The Ricci Flow: Techniques and Applications: Geometric Aspects (Mathematical Surveys and Monographs) (Pt. 1)

............ .....,. .....

MonoanPM Volume 135

The Ricci F·-low: Techniques and Applications Part I: Geometric Aspects Bennett Chow Sun-Chin Chu David Glickenstein Christine Guenther James Isenberg Tom Ivey Dan Knopf Peng Lu Feng Luo Lei Ni

•

American Mathematical Society

Mathematical Surveys and Monographs Volume 135

The Ricci Flow: Techniques and Applications Part I: Geometric Aspects

Bennett Chow Sun-Chin Chu David Glickenstein Christine Guenther James Isenberg Tom Ivey Dan Knopf Peng Lu Feng Luo Lei Ni

American Mathematical Society

EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood

Peter S. Landweber Michael P. Loss

J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 53C44, 53C25, 58J35, 35K55, 35K05.

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ISBN 978-0-8218-3946-1

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12 11 10 09 08 07

Contents Preface What this book is about Highlights of Part I Acknowledgments

xiii

Contents of Part I of Volume Two

xvii

IX

ix Xl

Chapter 1. Ricci Solitons 1. General solitons and their canonical forms 2. Differentiating the soliton equation - local and global analysis 3. Warped products and 2-dimensional solitons 4. Constructing the Bryant steady soliton 5. Rotationally symmetric expanding solitons 6. Homogeneous expanding solitons 7. When breathers and solitons are Einstein 8. Perelman's energy and entropy in relation to Ricci solitons 9. Buscher duality transformation of warped product solitons 10. Summary of results and open problems on Ricci solitons 11. Notes and commentary

1 2 6 11 17 26 32 41 44 46 50 52

Chapter 2. Kahler-Ricci Flow and Kahler-Ricci Solitons 1. Introduction to Kahler manifolds 2. Connection, curvature, and covariant differentiation 3. Existence of Kahler-Einstein metrics 4. Introduction to the Kahler-Ricci flow 5. Existence and convergence of the Kahler-Ricci flow 6. Survey of some results for the Kahler-Ricci flow 7. Examples of Kahler-Ricci solitons 8. Kahler-Ricci flow with nonnegative bisectional curvature 9. Matrix differential Harnack estimate for the Kahler-Ricci flow 10. Linear and interpolated differential Harnack estimates 11. Notes and commentary

55 55 62 70 74 80 95 97 103 109 118 124

Chapter 3. The Compactness Theorem for Ricci Flow 1. Introduction and statements of the compactness theorems 2. Convergence at all times from convergence at one time 3. Extensions of Hamilton's compactness theorem

127 127 132 138

v

CONTENTS

vi

4. 5.

Applications of Hamilton's compactness theorem Notes and commentary

142 148

Chapter 4. Proof of the Compactness Theorem 149 1. Outline of the proof 149 2. Approximate isometries, compactness of maps, and direct limits 150 3. Construction of good coverings by balls 158 4. The limit manifold (M~, 900) 165 5. Center of mass and nonlinear averages 175 6. Notes and commentary 187 Chapter 5. Energy, Monotonicity, and Breathers 1. Energy, its first variation, and the gradient flow 2. Monotonicity of energy for the Ricci flow 3. Steady and expanding breather solutions revisited 4. Classical entropy and Perelman's energy 5. Notes and commentary

189 190 197 203 214 219

Chapter 6. Entropy and No Local Collapsing 1. The entropy functional Wand its monotonicity 2. The functionals jJ, and 1/ 3. Shrinking breathers are shrinking gradient Ricci solitons 4. Logarithmic Sobolev inequality 5. No finite time local collapsing: A proof of Hamilton's little loop conjecture 6. Improved version of no local collapsing and diameter control 7. Some further calculations related to F and W 8. Notes and commentary

221 221 235 242 246

Chapter 7. The Reduced Distance 1. The C-Iength and distance for a static metric 2. The C-Iength and the L-distance 3. The first variation of C-Iength and existence of C-geodesics 4. The gradient and time-derivative of the L-distance function 5. The second variation formula for C and the Hessian of L 6. Equations and inequalities satisfied by Land f. 7. The f.-function on Einstein solutions and Ricci solitons 8. C-Jacobi fields and the C-exponential map 9. Weak solution formulation 10. Notes and commentary

285 286 288 296 306 312 322 335 345 363 379

251 264 273 284

Chapter 8. Applications of the Reduced Distance 381 1. Reduced volume of a static metric 381 2. Reduced volume for Ricci flow 386 3. A weakened no local collapsing theorem via the monotonicity of the reduced volume 399 4. Backward limit of ancient K-solution is a shrinker 406

