ) (γsj (τ )) > W (g) − δ, then let ∪i Σjs,i (τ ) be the collection of minimal spheres in (3.5). Combining (2.1), (3.5) with F = RicM , and Lemma 2.1 gives d d Areag(t) (γsj (τ )) ≤ Areag(t) (∪i Σjs,i (τ )) + C˜ ǫ kRicM kC 1 Areag(t) (γsj (τ )) dt t=τ dt t=τ Eg(τ ) (γsj (τ )) ≤ −4π − (3.11) min R(τ ) + C˜ ǫ M 2 3 ≤ −4π + max Eg(τ ) (γsj (τ )) + C˜ ǫ , 4(τ + C) s where the last inequality used the lower bound (3.7) for R(τ ). Since the metrics g(t) vary smoothly and every sweep–out γ j has uniformly bounded energy, it is easy to see that Eg(τ +h) (γsj (τ )) is a smooth function of h with a uniform C 2 bound independent of both j and s near h = 0 (cf. (2.1)). In particular, (3.11) and ¯ > 0 (independent of j) so that (3.8) holds for s with Taylor expansion gives h Eg(τ ) (γsj (τ )) > W (g) − δ. In the remaining case, we have E(γsj (τ )) ≤ W (g) − δ so the continuity of g(t) implies that (3.8) automatically holds after possibly shrinking ¯ h > 0. Finally, we claim that (1.3) implies finite extinction time. Namely, rewriting d (1.3) as dt W (g(t)) (t + C)−3/4 ≤ −4π (t + C)−3/4 and integrating gives h i (3.12) (T + C)−3/4 W (g(T )) ≤ C −3/4 W (g(0)) − 16 π (T + C)1/4 − C 1/4 .

Since W ≥ 0 by definition and the right hand side of (3.12) would become negative for T sufficiently large, the theorem follows. 4. Remarks on the reducible case

When M is reducible, then the factors in the prime decomposition must split off in a uniformly bounded time. This follows from a (easy) modification of the proof of Theorem 1.1. Namely, each (non–trivial) factor in the prime decomposition gives rise to a 2–sphere which does not bound a 3–ball in M and, hence, to a stable minimal 2–sphere in this isotopy class by [7]. Applying the argument of the proof


7

of Theorem 1.1 to these minimal 2–spheres, we see that the minimal area in this isotopy class must go to zero in finite time as claimed. Appendix A. Lower bounds for the rate of change of area of minimal 2–spheres The next lower bound is an adaptation of Hersch’s theorem; cf. [1]. Recall that Hersch’s theorem (see, for instance, [12]) states the sharp scale invariant inequality that for any metric on the 2–sphere λ1 times the area is bounded uniformly from above by the corresponding quantity on a round 2–sphere. Lemma A.1. If Σ ⊂ M 3 is an immersed minimal 2–sphere with index at most one, then Z Z (A.1) 8 π ≥ [|A|2 + RicM (n, n)] = 1 · LΣ 1 . Σ

Σ

Hence, by (2.7)

d Areag(t) (Σ) ≥ −16 π . dt t=0 Proof. If Σ is stable, i.e., if the index is zero, then for all φ Z (A.3) − φLφ ≥ 0, (A.2)

Σ

or equivalently (A.4)

Z

Σ

2

|∇φ| ≥

Z

[|A|2 + RicM (n, n)] φ2 ,

Σ

and thus by letting φ ≡ 1 in (A.4) we see that (A.1) holds. If the index is one, then we let η be an eigenfunction for LΣ with negative eigenvalue λ < 0. That is, (A.5)

LΣ η + λ η = 0 .

By a standard argument, then an eigenfunction corresponding to the first eigenvalue of a Schr¨ odinger operator (Laplacian plus potential) does notRchange sign and thus we may assume that η is everywhere positive. In particular, Σ η > 0. Since Σ has index one then it follows that (A.4) holds for all φ with Z (A.6) 0= ηφ. Σ

By the uniformization theorem, there exists a conformal diffeomorphism Φ : Σ → S2 ⊂ R3 . For i = 1, 2, 3 set φi = xi ◦ Φ. For x ∈ S2 let πx : S2 \ {x} → C be the stereographic projection and let ψx,t (y) = πx−1 (t(πx (y))), then for each t, x this can be extended to a conformal map on S2 . Define Ψ : S2 × [0, 1) → G, where G is the group of conformal transformations of S2 , by Ψ(x, t) = ψx,1/(1−t) . Since Ψ(x, 0) = idS2 for each x ∈ S2 , Ψ can be thought of as a continuous map on B1 (0) = S2 × [0, 1)/(x, 0) ≡ (y, 0). Set Z 1 R η xi ◦ Ψ(x, t) ◦ Φ , (A.7) A(Ψ(x, t)) = η Σ i=1,2,3 Σ where η is as in (A.5). It follows that (A.8)

A : B1 (0) → B1 (0) and

lim

(y,t)→(x,1)

A(Ψ(y, t)) = x .

8


In particular, it follows that A extends to ∂B1 (0) as the identity map. We can therefore, by elementary topology (after possibly replacing Φ by ψ ◦ Φ), assume that for each i Z (A.9) η φi = 0 ; Σ

that is each φi is orthogonal to η. It follows from (A.4) that for each i Z Z (A.10) |∇φi |2 ≥ [|A|2 + RicM (n, n)] φ2i . Σ

Σ

P3 Summing over i and using that Φ(Σ) ⊂ S2 so i=1 φ2i = 1 we get Z 3 Z X (A.11) |∇φi |2 ≥ [|A|2 + RicM (n, n)] . i=1

Σ

Σ

Now obviously, since Φ is conformal (so that it preserves energy) and since each xi is an eigenfunction for the Laplacian on S2 ⊂ R3 with eigenvalue λ1 (S2 ) = 2, we get Z Z Z (A.12) |∇φi |2 = |∇xi |2 = λ1 (S2 ) x2i . S2

Σ

S2

Combining (A.11) with (A.12) we get (A.13) (A.14)

2 Area(S2 ) = =

3 X

λ1 (S2 )

i=1 3 Z X i=1

Z

S2

Σ

x2i =

|∇φi |2 ≥

3 Z X i=1

Z

S2

|∇xi |2

[|A|2 + RicM (n, n)] .

Σ

Acknowledgement: We are grateful to John Lott for helpful comments. References 1. D. Christodoulou and S.T. Yau, Some remarks on the quasi-local mass, Mathematics and general relativity (Santa Cruz, CA, 1986), 9–14, Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988. 2. T.H. Colding and C. De Lellis, The min–max construction of minimal surfaces, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002), 75–107, Surv. Differ. Geom., VIII, Int. Press, Somerville, MA, 2003, math.AP/0303305. 3. T.H. Colding and W.P. Minicozzi II, Minimal surfaces, Courant Lecture Notes in Mathematics, 4. New York University, Courant Institute of Mathematical Sciences, New York, 1999. 4. R. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7–136, International Press, Cambridge, MA, 1995. 5. A. Hatcher, Notes on basic 3–manifold topology, www.math.cornell.edu/ hatcher/3M/3Mdownloads.html. 6. J. Jost, Two–dimensional geometric variational problems, J. Wiley and Sons, Chichester, N.Y. (1991). 7. W. Meeks III, L. Simon, and S.T. Yau, Embedded minimal surfaces, exotic spheres and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982) 621–659. 8. M.J. Micallef and J.D. Moore, Minimal two–spheres and the topology of manifolds with positive curvature on totally isotropic two–planes, Ann. of Math. (2) 127 (1988) 199–227. 9. G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three– manifolds, math.DG/0307245. 10. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math.DG/0211159.


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11. G. Perelman, Ricci flow with surgery on three–manifolds, math.DG/0303109. 12. R. Schoen and S.T. Yau, Lectures on differential geometry, International Press 1994. 13. R. Schoen and S.T. Yau, Lectures on harmonic maps, International Press 1997. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012 Department of Mathematics, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218 E-mail address: [email protected] and [email protected]

VOLUME COLLAPSED THREE-MANIFOLDS WITH A LOWER CURVATURE BOUND

arXiv:math/0304472v3 [math.DG] 15 Apr 2004

TAKASHI SHIOYA AND TAKAO YAMAGUCHI Abstract. In this paper we determine the topology of threedimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.

1. Introduction As a continuation of our investigation [9] of collapsing three-manifolds with a lower curvature bound and an upper diameter bound, we study the topology of a three-dimensional closed Riemannian manifold with a lower curvature bound whose volume is sufficiently small, where we assume no upper diameter bound. A closed three-manifold is called a graph manifold if it is a finite gluing of Seifert fibered spaces along their boundary tori. Theorem 1.1. There exist small positive numbers ǫ0 and δ0 such that if a closed orientable three-manifold M has a Riemannian metric with sectional curvature K ≥ −1 and vol(M) < ǫ0 , then one of the following holds: (1) M is homeomorphic to a graph manifold; (2) diam(M) < δ0 and M has finite fundamental group. It was shown in [9] that in the case of (2) in Theorem 1.1, M is homeomorphic to an Alexandrov space with nonnegative curvature. Theorem 1.1 determines the possible topological type of M if M has not so small diameter. In fact, from [2], every three-dimensional graph manifold M has a Riemannian metric gǫ with sectional curvature |Kgǫ | ≤ 1, diam(M, gǫ ) ≥ δ0 and vol(M, gǫ ) < ǫ for each ǫ > 0. In the bounded curvature case, it follows essentially from [3] that if a closed three-manifold has a Riemannian metric with |K| ≤ 1 whose volume is sufficiently small, then it is a graph manifold. The strategy of our proof is as follows: We assume M has large diameter which is the essential case. Applying our previous work [9], we obtain a local fiber structure on a neighborhood Bp of each point 2000 Mathematics Subject Classification. Primary 53C20, 53C23; Secondary 57N10, 57M99. Key words and phrases. the Gromov-Hausdorff convergence, Alexandrov spaces, topology of three-manifolds, graph manifolds. 1

p ∈ M over a metric ball Xp in some Alexandrov space with curvature bounded below, where dim Xp ∈ {1, 2}. If dim Xp = 2, we have a local S 1 -action on Bp . If dim Xp = 1, we have a (singular) sphere or torus bundle structure on Bp over the closed interval Xp , and Bp is homeomorphic to one of six compact three-manifolds, which will be called cylindrical if it is homeomorphic to either S 2 ×I or T 2 ×I, or cylindrical ˜ S 1 ×D2 or K 2 ×I, ˜ where with a cap if it is homeomorphic to D3 , P 2 ×I, ˜ indicates the twisted product. Using those local data, we decompose × M into two parts as M = U1 ∪ Uˆ2 , where U1 is a closed domain which ˆ2 is one which looks two-dimensional, in looks one-dimensional and U the Gromov-Hausdorff sense. More precisely, U1 is defined as the union of all Bp with area of Xp sufficiently small. Applying the critical point theory for distance functions, we conclude that each component of U1 is either cylindrical or cylindrical with a cap. We shall construct a local S 1 -action on the remaining piece Uˆ2 , from which a graph manifold structure on M is obtained. To do this, we need a gluing procedure, which is the main part of the present paper. To make the gluing procedure explicit and clear, we give quantitative descriptions of the local fibering Bp → Xp using the geometric properties of fibers in [10] and [11] over a regular part of Xp . Here the notion of strain radius comes in to control the behavior of the regular fibers. This forces us to obtain a sort of compactness of the set of regular parts of Xp ’s with Bp ˆ2 . This is the reason why in the decomposition M = U1 ∪ Uˆ2 meeting U a neighborhood Bp is included in the one-dimensional part U1 even if dim Xp = 2 when Xp has a small area. The organization of this paper is as follows: In Section 2, we first establish a uniform lower bound on the strain radii of regular parts ˆ2 , and then provide some basic properties of Xp ’s with Bp meeting U of Alexandrov surfaces to show that there are a lot of possibilities for the choices of the metric ball Xp with boundary having nice geometric properties. In Section 3, we describe the geometry and topology of the local fibering Bp → Xp in detail. In Section 4, using these fiber structure we have the decomposition M = U1 ∪ Uˆ2 and determine the topology of U1 . In Section 5, we provide a preliminary gluing ˆ2 . The gluing argument for the construction of local S 1 -action on U procedure is completed in Section 6. In Section 7, we discuss a thickthin decomposition of a closed orientable Riemannian three-manifold with a lower curvature bound. An announcement in Perelman’s paper [7] has recently been come to our attention. He claims that if a three-manifold collapses under a local lower sectional curvature bound, then it is a graph manifold (Theorem 7.4). This result also follows from the argument in our Theorem 1.1 without the extra assumption (3) there, since our gluing argument in 2

Sections 5 and 6 is only local (see also Section 8). The authors do not know his proof of the statement above, up to now. Acknowledgment. The authors would like to thank Grisha Perelman for correcting our claim in the first draft, on the relation between his Theorem 7.4 and our result, as above. The second author would like to thank John Morgan and Xiaochun Rong for the discussion at the American Institute of Mathematics, Palo Alto. 2. Strain radii and geometry of Alexandrov surfaces We discuss some basic properties of strain radii and metric balls in Alexandrov surfaces with curvature bounded below. See [1] for general facts on Alexandrov spaces. Let X be an m-dimensional complete Alexandrov space with curvature bounded below, say, curvature ≥ −1. For two points x, y in X, a minimal geodesic joining x to y is denoted by xy. The angle between minimal geodesics xy and xz is denoted by ∠yxz. For a geodesic ˜ triangle ∆xyz in X with vertices x, y and z, we denote by ∠xyz the corresponding angle at y˜ of a comparison triangle ∆˜ xy˜z˜ for ∆xyz in the hyperbolic plane of constant curvature −1. For δ > 0, the δ-regular set Rδ (X) is defined as the set of points p ∈ X such that there exists m pairs of points, (ai , bi ), 1 ≤ i ≤ m, called a δ-strainer at p, such that ˜ i pbi > π − δ, ∠a ˜ i paj > π/2 − δ, ∠a ˜ i pbj > π/2 − δ, ∠a ˜ i pbj > π/2 − δ, ∠b

for every i 6= j. The number min {d(ai , p), d(bi , p) | 1 ≤ i ≤ m} is called the length of the strainer. The δ-strain radius at p, denoted by δ-str. rad(p), is defined as the supremum of such r > 0 that there exists a δ-strainer at p of length r. For a closed domain D of Rδ (X), the δstrain radius of D, denoted by δ-str. rad(D), is defined as the infimum of δ-str. rad(p) when p runs over D. It should be noted that the notion of strain radius is a natural generalization of that of injectivity radius for Riemannian manifolds. For 1 ≤ n ≤ m, an (n, δ)-strainer at p is defined by n pairs of points, {(ai , bi )}, satisfying the same inequalities as above. For a subset C of X, we denote by B(C, r) or B(C, r; X) the closed metric r-ball around C and by S(C, r) or S(C, r; M) the metric r-sphere around C. For r < R, A(C; r, R) denotes the closure of B(C, R) − B(C, r). Lemma 2.1. For any m, a > 0, d > 0, r > 0 and δ > 0, there exists a positive number s = sm (a, d, r, δ) such that if B is a metric ball in an m-dimensional complete Alexandrov space X with curvature ≥ −1 satisfying (2.1)

area(B) ≥ a,

3

diam(B) ≤ d,

then the closure D of B − B(Sδ (X), r) has a definite lower bound for the strain radius: (2.2)

δ-str. rad(D) ≥ s.

Proof. Certainly we have a positive number sX depending on X with δstr. rad(D) ≥ sX . Since the set of all isometry classes of m-dimensional compact Alexandrov spaces satisfying (2.1) is compact with respect to the Gromov-Hausdorff distance, this provides a uniform positive lower bound sm (a, d, r, δ) for all sX . Thus the domain D has “bounded geometry” in the sense of (2.2). This elementary fact is important in our gluing argument in Section 5. The complement Sδ (X) := X − Rδ (X) is called the δ-singular set. Setting Sδ (int X) := Sδ (X) ∩ int X, we note that Sδ (X) = Sδ (int X) ∪ ∂X. Let ES(int X) denote the essential singular set of int X, i.e., the set of points p ∈ int X with radius rad(Σp ) := min max d(ξ, η) ≤ π/2. η∈Σp ξ∈Σp

From now on, we assume m = 2. Then X is known to be a topological two-manifold possibly with boundary. Moreover Sδ (int X) is discrete for any δ > 0. Lemma 2.2. For any p ∈ X, δ > 0 and D > 0, the number of elements of Sδ (int X) ∩ B(p, D) has a uniform upper bound Const(δ, D). In particular we have #(ES(int X) ∩ B(p, D)) ≤ Const(D).

This follows from an argument similar to Corollary 14.3 of [9], and hence the proof is omitted. For every p ∈ X, dp denotes the distance function from p. dp is called regular at a point q 6= p if there exists a ξ ∈ Σq such that the directional derivatives of dp satisfies d′p (ξ) > 0. Lemma 2.3 ([8]). For a fixed p ∈ X, there exists a set E ⊂ (0, ∞) of measure zero such that for every t ∈ (0, ∞) − E (1) t is a regular value of dp ; (2) B(p, t) is a topological manifold with (possibly empty) rectifiable boundary. As a consequence of Lemmas 2.2 and 2.3, we have Corollary 2.4. There exists a positive number σ = σ(δ) satisfying the following: For every Alexandrov surface X as above with diam(X) > 2, for every p ∈ X and for every t ∈ [1/2, 1], there exists ρ ∈ (t−10−2 , t+ 10−2 ) such that (1) B(p, ρ) is a topological manifold; (2) B(S(p, ρ), σ) ∩ int X ⊂ Rδ (X); (3) B(S(p, ρ), σ) is homeomorphic to S(p, ρ) × (0, 1). 4

Proof. The existence of σ satisfying (1) and (2) above is immediate from Lemmas 2.2 and 2.3. For (3), it suffices to prove that for every R > 0 there exists a positive constant const(R) such that if B(p, R) is a topological manifold, then the Euler number satisfies χ(B(p, R)) ≥ −const(R). Suppose this does not hold. Then we have a sequence (Xi , pi ) of pointed Alexandrov surfaces with curvature ≥ −1 with uniformly bounded Ri such that (1) B(pi , Ri ) is a topological manifold ; (2) χ(B(pi , Ri )) → −∞. We may assume that (Xi , pi) converges to a pointed Alexandrov surface (X, p). If dim X = 1, it is not hard to see that B(pi , Ri ) is either a cylinder or a Möbius band. If dim X = 2, then take an R with Ri ≤ R for every i and choose a regular value S of dp with S > R such that B(p, S) is a topological manifold. Then B(pi , S) is homeomorphic to B(p, S) by the stability result (see [6]). This is a contradiction. In what follows, we let δ ∗ := δ, which is a sufficiently small positive number determined later on in (5.2). We also denote the constant σ given in Corollary 2.4 by σ ∗ := σ(δ ∗ ).

(2.3)

3. Local structure A local S 1 -action ψ on a three-manifold M possibly with boundary consists of an open covering {Uα } of M and a nontrivial S 1 -action ψα on each Uα such that both the actions ψα and ψβ coincide up to orientation on the intersection Uα ∩ Uβ . Let X := M/S 1 , and π : M → X be the projection. X is a topological two-manifold (see [5] for instance). Set (3.1)

∂∗ X := ∂X − ∂0 X.

∂0 X := π(∂M),

The fixed point set of ψ coincides with ∂∗ X.

Lemma 3.1. If a compact three-manifold M admits a local S 1 -action with no singular orbits on ∂M, then it is a graph manifold. Proof. Note that each component C of ∂∗ X is a circle. Take a small collar neighborhood E(C) of C in X. Then N(C) := π −1 (E(C)) is a solid torus. Setting [ X0 := X − int N(C), M0 := π −1 (X0 ), C

we have the decomposition

M = M0 ∪

[ C

N(C) ,

where C runs over all the components of ∂∗ X. Since M0 is a Seifert fibered space over X0 , M is certainly a graph manifold. 5

In what follows, let M denote an orientable closed Riemannian manifold of dimension three satisfying (3.2)

K ≥ −1,

vol(M) < ǫ.

We shall determine the geometry and topology of local neighborhoods of M. First we recall the topological structure result for such an M when it has uniformly bounded diameter. Theorem 3.2 ([9]). For a given D > 0, there exists a positive constant ǫ(D) > 0 satisfying the following: If M satisfies diam(M) ≤ D and (3.2) with ǫ ≤ ǫ(D), then there exists a (possibly singular) fibration f : M → X, where X is a compact Alexandrov space with curvature ≥ −1 and dim X ≤ 2. The fiber structure of M can be described in more detail as follows: (1) If dim X = 2, then f is defined by a local S 1 -action on M with a possible exceptional orbit over a point in ES(int X). (2) Let dim X = 1. If X is a circle, then M is either a spherebundle or a torus-bundle over X. If X is a closed interval, then M is a gluing of U and V along their boundaries, where U ˜ or ones of S 1 × D2 and K 2 ×I. ˜ and V are ones of D3 and P 2×I (3) If dim X = 0, then a finite cover of M is homeomorphic to S 1 × S 2 , T 3 , a nilmanifold or a simply connected Alexandrov space with nonnegative curvature. The case when X is a circle was proved in [10], and the essential part of the case of dim X = 0 was proved in [4]. By Lemma 3.1, a three-manifold M satisfying one of the conclusions in Theorem 3.2 is a graph manifold except the case when M has finite fundamental group and dim X = 0. We also obtain some universal positive constants δ0 and ǫ0 such that if M satisfies diam(M) < δ0 and vol(M) < ǫ0 , then M is homeomorphic to one of the spaces in Theorem 3.2 (3). Thus Theorem 1.1 certainly holds in the bounded diameter case. Therefore from now we assume that M has large diameter: (3.3)

diam(M) ≫ 1.