CONTENTS

5.

vii

Perelman's Riemannian formalism in potentially infinite dimensions Notes and commentary

417 432

Chapter 9. Basic Topology of 3-Manifolds 1. Essential 2-spheres and irreducible 3-manifolds 2. Incompressible surfaces and the geometrization conjecture 3. Decomposition theorems and the Ricci flow 4. Notes and commentary

433 433 435 439 442

Appendix A. Basic Ricci Flow Theory 1. Riemannian geometry 2. Basic Ricci flow 3. Basic singularity theory for Ricci flow 4. More Ricci flow theory and ancient solutions 5. Classical singularity theory

445 445 456 465 470 474

Appendix B. Other Aspects of Ricci Flow and Related Flows 1. Convergence to Ricci solitons 2. The mean curvature flow 3. The cross curvature flow 4. Notes and commentary

477 477 482 490 500

Appendix C.

501

6.

Glossary

Bibliography

513

Index

531

Preface Trinity: It's the question that drives us, Neo. It's the question that brought you here. You know the question just as I did. Neo: What is the Matrix? ... Morpheus: Do you want to know what it is? The Matrix is everywhere .... Unfortunately, no one can be told what the Matrix is. You have to see it for yourself. - From the movie "The Matrix".

What this book is about This is the sequel to the book "The Ricci Flow: An Introduction" by two of the authors [108]. In the previous volume (henceforth referred to as Volume One) we laid some of the foundations for the study of Richard Hamilton's Ricci flow. The Ricci flow is an evolution equation which deforms Riemannian metrics by evolving them in the direction of minus the Ricci tensor. It is like a heat equation and tries to smooth out initial metrics. In some cases one can exhibit global existence and convergence of the Ricci flow. A striking example of this is the main result presented in Chapter 6 of Volume One: Hamilton's topological classification of closed 3-manifolds with positive Ricci curvature as spherical space forms. The idea of the proof is to show, for any initial metric with positive Ricci curvature, the normalized Ricci flow exists for all time and converges to a constant curvature metric as time approaches infinity. Note that on any closed 2-dimensional manifold, the normalized Ricci flow exists for all time and converges to a constant curvature metric. Many of the techniques used in Hamilton's original work in dimension 2 have influenced the study of the Ricci flow in higher dimensions. In this respect, of special note is Hamilton's 'meta-principle' of considering geometric quantities which either vanish or are constant on gmdient Ricci solitons. It is perhaps generally believed that the Ricci flow tries to make metrics more homogeneous and isotropic. However, for general initial metrics on closed manifolds, singularities may develop under the Ricci flow in dimensions as low as 3. 1 In Volume One we began to set up the study of singularities by discussing curvature and derivative of curvature estimates, looking at how generally dilations are done in all dimensions, and studying 1For noncompact manifolds, finite time singularities may even occur in dimension 2. ix

x

PREFACE

aspects of singularity formation in dimension 3. In this volume, we continue the study of the fundamental properties of the Ricci flow with particular emphasis on their application to the study of singularities. We pay particu-

lar attention to dimension 3, where we describe some aspects of Hamilton's and Perelman's nearly complete classification of the possible singularities. 2 As we saw in Volume One, Ricci solitons (Le., self-similar solutions), differential Harnack inequalities, derivative estimates, compactness theorems, maximum principles, and injectivity radius estimates play an important role in the study of the Ricci flow. The maximum principle was used extensively in the 3-dimensional results we presented. Some of the other techniques were presented only in the context of the Ricci flow on surfaces. In this volume we take a more detailed look at these general topics and also describe some of the fundamental new tools of Perelman which almost complete Hamilton's partial classification of singularities in dimension 3. In particular, we discuss Perelman's energy, entropy, reduced distance, and some applications. Much of Perelman's work is independent of dimension and leads to a new understanding of singularities. It is difficult to overemphasize the importance of the reduced distance function, which is a space-time distance-like function (not necessarily nonnegative!) which is intimately tied to the geometry of solutions of the Ricci flow and the understanding of forming singularities. We also discuss stability and the linearized Ricci flow. Here the emphasis is not just on one solution to the Ricci flow, but on the dependence of the solutions on their initial conditions. We hope that this direction of study may have applications to showing that certain singularity types are not generic. This volume is divided into two parts plus appendices. For the most part, the division is along the lines of whether the techniques are geometric or analytic. However, this distinction is rather arbitrary since the techniques in Ricci flow are often a synthesis of geometry and analysis. The first part is intended as an introduction to some basic geometric techniques used in the study of the singularity formation in general dimensions. Particular attention is paid to finite time singularities on closed manifolds, where the spatial maximum of the curvature tends to infinity in finite time. We also discuss some basic 3-manifold topology and reconcile this with some classification results for 3-dimensional finite time singularities. The partial classification of such singularities is used in defining Ricci flow with surgery. In particular, given a good enough understanding of the singularities which can occur in dimension 3, one can perform topological-geometric surgeries on solutions to the Ricci flow either right before or at the singularity time. One would then like to continue the solution to the Ricci flow until the next singularity and iterate this process. In the end one hopes to infer the existence of a geometric decomposition on the underlying 3-manifold. This is what Hamilton's program aims to accomplish and this is the same framework on which 2Not all singularity models have been classified, even for finite time solutions of the Ricci flow on closed 3-manifolds. Apparently this is independent of Hamilton's program for Thurston geometrization.

HIGHLIGHTS OF PART I

xi

Perelman's work is based. In view of the desired topological applications of Ricci flow, in Chapter 9 we give a more detailed review of 3-manifold topology than was presented in Volume One. We hope to discuss the topics of nonsingular solutions (and their variants), where one can infer the existence of a geometric decomposition, surgery techniques, and more advanced topics in the understanding of singularities elsewhere. 3 The second part of this volume emphasizes analytic and geometric techniques which are useful in the study of Ricci flow, again especially in regards to singularity analysis. We hope that the second part of this volume will not only be helpful for those wishing to understand analytic and geometric aspects of Ricci flow but that it will also provide tools for understanding certain technical aspects of Ricci flow. The appendices form an eclectic collection of topics which either further develop or support directions in this volume. We have endeavored to make each of the chapters as self-contained as possible. In this way it is hoped that this volume may be used not only as a text for self-study, but also as a reference for those who would like to learn any of the particular topics in Ricci flow. To aid the reader, we have included a detailed guide to the chapters and appendices of this volume and in the first appendix we have also collected the most relevant results from Volume One for handy reference. For the reader who would like to learn more about details of Perelman's work on Hamilton's program, we suggest the following excellent sources: Kleiner and Lott [231]' Sesum, Tian, and Wang [326], Morgan and Tian [273], Chen and Zhu [81], Cao and Zhu [56], and Topping [356]. For further expository accounts, please see Anderson [5], Ding [126], Milnor [267], and Morgan [272]. Part of the discussion of Perelman's work in this volume was derived from notes of four of the authors [102]. Finally a word about notation; if an unnumbered formula appears on p. 0 is necessarily compact (thanks to Myers's theorem) and, as the metric approaches zero, the manifold shrinks to a point in finite time. Einstein metrics are stationary points for the normalized Ricci flow on a closed manifold. Following up on our previous discussion of static Euclidean space, we have the following. EXAMPLE 1.9 (Gaussian soliton). Regard Euclidean space as a Ricci soliton in canonical form, so that (1.4) holds with CT(t) = 1 + ct for some c E R For c '" 0, the Euclidean solution is called the Gaussian soliton. By differentiating (1.4) and multiplying the result by CT(t), we obtain (1.15)

0 = -2 Rc(gcan) = D"fgcan + cgcan,

where (1.16)

f(x)=_c 1x I2 4

1. RICCI SOLITONS

6

is the exponent in the Gaussian function. Here we used the standard identity D"jgcan = 2,1''\1 f. 2. Differentiating the soliton equation analysis

local and global

2.1. Differentiating general solitons. Let (g, X) be a Ricci soliton structure (gradient or not) on Mn: (1.17)

2~j

+ 'ViXj + 'VjXi + Egij

=

O.

Condition (1.17) places a strong condition on 9 and X. For example, contracting (1.17) with 9 (tracing) gives (1.18)

R + div X

+ ~E

= 0,

where div X ~ gij'ViXj and R is the scalar curvature. If M is closed, then this implies 2r E = --,

(1.19)

n where r ~ iM R d{L/ vol(M) denotes the average scalar curvature. Furthermore, we also have the following. LEMMA 1.10. If (g, X) is a Ricci soliton structure on Mn, then (1.20) or more invariantly,

~Xi

+ ~£ lm Xm = 0,

~Xo +Rc (x o) = 0,

where Rc : T* M - T* M is defined by Rc (a)i ~ ~£ lmam . PROOF. Taking the divergence of (1.17) and applying the second contracted Bianchi identity (VI-3.13) and the Ricci identity (Vl-p. 286b), we obtain

£) mXm + ~Xi

= g3'k ( 'Vi'VjXk - Rjik£g

= -'ViR+~£lmxm+~Xi'

where we have used (1.18). The lemma follows from cancelling the -'ViR terms. 0 Lastly, computing the scalar curvature of the evolving metric (1.11) and comparing its time-derivative (at t = 0) with its evolution under the Ricci flow gives LEMMA 1.11. If (g, X) is a Ricci soliton structure on Mn, then (1.21)

~R + 21 Rc 12 = CxR - ER.