We now determine the topology of a local neighborhood of each point of M. A submersion f : M → N between Riemannian manifolds is called an ǫ-almost Riemannian submersion if (1) the diameter of every fiber of f is less than ǫ; (2) for every point p ∈ M and every tangent vector ξ at p that is normal to the fiber f −1 (f (p)), |df (ξ)| < ǫ. − 1 |ξ| 6

Note that an ǫ-almost Riemannian submersion is a fiber bundle map since it is proper. We denote by τ (ǫ) (resp. τ (r|ǫ)) a function of ǫ (resp. of r and ǫ) with limǫ→0 τ (ǫ) = 0 (resp. limǫ→0 τ (r|ǫ) = 0 for each fixed r). The following is a quantitative version of Theorem 3.2 (2) (see [10]). Corollary 3.3. There exists a positive number ǫ∗1 such that if a closed three-manifold M with K ≥ −1 satisfies dGH (M, I) < ǫ∗1 for some closed interval I of length ≥ 1/2, then there exists a singular fibration f : M → I as in Theorem 3.2 such that (1) the diameter of every fiber of f is less than τ (ǫ∗1 ); (2) the restriction of f to Ir is a τ (r|ǫ∗1 )-almost Riemannian submersion, where r > 0 and Ir := {x ∈ I | d(x, ∂I) ≥ r}. Later we shall take ǫ∗1 such as ǫ∗1 ≪ σ ∗ (see (4.5)). The final choice of ǫ∗1 will be determined at the end of Section 5. Let a∗ be a positive number such that if B is a metric ρ-ball with 1/10 ≤ ρ ≤ 1 in a complete Alexandrov surface X with curvature ≥ −1 and area(B) < a∗ , then (3.4)

dGH (B, I) < ǫ∗1 /2,

for some closed interval I. A surjective map f : M → X between Alexandrov spaces is called an ǫ-almost Lipschitz submersion if (1) the diameter of every fiber of f is less than ǫ; (2) for every p, q ∈ M, if θ is the infimum of ∠qpx when x runs over f −1 (f (p)), then d(f (p), f (q)) − sin θ < ǫ. d(p, q)

Remark that the notion of ǫ-almost Lipschitz submersion is a generalization of ǫ-almost Riemannian submersion. The following result was proved in Theorem 0.2 of [11] (see also Theorem 2.2 of [9]).

Theorem 3.4 ([11]). For given m and s > 0 there exists ν > 0 satisfying the following: Let X be an m-dimensional complete Alexandrov space with curvature ≥ −1 and with δ ∗ -str. rad(X) ≥ s. Then if the Gromov-Hausdorff distance between X and a complete Riemannian manifold M with K ≥ −1 is less than ν, then there exists a (τ (δ ∗ ) + τ (s|ν))-almost Lipschitz submersion f : M → X which is a locally trivial bundle map. The following is a localized and quantitative version of Theorem 3.2. Theorem 3.5. For every r > 0, there exists a positive constant ǫ0 = ǫ0 (a∗ , r, δ ∗ ) satisfying the following: For every M satisfying (3.2) and (3.3) with ǫ ≤ ǫ0 and for every p ∈ M, there exist closed domains Bp 7

ˆp around p and a pointed complete Alexandrov space (X, x0 ) with and B curvature ≥ −1 and dim X ∈ {1, 2} such that ˆp are small perturbations of metric balls around p and, (1) Bp and B ˆp ⊂ B(p, 1), B ˆp − int Bp ≃ ∂Bp × I; B(p, 1/2) ⊂ Bp ⊂ B ˆp have fiber structures over concentric metric balls (2) Bp and B ˆ p in X around x0 ; Xp ⊂ X

ˆ p ⊂ B(x0 , 1), ˆ p − Xp ≃ ∂0 Xp × I, (3) B(x0 , 1/2) ⊂ Xp ⊂ X X where ∂0 denotes the topological boundary. Moreover the fiber structure on Bp in (2) can be described as follows: ˆp , Bp ) → (X ˆ p , Xp ) be the fiber projection, and let Let πp : (B be defined as in (3.1). Case (A)

∂∗ Xp := ∂Xp − ∂0 Xp .

ˆ p = 1. (X ˆ p is a closed interval I in this case). dim X

ˆp , X ˆ p ) < ǫ∗ , and the diameter of every fiber of πp is less (a) dGH (B 1 ∗ than τ (ǫ1 ); (b) The restriction of πp to Ir is a τ (r|ǫ∗1 )-almost Riemannian submersion; (c) If ∂∗ Xp is empty, then Bp is homeomorphic to either I × S 2 or I × T 2; (d) If ∂∗ Xp is nonempty, then Bp is homeomorphic to one of D3 , ˜ , S 1 × D2 and K 2 ×I. ˜ P 2 ×I Case (B)

ˆ p = 2. dim X

ˆp , X ˆ p ) < τ (ǫ), and the length of every fiber of πp is less (a) dGH (B than τ (ǫ); ˆp ⊂ B(Bp , 2σ ∗ ); (b) B(Bp , σ ∗ ) ⊂ B (c) B(∂Xp , 2σ ∗ ) ∩ int X ⊂ Rδ∗ (X); (d) B(∂0 Xp , 2σ ∗ ) is homeomorphic to ∂0 Xp × (0, 1); ˆ p − B(Sδ∗ (X), r) satisfies (e) D := X ∗ (i) δ -str. rad(D) ≥ s, where s = s2 (a∗ , 1, r, δ ∗ ) is the constant as in Lemma 2.1; (ii) the restriction of πp to D is (τ (δ ∗ )+τ (s|ǫ))-almost Lipschitz submersion which is an S 1 -bundle; ˆp whose fixed point set corre(f ) πp gives a local S 1 -action on B ˆ p , where there is a possible exceptional fiber over sponds to ∂∗ X ˆ p only when x ∈ ES(int X). a point x ∈ X Proof. Suppose the theorem does not hold. Then there exist sequences ǫi → 0 and Mi satisfying (3.2) for ǫi and (3.3) such that for some pi ∈ Mi , B(pi , 1) does not contain closed domains satisfying the above conclusion. Passing to a subsequence if necessary, we may assume that (Mi , pi ) converges to a pointed complete Alexandrov space (X, x0 ) with 8

curvature ≥ −1. Observe 1 ≤ dim X ≤ 2. In view of Corollary 2.4 it is possible to take metric balls Y ⊂ Yˆ of X around x0 which are topological manifolds, satisfying (1) B(x0 , 1/2) ⊂ Y ⊂ B(Y, σ ∗ ) ⊂ Yˆ ⊂ B(Y, 2σ ∗ ) ⊂ B(x0 , 1); (2) B(∂0 Y, 2σ ∗ ) ∩ int X ⊂ Rδ∗ (X); (3) B(∂0 Y, 2σ ∗ ) is contained in a neighborhood of ∂0 Y homeomorphic to ∂0 Y × (0, 1); (4) Yˆ − Y ≃ ∂0 Y × I. If area(Yˆ ) < a∗ , then (Yˆ , Y ) are Gromov-Hausdorff close to some closed ˆ I), and we put X ˆ p := I, ˆ Xp := I in this case. If area(Yˆ ) ≥ intervals (I, ∗ ˆ p := Yˆ and Xp := Y . By Theorem 3.2, Corollary 3.3 a , then we put X and Theorem 3.4 together with Lemma 2.1, we obtain closed domains î with B(pi , 1/2) ⊂ Bi ⊂ B î ⊂ B(pi , 1) such that Bi and B î Bi and B are fiber spaces over Y and Yˆ respectively as described above satisfying all the conclusions, which is a contradiction. ˆp and Remark 3.6. (1) The several geometric properties of Bp ⊂ B ˆ p in Theorem 3.5 will be needed in the gluing argument Xp ⊂ X later on. ˆp (2) We will also need to consider a slight deformation of Xp ⊂ X according to requirements. From now on, we put (3.5)

r := σ ∗ /100,

s := s2 (a∗ , 1, r, δ ∗).

Then the constant ǫ0 = ǫ0 (a∗ , r, δ ∗ ) in Theorem 3.5 will become universal (see the end of Section 5). We now recall basic geometric properties of the regular fibers of πp . Definition 3.7. For a point p ∈ M suppose that there is a (2, δ ∗ /2)strainer {(aj , bj )} at p of length ≥ s/2. Then the subspace of the tangent space at p generated by the directions of minimal geodesics joining p to a1 and a2 is called a horizontal subspace at p. Let a small circle F in M be given in such a way that for every p ∈ F there is a (2, δ ∗ /2)-strainer at p of length ≥ s/2. For τ > 0, F is called τ perpendicular to horizontal subspaces if for each point p ∈ F , the angle θ between F and every horizontal subspace at p satisfies |θ − π/2| < τ.

ˆp → X ˆ p and D ⊂ X ˆ p (resp. Ir ⊂ X ˆp) Lemma 3.8 ([10],[11]). Let πp : B −1 be as in Theorem 3.5. For every x ∈ D (resp. x ∈ Ir ) and q ∈ πp (x), the following holds: (1) For every q ′ with d(q, q ′ ) ≥ r, the angle θ between πp−1 (x) and every minimal geodesic joining q to q ′ satisfies |θ − π/2| < τ (δ ∗ ) + τ (s|ǫ) 9

(resp. τ (r|ǫ∗1 ));

(2) The fiber πp−1 (x) is (τ (δ ∗ ) + τ (s|ǫ))-perpendicular to horizontal subspaces. 4. Decomposition Let M satisfy (3.2) with ǫ ≤ ǫ0 . Take points p1 , p2 , . . . , of M such ˆp } given by Theorem 3.5 are finite that the collections {Bpi } and {B i coverings of M. We may assume that (4.1) (4.2)

d(pi , pj ) ≥ 1/10 for every i 6= j;

{Bpi (1/10)} covers M,

where Bpi (1/10) := {x ∈ Bpi | d(x, ∂Bpi ) ≥ 1/10}. By the BishopGromov volume comparison theorem, we may assume that the maximal ˆp ’s having nonempty intersection is uniformly bounded number of B i ˆp above by a universal constant Q not depending on M. Let Xpi ⊂ X i ˆp . be chosen as in Theorem 3.5 for Bpi ⊂ B i For simplicity, we put î := B ˆp , Xi := Xp , X ˆ i := X ˆp . Bi := Bp , B i

i

i

i

1

If dim Xi = 2, then there exists a local S -action ψi on Bi such that î → X ˆ i is the projection. Bi /ψi ≃ Xi , where πi := πpi : B For each j ∈ {1, 2}, let Ij denote the set of all i with dim Xi = j, and consider [ Uj := Bi . i∈Ij

Let Bj := {Bi | dim Xi = j}, j ∈ {1, 2}. By Theorem 3.5, each element of B1 is either cylindrical or cylindrical with a cap (see Introduction).

Lemma 4.1. Each component of U1 is homeomorphic to one of D3 , ˜ S 2 × I, S 1 × D2 , K 2 ×I ˜ and T 2 × I unless U1 = M. P 2 ×I, If U1 = M, then M is homeomorphic to one of the spaces in Theorem 3.2 (2). Proof. Slightly enlarging closed domains Bi in B1 if necessary, we may assume that any two Bi , Bj in B1 has intersection Bi ∩ Bj which is either empty or else having diameter > σ ∗ . Since U1 is a part of M which looks one-dimensional in the Gromov-Hausdorff sense, it follows from (4.1) that U1 is a manifold. Suppose that Bi ∩ Bj is nonempty for two domains Bi , Bj in B1 . In the argument below, we may assume that Bi 6⊂ Bj and Bj 6⊂ Bi . Since Bi and Bj are either cylindrical or cylindrical with a cap, it follows from (3.3) that at least one of Bi and Bj , say Bj , has disconnected boundary. Consider the distance function dpi , where pi is the reference point of Bi . Letting F denote S 2 or T 2 , we know that Bi and Bj have F -fiber structures over I, which is singular at the top of the cap. By Lemma 3.8, one can construct a gradientlike vector field Vi for dpi on a neighborhood of Bj − Bi whose flow 10

curves are transversal to every fiber of Bj lying on a neighborhood of Bj − Bi . In view of (3.3), it follows that Bi ∪ Bj is homomorphic to Bi . Repeating the argument finitely many times, we obtain the conclusion of the lemma. From now on, we assume U1 6= M, and consider the decomposition of M M = U1 ∪ Uˆ2 , ˆ2 denotes the closure of M − U1 . where U For every fixed component L of ∂U1 , there exists a unique Bℓ ∈ B1 such that a component of ∂Bℓ coincides with L. Let B1L , . . . , BnL denote the set of all elements of B2 such that BiL (1/10) meets L, 1 ≤ i ≤ n. Since diam(L) < τ (ǫ∗1 ), BiL contains L. Let Li be the unique component of ∂BiL meeting Bℓ . Since the domain bounded by L and Li is one-dimensional in the Gromov-Hausdorff sense, it follows from the curvature condition that B1L , . . . , BnL lie in a linear order and for every 1 ≤ i 6= j ≤ n (4.3)

d(∂BiL , ∂BjL ) ≥ 1/20,

and we may assume that (4.4)

BnL ⊃ Li .

˜2 the union of all Bj in B2 which does not intersect U1 . We denote by U Obviously we have [ U2 = U˜2 ∪ (B1L ∪ · · · ∪ BnL ) , L

where L runs over all the components of ∂U1 .

Lemma 4.2. U2 is homeomorphic to Uˆ2 . ˆ L ⊃ BL, X ˆ L ⊃ XL Proof. Fix a component L of ∂U1 again. Let B n n n n L L L L L ˆ , B ) → (X ˆ , X ) be the orbit projection as in Theorem and πn : (B n n n n ˆ n denote the component of ∂ B ˆ L corresponding to Ln , and 3.5. Let L n let U be the domain bounded by L and Ln . Take a point x ∈ Uˆ2 with ˆ n . Since every fiber of π L meeting U has d(x, L) ≥ 1 and a point y ∈ L n ∗ diameter < τ (ǫ1 ), it follows that for every z ∈ U ˜ (4.5) ∠xzy > π − τ (σ ∗ |ǫ∗ ). 1

Let V be a gradient-like vector field for dx defined on a neighborhood of U.

Assertion 4.3. The flow curves of V are transversal to both L and Ln . Proof. Since the transversality to L is immediate from Lemma 3.8 (1), it suffices to check the transversality to Ln . For every p ∈ Ln , let φp (t) be the flow curve of V with φp (0) = p. Put q := expp σ ∗ V (p), 11

p¯ := πnL (p) and denote by ξ¯ the direction at p¯ defined by a minimal geodesic to πnL (q). In a way similar to Lemma 4.6 of [11], we have ¯ < t(τ (δ ∗ ) + τ (s|ǫ)), d(π L(φp (t)), exp tξ) n

p¯

for every sufficiently small t > 0. This implies that φp (t) makes an angle with Ln uniformly bounded away from zero. Now in view of (4.4), Assertion 4.3 implies U2 ≃ Uˆ2 . The proof of the following lemma is deferred to Sections 5 and 6.

Lemma 4.4. There exists a local S 1 -action defined on U2 , and hence on Uˆ2 . Proof of Theorem 1.1 assuming Lemma 4.4. Note that each component of ∂ Uˆ2 is homeomorphic to S 2 or T 2 . For each component L of ∂ Uˆ2 , let W (L) be the component of U1 containing L. Suppose first that L is homeomorphic to S 2 . Then one of the following holds: (1) ∂W (L) is connected and W (L) is homeomorphic to either D3 ˜ or P 2 ×I; (2) W (L) is homeomorphic to S 2 × I, and the other component of ∂W (L) is another component of Uˆ2 . Now consider the union [ V := Uˆ2 ∪ W (L) , L

where L runs over all the components of Uˆ2 homeomorphic to S 2 . Since an S 1 -action on S 2 is essentially by rotation, the local S 1 action on Uˆ2 extends to a local S 1 -action on V such that the orbit space W (L)/S 1 is a disk whose singular locus is one of (1) an interval on the boundary of W (L)/S 1 ( the case of W (L) ≃ D3 ); (2) the union of an interval on the boundary of W (L)/S 1 and a point in int W (L)/S 1 of type (2, 1)-singularity (the case of ˜ W (L) ≃ P 2 ×I); (3) the disjoint union of two intervals on the boundary of W (L)/S 1 (the case of W (L) ≃ S 2 × I). Note also that each component of ∂V is homeomorphic to T 2 and having no singular orbits. Therefore Lemma 3.1 implies that V is a graph manifold. From construction, for each component L of ∂V , one of the following holds: (a) ∂W (L) is connected and W (L) is homeomorphic to either S 1 × ˜ D2 or K 2 ×I; (b) W (L) is homeomorphic to T 2 × I, and the other component of ∂W (L) is another component of ∂V . Thus M is a graph manifold. 12

5. Gluing ˆ be closed domains in a closed orientable three-manifold M Let B ⊂ B ˆ be concentric closed with sectional curvature K ≥ −1, and let X ⊂ X ˆ metric balls of radii t < t in a two-dimensional complete Alexandrov space Z with curvature ≥ −1. Assume that ˆ and X ˆ (resp. B and (1) the Gromov-Hausdorff distance between B X) is sufficiently small; ˆ satisfy the conclusion of Case (B) in Theorem 3.5; (2) X ⊂ X (3) area(X) ≥ a∗ ; (4) 1/10 ≤ t ≤ t + s ≤ tˆ ≤ 1, ˆ := X ˆ − where s is as in (3.5). Let D := X − B(Sδ∗ (Z), r), D ∗ ˆ ≥ s. Applying Theorem 3.4, we B(Sδ∗ (Z), r). Note that δ -str. rad(D) ˆ and N of B ˆ and B respectively, and an almost have closed domains N ˆ ˆ D), which is an S 1 -bundle. Lipschitz submersion π : (N , N) → (D, First we need to establish the uniform boundedness of length ratio for the fibers of π : N → D.

Lemma 5.1. There exists a ζ = ζ(a∗ , s∗ , δ ∗ ) > 0 such that the following holds: Suppose that ˆ the length ℓ(x) of the fiber π −1 (x) is less than (1) for every x ∈ D ζ; (2) for every x ∈ Dand p ∈ π −1 (x), letting θ(p) denote the angle between π −1 (x) and a horizontal subspace at p, we have Then

|θ(p) − π/2| < ζ.

ℓ(x) < c, ℓ(y) for every x, y ∈ D, where c = c(a∗ , s) is a uniform positive constant. c−1
0 and ζ = ζ(ν) > 0 such that the following holds: Let π : B → X, ˆ ′ , B ′ ) → (X ˆ ′ , X ′ ) be as above satisfying π ′ : (B (a) every fiber F of π contained in N1 and F ′ of π ′ contained in N ′ are ζ-perpendicular to horizontal subspaces; (b) any two orbits of π in N with distance ≤ 1 have length ratio uniformly bounded as in Lemma 5.1. We also assume that (5.1)

π ′ (∂B ′ ∩ B) ⊂ D′ . 14

Then we have a local S 1 -action ψ ′′ on a small perturbation B ∪ B ′ of B ∪ B ′ and a topological two-manifold X ′′ with the orbit projection π ′′ : B ∪ B ′ → X ′′ satisfying (1) B ∪ B ′ is a manifold with boundary, and (B ∪ B ′ )(10ℓ′) ⊂ B ∪ B ′ ⊂ B(B ∪ B ′ , 10ℓ′),

(2)

where ℓ′ denotes the maximal length of fibers of π ′ meeting ∂B ′ ∩ B, and (B ∪ B ′ )(10ℓ′ ) = {x ∈ B ∪ B ′ | d(x, ∂(B ∪ B ′ )) ≥ 10ℓ′ }; ( ψ ψ ′′ = ψ′

on B ∪ B ′ − B(B ′ , 10ℓ′ ) on B ′ ;

(3) each orbit of ψ ′′ has length < 2ℓ′′ , where ℓ′′ denotes the maximal length of all fibers of π and π ′ intersecting 10ℓ′ -neighborhood of ∂B ′ ∩ B; (4) every fiber of π ′′ contained in N1′′ is ν-perpendicular to horizontal subspaces, where N1′′ is the set of points p of B ∪ B ′ such that there is a (2, 2δ ∗)-strainer at p of length ≥ s/2.

ˆ B) and Remark 5.4. Under the situation of Lemma 5.3, if both (B, ′ ′ ˆ , B ) are as in Lemma 5.1, then by Lemmas 5.1 and 5.2, B → X (B ˆ ′ , B ′ ) → (X ˆ ′ , X ′) satisfy the assumptions of Lemma 5.3 and π ′ : (B except (5.1) if τ (2δ ∗ ) + τ (s/2|ǫ) < ζ which is realized by δ ∗ ≪ 1 and ǫ ≪ δ ∗ . Note that the proof of Lemma 5.1 goes through for N1 as well in place of N. For the proof of Lemma 5.3, we need a sublemma. Let A be a small neighborhood of π ′ (∂B ′ ∩ B) in π ′ (∂B ′ ), and let C be the closure of the intersection of int X with the boundary of the 10ℓ′ neighborhood of π(B ∩ B ′ ). Slightly perturbing A and C if necessary, we may assume that both are one-manifolds. Fix any x, xˆ ∈ A with 10ℓ′ ≤ d(x, xˆ) ≤ 20ℓ′ . Taking a nearest point z of π ′ (π −1 (C)) from x, choose any point y ∈ π((π ′ )−1 (z)). Similarly we choose yˆ ∈ C for xˆ. Put F ′ := (π ′ )−1 (x), Fˆ ′ := (π ′ )−1 (ˆ x), F := (π)−1 (y) and Fˆ := (π)−1 (ˆ y ). Sublemma 5.5. Under the situation above, there exist δ ∗ = δ ∗ (ν) > 0 and ζ = ζ(ν) > 0 satisfying the following: (1) ℓ(F ) ℓ(F ′ ) − 1 < ν,

where ℓ(F ) denotes the length of F . ˆ in M equipped with an (2) There exists an annulus E (resp. E) 1 S -fiber structure via a bi-Lipschitz homeomorphism h : [0, 1] × ˆ : [0, 1] × S 1 → E) ˆ such that S 1 → E (resp. h 15

ˆ × S 1) (a) F = h(0 × S 1 ) and F ′ = h(1 × S 1 ) (resp. Fˆ = h(0 ˆ × S 1 )); and Fˆ ′ = h(1 ˆ × S 1 ) ) is ν(b) for each t ∈ [0, 1], h(t × S 1 ) (resp. h(t perpendicular to horizontal subspaces. (3) Let [x, xˆ] and [y, yˆ] be the subarcs of A and C respectively, and ˆ Then let T be the union of (π ′ )−1 ([x, xˆ]), π −1 ([y, yˆ]), E and E. 1 the domain D bounded by T has an S -fiber structure via a biLipschitz homeomorphism k : D2 × S 1 → D such that (a) for each x ∈ ∂D2 , k(x × S 1 ) coincides with a fiber on T ; (b) for each x ∈ D2 , k(x × S 1 ) is ν-perpendicular to horizontal subspaces.