2.

DIFFERENTIATING THE SOLITON EQUATION

7

PROOF. The left-hand side (LHS) is just the usual expression for ~~ under the Ricci flow. To obtain the equality, we compute ~~ by way of the equality R(Tl.p*g) = -r- 1 O. If R == 0, then (1.22) would imply Rc == 0, contradicting the assumption that (g, X) is shrinking. Hence R> O. D We will return to the consideration of shrinking solitons on closed manifolds in Section 7 of this chapter.

2.2. Differentiating gradient solitons. The gradient Ricci soliton structure equation (1.24)

relates the Ricci tensor to the Hessian of f. First note that tracing this gives nc (1.25) R + b.f + 2 = o.

1. RICCI SOLITONS

8

Now we proceed to systematically differentiate the function f up to fourth order, commute pairs of covariant derivatives, trace, and apply (1.24) and the second Bianchi identity to relate derivatives of the curvature and f. We begin with considering three derivatives of f since for one derivative there is nothing to do, and a pair of covariant derivatives acting on a function commute. Since ''ih [\7 i , \7 j ] f = 0, the only nontrivial commutator is \7 i \7j\7kf - \7j\7i\7d = [\7 i , \7j ] \7kf.

By (1.24), the commutator formula, and \7g = 0, we have (1.26) Taking the trace by multiplying by gjk and using the contracted second Bianchi identity, we get (1.27) Note that (1.26) is antisymmetric in i and j, and in particular, (1.27) is the only equation obtained by tracing (1.26). Next we consider equations obtained by commuting four derivatives of f. The only essentially new equation is obtained by considering \7 i [\7 j, \7 k] \7 d. The quantity \7 i \7 j [\7 k, \7 e] f is zero and [\7 i, \7 j] \7 k \7 d yields only a standard commutator formula. We have

which implies

and hence (1.28) First we trace by gij. Commuting derivatives and applying the contracted second Bianchi identity yield b.Rke -

1

"2 \7 k \7 eR + 2~kem~m -

E

RkmRme

+ "2 Rke

= \7 iRikem \7 mf = (-\7eRkm + \7 mRke) \7 mf. Since (1.28) is antisymmetric in j and k, we have that tracing (1.28) by gjk is zero, whereas tracing by gik is equivalent to tracing by gij. The remaining trace is by gil. We leave it as an exercise for the reader to show that taking this trace yields (-\7jRkm + \7kRjm) \7 m f = 0, which is nothing new since it follows directly from (1.26).

2.

DIFFERENTIATING THE SOLITON EQUATION

9

In conclusion, the new identities we obtain by differentiating fourth order are (1.29) (1.30)

-\1iRjk

+ \1jRik =

f

up to

-Rijke\1d,

\1i\1jRkf - \1i\1kRje = \1i R jkem \1 mf - Rjkem (Rim

+ ~gim)

and the traces are 1

"2 \1i R =

(1.31 ) (1.32) (1.33)

Rie\1d,

+ \1mRkf) \1mf,

M ke = (-\1f Rkm tlR + 21Rcl 2

+ eR =

\1f· \1R,

where

1 Mkf ~ tlRkf - "2\1k \1e R

+ 2RikfmRim -

RkmRmf

e

+ "2Rkf'

The last equation, which is the trace of the equation above it, is a special case of (1.21). REMARK 1.14. The quantities in (1.29) and (1.32) appear in the matrix Harnack quadratic discussed in Part II of this volume (see also subsection 4.2 of Appendix A). We will finish this subsection with an application of Lemma 1.10 to gradient solitons. Substituting X = \1 f in (1.20) yields tl(\1d)

+ Rifg fm \1 mf = o.

On the other hand, commuting covariant derivatives gives tl(\1d) = \1i(tlf)

+ Riklf\1ff.

Combining these equations with (1.25) and (1.9) yields 0= \1iR + 2gkf\1 i \1 kf\1d

+ e\1d =

\1 i (R + 1\1 fl2

+ ef),

proving the following. PROPOSITION 1.15 (Constant gradient quantity on solitons). 1f(g, \1 f, e) is a gradient Ricci soliton structure on a manifold Mn, then (1.34)

R + 1\1 fl2

+ ef == const

is constant in space. Consequently, by (1.25), (1.35)

R + 2tlf - 1\1 fl2 - ef

== const.