F F

E D E

π−1(C)

F’ F’ −1

(π’) (A)

B

B’

y

x

y

x’

C

A

X

X’

Figure 1. Proof. We prove it by contradiction. If the conclusion does not hold, we would have a sequence of closed three-manifolds Mi with K ≥ −1 ˆ ′ , B ′ ) → (X ˆ ′ , X ′) satisfying for which there are πi : Bi → Xi , πi′ : (B i i i i the assumptions of Sublemma 5.5 for δi∗ → 0 and ζi → 0, but not 16

satisfying the conclusions for ζ. Let Ai ⊂ ∂Xi′ , Ci ⊂ Xi , xi , x î ∈ Ai and yi , yî ∈ Ci be defined as above. In particular, 10ℓ′i ≤ d(xi , x î ) ≤ 20ℓ′i , where ℓ′i is defined in a way similar to ℓ′ (see Lemma 5.3 (1)). Put Fi′ := (πi′ )−1 (xi ), Fî′ := (πi′ )−1 (ˆ xi ), Fi := (πi )−1 (yi ) and Fî := (πi )−1 (ˆ yi ). ′ Let ℓi (yi ) and ℓi (xi ) denote the length of Fi and Fi′ respectively. Passing 1 Mi , xi ) converges to a to a subsequence, we may assume that ( ℓ′ (x i i) pointed space (W, w0 ), where W is a complete Alexandrov space with nonnegative curvature. From assumption, we see that W is actually isometric to R2 × S11 , where S11 denotes the circle of length 1. Thus 1 for any fixed R ≫ 1, B(xi , R; ℓ′ (x Mi ) is almost isometric to B(w0 , R). i i) This together with the condition (a) of Lemma 5.3 implies that ℓ′i (xi ) and the length of any orbit of πi nearby (πi′ )−1 (xi ) are comparable in the sense of Lemma 5.1. Then by (5.1), ℓ′i (xi ) and ℓi (yi ) are comparable. (yi ) → 1, which proves (1). The above convergence then yields ℓℓ′i(x i i) Let F , Fˆ , F ′ and Fˆ ′ be the limits of Fi , Fî , Fi′ and Fî′ respectively under the above convergence. Since F and F ′ (resp. Fˆ and Fˆ ′ ) can be ˆ joined by one-parameter family of parallel circles, say E (resp. say E) ′ of length 1, Fi and Fi can be joined by one-parameter family, say Ei î ) of circles each of which is ν-perpendicular to horizontal (resp. say E 1 subspaces for sufficiently large i. Let ϕi : B(w0 , R) → B(xi , R; ℓ′ (x Mi ) i i) be an almost isometry. Note that the closed domain Di bounded by πi−1 ([yi , yî]), (πi′ )−1 ([xi , x î ]), Ei and Eî is mapped via ϕ−1 onto a doi −1 −1 ˆ main D bounded by E, E, π ([x, xˆ]) and π ([y, yˆ]). Note that ϕi maps horizontal subspaces to horizontal subspaces (see [10] for the details). Since D is isometric to a product H × S11 for a rectangle H, this gives a compatible S 1 -fiber structure on Di each of whose fibers is ν-perpendicular to horizontal subspaces. This is a contradiction. Proof of Lemma 5.3. We shall carry out the required gluing procedure on each component, say U, of B ∩ B ′ . Let A0 be any component of A ∩ π ′ (U), and take consecutive points x1 , . . . , xN of A0 with 10ℓ′ ≤ d(xα , xα+1 ) ≤ 20ℓ′ for each 1 ≤ α ≤ N − 1. First consider Case (A) ∂B does not meet ∂B ′ on U. In this case, both A0 and the component C0 of C corresponding to A0 are circles. Applying Sublemma 5.5 to x := xα and xˆ := xα+1 , we obtain a closed domain D bounded by π −1 (C0 ) and (π ′ )−1 (A0 ) having an S 1 -bundle structure via a bi-Lipschitz homeomorphism k : (I ×S 1 )× S 1 → D such that (1) for each x ∈ ∂I × S 1 , k(x × S 1 ) coincides with a fiber on π −1 (C0 ) ∪ (π ′ )−1 (A0 ); (2) for each x ∈ I × S 1 , k(x × S 1 ) is ν-perpendicular to horizontal subspaces. 17

B

D

U (π’)−1 (A0 )

π−1(C0 )

π(U)

X C0

Figure 2. Case (A) Case (B) ∂B meets ∂B ′ on U. In this case A0 is an arc. In a way similar to Case (A), we apply Sublemma 5.5 to obtain a closed domain D bounded by π −1 (C0 ), (π ′ )−1 (A0 ) and two annuli joining (π ′ )−1 (∂A0 ) and (π ′ )−1 (∂C0 ) which has an S 1 bundle structure via a bi-Lipschitz homeomorphism k : I 2 × S 1 → D such that (1) for each x ∈ ∂I 2 , k(x × S 1 ) coincides with a fiber on ∂D; (2) for each x ∈ I 2 , k(x × S 1 ) is ν-perpendicular to horizontal subspaces. Thus we obtain the conclusion of Lemma 5.3.

Let ν > 0 be sufficiently small like ν = 10−10 , and let δ ∗ = δ ∗ (ν) and ζ = ζ(ν) be the constants given in Lemma 5.3. Letting Q be the positive integer in Section 4 and setting Q-times

(5.2)

z }| { ζ := ζ(ζ(· · · (ζ(ν)) · · · )), ∗

δ ∗ := δ ∗ (ζ ∗ ),

we choose ǫ in (3.2) satisfying (5.3)

τ (2δ ∗ ) + τ (s/2|ǫ) < ζ ∗, 18

D −1

(π’) (A ) 0

π−1(C ) 0

B

B’

π(U) C

0

X

Figure 3. Case (B) where r and σ ∗ are defined as in (3.5) and (2.3). Note that ζ ∗ , δ ∗ , σ ∗ and s are universal constants. We also choose a universal constant ǫ∗1 in Corollary 3.3 like ǫ∗1 ≪ σ ∗ . 6. Proof of Lemma 4.4 In this section, we shall prove Lemma 4.4. dM H denotes the Hausdorff distance in M. Put for i1 , . . . , ik ∈ I2 .

Bi1 ,...,ik := Bi1 ∪ · · · ∪ Bik ,

Assertion 6.1. We assume that Bij (1/10) meets Bi1 ,...,ij−1 for every 2 ≤ j ≤ k. Then there exist a local S 1 -action ψi1 ,...,ik on Bi1 ,...,ik and a topological two-manifold Xi1 ,...,ik with the orbit projection πi1 ,...,ik : Bi1 ,...,ik → Xi1 ,...,ik satisfying the following : 19

(1) ψi1 ,...,ik =

( ψi1 ,...,ik−1

on Bi1 ∪ . . . ∪ Bik−1 − Bik

on Bik ;

ψik

(2) Bi1 ,...,ik is a manifold with boundary, and dM H (Bi1 ,...,ik , Bi1 ,...,ik ) < τ (ǫ). (3) Each orbit of ψi1 ,...,ik has diameter < τ (ǫ); (4) There are no singular orbits of ψi1 ,...,ik over ∂Xi1 ,...,ik −∂∗ Xi1 ,...,ik , where ∂∗ Xi1 ,...,ik is defined as in (3.1). (5) Let Ni1 ,...,ik ;1 ⊂ Bi1 ,...,ik be the set of points p ∈ Bi1 ,...,ik such that there is a (2, 2δ ∗ )-strainer at p of length ≥ s/2. Then every fiber F of ψi1 ,...,ik contained in Ni1 ,...,ik ;1 is ζ Q−n(ν)-perpendicular to horizontal subspaces, where (Q − n)-times

ζ

Q−n

z }| { (ν) := ζ(· · · (ζ (ν)) · · · )),

î containing F . and n denotes the number of B Proof. We prove it by induction. Assertion 6.1 certainly holds for k = 1 by Theorem 3.5, Lemmas 3.8(2) and 5.2. Assume that a local S 1 -action ψi1 ,...,ik−1 on Bi1 ,...,ik−1 satisfying Assertion 6.1 has been constructed. For simplicity, we set B := Bi1 ,...,ik−1 , ˆ ′ := B î , B k

X := Xi1 ,...,ik−1 , ˆ ′ := X î , X k

B ′ := Bik ,

ψ := ψi1 ,...,ik−1 ,

ψ ′ := ψik

π := πi1 ,...,ik−1 ,

π ′ := πik .

X ′ := Xik ,

We shall carry out a gluing procedure on each component, say U, of B ∩ B ′ using Lemma 5.3. ∂∗ X and ∂∗ X ′ are defined as in (3.1). Let A ⊂ ∂X ′ , A0 , C ⊂ X and C0 be defined as in the proof of Lemma 5.3. First consider the case when A0 does not meet B(Sδ∗ (Z ′ ), r), where Z ′ is the Alexandrov space containing X ′ , which implies (π ′ )−1 (A0 ) ⊂ N ′ . Thus every π ′ -fiber in (π ′ )−1 (A0 ) is ζ Q (ν)-perpendicular to horizontal subspaces. Since π −1 (C0 ) is close to (π ′ )−1 (A0 ), we obtain that π −1 (C0 ) ⊂ Ni1 ,...,ik−1 ;1 . Condition (5) of Assertion 6.1 for ψi1 ,...,ik−1 then implies that every π-fiber in (π)−1 (C0 ) is ζ Q−n+1(ν)-perpendicular to horizontal subspaces. Therefore we can apply Lemma 5.3 to get the required gluing of ψ and ψ ′ on a neighborhood joining (π ′ )−1 (A0 ) and π −1 (C0 ). Next consider the other case when A0 meets B(Sδ∗ (Z ′ ), r). In this case it follows from the condition (c) of Case (B) of Theorem 3.5 that an endpoint of A0 must be contained in B(∂∗ X ′ , r). Take x, xˆ ∈ A0 such that the subarc [x, xˆ] of A0 is contained in ′ X −B(Sδ∗ (Z ′ ), r) and A0 −[x, xˆ] as well as x is contained in B(∂∗ X ′ , r). 20

−1

(π’) (A ) 0

E

U

−1

π (C ) 0

E

B

B’

X* π(U) C

0

X

Figure 4. The case xˆ ∈ B(∂∗ X ′ , r) may happen. Let y, yˆ ∈ C0 be defined as in the proof of Lemma 5.3 for x, xˆ. Applying (the proof of) Lemma 5.3, we have a domain D bounded by (π ′ )−1 ([x, xˆ]), π −1 ([y, yˆ]) and two ˆ having a compatible S 1 -fiber structure via a biannuli, say E and E, Lipschitz homeomorphism k : D2 × S 1 → D such that for each x ∈ D2 , k(x × S 1 ) is ζ Q−n (ν)-perpendicular to horizontal subspaces. Note that each component, say W , of the domain bounded by (π ′ )−1 (A0 − [x, xˆ]), ˆ is a three-disk. Therefore one π −1 (C0 − [y, yˆ]) and E (and possibly E) 1 can put a compatible structure of S -action on W all of whose orbit has diameter < τ (ǫ). From the gluing constructions above, we obtain Bi1 ,...,ik and a local 1 S -action ψi1 ,...,ik on it satisfying the conclusion of the assertion. This completes the proof of Assertion 6.1. From (4.4) and the gluing argument used in the proof of Assertion 6.1, it is now obvious that U2 ≃ U2 . Thus we have completed the proof of Lemma 4.4. 7. Thick-thin decomposition For a three-manifold N, we denote by Cap(N) a three-manifold obtained by gluing of N and some copies of D3 along all the spherecomponents of ∂N. Theorem 7.1. If M is a closed orientable Riemannian three-manifold with sectional curvature K ≥ −1, then we have a decomposition M = Mthick ∪ Mthin , 21

satisfying the following: Let ǫ0 and δ0 be positive numbers given Theorem 1.1. (1) For every p ∈ Mthick , vol(B(p, 1)) ≥ ǫ0 /2. (2) Cap(Mthin ) is homeomorphic to a graph manifold if diam(M) ≥ δ0 . Roughly speaking, Mthin is a piece of M which collapses. Theorem 7.1 is closely related with a result in [7], where a thick-thin decomposition of a closed three-manifold in connection with Ricci flow is announced. Proof of Theorem 7.1. If diam(M) < δ0 , then we put Mthin := M. For the proof of Theorem 7.1, we may assume diam(M) ≥ δ0 . For every point p ∈ M, one of the following holds:

(1) vol B(p, 1) ≥ ǫ0 . In this case, we put Bp := B(p, 1) and Xp := B(p, 1). (2) vol B(p, 1) < ǫ0 . In this case, by Theorem 3.5 there exist a small perturbation Bp of B(p, 1) and a metric ball Xp in some complete Alexandrov space X with curvature ≥ −1 and 1 ≤ dim X ≤ 2 such that Bp has a fiber structure over Xp . Take points p1 , p2 , . . . , of M as in Section 4 such that the collection {Bpi } given as above is a covering of M. Let Xpi be also chosen as above. For simplicity, we put Bi := Bpi ,

Xi := Xpi .

If dim Xi = 2, then there exists a local S 1 -action ψi on Bi such that Bi /ψi ≃ Xi . For each 1 ≤ j ≤ 3, let Ij denote the set of all i with dim Xi = j, and consider [ Uj := Bi . i∈Ij

By Lemm 4.1, each component of U1 is either cylindrical or cylindrical with a cap. By Lemma 4.4, we have a small perturbation U2 of U2 on which one can construct a local S 1 -action. Note that each component of Mthin := U1 ∪U2 is homeomorphic to S 2 or T 2 . The argument in Section 4 shows that Cap(Mthin ) is homeomorphic to a graph manifold. Now it is obvious that for any point p in Mthick := M − Mthin , vol(B(p, 1)) ≥ ǫ0 /2. 8. Appendix: Collapsing under a local lower curvature bound In this appendix, we give a short description about collapsing threemanifold under a local lower curvature bound, which is discussed in [7]. 22

For a positive number ǫ, a closed Riemannian n-manifold (M, g) is called ǫ-collapsed under a local lower curvature bound if for each x ∈ M, there exists a ρ, 0 < ρ ≤ diam(M, g), with (8.1)

vol B(x, ρ) ≤ ǫρn ,

K ≥ −ρ−2

on B(x, ρ).

Theorem 1.1 extends to the following:

Theorem 8.1 (Theorem 7.4 [7]). Let ǫ0 be a positive number given in Theorem 1.1. If a closed orientable Riemannian three-manifold (M, g) is ǫ0 -collapsed under a local lower curvature bound, then it is homeomorphic to a graph manifold. Let ρ(x), 0 < ρ(x) ≤ diam(M, g), be the supremum of ρ > 0 satisfying (8.1). By Theorem 3.5, a small perturbation Bx of B(x, ρ(x)) has a singular fibration over a metric ball in some Alexandrov space with curvature ≥ −1 and dimension one or two. Choose a covering {Bxi } of M such that {B(xi , ρ(xi )/10)} is a maximal disjoint family. Let U1 , U2 and M = U1 ∪ Uˆ2 be as in Section 4. In a way similar to Lemma 4.1, one can prove that each component of U1 is either cylindrical or cylindrical with a cap. Let B2 be defined as in Section 4. Lemma 8.2. Suppose Bx and By in B2 satisfy that B(x, ρ(x)/4) meets B(y, ρ(y)/4). Then ρ(x) C −1 ≤ ≤ C, ρ(y) for some universal positive number C. Proof. Assuming ρ(x) < ρ(y), we put ρ(y) = Rρ(x). By triangle inequality, B(x, R1 ρ(x)) ⊂ B(y, ρ(y)), where R1 := R/2, which implies that (8.2) Next we show

K ≥ −(R1 ρ(x))−2

on B(x, R1 ρ(x)).

vol B(x, R1 ρ(x)) ≤ ǫ0 (R1 ρ(x))3

(8.3)

if R1 is larger than some uniform positive constatn. In view of the maximality of ρ(x), (8.2) and (8.3) yield the conclusion of the lemma. First note that large parts of Bx and By have S 1 -fiber structures. Let ℓ denote the length of a regular circle fiber F contained in B(x, ρ(x)/4)∩ B(y, ρ(y)/4). Let gx := ρ(x)−2 g. Since B(y, ρ(y)/8) ⊂ B(x, R1 ρ(x)) ⊂ B(y, ρ(y)), Lemma 5.1 implies C1−1

ℓ ℓ ≤ volgy B(x, R1 ρ(x)) ≤ C1 , ρ(y) ρ(y)

for some uniform positive number C1 . It follows that 4C1−1 ℓR12 ρ(x)2 ≤ volg B(x, R1 ρ(x)) ≤ 4C1 ℓR12 ρ(x)2 . 23

Hence (8.3) holds if 4C1 ℓ < ǫ0 R1 ρ(x). On the other hand, it follows from the assumption and Lemma 5.1 that ℓ C1−1 ≤ volgx B(x, ρ(x)) < ǫ0 . ρ(x) Therefore we obtain (8.3) for R1 > 4C12 . We conclude that the constant C in the lemma is given by C = 8C12 . Lemma 8.2 together with the Bishop-Gromov comparison theorem yields a uniform upper bound on the maximal number of intersections among the metric balls in B2 . Therefore our local gluing argument in Sections 5 and 6 goes through the present context as well to complete the proof of Theorem 8.1. References 1. Yu. Burago, M. Gromov, and G. Perel’man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222, translation in Russian Math. Surveys 47 (1992), no. 2, 1–58. 2. J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded I, J. Differential Geom. 23 (1986), no. 3, 309–346. , Collapsing Riemannian manifolds while keeping their curvature 3. bounded II, J. Differential Geom. 32 (1990), no. 1, 269–298. 4. K. Fukaya and T. Yamaguchi, The fundamental groups of almost nonnegatively curved manifolds, Ann. of Math. (2) 136 (1992), 253–333. 5. P. Orlik, Seifert manifolds, Lecture Notes in Math., no. 291, Springer-Verlag, Berlin-New York, 1972. 6. G. Perelman, A. D. Alexandrov’s spaces with curvatures bounded from below II, preprint. , Ricci flow with surgery on three-manifolds, arXiv:math. DG / 0303109. 7. 8. K. Shiohama and M. Tanaka, Cut loci and distance spheres on Alexandrov surfaces, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 531–559. 9. T. Shioya and T. Yamaguchi, Collapsing three-manifolds under a lower curvature bound, J. Differential Geom. 56 (2000), no. 1, 1–66. 10. T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. (2) 133 (1991), 317–357. , A convergence theorem in the geometry of Alexandrov spaces, Actes 11. de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 601–642. Mathematical Institute, Tohoku University, Sendai 980-8578, JAPAN Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, JAPAN E-mail address, T. Shioya: [email protected] E-mail address, T. Yamaguchi: [email protected]

24

WIDTH AND FINITE EXTINCTION TIME OF RICCI FLOW

arXiv:0707.0108v1 [math.DG] 1 Jul 2007


0. Introduction This is an expository article with complete proofs intended for a general non-specialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2-spheres. For instance, when M is a homotopy 3-sphere, the width is loosely speaking the area of the smallest 2-sphere needed to “pull over” M. Second, we use this to conclude that Hamilton’s Ricci flow becomes extinct in finite time on any homotopy 3-sphere. We have chosen to write this since the results and ideas given here are quite useful and seem to be of interest to a wide audience. Given a Riemannian metric on a closed manifold M, sweep M out by a continuous oneparameter family of maps from S2 to M starting and ending at point maps. Pull the sweepout tight by, in a continuous way, pulling each map as tight as possible yet preserving the sweepout. We show the following useful property (see Theorem 1.14 below); cf. 12.5 of [Al], proposition 3.1 of [Pi], proposition 3.1 of [CD], [CM3], and [CM1]: Each map in the tightened sweepout whose area is close to the width (i.e., the maximal energy of the maps in the sweepout) must itself be close to a collection of harmonic maps. In particular, there are maps in the sweepout that are close to a collection of immersed minimal 2-spheres. This useful property that all almost maximal slices are close to critical points is virtually always implicit in any sweepout construction of critical points for variational problems yet it is not always recorded since most authors are only interested in existence of a critical point. Similar results hold for sweepouts by curves1 instead of 2-spheres; cf. [CM3] where sweepouts by curves are used to estimate the rate of change of a 1-dimensional width for convex hypersurfaces in Euclidean space flowing by positive powers of their mean curvatures. The ideas are essentially the same whether one sweeps out by curves or 2-spheres, though the techniques in the curve case are purely ad hoc whereas for sweepouts by 2-spheres additional techniques, developed in the 1980s, have to be used to deal with energy concentration (i.e., “bubbling”); cf. [SaU] and [Jo]. The basic idea in each of the two cases is a local replacement process that can be thought of as a discrete gradient flow. For curves, this is now known as Birkhoff’s curve shortening process; see [B1], [B2]. The authors were partially supported by NSF Grants DMS 0606629 and DMS 0405695. 1 Finding closed geodesics on the 2-sphere by using sweepouts goes back to Birkhoff in 1917; see [B1], [B2], section 2 in [Cr], and [CM3]. In the 1980s Sacks-Uhlenbeck, [SaU], found minimal 2-spheres on general manifolds using Morse theoretic arguments that are essentially equivalent to sweepouts; a few years later, Jost explicitly used sweepouts to obtain minimal 2-spheres in [Jo]. The argument given here works equally well on any closed manifold, but only produces non-trivial minimal objects when the width is positive. 1

2


Local replacement had already been used by H.A. Schwarz in 1870 to solve the Dirichlet problem in general domains, writing the domain as a union of overlapping balls, and using that a solution can be found explicitly on balls by, e.g., the Poisson formula; see [Sc1] and [Sc2]. His method, which is now known as Schwarz’s alternating method, continues to play an important role in applied mathematics, in part because the replacements converge rapidly to the solution. The underlying reason why both Birkhoff’s method of finding closed geodesics and Schwarz’s method of solving the Dirichlet problem converge is convexity. We will deviate slightly from the usual local replacement argument and prove a new convexity result for harmonic maps. This allows us to make replacements on balls with small energy, as opposed to balls with small C 0 oscillation. It is, in our view, much more natural to make the replacement based on energy and gives, as a bi-product, a new uniqueness theorem for harmonic maps since already in dimension two the Sobolev embedding fails to control the C 0 norm in terms of the energy; see Figure 1. The second thing we do is explain how to use this property of the width to show that on a homotopy 3-sphere, or more generally closed 3-manifolds without aspherical summands, the Ricci flow becomes extinct in finite time. This was shown by Perelman in [Pe] and by ColdingMinicozzi in [CM1]; see also [Pe] for applications to the elliptic part of geometrization.

W

The min-max surface. Figure 1. A conformal map to a long thin surface with small area has little energy. In fact, for a conformal map, the part of the map that goes to small area tentacles contributes little energy and will be truncated by harmonic replacement.

Figure 2. The sweepout, the min–max surface, and the width W.

We would like to thank Frédéric Hélein, Bruce Kleiner, and John Lott for their comments. 1. Width and finite extinction On a homotopy 3-sphere there is a natural way of constructing minimal surfaces and that comes from the min-max argument where the minimal of all maximal slices of sweepouts is a minimal surface. In [CM1] we looked at how the area of this min-max surface changes under the flow. Geometrically the area measures a kind of width of the 3-manifold (see Figure 2) and for 3-manifolds without aspherical summands (like a homotopy 3-sphere) when the


3

metric evolve by the Ricci flow, the area becomes zero in finite time corresponding to that the solution becomes extinct in finite time.2 1.1. Width. Let Ω be the set of continuous maps σ : S2 × [0, 1] → M so that for each t ∈ [0, 1] the map σ(·, t) is in C 0 ∩ W 1,2 , the map t → σ(·, t) is continuous from [0, 1] to C 0 ∩ W 1,2 , and finally σ maps S2 × {0} and S1 × {1} to points. Given a map β ∈ Ω, the homotopy class Ωβ is defined to be the set of maps σ ∈ Ω that are homotopic to β through maps in Ω. We will call any such β a sweepout; some authors use a more restrictive notion where β must also induce a degree one map from S3 to M. We will, in fact, be most interested in the case where β induces a map from S3 to M in a non-trivial class3 in π3 (M). The reason for this is that the width is positive in this case and, as we will see, equal to the area of a non-empty collection of minimal 2-spheres. The (energy) width WE = WE (β, M) associated to the homotopy class Ωβ is defined by taking the infimum of the maximum of the energy of each slice. That is, set (1.1)

WE = inf max E (σ(·, t)) , σ∈Ωβ t∈[0,1]

where the energy is given by

Z 1 (1.2) E (σ(·, t)) = |∇x σ(x, t)|2 dx . 2 S2 Even though this type of construction is always called min-max, it is really inf-max. That is, for each (smooth) sweepout one looks at the maximal energy of the slices and then takes the infimum over all sweepouts in a given homotopy class. The width is always non-negative by definition, and positive when the homotopy class of β is non-trivial. Positivity can, for instance, be seen directly using [Jo]. Namely, page 125 in [Jo] shows that if maxt E(σ(·, t)) is sufficiently small (depending on M), then σ is homotopically trivial.4 One could alternatively define the width using area rather than energy by setting (1.3)

WA = inf max Area (σ(·, t)) . σ∈Ωβ t∈[0,1]

The area of a W 1,2 map u : S2 → RN is by definition the integral of the Jacobian Ju = p det (duT du), where du is the differential of u and duT is its transpose. That is, if e1 , e2 is 1

an orthonormal frame on D ⊂ S2 , then Ju = (|ue1 |2 |ue2 |2 − hue1 , ue2 i2 ) 2 ≤ Z (1.4) Area(u D ) = Ju ≤ E(u D ) .

1 2

|du|2 and

D

Consequently, area is less than or equal to energy with equality if and only if hue1 , ue2 i and |ue1 |2 − |ue2 |2 are zero (as L1 functions). In the case of equality, we say that u is almost conformal. As in the classical Plateau problem (cf. Section 4 of [CM2]), energy is somewhat 2It

may be of interest to compare our notion of width, and the use of it, to a well-known approach to the Poincaré conjecture. This approach asks to show that for any metric on a homotopy 3-sphere a min-max type argument produces an embedded minimal 2-sphere. Note that in the definition of the width it play no role whether the minimal 2-sphere is embedded or just immersed, and thus, the analysis involved in this was settled a long time ago. This well-known approach has been considered by many people, including Freedman, Meeks, Pitts, Rubinstein, Schoen, Simon, Smith, and Yau; see [CD]. 3For example, when M is a homotopy 3-sphere and the induced map has degree one. 4See the remarks after Corollary 3.4 for a different proof.

4


easier to work with in proving the existence of minimal surfaces. The next proposition, proven in Appendix D, shows that WE = WA as for the Plateau problem (clearly, WA ≤ WE by the discussion above). Therefore, we will drop the subscript and just write W . Proposition 1.5. WE = WA . 1.2. Finite extinction. Let M 3 be a smooth closed orientable 3-manifold and g(t) a oneparameter family of metrics on M evolving by Hamilton’s Ricci flow, [Ha1], so ∂t g = −2 RicMt .

(1.6)

When M is prime and non-aspherical, then it follows by standard topology that π3 (M) is non-trivial (see, e.g., [CM1]). For such an M, fix a non-trivial homotopy class β ∈ Ω. It follows that the width W (g(t)) = W (β, g(t)) is positive for each metric g(t). This positivity is the only place where the assumption on the topology of M is used in the theorem below giving an upper bound for the derivative of the width under the Ricci flow. As a consequence, we get that the solution of the flow becomes extinct in finite time (see paragraph 4.4 of [Pe] for the precise definition of extinction time when surgery occurs). Theorem 1.7. [CM1]. Let M 3 be a closed orientable prime non-aspherical 3-manifold equipped with a metric g = g(0). Under the Ricci flow, the width W (g(t)) satisfies d 3 W (g(t)) ≤ −4π + W (g(t)) , dt 4(t + C)

(1.8)

in the sense of the limsup of forward difference quotients. Hence, g(t) becomes extinct in finite time. The 4π in (1.8) comes from the Gauss-Bonnet theorem and the 3/4 comes from the bound on the minimum of the scalar curvature that the evolution equation implies. Both of these constants matter whereas the constant C > 0 depends on the initial metric and the actual value is not important. To see that (1.8) implies finite extinction time rewrite (1.8) as (1.9) and integrate to get (1.10)

d W (g(t)) (t + C)−3/4 ≤ −4π (t + C)−3/4 dt

(T + C)−3/4 W (g(T )) ≤ C −3/4 W (g(0)) − 16 π (T + C)1/4 − C 1/4 .

Since W ≥ 0 by definition and the right hand side of (1.10) would become negative for T sufficiently large, we get the claim. Theorem 1.7 shows, in particular, that the Ricci flow becomes extinct for any homotopy 3-sphere. In fact, we get as a corollary finite extinction time for the Ricci flow on all 3manifolds without aspherical summands (see 1.5 of [Pe] or section 4 of [CM1] for why this easily follows): Corollary 1.11. ([CM1], [Pe]). Let M 3 be a closed orientable 3-manifold whose prime decomposition has only non-aspherical factors and is equipped with a metric g = g(0). Under the Ricci flow with surgery, g(t) becomes extinct in finite time.


5

Part of Perelman’s interest in the question about finite time extinction comes from the following: If one is interested in geometrization of a homotopy 3-sphere (or, more generally, a 3-manifold without aspherical summands) and knew that the Ricci flow became extinct in finite time, then one would not need to analyze what happens to the flow as time goes to infinity. Thus, in particular, one would not need collapsing arguments. One of the key ingredients in the proof of Theorem 1.7 is the existence of a sequence of good sweepouts of M, where each map in the sweepout whose area is close to the width (i.e., the maximal energy of any map in the sweepout) must itself be close to a collection of harmonic maps. This will be given by Theorem 1.14 below, but we will first need a notion of closeness and a notion of convergence of maps from S2 into a manifold. 1.3. Varifold convergence. Fix a closed manifold M and let Π : Gk M → M be the Grassmanian bundle of (un-oriented) k-planes, that is, each fiber Π−1 (p) is the set of all k-dimensional linear subspaces of the tangent space of M at p. Since Gk M is compact, we can choose a countable dense subset {hn } of all continuous functions on Gk M with supremum norm at most one (dense with respect to the supremum norm).5 If (X0 , F0 ) and (X1 , F1 ) are two compact (not necessarily connected) surfaces X0 , X1 with measurable maps Fi : Xi → Gk M so that each fi = Π ◦ Fi is in W 1,2 (Xi , M) and Jfi is the Jacobian of fi , then the varifold distance between them is by definition Z Z X −n hn ◦ F0 Jf0 − hn ◦ F1 Jf1 . (1.12) dV (F0 , F1 ) = 2 n

X0

X1

It follows easily that a sequence Xi = (Xi , Fi ) with uniformly bounded areas R converges to 0 (X, F ), iff it converges weakly, that is, if for all h ∈ C (G2 M) we have Xi h ◦ Fi Jfi → R h ◦ F Jf . For instance, when M is a 3-manifold, then G2 M, G1 M, and T 1 M/{±v} are X isomorphic. (Here T 1 M is the unit tangent bundle.) If Σi is a sequence of closed immersed surfaces in M converging to a closed surface Σ in the usual C k topology, then we can think of each surface as being embedded in T 1 M/{±v} ≡ G2 M by mapping each point to plusminus the unit normal vector, ±n, to the surface. It follows easily that the surfaces with these inclusion maps converges in the varifold distance. More generally, if X is a compact surface and f : X → M is a W 1,2 map, where M is no longer assumed to be 3-dimensional, then we let F : X → G2 M be given by that F (x) is the linear subspace df (Tx X). (When M is 3-dimensional, then we may think of the image of this map as lying in T 1 M/{±v}.) Strictly speaking, this is only defined on the measurable space, where Jf is non-zero; we extend R it arbitrarily to all of X since the corresponding Radon measure on G2 M given by h → X h ◦ F Jf is independent of the extension. 1.4. Existence of good sweepouts. A W 1,2 map u on a smooth compact surface D with boundary ∂D is energy minimizing to M ⊂ RN if u(x) is in M for almost every x and (1.13)

E(u) = inf {E(w) | w ∈ W 1,2 (D, M) and (w − u) ∈ W01,2 (D)} .

The map u is said to be weakly harmonic if u is a W 1,2 weak solution of the harmonic map equation ∆u ⊥ T M; see, e.g., lemma 1.4.10 in [He1]. The next result gives the existence of a sequence of good sweepouts. 5This

is a corollary of the Stone-Weierstrass theorem; see corollary 35 on page 213 of [R].

6


Theorem 1.14. Given a metric g on M and a map β ∈ Ω representing a non-trivial class in π3 (M), there exists a sequence of sweepouts γ j ∈ Ωβ with maxs∈[0,1] E(γsj ) → W (g), and so that given ǫ > 0, there exist ¯j and δ > 0 so that if j > ¯j and Area(γ j (·, s)) > W (g) − δ ,

(1.15)

then there are finitely many harmonic maps ui : S2 → M with dV (γ j (·, s), ∪i{ui }) < ǫ .

(1.16)

One immediate consequence of Theorem 1.14 is that if sj is any sequence with Area(γ j (·, sj )) converging to the width W (g) as j → ∞, then a subsequence of γ j (·, sj ) converges to a collection of harmonic maps from S2 to M. In particular, the sum of the areas of these maps is exactly W (g) and, since the maps are automatically conformal, the sum of the energies is also W (g). The existence of at least one non-trivial harmonic map from S2 to M was first proven in [SaU], but they allowed for loss of energy in the limit; cf. also [St]. This energy loss was ruled out by Siu and Yau, using also arguments of Meeks and Yau (see Chapter VIII in [SY]). This was also proven later by Jost in theorem 4.2.1 of [Jo] which gives at least one min-max sequence converging to a collection of harmonic maps. The convergence in [Jo] is in a different topology that, as we will see in Appendix A, implies varifold convergence. 1.5. Upper bounds for the rate of change of width. Throughout this subsection, let M 3 be a smooth closed prime and non-aspherical orientable 3-manifold and let g(t) be a one-parameter family of metrics on M evolving by the Ricci flow. We will prove Theorem 1.7 giving the upper bound for the derivative of the width W (g(t)) under the Ricci flow. To do this, we need three things. One is that the evolution equation for the scalar curvature R = R(t), see page 16 of [Ha2], 2 2 R , 3 implies by a straightforward maximum principle argument that at time t > 0

(1.17)

(1.18)

∂t R = ∆R + 2|Ric|2 ≥ ∆R +

R(t) ≥

1 3 =− . 1/[min R(0)] − 2t/3 2(t + C)

The curvature is normalized so that on the unit S3 the Ricci curvature is 2 and the scalar curvature is 6. In the derivation of (1.18) we implicitly assumed that min R(0) < 0. If this was not the case, then (1.18) trivially holds for any C > 0, since, by (1.17), min R(t) is always non-decreasing. This last remark is also used when surgery occurs. This is because by construction any surgery region has large (positive) scalar curvature. The second thing that we need in the proof is the observation that if {Σi } is a collection of branched minimal 2-spheres and f ∈ W 1,2 (S2 , M) with dV (f, ∪i Σi ) < ǫ, then for any smooth quadratic form Q on M we have (the unit normal nf is defined where Jf 6= 0) Z Z X [Tr(Q) − Q(nΣi , nΣi )] < C ǫ kQkC 1 Area(f ) . (1.19) [Tr(Q) − Q(nf , nf )] − f Σi i

The last thing is an upper bound for the rate of change of area of minimal 2-spheres. Suppose that X is a closed surface and f : X → M is a W 1,2 map, then using (1.6) an easy


7

calculation gives (cf. pages 38–41 of [Ha2]) Z d Areag(t) (f ) = − [R − RicM (nf , nf )] . (1.20) dt t=0 f

If Σ ⊂ M is a closed immersed minimal surface, then Z Z d 1 (1.21) Areag(t) (Σ) = − KΣ − [|A|2 + R] . dt t=0 2 Σ Σ

Here KΣ is the (intrinsic) curvature of Σ, A is the second fundamental form of Σ, and |A|2 is the sum of the squares of the principal curvatures. To get (1.21) from (1.20), we used that if KM is the sectional curvature of M on the two-plane tangent to Σ, then the Gauss equations and minimality of Σ give KΣ = KM − 21 |A|2 . The next lemma gives the upper bound.

Lemma 1.22. If Σ ⊂ M 3 is a branched minimal immersion of the 2-sphere, then Areag(0) (Σ) d min R(0) . Areag(t) (Σ) ≤ −4π − (1.23) M dt t=0 2 Proof. Let {pi } be the set of branch points of Σ and bi > 0 the order of branching. By (1.21) Z Z Z X d 1 1 (1.24) Areag(t) (Σ) ≤ − KΣ − R = −4π − 2π bi − R, dt t=0 2 Σ 2 Σ Σ where the equality used the Gauss-Bonnet theorem with branch points (this equality also follows from the Bochner type formula for harmonic maps between surfaces given on page 10 of [SY] and the second displayed equation on page 12 of [SY] that accounts for the branch points). Note that branch points only help in the inequality (1.23). Using these three things, we can show the upper bound for the rate of change of the width. Proof. (of Theorem 1.7) Fix a time τ . Below C˜ denotes a constant depending only on τ but will be allowed to change from line to line. Let γ j (τ ) be the sequence of sweepouts for the metric g(τ ) given by Theorem 1.14. We will use the sweepout at time τ as a comparison to get an upper bound for the width at times t > τ . The key for this is the following claim: ¯ > 0 so that if j > ¯j and 0 < h < h, ¯ then Given ǫ > 0, there exist ¯j and h Areag(τ +h) (γsj (τ )) − max Areag(τ ) (γsj0 (τ )) s0

(1.25)

≤ [−4π + C˜ ǫ +

3 max Areag(τ ) (γsj0 (τ ))] h + C˜ h2 . 4(τ + C) s0

To see why (1.25) implies (1.8), use the equivalence of the two definitions of widths to get (1.26)

W (g(τ + h)) ≤ max Areag(τ +h) (γsj (τ )) , s∈[0,1]

and take the limit as j → ∞ (so that6 maxs0 Areag(τ ) (γsj0 (τ )) → W (g(τ ))) in (1.25) to get (1.27)

W (g(τ + h)) − W (g(τ )) 3 ≤ −4π + C˜ ǫ + W (g(τ )) + C˜ h . h 4(τ + C)

Taking ǫ → 0 in (1.27) gives (1.8). 6This

follows by combining that Areag(τ ) (γsj0 (τ )) ≤ Eg(τ ) (γsj0 (τ )) by (1.4), maxs0 Eg(τ ) (γsj0 (τ )) → W (g(τ )), and W (g(τ )) ≤ maxs0 Areag(τ ) (γsj0 (τ )) by the equivalence of the two definitions of width.

8


It remains to prove (1.25). First, let δ > 0 and ¯j, depending on ǫ (and on τ ), be given by Theorem 1.14. If j > ¯j and Areag(τ ) (γsj (τ )) > W (g) − δ, then let ∪i Σjs,i (τ ) be the collection of minimal spheres given by Theorem 1.14. Combining (1.20), (1.19) with Q = RicM , and Lemma 1.22 gives d d Areag(t) (γsj (τ )) ≤ Areag(t) (∪i Σjs,i(τ )) + C˜ ǫ kRicM kC 1 Areag(τ ) (γsj (τ )) dt t=τ dt t=τ Areag(τ ) (γsj (τ )) ≤ −4π − (1.28) min R(τ ) + C˜ ǫ M 2 3 max Areag(τ ) (γsj0 (τ )) + C˜ ǫ , ≤ −4π + 4(τ + C) s0 where the last inequality used the lower bound (1.18) for R(τ ). Since the metrics g(t) vary smoothly and every sweepout γ j has uniformly bounded energy, it is easy to see that Areag(τ +h) (γsj (τ )) is a smooth function of h with a uniform C 2 bound independent of both ¯ > 0 j and s near h = 0 (cf. (1.20)). In particular, (1.28) and Taylor expansion give h j (independent of j) so that (1.25) holds for s with Areag(τ ) (γs (τ )) > W (g) − δ. In the remaining case, we have Area(γsj (τ )) ≤ W (g) − δ so the continuity of g(t) implies that (1.25) ¯ > 0. automatically holds after possibly shrinking h 1.6. Parameter spaces. Instead of using the unit interval, [0, 1], as the parameter space for the maps in the sweepout and assuming that the maps start and end in point maps, we could have used any compact finite dimensional topological space P and required that the maps are constant on ∂P (or that ∂P = ∅). In this case, let ΩP be the set of continuous maps σ : S2 × P → M so that for each t ∈ P the map σ(·, t) is in C 0 ∩ W 1,2 (S2 , M), the map t → σ(·, t) is continuous from P to C 0 ∩ W 1,2 (S2 , M), and finally σ maps ∂P to point maps. Given a map σ ˆ ∈ ΩP , the homotopy class ΩPσˆ ⊂ ΩP is defined to be the set of maps P σ ∈ Ω that are homotopic to σˆ through maps in ΩP . Finally, the width W = W (ˆ σ ) is inf σ∈ΩσPˆ maxt∈P E (σ(·, t)). With only trivial changes, the same proof yields Theorem 1.14 for these general parameter spaces.7 2. The energy decreasing map and its consequences To prove Theorem 1.14, we will first define an energy decreasing map from Ω to itself that preserves the homotopy class (i.e., maps each Ωβ to itself) and record its key properties. This should be thought of as a generalization of Birkhoff’s curve shortening process that plays a similar role when tightening a sweepout by curves; see [B1], [B2], [Cr], and [CM3]. Throughout this paper, by a ball B ⊂ S2 , we will mean a subset of S2 and a stereographic projection ΠB so that ΠB (B) ⊂ R2 is a ball. Given ρ > 0, we will let ρ B ⊂ S2 denote Π−1 B of the ball with the same center as ΠB (B) and radius ρ times that of ΠB (B). Theorem 2.1. There is a constant ǫ0 > 0 and a continuous function Ψ : [0, ∞) → [0, ∞) with Ψ(0) = 0, both depending on M, so that given any γ˜ ∈ Ω without non-constant harmonic slices and W > 0, there exists γ ∈ Ωγ˜ so that E(γ(·, t)) ≤ E(˜ γ (·, t)) for each t and so for each t with E(˜ γ (·, t)) ≥ W/2: 7The main change is in Lemma 3.39 below where the bound 2 for the multiplicity

in (1) becomes dim(P)+1. This follows from the definition of (covering) dimension; see pages 302 and 303 in [Mu].


9

R (BΨ ) If B is any finite collection of disjoint closed balls in S2 with ∪B B |∇γ(·, t)|2 < ǫ0 and v : ∪B 18 B → M is an energy minimizing map equal to γ(·, t) on ∪B 18 ∂ B, then Z |∇γ(·, t) − ∇v|2 ≤ Ψ [E(˜ γ (·, t)) − E(γ(·, t))] . ∪B 81 B

The proof of Theorem 2.1 is given in Section 3. The second ingredient that we will need to prove Theorem 1.14 is a compactness result that generalizes compactness of harmonic maps to maps that are closer and closer to being harmonic (this is Proposition 2.2 below and will be proven in Appendix B). 2.1. Compactness of almost harmonic maps. Our notion of almost harmonic relies on two important properties of harmonic maps from S2 to M. The first is that harmonic maps from S2 are conformal and, thus, energy and area are equal; see (A) below. The second is that any harmonic map from a surface is energy minimizing when restricted to balls where the energy is sufficiently small; see (B) below. In the proposition, ǫSU > 0 (depending on M) is the small energy constant from lemma 3.4 in [SaU], so that we get interior estimates for harmonic maps with energy at most ǫSU . In particular, any non-constant harmonic map from S2 to M has energy greater than ǫSU . Proposition 2.2. Suppose that ǫ0 , E0 > 0 are constants with ǫSU > ǫ0 and uj : S2 → M is a sequence of C 0 ∩ W 1,2 maps with E0 ≥ E(uj ) satisfying: (A) Area(uj ) > E(uj ) − 1/j. R (B) For any finite collection B of disjoint closed balls in S2 with ∪B B |∇uj |2 < ǫ0 there is an energy minimizing map v : ∪B 81 B → M that equals uj on ∪B 81 ∂B with Z j ∇u − ∇v 2 ≤ 1/j . ∪B 18 B

If (A) and (B) are satisfied, then a subsequence of the uj ’s varifold converges to a collection of harmonic maps v 0 , . . . , v m : S2 → M.

One immediate consequence of Proposition 2.2 is a compactness theorem for sequences of harmonic maps with bounded energy. This was proven by Jost in lemma 4.3.1 in [Jo]. In fact, Parker proved compactness of bounded energy harmonic maps in a stronger topology, with C 0 convergence in addition to W 1,2 convergence; see theorem 2.2 in [Pa]. Therefore, it is perhaps not surprising that a similar compactness holds for sequences that are closer and closer to being harmonic in the sense above. However, it is useful to keep in mind that Parker has constructed sequences of maps where the Laplacian is going to zero in L1 and yet there is no convergent subsequence (see proposition 4.2 in [Pa]). Finally, we point out that Proposition 2.2 can be thought of as a discrete version of Palais-Smale Condition (C). Namely, if we have a sequence of maps where the maximal energy decrease from harmonic replacement goes to zero, then a subsequence converges to a collection of harmonic maps. 2.2. Constructing good sweepouts from the energy decreasing map on Ω. Given Theorem 2.1 and Proposition 2.2, we will prove Theorem 1.14. Let G W +1 be the set of collections of harmonic maps from S2 to M so that the sum of the energies is at most W + 1.

10


Proof. (of Theorem 1.14.) Choose a sequence of maps γ˜j ∈ Ωβ with max E (˜ γ j (·, t)) < W +

(2.3)

t∈[0,1]

1 , j

and so that γ˜j (·, t) is not harmonic unless it is a constant map.8 We can assume that W > 0 since otherwise Area(˜ γ j (·, t)) ≤ E(˜ γ j (·, t)) → 0 and the theorem follows trivially. Applying Theorem 2.1 to the γ˜ j ’s gives a sequence γ j ∈ Ωβ where each γ j (·, t) has energy at most that of γ˜j (·, t). We will argue by contradiction to show that the γ j ’s have the desired property. Suppose, therefore, that there exist jk → ∞ and sk ∈ [0, 1] with dV (γ jk (·, sk ), G W +1) ≥ ǫ > 0 and Area(γ jk (·, sk )) > W − 1/k. Thus, by (2.3) and the fact that E(·) ≥ Area(·), we get (2.4)

E(˜ γ jk (·, sk )) − E(γ jk (·, sk )) ≤ E(˜ γ jk (·, sk )) − Area(γ jk (·, sk )) ≤ 1/k + 1/jk → 0 ,

and, similarly, E (γ jk (·, sk )) − Area (γ jk (·, sk )) → 0. Using (2.4)Rin Theorem 2.1 gives (B) If B is any collection of disjoint closed balls in S2 with ∪B B |∇γ jk (·, sk )|2 < ǫ0 and v : ∪B 18 B → M is an energy minimizing map that equals γ jk (·, sk ) on ∪B 81 ∂B, then Z j ∇γ k (·, sk ) − ∇v 2 ≤ Ψ(1/k + 1/jk ) → 0 . (2.5) ∪B 18 B

Therefore, we can apply Proposition 2.2 to get that a subsequence of the γ jk (·, sk )’s varifold converges to a collection of harmonic maps. However, this contradicts the lower bound for the varifold distance to G W +1 , thus completing the proof. 3. Constructing the energy decreasing map

3.1. Harmonic replacement. The energy decreasing map from Ω to itself will be given by a repeated replacement procedure. At each step, we replace a map u by a map H(u) that coincides with u outside a ball and inside the ball is equal to an energy-minimizing map with the same boundary values as u. This is often referred to as harmonic replacement. One of the key properties that makes harmonic replacement useful is that the energy functional is strictly convex on small energy maps. Namely, Theorem 3.1 below gives a uniform lower bound for the gap in energy between a harmonic map and a W 1,2 map with the same boundary values; see Appendix C for the proof. Theorem 3.1. There exists a constant ǫ1 > 0 (depending on M) so that if u and v are W 1,2 maps from B1 ⊂ R2 to M, u and v agree on ∂B1 , and v is weakly harmonic with energy at 8To

do this, first use Lemma D.1 (density of C 2 -sweepouts) to choose γ˜1j ∈ Ωβ so t → γ˜1j (·, t) is continuous 1 . Using stereographic projection, we can view γ˜1j (·, t) as from [0, 1] to C 2 and maxt∈[0,1] E (˜ γ1j (·, t)) < W + 2j γ1j (·, t)|2 ≤ C a map from R2 . Now fix a j. The continuity in C 2 gives a uniform bound supt∈[0,1] supB1 |∇˜ 2 2 2 for some C. Choose R > 0 with 4πC R ≤ 1/(2j). Define a map Φ : R → R in polar coordinates by: Φ(r, θ) = (2r, θ) for r < R/2, Φ(r, θ) = (R, θ) for R/2 ≤ r ≤ R, and Φ(r, θ) = (r, θ) for R < r. Note that Φ is homotopic to the identity, is conformal away from the annulus BR \ BR/2 , and on BR \ BR/2 has |∂r Φ| = 0 and |dΦ| ≤ 2. It follows that γ˜ j (·, t) = γ˜1j (·, t) ◦ Φ is in Ωβ , satisfies (2.3), and has ∂r γ˜ j (·, t) = 0 on BR \ BR/2 . Since harmonic maps from S2 are conformal (corollary 1.7 in [SaU]), any harmonic γ˜ j (·, t) is constant on BR \ BR/2 and, thus, constant on S2 by unique continuation (theorem 1.1 in [Sj]).


most ǫ1 , then

Z

Z

1 |∇u| − |∇v| ≥ 2 B1 B1

(3.2)

2

2

Z

B1

11

|∇v − ∇u|2 .

An immediate corollary of Theorem 3.1 is uniqueness of solutions to the Dirichlet problem for small energy maps (and also that any such harmonic map minimizes energy). Corollary 3.3. Let ǫ1 > 0 be as in Theorem 3.1. If u1 and u2 are W 1,2 weakly harmonic maps from B1 ⊂ R2 to M, both with energy at most ǫ1 , and they agree on ∂B1 , then u1 = u2 .

3.2. Continuity of harmonic replacement on C 0 (B1 ) ∩ W 1,2 (B1 ). The second consequence of Theorem 3.1 is that harmonic replacement is continuous as a map from C 0 (B1 ) ∩ W 1,2 (B1 ) to itself if we restrict to small energy maps. (The norm on C 0 (B1 ) ∩ W 1,2 (B1 ) is the sum of the sup norm and the W 1,2 norm.) Corollary 3.4. Let ǫ1 > 0 be as in Theorem 3.1 and set M = {u ∈ C 0 (B1 , M) ∩ W 1,2 (B1 , M) | E(u) ≤ ǫ1 } .

(3.5)

Given u ∈ M, there is a unique energy minimizing map w equal to u on ∂B1 and w is in M. Furthermore, there exists C depending on M so that if u1 , u2 ∈ M with corresponding energy minimizing maps w1 , w2 , and we set E = E(u1 ) + E(u2 ), then (3.6)

|E(w1 ) − E(w2 )| ≤ C ||u1 − u2||C 0 (B1 ) E + C ||∇u1 − ∇u2 ||L2 (B1 ) E1/2 .

Finally, the map from u to w is continuous as a map from C 0 (B1 ) ∩ W 1,2 (B1 ) to itself.

In the proof, we will use that since M is smooth, compact and embedded, there exists a δ > 0 so that for each x in the δ-tubular neighborhood Mδ of M in RN , there is a unique closest point Π(x) ∈ M and so the map x → Π(x) is smooth. Π is called nearest point projection. Furthermore, for any x ∈ M, we have |dΠx (V )| ≤ |V |. Therefore, there is a constant CΠ depending on M so that for any x ∈ Mδ , we have |dΠx (V )| ≤ (1 + CΠ |x − Π(x)|) |V |. In particular, we can choose δˆ ∈ (0, δ) so that |dΠx (V )|2 ≤ 2 |V |2 for any x ∈ Mδˆ and V ∈ RN .

Proof. (of Corollary 3.4.) The existence of an energy minimizing map w ∈ W 1,2 (B1 ) was proven by Morrey in [Mo1]; by Corollary 3.3, w is unique. The continuity of w on B1 is the main theorem of [Q].9 It follows that w ∈ M. ˆ since Step 1: E(w) is uniformly continuous. We can assume that ||u1 − u2 ||C 0 (B1 ) ≤ δ, ˆ Define a map v1 by ≥ δ. (3.6) holds with C = 1/δˆ if ||u1 − u2 || 0 C (B1 )

(3.7)

v1 = Π ◦ (w2 + (u1 − u2 )) ,

so that v1 maps to M and agrees with u1 on ∂B1 . Using that |dΠx (V )| ≤ |V | for x ∈ M and w2 maps to M, we can estimate the energy of v1 by (3.8) E(v1 ) ≤ (1 + CΠ ||u1 − u2||C 0 (B1 ) )2 E(w2 ) + 2(E(w2 ) E(u1 − u2 ))1/2 + E(u1 − u2 ) ,

where CΠ is the Lipschitz norm of dΠ in Mδˆ. Since v1 and w1 agree on ∂B1 , Corollary 3.3 yields E(w1 ) ≤ E(v1 ). By symmetry, we can assume that E(w2 ) ≤ E(w1 ) so that (3.8) implies (3.6). 9Continuity

also essentially follows from the boundary regularity of Schoen and Uhlenbeck, [SU2], except that [SU2] assumes C 2,α regularity of the boundary data.

12


Step 2: The continuity of u → w. Suppose that u, uj are in M with uj → u in C 0 (B1 ) ∩ W 1,2 (B1 ) and w and wj are the corresponding energy minimizing maps. We will first show that wj → w in W 1,2 (B1 ). To do this, set (3.9)

vj = Π ◦ (w + (uj − u)) ,

so that vj maps to M and agrees with uj on ∂B1 . Arguing as in (3.8) and using that E(wj ) → E(w) by Step 1, we get that [E(vj ) − E(wj )] → 0. Therefore, applying Theorem 3.1 to wj , vj gives that ||wj − vj ||W 1,2 (B1 ) → 0. Since ||uj − u||C 0 (B1 )∩W 1,2 (B1 ) → 0 and Π ◦ w = w, it follows that ||w − vj ||W 1,2 (B1 ) → 0. The triangle inequality gives ||w − wj ||W 1,2(B1 ) → 0. Finally, we will argue by contradiction to see that wj → w in C 0 (B1 ). Suppose instead that there is a subsequence (still denoted wj ) with (3.10)

||wj − w||C 0(B1 ) ≥ ǫ > 0 .

Using the uniform energy bound for the wj ’s together with interior estimates for energy minimizing maps of [SU1] (and the Arzela-Ascoli theorem), we can pass to a further subsequence so that the wj ’s converge uniformly in C 2 on any compact subset K ⊂ B1 . Finally, as remarked in the proof of the main theorem in [Q], proposition 1 and remark 1 of [Q] imply that the wj ’s are also equicontinuous near ∂B1 , so Arzela-Ascoli gives a further subsequence that converges uniformly on B1 to a harmonic map w∞ that agrees with w on the boundary. However, (3.10) implies that ||w − w∞ ||C 0(B1 ) ≥ ǫ > 0 which contradicts the uniqueness of small energy harmonic maps. This completes the proof. Corollary 3.4 gives another proof that the width is positive when the homotopy class is non-trivial or, equivalently, that if maxt E(σ(·, t)) is sufficiently small (depending on M), then σ is homotopically trivial. Namely, since t → σ(·, t) is continuous from [0, 1] to C 0 , we can choose r > 0 so that σ(·, t) maps the ball Br (p) ⊂ S2 into a convex geodesic ball B t in M for every t. If each σ(·, t) has energy less than ǫ1 > 0 given by Corollary 3.4, then replacing σ(·, t) outside Br (p) by the energy minimizing map with the same boundary values gives a homotopic sweepout σ ˜ . Moreover, the entire image of σ˜ (·, t) is contained in the convex ball B t by the maximum principle.10 It follows that σ ˜ is homotopically trivial by contracting each σ˜ (·, t) to the point σ(p, t) via a geodesic homotopy. 3.3. Uniform continuity of energy improvement on W 1,2 . It will be convenient to introduce some notation for the next lemma. Namely, given a C 0 ∩ W 1,2 map u from S2 to M and a finite collection B of disjoint closed balls in S2 so the energy of u on ∪B B is at most ǫ1 /3, let H(u, B) : S2 → M denote the map that coincides with u on S2 \ ∪B B and on ∪B B is equal to the energy minimizing map from ∪B B to M that agrees with u on ∪B ∂B. To keep the notation simple, we will set H(u, B1 , B2 ) = H(H(u, B1), B2 ). Finally, if α ∈ (0, 1], then αB will denote the collection of concentric balls but whose radii are shrunk by the factor α. In general, H(u, B1 , B2 ) is not the same as H(u, B2, B1 ). This matters in the proof of Theorem 2.1, where harmonic replacement on either 21 B1 or 21 B2 decreases the energy of u by a definite amount. The next lemma (see (3.12)) shows that the energy goes down a definite amount regardless of the order that we do the replacements. The second inequality bounds 10This

follows from lemma 4.1.3 in [Jo] which requires that σ(·, t) is homotopic to a map in B t and this follows from the small energy bound and the uniform lower bound for the energy of any homotopically non-trivial map from S2 given, e.g., in the first line of the proof of proposition 2 on page 143 of [SY].


13

the possible decrease in energy from applying harmonic replacement on H(u, B1) in terms of the possible decrease from harmonic replacement on u. Lemma 3.11. There is a constant κ > 0 (depending on M) so that if u : S2 → M is in C 0 ∩ W 1,2 and B1 , B2 are each finite collections of disjoint closed balls in S2 so that the energy of u on each ∪Bi B is at most ǫ1 /3, then 2 1 . (3.12) E(u) − E [H(u, B1, B2 )] ≥ κ E(u) − E H(u, B2 ) 2 Furthermore, for any µ ∈ [1/8, 1/2], we have

(E(u) − E [H(u, B1)])1/2 (3.13) +E(u)−E [H(u, 2 µB2)] ≥ E [H(u, B1 )]−E [H(u, B1 , µ B2 )] . κ We will prove Lemma 3.11 by constructing comparison maps with the same boundary values and using the minimizing property of small energy harmonic maps to get upper bounds for the energy. The following lemma will be used to construct the comparison maps. Lemma 3.14. There exists τ > 0 (depending on M) so that if f, g : ∂BR → M are C 0 ∩W 1,2 maps that agree at one point and satisfy Z (3.15) R |f ′ − g ′ |2 ≤ τ 2 , ∂BR

then there exists some ρ ∈ (0, R/2] and a C 0 ∩ W 1,2 map w : BR \ BR−ρ → M so that

(3.16) w(R − ρ, θ) = f (R, θ) and w(R, θ) = g(R, θ) , 1/2 R 1/2 √ R R and BR \BR−ρ |∇w|2 ≤ 17 2 R ∂BR (|f ′|2 + |g ′|2 ) R ∂BR |f ′ − g ′ |2 .

Proof. √ Let Π and δ > δˆ > 0 (depending on M) be as in the proof of Corollary 3.4 and set ˆ ˆ 2π. Since f −g vanishes somewhere on ∂BR , integrating (3.15) gives max |f −g| ≤ δ. τ = δ/ 2 the statement is scale-invariant, it suffices to prove the case R = 1. Set ρ = R Since R ′ ′ 2 ′ 2 |f − g | /[8 S1 (|f | + |g ′ |2 )] ≤ 1/4 and define wˆ : B1 \ B1−ρ → RN by S1 r+ρ−1 (3.17) w(r, ˆ θ) = f (θ) + (g(θ) − f (θ)) . ρ

Observe that wˆ satisfies (3.16). Furthermore, on S1 , we can R R since f −′ 2g vanishes somewhere R 2 use Wirtinger’s inequality S1 |f − g| ≤ 4 S1 |(f − g) | to bound B1 \B1−ρ |∇w| ˆ 2 by Z Z 1 Z 2π Z 2π 1 1 2 2 ′ 2 ′ 2 |∇w| ˆ ≤ |f − g| (θ) dθ + 2 (|f | + |g | )(θ) dθ r dr 2 r 0 0 B1 \B1−ρ 1−ρ ρ Z Z 2π 4 2π ′ ′ 2 (3.18) |f − g | (θ) dθ + 2ρ (|f ′ |2 + |g ′|2 )(θ) dθ ≤ ρ 0 0 Z 1/2 Z √ ′ ′ 2 ′ 2 ′ 2 = 17/ 2 |f − g | (|f | + |g | ) . S1

S1

ˆ the image of wˆ is contained in Mˆ where we have |dΠ|2 ≤ 2. Therefore, if Since |f − g| ≤ δ, δ we set w = Π ◦ w, ˆ then the energy of w is at most twice the energy of w. ˆ

14


Proof. (of Lemma 3.11.) We will index the balls in B1 by α and use j for the balls in B2 ; i.e., let B1 = {Bα1 } and B2 = {Bj2 }. The key point is that, by Corollary 3.4, small energy harmonic maps minimize energy. Using this, we get upper bounds for the energy of the harmonic replacement by cutting and pasting to construct comparison functions with the same boundary values. Observe that the total energy of u on the union of the balls in B1 ∪ B2 is at most 2ǫ1 /3. Since harmonic replacement on B1 does not change the map outside these balls and is energy non-increasing, it follows that the total energy of H(u, B1) on B2 is at most 2ǫ1 /3. The proof of (3.12). We will divide B2 into two disjoint subsets, B2,+ and B2,− , and argue separately, depending on which of these accounts for more of the decrease in energy after harmonic replacement. Namely, set 1 (3.19) B2,+ = {Bj2 ∈ B2 | Bj2 ⊂ Bα1 for some Bα1 ∈ B1 } and B2,− = B2 \ B2,+ . 2 Since the balls in B2 are disjoint, it follows that 1 1 1 (3.20) E(u) − E(H(u, B2 )) = E(u) − E(H(u, B2,− )) + E(u) − E(H(u, B2,+ )) . 2 2 2 Case 1. Suppose that E(u) − E H(u, 12 B2,+ ) ≥ E(u) − E H(u, 12 B2 ) /2. Since the balls in 12 B2,+ are contained in balls in B1 and harmonic replacements minimize energy, we get 1 (3.21) E(H(u, B1 , B2 )) ≤ E(H(u, B1)) ≤ E(H(u, B2,+ )) , 2 1 1 so that E(u) − E H(u, 2 B2 ) /2 ≤ E(u) − E(H(u, 2 B2,+ )) ≤ E(u) − E(H(u, B1, B2 )). Case 2. Suppose now that 1 1 1 (3.22) E(u) − E(H(u, B2,− )) ≥ E(u) − E(H(u, B2 )) . 2 2 2 Let τ > 0 be given by Lemma 3.14. We can assume that Z (3.23) 9 |∇H(u, B1) − ∇u|2 ≤ τ 2 , S2

since otherwise Theorem 3.1 gives (3.12) with κ = τ 2 /ǫ21 . The key is to show for Bj2 ∈ B2,− that 2 Z Z Z Z 2 1 2 2 2 2 ∇H(u, Bj ) ∇H(u, B1, Bj ) ≥ (3.24) |∇u| − |∇H(u, B1 )| − 1 2 1 2 2 2 2 Bj

Bj

−C

Z

Bj2

|∇u|2 + |∇H(u, B1)|2

!1/2

2

Z

Bj

Bj2

2

Bj

|∇(u − H(u, B1))|2

!1/2

,

where C is a universal constant. Namely, summing (3.24) over B2,− and using the inequality P P 2 1/2 P 2 1/2 | aj bj | ≤ aj bj , the bound for the energy of u in B1 ∪ B2 , and Theorem 3.1 to relate the energy of u − H(u, B1 ) to E(u) − E(H(u, B1 )) gives 1 1/2 E(u) − E(H(u, B2,− )) ≤ E(H(u, B1 )) − E(H(u, B1, B2,− )) + C ǫ1 (E(u) − E[H(u, B1 )])1/2 2 1/2 1/2 1/2 1/2 (3.25) ≤ δE + C ǫ1 δE ≤ (C + 1) ǫ1 δE ,


15

where we have set δE = E(u) − E(H(u, B1, B2 )) in the last line and the last inequality used that δE ≤ 2ǫ1 /3 < ǫ1 . Combining (3.22) with (3.25) gives (3.12). To complete Case 2, we must prove (3.24). After translation, we can assume that Bj2 is the ball BR of radius R about 0 in R2 . Set u1 = H(u, B1) and apply the co-area formula to get r ∈ [3R/4, R] (in fact, a set of r’s of measure at least R/36) with 9 |∇u1 − ∇u| ≤ R ∂Br

Z

9 (|∇u1 | + |∇u| ) ≤ R ∂Br

Z

Z

(3.26) (3.27) Z

2

2

R

3R/4

2

R 3R/4

Z

∂Bs

Z

|∇u1 − ∇u| 2

∂Bs

2

|∇u1 | + |∇u|

2

9 ds ≤ r

Z

BR

9 ds ≤ r

|∇u1 − ∇u|2 ,

Z

BR

(|∇u1|2 + |∇u|2 ) .

Since Bj2 ∈ B2,− and r > R/2, the circle ∂Br is not contained in any of the balls in B1 . It follows that ∂Br contains at least one point outside ∪B1 B and, thus, there is a point in ∂Br where u = u1 . This and (3.23) allow us to apply Lemma 3.14 to get ρ ∈ (0, r/2] and a map w : Br \ Br−ρ → M with w(r, θ) = u1 (r, θ), w(r − ρ, θ) = u(r, θ), and Z

(3.28)

Br \Br−ρ

|∇w|2 ≤ C

Z

Bj2

|∇u|2 + |∇H(u, B1 )|2

!1/2

Z

Bj2

|∇(u − H(u, B1))|2

!1/2

.

Observe that the map x → H(u, Br )(r x/(r − ρ)) maps Br−ρ to M and agrees with w on ∂Br−ρ . Therefore, the map from BR to M which is equal to u1 on BR \ Br , is equal to w on Br \ Br−ρ , and is equal to H(u, Br )(r · /(r − ρ)) on Br−ρ gives an upper bound for the energy of H(u1, BR ) Z

(3.29)

2

BR

|∇H(u1, BR )| ≤

Z

2

BR \Br

|∇u1 | +

Z

2

Br \Br−ρ

|∇w| +

Z

|∇H(u, Br )|2 .

Br

Using (3.28) and that ||∇u1|2 − |∇u|2 | ≤ (|∇u| + |∇u1 |) |∇(u − u1 )|, we get Z

2

BR

|∇u1 | −

≥

Z

Z

BR

2

Br

|∇u| −

2

|∇H(u1 , BR )| ≥ Z

Br

2

Z

Br

|∇H(u, Br )| − C

2

|∇u1 | − Z

Z

2

Br 2

Br

|∇H(u, Br )| −

|∇u| + |∇u1 |

2

1/2 Z

Z

Br \Br−ρ

Br

|∇w|2

|∇(u − u1 )|

2

1/2

.

R R R Since BR/2 |∇H(u, BR/2 )|2 ≤ BR/2 \Br |∇u|2 + Br |∇H(u, Br )|2 , we get (3.24). The proof of (3.13). We will argue similarly with a few small modifications that we will describe. This time, let B2,+ ⊂ B2 be the balls Bj2 with µBj2 contained in some Bα1 ∈ B1 . It follows that harmonic replacement on µB2,+ does not change H(u, B1 ) and, thus, (3.30)

E [H(u, B1)] = E [H(u, B1, µB2,+ )] .

16


Again, we can assume that (3.23) holds. Suppose now that Bj2 ∈ B2,− . Arguing as in the proof of (3.24) (switching the roles of u and H(u, B1)), we get Z Z Z Z 2 2 2 2 ∇H(u, B1 , µ B2 ) 2 (3.31) |∇u| − ∇H(u, 2µBj ) ≥ |∇H(u, B1)| − j Bj2

Bj2

−C

Z

µBj2

2

Bj2

|∇u| + |∇H(u, B1 )|

2

!1/2

µBj2

Z

Bj2

2

|∇(u − H(u, B1 ))|

!1/2

.

Summing this over B2,− and arguing as for (3.25) gives Z Z Z Z 2 2 2 (3.32) |∇u| − |∇H(u, 2µB2)| ≥ |∇H(u, B1)| − |∇H(u, B1, µ B2,− )|2 1/2

− C ǫ1

(E(u) − E[H(u, B1)])1/2 .

Combining (3.30) and (3.32) completes the proof.

3.4. Constructing the map from γ˜ to γ. We will construct γ(·, t) from γ˜(·, t) by harmonic replacement on a family of balls in S2 varying continuously in t. The balls will be chosen in Lemma 3.39 below. Throughout this subsection, ǫ1 > 0 will be the small energy constant (depending on M) given by Theorem 3.1. Given σ ∈ Ω and ǫ ∈ (0, ǫ1 ], define the maximal improvement from harmonic replacement on families of balls with energy at most ǫ by 1 (3.33) eσ,ǫ (t) = sup {E(σ(·, t)) − E(H(σ(·, t), B))} , 2 B where the supremum is over all finite collections B of disjoint closed balls where the total energy of σ(·, t) on B is at most ǫ. Observe that eσ,ǫ (t) is nonnegative, monotone nondecreasing in ǫ, and is positive if σ(·, t) is not harmonic. Lemma 3.34. If σ(·, t) is not harmonic and ǫ ∈ (0, ǫ1 ], then there is an open interval I t containing t so that eσ,ǫ/2 (s) ≤ 2 eσ,ǫ (t) for all s in the double interval 2I t .

Proof. By (3.6) in Corollary 3.4, there exists δ1 > 0 (depending on t) so that if (3.35)

||σ(·, t) − σ(·, s)||C 0∩W 1,2 < δ1

and B is a finite collection of disjoint closed balls where both σ(·, t) and σ(·, s) have energy at most ǫ1 , then 1 1 (3.36) E(H(σ(·, s), 2 B)) − E(H(σ(·, t), 2 B)) ≤ eσ,ǫ (t)/2 .

Here we have used that eσ,ǫ (t) > 0 since σ(·, t) is not harmonic. Since t → σ(·, t) is continuous as a map to C 0 ∩ W 1,2 , we can choose I t so that for all s ∈ 2 I t (3.35) holds and Z 1 |∇σ(·, t)|2 − |∇σ(·, s)|2 ≤ min { ǫ , eσ,ǫ (t) } . (3.37) 2 S2 2 2 t Suppose now that s ∈ 2I and the energy of σ(·, s) is at most ǫ/2 on a collection B. It follows from (3.37) that the energy of σ(·, t) is at most ǫ on B. Combining (3.36) and (3.37) gives 1 1 (3.38) E(σ(·, s)) − E(H(σ(·, s), 2 B)) − E(σ(·, t)) + E(H(σ(·, t), 2 B)) ≤ eσ,ǫ (t) .


17

Since this applies to any such B, we get that eσ,ǫ/2 (s) ≤ 2 eσ,ǫ (t).

Given a sweepout with no harmonic slices, the next lemma constructs finitely many collections of balls so that harmonic replacement on at least one of these collections strictly decreases the energy. In addition, each collection consists of finitely many pairwise disjoint closed balls. Lemma 3.39. If W > 0 and γ˜ ∈ Ω has no non-constant harmonic slices, then we get an integer m (depending on γ˜ ), m collections of balls B1 , . . . , Bm in S2 , and continuous functions r1 , . . . , rm : [0, 1] → [0, 1] so that for each t: R P (1) At most two rj (t)’s are positive and B∈Bj 12 rj (t)B |∇˜ γ (·, t)|2 < ǫ1 /3 for each j. (2) If E(˜ γ (·, t)) ≥ W/2, then there exists j(t) so that harmonic replacement on decreases energy by at least eγ˜,ǫ1 /8 (t)/8.

rj(t) 2

Bj(t)

Proof. Since the energy of the slices is continuous in t, the set I = {t | E(˜ γ (·, t)) ≥ W/2} t is Rcompact. For each t ∈ I, choose a finite collection B of disjoint closed balls in S2 with 1 |∇˜ γ (·, t)|2 ≤ ǫ1 /4 so 2 ∪ t B

eγ˜,ǫ1 /4 (t) 1 t B )) ≥ > 0. 2 2 Lemma 3.34 gives an open interval I t containing t so that for all s ∈ 2I t

(3.40)

(3.41)

E(γ(·, t)) − E(H(γ(·, t),

eγ˜,ǫ1 /8 (s) ≤ 2 eγ˜,ǫ1 /4 (t) .

Using the continuity of γ˜ (·, s) in C 0 ∩ W 1,2 and Corollary 3.4, we can shrink I t so that γ˜ (·, s) has energy at most ǫ1 /3 in Bt for s ∈ 2I t and, in addition, 1 t 1 t eγ˜,ǫ1 /4 (t) (3.42) . E(γ(·, s)) − E(H(γ(·, s), 2 B )) − E(γ(·, t)) + E(H(γ(·, t), 2 B )) ≤ 4 Since I is compact, we can cover I by finitely many I t ’s, say I t1 , . . . , I tm . Moreover, after discarding some of the intervals, we can arrange that each t is in at least one closed interval I tj , each I tj intersects at most two other I tk ’s, and the I tk ’s intersecting I tj do not intersect each other.11 For each j = 1, . . . m, choose a continuous function rj : [0, 1] → [0, 1] so that / 2I tj . • rj (t) = 1 on I tj and rj (t) is zero for t ∈ • rj (t) is zero on the intervals that do not intersect I tj . Property (1) follows directly and (2) follows from (3.40), (3.41), and (3.42).

Proof. (of Theorem 2.1). Let B1 , . . . , Bm and r1 , . . . , rm : [0, 1] → [0, π) be given by Lemma 3.39. We will use an m step replacement process to define γ. Namely, first set γ 0 = γ˜ and then, for each k = 1, . . . , m, define γ k by applying harmonic replacement to γ k−1 (·, t) on the k-th family of balls rk (t) Bk ; i.e, set γ k (·, t) = H(γ k−1(·, t), rk (t) Bk ). Finally, we set γ = γ m . A key point in the construction is that property (1) of the family of balls gives that only two rk (t)’s are positive for each t. Therefore, the energy bound on the balls given by property 11We

will give a recipe for doing this. First, if I t1 is contained in the union of two other intervals, then throw it out. Otherwise, consider the intervals whose left endpoint is in I t1 , find one whose right endpoint is largest and discard the others (which are anyway contained in these). Similarly, consider the intervals whose right endpoint is in I t1 and throw out all but one whose left endpoint is smallest. Next, repeat this process on I t2 (unless it has already been discarded), etc. After at most m steps, we get the desired cover.

18


(1) implies that each energy minimizing map replaces a map with energy at most 2ǫ1 /3 < ǫ1 . Hence, Corollary 3.4 implies that these depend continuously on the boundary values, which are themselves continuous in t, so that the resulting map γ˜ is also continuous in t. Finally, it is clear that γ˜ is homotopic to γ since continuously shrinking the disjoint closed balls on which we make harmonic replacement gives an explicit homotopy. Thus, γ ∈ Ωγ˜ as claimed. For each t with E(˜ γ (·, t)) ≥ W/2, property (2) of the family of balls gives some j(t) so that e (t) r (t) harmonic replacement for γ˜ (·, t) on j2 Bj(t) decreases the energy by at least γ˜,ǫ18/8 . Thus, even in the worst case where rj (t) Bj(t) is the second family of balls that we do replacement on at t, (3.12) in Lemma 3.11 gives eγ˜,ǫ1 /8 (t) 2 . (3.43) E(˜ γ (·, t)) − E(γ(·, t)) ≥ κ 8 To establish (BΨ ), suppose that B is a finite collection of disjoint closed balls in S2 so that the energy of γ(·, t) on B is at most ǫ1 /12. We can assume that γ k (·, t) has energy at most ǫ1 /8 on B for every k since otherwise Theorem 3.1 implies a positive lower bound for E(˜ γ (·, t)) − E(γ(·, t)). Consequently, we can apply (3.13) in Lemma 3.11 twice (first with µ = 1/8 and then with µ = 1/4) to get 1 1 2 E(γ(·, t)) − E H(γ(·, t), B) ≤ E(˜ γ (·, t)) − E H(˜ γ (·, t), B) + (E(˜ γ (·, t)) − E(γ(·, t)))1/2 8 2 κ 2 (3.44) γ (·, t)) − E(γ(·, t)))1/2 . ≤ eγ˜,ǫ1 /8 (t) + (E(˜ κ Combining (3.43) and (3.44) with Theorem 3.1 gives (BΨ ) and, thus, completes the proof. Appendix A. Bubble convergence implies varifold convergence A.1. Bubble convergence and the topology on Ω. We will need a notion of convergence for a sequence v j of W 1,2 maps to a collection {u0 , . . . , um} of W 1,2 maps which is similar in spirit to the convergence in Gromov’s compactness theorem for pseudo holomorphic curves, [G]. The notion that we will use is a slight weakening of the bubble tree convergence developed by Parker and Wolfson for J-holomorphic curves in [PaW] and used by Parker for harmonic maps in [Pa]. In our applications, the v j ’s will be approximately harmonic while the limit maps ui will be harmonic. We will need the next definition to make this precise. S + and S − will denote the northern and southern hemispheres in S2 and p+ = (0, 0, 1) and p− = (0, 0, −1) the north and south poles. Definition A.1. Given a ball Br (x) ⊂ S2 , the conformal dilation taking Br (x) to S − is the composition of translation x → p− followed by dilation of S2 about p− taking Br (p− ) to S − . The standard example of a conformal dilation comes from applying stereographic projection Π : S2 \ {(0, 0, 1)} → R2 , then dilating R2 by a positive λ 6= 1, and applying Π−1 . In the definition below of convergence, the map u0 will be the standard W 1,2 -weak limit of the v j ’s (see (B1)), while the other ui ’s will arise as weak limits of the composition of the v j ’s with a divergent sequence of conformal dilations of S2 (see (B2)). The condition (B3) guarantees that these limits all arise in genuinely distinct ways, and the condition (B4) means that together the ui’s account for all of the energy.


19

Definition A.2. Bubble convergence. We will say that a sequence v j : S2 → M of W 1,2 maps converges to a collection of W 1,2 maps u0 , . . . , um : S2 → M if the following hold:

(B1) The v j ’s converge weakly to u0 in W 1,2 and there is a finite set S0 = {x10 , . . . , xk00 } ⊂ S2 so that the v j ’s converge strongly to u0 in W 1,2 (K) for any compact K ⊂ S2 \ S0 . (B2) For each i > 0, we get a point xℓi ∈ S0 and a sequence of balls Bri,j (yi,j ) with yi,j → xℓi and ri,j → 0. Furthermore, if Di,j : S2 → S2 is the conformal dilation taking the southern hemisphere to Bri,j (yi,j ), then the maps v j ◦ Di,j converge to ui as in (B1). Namely, v j ◦ Di,j → ui weakly in W 1,2 (S2 ) and there is a finite set Si so that the v j ◦ Di,j ’s converge strongly in W 1,2 (K) for any compact K ⊂ S2 \ Si . r r |y −y |2 (B3) If i1 6= i2 , then rii1 ,j,j + rii2 ,j,j + ir1i,j,j rii2,j,j → ∞. 2 1 P 1 1 j (B4) We get the energy equality m i=0 E(ui ) = limj→∞ E(v ) .

A.2. Two simple examples of bubble convergence. The simplest non-trivial example of bubble convergence is when each map v j = u ◦ Ψj is the composition of a fixed harmonic map u : S2 → M with a divergent sequence of dilations Ψj : S2 → S2 . In this case, the v j ’s converge to the constant map u0 = u(p+ ) on each compact set of S2 \ {p− } and all of the energy concentrates at the single point p− = S0 . Composing the v j ’s with the divergent sequence Ψ−1 j of conformal dilations gives the limit u1 = u. For the second example, let Π : S2 \ {(0, 0, 1)} → R2 be stereographic projection and 2 , let z = x + iy be complex coordinates on R2 = C. If we set fj (z) = 1/(jz) + z = z +1/j z then the maps v j = Π−1 ◦ fj ◦ Π : S2 → S2 are conformal and, therefore, also harmonic. Since each v j is a rational map of degree two, we have E(v j ) = Area(v j ) = 8π. Moreover, the v j ’s converge away from 0 to the identity map which has energy 4π. The other 4π of energy disappears at 0 but can be accounted for by a map u1 by composing with a divergent sequence of conformal dilations; u1 must also have degree one. In this case, the conformal dilations take fj to f˜j (z) = fj (z/j) = 1/z + z/j which converges to the conformal inversion about the circle of radius one. A.3. Bubble convergence implies varifold convergence. Proposition A.3. If a sequence v j of W 1,2 (S2 , M) maps bubble converges to a finite collection of smooth maps u0 , . . . , um : S2 → M, then it also varifold converges. Before getting to the proof, recall that a sequence of functions fj is said to converge in measure to a function f if for all δ > 0 the measure of {x | |fj − f |(x) > δ} goes to zero as j → ∞; see [R], page 95. Clearly, L1 convergence implies convergence in measure. Furthermore, if fj → f in measure and h is uniformly continuous, then h ◦ fj → h ◦ f in measure. Finally, we will use the following general version of the dominated convergence theorem which combines theorem 17 on page 92 of [R] and proposition 20 on page 96 of [R]: R R (DCT) If fj → f in measure, gj → g in L1 , and |fj | ≤ gj , then fj → f . We will also use that the map ∇u → Ju is continuous as a map from L2 to L1 and, thus, Area(u) is continuous with respect to E(u). To be precise, if u, v ∈ W 1,2 (S2 , M), then √ (A.4) |Ju − Jv | ≤ 2 |∇u − ∇v|1/2 max{|∇u|3/2 , |∇v|3/2 } .

20


This follows from the linear algebra fact12 that if S and T are N × 2 matrices, then det S T S − det T T T ≤ 2 |T − S| max{|S|3 , |T |3 } , (A.5) where |S|2 is the sum of the squares of the entries of S and S T is the transpose.

Proof. (of Proposition A.3.) For each v j , we will let V j denote the corresponding map to G2 M. Similarly, for each ui, let Ui denote the corresponding map to G2 M. It follows from (B1)-(B4) that we can choose m + 1 sequences of domains Ωj0 , . . . , Ωjm ⊂ S2 −1 that are pairwise disjoint for each j and so that for each i = 0, . . . , m applying Di,j to Ωji gives a sequence of domains converging to S2 \ Si and accounts for all the energy, that is, Z |∇v j |2 = 0 . (A.6) lim j→∞ S2 \ ∪ Ωj ( i i) By (A.6), the proposition follows from showing for each i and any h in C 0 (G2 M) that Z Z Z j h ◦ V j ◦ Di,j J(vj ◦Di,j ) , h ◦ V Jvj = lim (A.7) h ◦ Ui Jui = lim j j j→∞ −1 j→∞ Di,j (Ωi ) S2 Ωi where the last equality is simply the change of variables formula for integration. To simplify notation in the proof of (A.7), for each i and j, let vij denote the restriction −1 of v j ◦ Di,j to Di,j Ωji and let Vij denote the corresponding map to G2 M. Observe first that Jvj → Jui in L1 (S2 ) by (A.4). Given ǫ > 0 and i, let Ωiǫ be the set where i Jui ≥ ǫ. Since h is bounded and Jvj → Jui in L1 (S2 ), (A.7) would follow from i Z Z j (A.8) lim h ◦ Vi Jv j = h ◦ Ui Jui . j→∞

Ωiǫ

i

Ωiǫ

However, given any δ > 0, W 1,2 convergence implies that the measure of ǫ (A.9) {x ∈ Ωiǫ | Jvj ≥ and |Vij − Ui | ≥ δ} i 2 1 goes to zero as j → ∞. Since L convergence of Jacobians implies that the measure of {x ∈ Ωiǫ | Jvj < 2ǫ } goes to zero, it follows that the maps Vij converge in measure to Ui on i

Ωiǫ . Therefore, the h ◦ Vij ’s converge in measure to h ◦ Ui on Ωiǫ . Consequently, the general version of the dominated convergence theorem (DCT) gives (A.8) and, thus, also (A.7). Appendix B. The proof of Proposition 2.2 The proof of Proposition 2.2 will follow the general structure developed by Parker and Wolfson in [PaW] and used by Parker in [Pa] to prove compactness of harmonic maps with bounded energy. The main difficulty is to rule out loss of energy in the limit (see (B4) in the definition of bubble convergence). The rough idea to deal with this is that energy loss only occurs when there are very small annuli where the maps are “almost” harmonic and the ratio between the inner and outer radii of the annulus is enormous. We will use Proposition that |S T T | ≤ |S| |T |, Tr (S T T ) ≤ |S| |T |, and if Xt is a path of 2 × 2 matrices, then ∂t det Xt = Tr (Xtc ∂t Xt ) where Xtc is the cofactor matrix given by swapping diagonal entries and mulT tiplying off-diagonals by −1. Applying this to Xt = (S + t (T − S)) (S + t (T − S)) and using the mean value theorem gives (A.5). 12Note


21

B.29 to show that the map must be “far” from being conformal on such an annulus and, thus, condition (A) allows us to rule out energy loss. Here “far” from conformal will mean that the θ-energy of the map is much less than the radial energy. To make this precise, it is convenient to replace an annulus Ber2 \ Ber1 in R2 by the conformally equivalent cylinder [r1 , r2 ]×S1 . The (non-compact) cylinder R×S1 with the flat product metric and coordinates t and θ will be denoted by C. For r1 < r2 , let Cr1 ,r2 ⊂ C be the product [r1 , r2 ] × S1 . B.1. Harmonic maps on cylinders. The main result of this subsection is that harmonic maps with small energy on long cylinders are almost radial. This implies that a sequence of such maps with energy bounded away from zero is uniformly far from being conformal and, thus, cannot satisfy (A) in Proposition 2.2. It will be used to prove a similar result for “almost harmonic” maps in Proposition B.29 and eventually be used when we show that energy will not be lost. Proposition B.1. Given δ > 0, there exist ǫ2 > 0 and ℓ ≥ 1 depending on δ (and M) so that if u is a (non-constant) C 3 harmonic map from the flat cylinder C−3ℓ,3ℓ = [−3ℓ, 3ℓ] × S1 to M with E(u) ≤ ǫ2 , then Z Z 2 (B.2) |uθ | < δ |∇u|2 . C−ℓ,ℓ

C−2ℓ,2ℓ

To show this proposition, we show a differential inequality which leads to exponential growth for the θ-energy of the harmonic map on the level sets of the cylinder. Once we have that, the proposition follows. Namely, if the θ-energy in the “middle” of the cylinder was a definite fraction of the total energy over the double cylinder, then the exponential growth would force the θ-energy of near the boundary of the cylinder to be too large. The following standard lemma is the differential inequality for the θ-energy that leads to exponential growth through Lemma B.8 below. Lemma B.3. For a C 3 harmonic map u from Cr1 ,r2 ⊂ C to M ⊂ RN Z Z Z 3 2 2 2 2 |uθ | − 2 sup |A| |∇u|4 . (B.4) ∂t |uθ | ≥ 2 M t t t R 2 Proof. Differentiating t |uθ | and integrating by parts in θ gives Z Z Z Z Z Z Z 1 2 2 2 2 2 ∂ |uθ | = |utθ | + huθ , uttθ i = |utθ | − huθθ , utt i = |utθ | − huθθ , (∆u − uθθ )i 2 t t t t t t Zt Zt Z (B.5) ≥ |utθ |2 + |uθθ |2 − sup |A| |uθθ | |∇u|2 , t

t

M

t

where the last inequality used that |∆u| ≤ |∇u|2 supM |A| by the harmonic map equation.13 2 2 The lemma follows from applying the absorbing R R inequality R 2ab2 ≤ a /2 + 2b and noting that 2 u = 0 so that Wirtinger’s inequality gives t |uθ | ≤ t |uθθ | . t θ 13If

ui are the components of the harmonic map u, gjk is the metric on B, and Aiu(x) is the i-th component of the second fundamental form of M at the point u(x), then page 157 of [SY] gives (B.6)

∆M ui = g jk Aiu(x) (∂j u, ∂k u) .

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Remark B.7. The differential inequality in Lemma B.3 immediately implies that Proposition B.1 holds for harmonic functions, i.e., when |A| ≡ 0, even without the small energy assumption. The general case will follow by using the small energy assumption to show that the perturbation terms are negligible. We will need a simple ODE comparison lemma: Lemma B.8. Suppose that f is a non-negative C 2 function on [−2ℓ, 2ℓ] ⊂ R satisfying f ′′ ≥ f − a ,

(B.9)

for some constant a > 0. If max[−ℓ,ℓ] f ≥ 2a, then Z 2ℓ √ √ (B.10) f ≥ 2 2 a sinh(ℓ/ 2) . −2ℓ

Proof. Fix some x0 ∈ [−ℓ, ℓ] where f achieves its maximum on [−ℓ, ℓ]. Since the lemma is invariant under reflection x → −x, we can assume that x0 ≥ 0. If x0 is an interior point, then f ′ (x0 ) = 0; otherwise, if x0 = ℓ, then f ′ (x0 ) ≥ 0. In either case, we get f ′ (x0 ) ≥ 0. Since f (x0 ) ≥ 2a, (B.9) gives f ′′ (x0 ) ≥ a > 0 and, hence, f ′ is strictly increasing at x0 . We claim that f ′ (x) > 0 for all x in (x0 , 2ℓ]. If not, then there would be a first point y > x0 with f ′ (y) = 0. It follows that f ′ ≥ 0 on [x0 , y] so that f ≥ f (x0 ) ≥ 2a on [x0 , y] and, thus, that f ′′ ≥ a > 0 on [x0 , y], contradicting that f ′ (y) ≤ f ′ (x0 ). By the claim, f is monotone increasing on [x0 , 2ℓ] so that (B.9) gives f ′′ ≥

(B.11)

1 f on [x0 , 2ℓ] . 2

By a standard Riccati comparison argument using f ′ (x0 ) ≥ 0 and (B.11) (see, e.g., corollary A.9 in [CDM]), we get for t ∈ [0, 2ℓ − x0 ] √ √ (B.12) f (x0 + t) ≥ f (x0 ) cosh(t/ 2) ≥ 2 a cosh(t/ 2) . Finally, integrating (B.12) on [0, ℓ] gives (B.10).

Proof. (of Proposition B.1.) Since we will choose ℓ ≥ 1 and ǫ2 < ǫSU , the small-energy interior estimates for harmonic maps (see lemma 3.4 in [SaU]; cf. [SU1]) imply that Z 2 (B.13) sup |∇u| ≤ CSU |∇u|2 ≤ CSU ǫ2 . R

C−2ℓ,2ℓ

C−3ℓ,3ℓ

|uθ |2 . It follows from Lemma B.3 that Z Z 3 2 2 2 ′′ (B.14) f (t) ≥ f (t) − 2 sup |A| CSU ǫ2 (|uθ | + |ut | ) ≥ f (t) − C ǫ2 (|ut |2 − |uθ |2 ) , 2 M t t Set f (t) =

t

2 where C = 2 CSU sup have assumed that C ǫ2 ≤ 1/4 in the second inequality. R M |A|2 and we 2 We will use that t (|ut | − |uθ | ) is constant in t. To see this, differentiate to get Z Z Z 1 2 2 (B.15) ∂t (|ut | − |uθ | ) = (hut, utt i − huθ , utθ i) = hut , (utt + uθθ )i = 0 , 2 t t t


23

where the second equality used integration by parts in θ and the last equality used that utt + uθθ = ∆u is normal to M while ut is tangent.14 Bound this constant by Z Z Z 1 1 2 2 2 2 (B.16) (|ut | − |uθ | ) = (|ut | − |uθ | ) ≤ |∇u|2 . 4ℓ 4ℓ t C−2ℓ,2ℓ C−2ℓ,2ℓ R By (B.14) and (B.16), Lemma B.8 with a = C4ℓǫ2 C−2ℓ,2ℓ |∇u|2 implies that either Z C ǫ2 |∇u|2 , (B.17) max f < 2 [−ℓ,ℓ] 4ℓ C−2ℓ,2ℓ or √ Z Z Z 2ℓ √ sinh(ℓ/ 2) 2 (B.18) |uθ | = f (t) dt ≥ 2 2 C ǫ2 |∇u|2 . 4ℓ C−2ℓ,2ℓ −2ℓ C−2ℓ,2ℓ The second possibility cannot occur as long as ℓ is sufficiently large so that we have √ √ sinh(ℓ/ 2) (B.19) 2 2 C ǫ2 > 1. 4ℓ Using the upper bound (B.17) for f on [−ℓ, ℓ] to bound the integral of f gives Z Z 2 (B.20) |uθ | ≤ 2ℓ max f < C ǫ2 |∇u|2 . C−ℓ,ℓ

[−ℓ,ℓ]

C−2ℓ,2ℓ

The proposition follows by choosing ǫ2 > 0 so that C ǫ2 < min{1/4, δ} and then choosing ℓ so that (B.19) holds. B.2. Weak compactness of almost harmonic maps. We will need a compactness theorem for a sequence of maps uj in W 1,2 (S2 , M) which have uniformly bounded energy and are locally well-approximated by harmonic maps. Before stating this precisely, it is useful to recall the situation for harmonic maps. Suppose therefore that uj : S2 → M is a sequence of harmonic maps with E(uj ) ≤ E0 for some fixed E0 . After passing to a subsequence, we can assume that the measures |∇uj |2 dx converge and there is a finite set S of points where the energy concentrates so that: Z j 2 |∇u | ≥ ǫSU . (B.21) If x ∈ S, then inf lim r>0 j→∞ B (x) r Z j 2 (B.22) If x ∈ / S, then inf lim |∇u | < ǫSU . r>0

j→∞

Br (x)

The constant ǫSU > 0 comes from [SaU], so that (B.22) implies uniform C 2,α estimates on the uj ’s in some neighborhood of x. Hence, Arzela-Ascoli and a diagonal argument give a further subsequence of the uj ’s C 2 -converging to a harmonic map on every compact subset of S2 \ S. We will need a more general version of this, where uj : S2 → M is a sequence of W 1,2 maps with E(uj ) ≤ E0 that are ǫ0 -almost harmonic in the following sense: 14In

fact, something much stronger is true: The complex-valued function φ(t, θ) = (|ut |2 − |uθ |2 ) − 2 i hut , uθ i

is holomorphic on the cylinder (see page 6 of [SY]). This is usually called the Hopf differential.

24


R (B0 ) If B ⊂ S2 is any ball with B |∇uj |2 < ǫ0 , then there is an energy minimizing map v : 81 B → M with the same boundary values as uj on ∂ 18 B with Z j ∇u − ∇v 2 ≤ 1/j . 1 B 8

Lemma B.23. Let ǫ0 > 0 be less than ǫSU . If uj : S2 → M is a sequence of W 1,2 maps satisfying (B0 ) and with E(uj ) ≤ E0 , then there exists a finite collection of points {x1 , . . . , xk }, a subsequence still denoted by uj , and a harmonic map u : S2 → M so that uj → u weakly in W 1,2 and if K ⊂ S2 \ {x1 , . . . , xk } is compact, then uj → u in W 1,2 (K). Furthermore, the measures |∇uj |2 dx converge to a measure ν with ǫ0 ≤ ν(xi ) and ν(S2 ) ≤ E0 . Proof. After passing to a subsequence, we can assume that: • The uj ’s converge weakly in W 1,2 to a W 1,2 map u : S2 → M. • The measures |∇uj |2 dx converge to a limiting measure ν with ν(S2 ) ≤ E0 .

It follows that there are at most E0 /ǫ0 points x1 , . . . , xk with limr→0 ν (Br (xj )) ≥ ǫ0 . We will show next that away from the xi ’s the convergence is strong in W 1,2 and u is harmonic. To see a point x ∈ / {x1 , . . . , xk }. By definition, there exist rx > 0 R this, consider j 2 and Jx so that Br (x) |∇u | < ǫ0 for j ≥ Jx . In particular, (B0 ) applies so we get energy x minimizing maps vxj : 18 Brx (x) → M that agree with uj on ∂ 18 Brx (x) and satisfy Z j ∇vx − ∇uj 2 ≤ 1/j . (B.24) 1 8

Brx (x)

(Here 81 Brx (x) is the ball in S2 centered at x so that the stereographic projection Πx which takes x to 0 ∈ R2 takes 18 Brx (x) and Brx (x) to balls centered at 0 whose radii differ by a factor of 8.) Since E(vxj ) ≤ ǫ0 ≤ ǫSU , it follows from lemma 3.4 in [SaU] (cf. [SU1]) that a subsequence of the vxj ’s converges strongly in W 1,2 ( 91 Brx (x)) to a harmonic map vx : 19 Brx (x) → M. Combining with the triangle inequality and (B.24), we get Z Z Z j 2 j j j 2 ∇v − ∇vx 2 → 0 . (B.25) ∇u − ∇vx ≤ 2 ∇u − ∇vx + 2 x 1 9

1 9

Brx (x)

1 9

Brx (x)

Brx (x)

Similarly, this convergence, the triangle inequality, (B.24), and the Dirichlet Poincaré inequality (theorem 3 on page 265 of [E]; this applies since vxj equals uj on ∂ 18 Brx (x)) give Z Z Z j 2 j j j 2 v − vx 2 → 0 . (B.26) u − vx ≤ 2 u − vx + 2 x 1 9

Brx (x)

1 8

Brx (x)

1 9

Brx (x)

j Combining (B.25) in W 1,2 ( 91 Brx (x)). and (B.26), we see that the u ’s converge to vx strongly In particular, u 1 Br (x) = vx . We conclude that u is harmonic on S2 \ {x1 , . . . , xk }. Further9

x

more, since any compact K ⊂ S2 \ {x1 , . . . , xk } can be covered by a finite number of such ninth-balls, we get that uj → u strongly in W 1,2 (K). Finally, since u has finite energy, it must have removable singularities at each of the xi ’s and, hence, u extends to a harmonic map on all of S2 (see theorem 3.6 in [SaU]).


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B.3. Almost harmonic maps on cylinders. The main result of this subsection, Proposition B.29 below, extends Proposition B.1 from harmonic maps to “almost harmonic” maps. Here “almost harmonic” is made precise in Definition B.27 below and roughly means that harmonic replacement on certain balls does not reduce the energy by much. Definition B.27. Given ν > 0 and a cylinder Cr1 ,r2 , we will say that a W 1,2 (Cr1 ,r2 , M) map u is ν-almost harmonic if for any finite collection of disjoint closed balls B in the conformally equivalent annulus Ber2 \ Ber1 ⊂ R2 there is an energy minimizing map v : ∪B 81 B → M that equals u on ∪B 18 ∂B and satisfies Z Z ν 2 |∇u|2 . |∇u − ∇v| ≤ (B.28) 2 Cr1 ,r2 ∪B 81 B We have used a slight abuse of notation, since our sets will always be thought of as being subsets of the cylinder; i.e., we identify Euclidean balls in the annulus with their image under the conformal map to the cylinder. In this subsection and the two that follow it, given δ > 0, the constants ℓ ≥ 1 and ǫ2 > 0 will be given by Proposition B.1; these depend only on M and δ. Proposition B.29. Given δ > 0, there exists ν > 0 (depending on δ and M) so that if m is a positive integer and u is ν-almost harmonic from C−(m+3)ℓ,3ℓ to M with E(u) ≤ ǫ2 , then Z Z 2 (B.30) |uθ | ≤ 7 δ |∇u|2 . C−mℓ,0

C−(m+3)ℓ,3ℓ

We will prove Proposition B.29 by using a compactness argument to reduce it to the case of harmonic maps and then appeal to Proposition B.1. A key difficulty is that there is no upper bound on the length of the cylinder in Proposition B.29 (i.e., no upper bound on m), so we cannot directly apply the compactness argument. This will be taken care of by dividing the cylinder into subcylinders of a fixed size and then using a covering argument. B.4. The compactness argument. The next lemma extends Proposition B.1 from harmonic maps on C−3ℓ,3ℓ to almost harmonic maps. The main difference from Proposition B.29 is that the cylinder is of a fixed size in Lemma B.31. Lemma B.31. Given δ > 0, there exists µ > 0 (depending on δ and M) so that if u is a µ-almost harmonic map from C−3ℓ,3ℓ to M with E(u) ≤ ǫ2 , then Z Z 2 (B.32) |uθ | ≤ δ |∇u|2 . C−ℓ,ℓ

C−3ℓ,3ℓ

Proof. We will argue by contradiction, so suppose that there exists a sequence uj of 1/jalmost harmonic maps from C−3ℓ,3ℓ to M with E(uj ) ≤ ǫ2 and Z Z j 2 (B.33) |uθ | > δ |∇uj |2 . C−ℓ,ℓ

C−3ℓ,3ℓ

We will show that a subsequence of the uj ’s converges to a non-constant harmonic map that contradicts Proposition B.1. We will consider two separate cases, depending on whether or not E(uj ) goes to 0.

26


Suppose first that lim supj→∞ E(uj ) > 0. The upper bound on the energy combined with being 1/j-almost harmonic (and the compactness of M) allows us to argue as in Lemma B.23 to get a subsequence that converges in W 1,2 on compact subsets of C−3ℓ,3ℓ to a non-constant harmonic map u˜ : C−3ℓ,3ℓ → M. Furthermore, using the W 1,2 convergence on CR−ℓ,ℓ together R with the lower semi-continuity of energy, (B.33) implies that C−ℓ,ℓ |˜ uθ |2 ≥ δ C−3ℓ,3ℓ |∇˜ u|2 . This contradicts Proposition B.1. Suppose now that E(uj ) → 0. Replacing uj by v j = (uj −uj (0))/(E(uj ))1/2Rgives a sequence of maps to Mj = (M − uj (0))/E(uj ))1/2 with E(v j ) = 1 and, by (B.33), C−ℓ,ℓ |vθj |2 > δ > 0. Furthermore, the v j ’s are also 1/j-almost harmonic (this property is invariant under dilation), so we can still argue as in Lemma B.23 to get a subsequence that converges in W 1,2 on compact subsets of C−3ℓ,3ℓ to a harmonic map v : S2 → RN (we are using here that a subsequence (B.33) implies that j ’s converges to an affine space). As before, R R of the M R 2 2 |vθ | ≥ δ C−3ℓ,3ℓ |∇v| . This time our normalization gives C−ℓ,ℓ |vθ |2 ≥ δ so that v C−ℓ,ℓ contradicts Proposition B.1 (see Remark B.7), completing the proof. B.5. The proof of Proposition B.29. Proof. (of Proposition B.29). For each integer j = 0, . . . , m, let C(j) = C−(j+3)ℓ,(3−j)ℓ and let µ > 0 be given by Lemma B.31. We will say that the j-th cylinder C(j) is good if the restriction of u to C(j) is µ-almost harmonic; otherwise, we will say that C(j) is bad . On each good C(j), we apply Lemma B.31 to get Z Z 2 (B.34) |uθ | ≤ δ |∇u|2 , C−(j+1)ℓ,(1−j)ℓ

C(j)

so that summing this over the good j’s gives X Z X Z 2 (B.35) |uθ | ≤ δ j good

C−(j+1)ℓ,(1−j)ℓ

j good

2

C(j)

|∇u| ≤ 6 δ

Z

C−(m+3)ℓ,3ℓ

|∇u|2 ,

where the last inequality used that each Ci,i+1 is contained in at most 6 of the C(j)’s. We will complete the proof by showing that the total energy (not just the θ-energy) on the bad C(j)’s is small. By definition, for each bad C(j), we can choose a finite collection of disjoint closed balls Bj in C(j) so that if v : 18 Bj → M is any energy-minimizing map that equals u on ∂ 18 Bj , then Z Z 2 |∇u − ∇v| ≥ aj > µ |∇u|2 . (B.36) 1 B 8 j

C(j)

Since the interior of each C(j) intersects only the C(k)’s with 0 < |j − k| ≤ 5, we can divide the bad C(j)’s into ten subcollections so that the interiors of the C(j)’s in each subcollection are pair-wise disjoint. In particular, one of these disjoint subcollections, call it Γ, satisfies Z X 1 X 1 X (B.37) aj ≥ aj ≥ µ |∇u|2 , 10 10 C(j) j∈Γ j bad j bad where the last inequality used (B.36).


27

However, since ∪j∈Γ Bj is itself a finite collection of disjoint closed balls in the entire cylinder C−(m+3)ℓ,3ℓ and u is ν-almost harmonic on C−(m+3)ℓ,3ℓ , we get that Z Z µ X 2 (B.38) |∇u| ≤ ν |∇u|2 . 10 C(j) C−(m+3)ℓ,3ℓ j bad

To get the proposition, combine (B.35) with (B.38) to get Z Z 10 ν 2 |∇u|2 . (B.39) |uθ | ≤ 6 δ + µ C−(m+3)ℓ,3ℓ C−mℓ,0

Finally, choosing ν sufficiently small completes the proof.

B.6. Bubble compactness. We will now prove Proposition 2.2 using a variation of the renormalization procedure developed in [PaW] for pseudo-holomorphic curves and later used in [Pa] for harmonic maps. A key point in the proof will be that the uniform energy bound, (A), and (B) are all dilation invariant, so they apply also to the compositions of the uj ’s with any sequence of conformal dilations of S2 . Proof. (of Proposition 2.2). We will use the energy bound and (B) to show that a subsequence of the uj ’s converges in the sense of (B1), (B2), and (B3) of Definition A.2 to a collection of harmonic maps. We will then come back and use (A) and (B) to show that the energy equality (B4) also holds. Hence, the subsequence bubble converges and, thus by Proposition A.3, also varifold converges. Set δ = 1/21 and let ℓ ≥ 1 and ǫ2 > 0 be given by Proposition B.1. Set ǫ3 = min{ǫ0 /2, ǫ2 }. Step 1: Initial compactness. Lemma B.23 gives a finite collection of singular points S0 ⊂ S2 , a harmonic map v0 : S2 → M, and a subsequence (still denoted uj ) that converges to v0 weakly in W 1,2 (S2 ) and strongly in W 1,2 (K) for any compact subset K ⊂ S2 \ S0 . Furthermore, the measures |∇uj |2 dx converge to a measure ν0 with ν0 (S2 ) ≤ E0 and each singular point in x ∈ S0 has ν0 (x) ≥ ǫ0 . Step 2: Renormalizing at a singular point. Suppose that x ∈ S0 is a singular point from the Fix a radius ρ > 0 so that x is the only singular point in B2ρ (x) and R first step. 2 |∇v | < ǫ 0 3 /3. For each j, let rj > 0 be the smallest radius so that Bρ (x) Z |∇uj |2 = ǫ3 , (B.40) inf y∈Bρ−rj (x)

Bρ (x)\Brj (y)

and choose a ball Brj (yj ) ⊂ Bρ (x) with

R

Bρ (x)\Brj (yj )

|∇uj |2 = ǫ3 . Since the uj ’s converge

to v0 on compact subsets of Bρ (x) \ {x}, we get that yj → x and rj → 0. For each j, let Ψj : R2 → R2 be the “dilation” that takes Brj (yj ) to the unit ball B1 (0) ⊂ R2 . By dilation invariance, the dilated maps u˜j1 = uj ◦ Ψ−1 j still satisfy (B) and have the same energy. Hence, Lemma B.23 gives a subsequence (still denoted by u˜j1 ), a finite singular set S1 , and a harmonic map v1 so that the u˜j1 ◦ Π’s converge to v1 weakly in W 1,2 (S2 ) and strongly in W 1,2 (K) for any compact subset K ⊂ S2 \ S1 . Moreover, the measures |∇˜ uj1 ◦ Π|2 dx’s converge to a measure ν1 . The choice of the balls Brj (yj ) guarantees that ν1 (S2 \ {p+ }) ≤ ν0 (x) and ν1 (S − ) ≤ ν0 (x) − ǫ3 . (Recall that stereographic projection Π takes the open southern hemisphere S − to the open unit ball in R2 .) The key point for iterating this is the following claim:

28


(⋆) The maximal energy concentration at any y ∈ S1 \ {p+ } is at most ν0 (x) − ǫ3 /3. Since the energy at a singular point or the energy for a non-trivial harmonic map is at least ǫ0 > ǫ3 , the only one way that (⋆) could possibly fail is if v1 is constant, S1 is exactly two points p+ and y, and at mostRǫ3 /3 of ν0 (x) escapes at p+ . However, this would imply that t all but at most 2ǫ3 /3 of the Bρ (x) |∇uj |2 is in Btj (yj ) with rjj → 0 which contradicts the minimality of rj . Step 3: Repeating this. We repeat this blowing up construction at the remaining singular points in S0 , as well as each of the singular points S1 in the southern hemisphere, etc., to get new limiting harmonic maps and new singular points to blow up at. It follows from (⋆) that this must terminate after at most 3 E0 /ǫ3 steps. Step 4: The necks. We have shown that the uj ’s converge to a collection of harmonic maps in the sense of (B1), (B2), and (B3). It remains to show (B4), i.e., that the vk ’s accounted for all of the energy in the sequence uj and no energy was lost in the limit. To understand how energy could be lost, it is useful to re-examine what happens to the energy during the blow up process. At each stage in the blow up process, energy is “taken from” a singular point x and then goes to one of two places: • It can show up in the new limiting harmonic map of to a singular point in S2 \ {p+ }. • It can disappear at the north pole p+ (i.e., ν1 (S2 \ {p+ }) < ν0 (x)). In the first case, the energy is accounted for in the limit or survives to a later stage. However, in the second case, the energy is lost for good, so this is what we must rule out. We will argue by contradiction, so suppose that ν1 (S2 \ {p+ }) < ν0 (x) − δˆ for some δˆ > 0. (Note that we must have δˆ ≤ ǫ3 .) Using the notation in Step 1, suppose therefore that Aj = Bsj (yj ) \ Btj (yj ) are annuli with: Z tj (B.41) sj → 0 , → ∞ , and |∇uj |2 ≥ δˆ > 0 . rj Aj

There is obviously quite a bit of freedom in choosing sj and tj . In particular, we can choose a sequence λj → ∞ so that the annuli A˜j = Bρ/2 (yj ) \ Btj /λj (yj ) also satisfies this, i.e., λ R j sj →j 02 and tj /(λj rj ) → ∞. It follows from (B.41) and the definition of the rj ’s that ˜j |∇u | ≤ ǫ3 ≤ ǫ2 . However, combining this with Proposition B.29 (with δ = 1/21) shows A that the area must be strictly less than the energy for j large, contradicting (A), and thus completing the proof. Appendix C. The proof of Theorem 3.1

C.1. An application of the Wente lemma. The proof of Theorem 3.1 will use the following L2 estimate for h ζ where ζ is a L2 (B1 ) holomorphic function and h is a W 1,2 function vanishing on ∂B1 . Proposition C.1. If ζ is a holomorphic function on B1 ⊂ R2 and h ∈ W01,2 (B1 ), then Z Z Z 2 2 2 2 (C.2) h |ζ| ≤ 8 |∇h| |ζ| . B1

B1

B1

The estimate (C.2) does not follow from the Sobolev embedding theorem as the product of functions in L2 and W 1,2 is in Lp for p < 2, but not necessarily for p = 2. To get around this, we will use the following lemma of H. Wente (see [W]; cf. theorem 3.1.2 in [He1]).


29

Lemma C.3. If B1 ⊂ R2 and u, v ∈ W 1,2(B1 ), then there exists φ ∈ C 0 ∩ W01,2 (B1 ) with ∆φ = h(∂x1 u, ∂x2 u), (−∂x2 v, ∂x1 v)i so that ||φ||C 0 + ||∇φ||L2 ≤ ||∇u||L2 ||∇v||L2 .

(C.4)

Proof. (of Proposition C.1.) Let f and g be the real and imaginary parts, respectively, of the holomorphic function ζ, so that the Cauchy-Riemann equations give ∂x1 f = ∂x2 g and ∂x2 f = −∂x1 g .

(C.5)

Since B1 is simply connected, (C.5) gives functions u and v on B1 with ∇u = (g, f ) and ∇v = (f, −g). We have (C.6)

|∇u|2 = |∇v|2 = h(∂x1 u, ∂x2 u), (−∂x2 v, ∂x1 v)i = |ζ|2 .

Therefore, Lemma C.3 gives φ with ∆φ = |ζ|2, φ|∂B1 = 0, and Z (C.7) ||φ||C 0 + ||∇φ||L2 ≤ |ζ|2 .

Applying Stokes’ theorem to div(h2 ∇φ) and using Cauchy-Schwarz gives Z 1/2 Z Z Z 2 2 2 2 2 2 (C.8) h |ζ| = h ∆φ ≤ |∇h | |∇φ| ≤ 2 ||∇h||L2 . h |∇φ|

Applying Stokes’ theorem to div(h2 φ∇φ), noting that ∆φ ≥ 0, and using (C.8) gives Z 1/2 Z Z 2 2 2 2 2 2 (C.9) h |∇φ| ≤ |φ| h ∆φ + |∇h | |∇φ| ≤ 4 ||φ||C 0 ||∇h||L2 h |∇φ| ,

1/2 R 2 h |∇φ|2 ≤ 4 ||∇h||L2 ||φ||C 0 . Finally, substituting this bound back into (C.8) so that and using (C.7) to bound ||φ||C 0 gives the proposition.

C.2. An application to harmonic maps. Proposition C.10. Suppose that M ⊂ RN is a smooth closed isometrically embedded manifold. There exists a constant ǫ0 > 0 (depending on M) so that if v : B1 → M is a W 1,2 weakly harmonic map with energy at most ǫ0 , then v is a smooth harmonic map. In addition, for any h ∈ W01,2 (B1 ), we have Z Z Z 2 2 2 2 (C.11) |h| |∇v| ≤ C |∇h| |∇v| . B1

B1

B1

Proof. The first claim follows immediately from F. Hélein’s 1991 regularity theorem for weakly harmonic maps from surfaces; see [He2] or theorem 4.1.1 in [He1]. We will show that (C.11) follows by combining estimates from the proof of theorem 4.1.1 in [He1]15 with Proposition C.1. Following [He1], we can assume that the pull-back v ∗ (T M) of the tangent bundle of M has orthonormal frames on B1 and, moreover, that there is a finite energy harmonic section e1 , . . . , en of the bundle of orthonormal frames for v ∗ (T M) 15Alternatively,

one could use the recent results of T. Rivière, [Ri].

30


(the frame e1 , . . . , en is usually called a Coulomb gauge). Set αj = h∂x1 v, ej i − i h∂x2 v, ej i for j = 1, . . . , n. Since e1 , . . . , en is an orthonormal frame for v ∗ (T M), we have 2

|∇v| =

(C.12)

n X j=1

|αj |2 .

On pages 181 and 182 of [He1], Hélein uses that the frame e1 , . . . , en is harmonic to construct an n × n matrix-valued function β (i.e., a map β : B1 → GL(n, C)) with |β| ≤ C, |β −1| ≤ C, and with ∂z¯ (β −1 α) = 0 (where the constant C depends only on M and the bound for the energy of v; see also lemma 3 on page 461 in [Q] where this is also stated). In particular, we get an n-tuple of holomorphic functions (ζ 1, . . . , ζ n ) = ζ = β −1 α, so that (C.13)

C −2 |ζ|2 ≤ |α|2 = |β ζ|2 ≤ C 2 |ζ|2 .

The claim (C.11) now follows from Proposition C.1. Namely, using (C.12), the second inequality in (C.13), and then applying Proposition C.1 to the n holomorphic functions ζ 1 , . . . , ζ n gives Z Z Z Z Z Z 2 2 2 2 2 2 2 2 4 2 (C.14) |h| |∇v| ≤ C |h| |ζ| ≤ 8 C |∇h| |ζ| ≤ 8 C |∇h| |∇v|2 , where the last inequality used the first inequality in (C.13) and (C.12).

C.3. The proof of Theorem 3.1. Proof. (of Theorem 3.1.) Use Stokes’ theorem and that u and v are equal on ∂B1 to get Z Z Z Z 2 2 2 (C.15) |∇u| − |∇v| − |∇(u − v)| = −2 h(u − v), ∆vi ≡ Ψ . R To show (3.2), it suffices to bound |Ψ| by 12 |∇v − ∇u|2 . The harmonic map equation (B.6) implies that ∆v is perpendicular to M and (C.16)

|∆v| ≤ |∇v|2 sup |A| . M

We will need the elementary geometric fact that there exists a constant C depending on M so that whenever x, y ∈ M, then (x − y)N ≤ C |x − y|2 , (C.17)

where (x − y)N denotes the normal part of the vector (x − y) at the point x ∈ M (the same bound holds at y by symmetry). The point is that either |x − y| ≥ 1/C so (C.17) holds trivially or the vector (x − y) is “almost tangent” to M. Using that u and v both map to M, we can apply (C.17) to get (u − v)N ≤ C |u − v|2 , where the normal projection is at the point v(x) ∈ M. Putting all of this together gives Z (C.18) |Ψ| ≤ C |v − u|2 |∇v|2 ,

where C depends on supM |A|. As long as ǫ1 is less than ǫ0 , we can apply Proposition C.10 with h = |u − v| to get Z Z Z Z 2 2 ′ 2 2 ′ (C.19) |v − u| |∇v| ≤ C |∇|u − v|| |∇v| ≤ C ǫ1 |∇u − ∇v|2 .


31

The lemma follows by combining (C.18) and (C.19) and then taking ǫ1 sufficiently small. Combining Corollary 3.3 and the regularity theory of [Mo1], or [SU1], for energy minimizing maps recovers Hélein’s theorem that weakly harmonic maps from surfaces are smooth. Note, however, that we used estimates from [He1] in the proof of Theorem 3.1. Appendix D. The equivalence of energy and area By (1.4), Proposition 1.5 follows once we show that WE ≤ WA . The corresponding result for the Plateau problem is proven by taking a minimizing sequence for area and reparametrizing to make these maps conformal, i.e., choosing isothermal coordinates. There are a few technical difficulties in carrying this out since the pull-back metric may be degenerate and is only in L1 , while the existence of isothermal coordinates requires that the induced metric be positive and bounded; see, e.g., proposition 5.4 in [SW]. We will follow the same approach here, the difference is that we need the reparametrizations to vary continuously with t. D.1. Density of smooth mappings. The next lemma observes that the regularization using convolution of Schoen-Uhlenbeck in the proposition in section 4 of [SU2] is continuous. Lemma D.1. Given γ ∈ Ω and ǫ > 0, there exists a regularization γ˜ ∈ Ωγ so that (D.2)

max ||˜ γ (·, t) − γ(·, t)||W 1,2 ≤ ǫ , t

each slice γ˜ (·, t) is C 2 , and the map t → γ˜ (·, t) is continuous from [0, 1] to C 2 (S2 , M). Proof. Since M is smooth, compact and embedded, there exists a δ > 0 so that for each x in the δ-tubular neighborhood Mδ of M in RN , there is a unique closest point Π(x) ∈ M and so the map x → Π(x) is smooth. Π is called nearest point projection. x−y . Since Given y in the open ball B1 (0) ⊂ R3 , define Ty : S2 → S2 by Ty (x) = |x−y| each Ty is smooth and these maps depend smoothly on y, it follows that the map (y, f ) → f ◦ Ty is continuous from B1 (0) × C 0 ∩ W 1,2 (S2 , RN ) → C 0 ∩ W 1,2 (S2 , RN ) (this is clear for f ∈ C 1 and follows for C 0 ∩ W 1,2 by density). Therefore, since T0 is the identity, given f ∈ C 0 ∩W 1,2 (S2 , RN ) and µ > 0, there exists r > 0 so that sup|y|≤r ||f ◦Ty −f ||C 0 ∩W 1,2 < µ. Applying this to γ(·, t) for each t and using that t → γ(·, t) is continuous to C 0 ∩ W 1,2 and [0, 1] is compact, we get r¯ > 0 with (D.3)

sup sup ||Ty γ(·, t) − γ(·, t)||C 0∩W 1,2 < µ .

t∈[0,1] |y|≤¯ r

Next fix a smooth radial mollifier φ ≥ 0 with integral one and compact support in the unit ball in R3 . For each r ∈ (0, 1), define φr (x) = r −3 φ(x/r) and set Z Z y (D.4) γr (x, t) = φr (y)γ(Ty (x) , t) dy = φr (x − y)γ( , t) dy . |y| Br (0) Br (x) We have the following standard properties of convolution with a mollifier (see, e.g., section 5.3 and appendix C.4 in [E]): First, each γr (·, t) is smooth and for each k the map t → γr (·, t)

32


is continuous from [0, 1] to C k (S2 , RN ). Second, (D.5)

||γr (·, t) − γ(·, t)||2C 0 ≤ sup ||Ty γ(·, t) − γ(·, t)||2C 0 , ||∇γr (·, t) −

∇γ(·, t)||2L2

|y|≤r

≤ sup ||Ty γ(·, t) − γ(·, t)||2L2 . |y|≤r

It follows from (D.5) and (D.3) that for r ≤ r¯ and all t we have (D.6)

||γr (·, t) − γ(·, t)||C 0∩W 1,2 < µ .

The map γr (·, t) may not land in M, but it is in Mδ when µ is small by (D.6). Hence, the map γ˜ (x, t) = Π ◦ γr (x, t) satisfies (D.2), each slice γ˜ (·, t) is C 2 , and t → γ˜ (·, t) is continuous from [0, 1] to C 2 (S2 , M). Finally, s → γ˜sr is an explicit homotopy connecting γ˜ and γ. D.2. Equivalence of energy and area. We will also need the existence of isothermal coordinates, taking special care on the dependence on the metric. Let S2g0 denote the round metric on S2 with constant curvature one. Lemma D.7. Given a C 1 metric g˜ on S2 , there is a unique orientation preserving C 1,1/2 conformal diffeomorphism hg˜ : S2g0 → S2g˜ that fixes 3 given points. Moreover, if g˜1 and g˜2 are two C 1 metrics that are both ≥ ǫ g0 for some ǫ > 0, then (D.8)

g1 − g˜2 ||C 0 , ||hg˜1 − hg˜2 ||C 0∩W 1,2 ≤ C ||˜

where the constant C depends on ǫ and the maximum of the C 1 norms of the g˜i’s. Proof. The Riemann mapping theorem for variable metrics (see theorem 3.1.1 and corollary 3.1.1 in [Jo]; cf. [ABe] or [Mo2]) gives the conformal diffeomorphism hg˜ : S2g0 → S2g˜ . We will separately bound the C 0 and W 1,2 norms. First, lemma 17 in [ABe] gives (D.9)

||hg˜1 − hg˜2 ||C 0 ≤ C1 ||˜ g1 − g˜2 ||C 0 ,

where C1 depends on ǫ and the C 0 norms of the metrics. Second, theorem 8 in [ABe] gives a uniform Lp bound for ∇(hg˜1 − hg˜2 ) on any unit ball in S2 where p > 2 by (8) in [ABe] (D.10)

||∇(hg˜1 − hg˜2 )||Lp (B1 ) ≤ C2 ||˜ g1 − g˜2 ||C 0 (S2 ) ,

where C2 depends on ǫ and the C 0 norms of the metrics. Covering S2 by a finite collection of unit balls and applying Hölder’s inequality gives the desired energy bound. We can now prove the equivalence of the two widths. Proof. (of Proposition 1.5). By (1.4), we have that WA ≤ WE . To prove that WE ≤ WA , given ǫ > 0, let γ ∈ Ωβ be a sweepout with maxt∈[0,1] Area (γ(·, t)) < WA + ǫ/2. By Lemma D.1, there is a regularization γ˜ ∈ Ωβ so that each slice γ˜(·, t) is C 2 , the map t → γ˜ (·, t) is continuous from [0, 1] to C 2 (S2 , M), and (also by (A.4)) (D.11)

max Area (˜ γ (·, t)) < WA + ǫ . t

The maps γ˜(·, t) induce a continuous one-parameter family of pull-back (possibly degenerate) C 1 metrics g(t) on S2 . Lemma D.7 requires that the metrics be non-degenerate, so define perturbed metrics g˜(t) = g(t) + δ g0 . For each t, Lemma D.7 gives C 1,1/2 conformal diffeomorphisms ht : S2g0 → S2g˜(t) that vary continuously in C 0 ∩ W 1,2 (S2 , S2 ). The continuity


33

of t → γ˜ (·, t) ◦ ht as a map from [0, 1] to C 0 ∩ W 1,2 (S2 , M) follows from this, the continuity of t → γ˜ (·, t) in C 2 , and the chain rule. We will now use the conformality of the map ht to control the energy of the composition as in proposition 5.4 of [SW]. Namely, we have that E (˜ γ (·, t) ◦ ht ) = E (ht : S2g0 → S2g(t) ) ≤ E (ht : S2g0 → S2g˜(t) ) Z 2 (D.12) = Area (Sg˜(t) ) = [det(g0−1 g(t)) + δ Tr(g0−1 g(t)) + δ 2 ]1/2 dvolg0 S2

≤

Area (S2g(t) )

+ 4π [δ 2 + 2 δ sup |g0−1 g(t)|]1/2 . t

Choose δ > 0 so that 4π [δ 2 + 2 δ supt |g0−1 g(t)|]1/2 < ǫ. We would be done if γ˜ (·, t) ◦ ht was homotopic to γ˜ . However, the space of orientation preserving diffeomorphisms of S2 is homotopic to RP3 by Smale’s theorem. To get around this, note that t → ||˜ γ (·, t)||C 2 is continuous and zero when t = 1, thus for some τ < 1 ǫ (D.13) sup ||˜ γ (·, t)||C 2 ≤ . supt∈[0,1] ||ht ||2W 1,2 t≥τ ˜ t equal to ht ≡ h(t) on [0, τ ] and equal to h(τ (1 − t)/(1 − τ )) on Consequently, if we set h ˜ t ) ≤ WA + 2 ǫ . Moreover, the [τ, 1], then (D.12) and (D.13) imply that maxt∈[0,1] E (˜ γ (·, t) ◦ h ˜ t is also in Ω. Finally, replacing τ by sτ and taking s → 0 gives an explicit map γ˜ (·, t) ◦ h ˜ t to γ˜ (·, t). homotopy in Ω from γ˜(·, t) ◦ h References [ABe] L. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960) 385–404. [Al] F.J. Almgren, The theory of varifolds, Mimeographed notes, Princeton, 1965. [B1] G.D. Birkhoff, Dynamical systems with two degrees of freedom. TAMS 18 (1917), no. 2, 199–300. [B2] G.D. Birkhoff, Dynamical systems, AMS Colloq. Publ. vol 9, Providence, RI, 1927. [ChYa] D. Christodoulou and S.T. Yau, Some remarks on the quasi-local mass, Mathematics and general relativity (Santa Cruz, CA, 1986), 9–14, Contemp. Math., vol. 71, AMS, Providence, RI, 1988. [CD] T.H. Colding and C. De Lellis, The min–max construction of minimal surfaces, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002), 75–107, Int. Press, Somerville, MA, 2003, math.AP/0303305. [CDM] T.H. Colding, C. De Lellis, and W.P. Minicozzi II, Three circles theorems for Schrödinger operators on cylindrical ends and geometric applications, CPAM , to appear, math.DG/0701302. [CM1] T.H. Colding and W.P. Minicozzi II, Estimates for the extinction time for the Ricci flow on certain 3–manifolds and a question of Perelman, JAMS , 18 (2005), no. 3, 561–569, math.AP/0308090. [CM2] T.H. Colding and W.P. Minicozzi II, Minimal surfaces, Courant Lecture Notes in Mathematics, 4. New York University, Courant Institute of Mathematical Sciences, New York, 1999. [CM3] T.H. Colding and W.P. Minicozzi II, Width and mean curvature flow, preprint, math.DG/0705.3827. [Cr] C.B. Croke, Area and the length of the shortest closed geodesic, JDG, 27 (1988), no. 1, 1–21. [E] L.C. Evans, Partial differential equations. Graduate Studies in Math., 19. AMS, Providence, RI, 1998. [G] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math., 85 (1985) 307–347. [Ha1] R. Hamilton, Three-manifolds with positive Ricci curvature, JDG 17 (1982) 255-306. [Ha2] R. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7–136, Int. Press, Cambridge, MA, 1995. [He1] F. Hélein, Harmonic maps, conservation laws, and moving frames. Cambridge Tracts in Mathematics, 150. Cambridge University Press, Cambridge, 2002.

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[He2] F. Hélein, Régularité des applications harmoniques entre une surface et une variété riemannienne, CRAS , 312 (1991) 591–596. [Jo] J. Jost, Two–dimensional geometric variational problems, J. Wiley and Sons, Chichester, N.Y. (1991). [MeSiYa] W. Meeks III, L. Simon, and S.T. Yau, Embedded minimal surfaces, exotic spheres and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982) 621–659. [Mo1] C. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966. [Mo2] C. Morrey, On the solutions of quasi-linear elliptic partial differential equations, TAMS 43 (1938) 126–166. [Mu] J. R. Munkres, Topology: A First Course, Prentice-Hall, 1975. [Pa] T. Parker, Bubble tree convergence for harmonic maps, JDG 44 (1996) 595–633. [PaW] T. Parker and J. Wolfson, Pseudo-holomorphic maps and bubble trees, JGA 3 (1993) 63–98. [Pe] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three–manifolds, math.DG/0307245. [Pi] J.T. Pitts, Existence and regularity of minimal surfaces on Riemannian manfolds, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo 1981. [Q] J. Qing, Boundary regularity of weakly harmonic maps from surfaces, JFA, 114 (1993) 458–466. [Ri] T. Rivière, Conservation laws for conformally invariant variational problems, Invent. Math. 168(1) (2007), 1–22. [R] H. Royden, Real Analysis, 3rd edition, Macmillan, London, 1988. [SaU] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981) 1–24. [Sj] J. H. Sampson, Some properties and applications of harmonic mapings, Ann. Scient. Ec. Norm. Sup. 11 (1977) 211–228. [SU1] R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, JDG 17 (1982), no. 2, 307–335. [SU2] R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, JDG 18 (1983), no. 2, 253–268. [SW] R. Schoen and J. Wolfson, Minimizing area among Lagrangian surfaces: the mapping problem, JDG 58 (2001) 1–86. [SY] R. Schoen and S.T. Yau, Lectures on harmonic maps, Int. Press (1997). ¨ [Sc1] H. A. Schwarz, Uber einen Grenzbergang durch altirneirendes Verfahren, Vierteljahrsschrift der Naturforschenden Gesellschaft in Z¨ urich, (1870) 15, 272-286. [Sc2] H. A. Schwarz, Gesammelete Mathematische Abhandlungen, Springer, Berlin, 1890. [SiY] Y.-T. Siu and S.T. Yau, Compact K¨ ahler manifolds of positive bisectional curvature, Invent. Math, 59 (1980) 189–201. [St] M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, CMH 60 (1985) 558–581. [W] H. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl., 26 (1969), 318–344. MIT, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, and Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012. Department of Mathematics, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218 E-mail address: [email protected] and [email protected]

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