Formula (1.34) is used in the study of the geometric properties of gradient Ricci solitons; see Chapter 9 of [111] for an exposition. A significance of the quantity on the LHS of (1.35) will be exhibited in the use of (5.42) and (5.43) to prove energy monotonicity in Chapter 5. With additional hypotheses, we can determine the constant in the proposition above (see also Theorem 20.1 in [186]).

10

1. RICCI SOLITONS

COROLLARY 1.16. If (g, V J) is a steady gradient Ricci soliton structure on Mn with positive Ricci curvature and if the scalar curvature attains its maximum at a point 0, then

R+

IVfl 2 = R(O).

PROOF. We have Rij = -ViVjf > 0 and by Proposition 1.15, R+IV fl2 is constant. On the other hand, at 0 we have 0 = Vi IV fl 2 = -2RijgJ'k Vkf, which implies that V f (0) = O. 0 REMARK 1.17. If we only assume the Ricci curvature is nonnegative, then there is a counterexample by simply taking the flat metric on the xyplane and letting f be the x-coordinate, in which case Rij = - Vi V j f = 0, so that R + IV fl2 = const > R(O) = O. Quantities which are constant on gradient Ricci solitons are useful in the study of their geometries; see Chapter 9 of [111] for an exposition of some examples of this. 2.3. Ricci soliton structures and exterior differential systems. We end this section with a local analysis of Ricci soliton structures. From (1.17), which defines a Ricci soliton structure (g, X), one might hope to obtain more stringent tensoriallocal conditions on 9 and/or X by a combination of differentiation, contracting, and equating mixed partials. However, no further lower-order (Le., first-order in X and second-order in g) identities arise this way. This follows from applying the machinery of exterior differential systems to the soliton equation (1.17), written in local coordinates as a system of PDE that is second-order in the entries of 9 and first-order in the components of X. The theory of exterior differential systems - in particular, the Cartan-K8.hler Theorem3 - is able to predict when a system of PDE has solutions, and how large the solution space is, provided the system passes a test indicating that it is involutive. 4 In addition, if the system is involutive, then any k-jet of a solution can be extended to a (k + I)-jet of a solution, and so on to a convergent power series solution. In the case of (1.17), in order to obtain an involutive system, one has to reduce the enormous size of the solution space (which is due to the diffeomorphism invariance of the soliton condition) by adding the requirement that the local coordinates are harmonic functions with respect to the metric g. Since ~ = gij (OiOj - rfj Ok) , this is a first-order condition on the entries of 9 and takes the form (1.36)

gijrfj = 0,

where rfj are the Christoffel symbols. This is just another variation on DeTurck's trick for proving short-time existence for the Ricci flow (see 3See Theorem III.2.2 and Corollary II1.2.3 on pp. 80-88 in [36] or Theorem 7.3.3 on pp. 254-256 in [223]. 4See Chapter III, pp. 58-65 of [36] or Chapter 7, p. 256 in [223].

3. WARPED PRODUCTS AND 2-DIMENSIONAL SOLITONS

11

Sections 4.3 and 5 of Chapter 3 of [108]). Once this condition is added to (1.17), the system becomes involutive (see [221] for the proof). Furthermore, involutivity implies that Cauchy problems for the augmented system (1.17) and (1.36) are locally solvable; for example, the I-jet of 9 may be arbitrarily prescribed along a hypersurface transverse to a given vector field X. Thus, Ricci soliton structures (g, X) locally depend, modulo diffeomorphisms, on n(n + 1) functions of n - 1 variables (i.e., the components of g, and their transverse derivatives, along the hypersurface). 3. Warped products and 2-dimensional solitons

In this section, we review some of the examples of 2-dimensional Ricci solitons, such as the cigar, which have been constructed to date; the Bryant soliton is discussed in Section 4 of this chapter and examples of Kahler-Ricci solitons will be discussed in the next chapter. In Section 1 of Appendix B we will give some conditions under which the Ricci flow converges to some of the solitons discussed here. Recall from Lemma 5.96 on p. 168 of Volume One that any ancient solution of the Ricci flow on a surface of positive curvature that attains its maximum curvature in space and time is the cigar. Similarly, any ancient solution of the Ricci flow on a manifold of positive curvature operator that attains its maximum curvature in space and time is a steady gradient Ricci soliton. In dimension 3, it is conjectured that such a soliton is the Bryant soliton. This is one reason for focusing our attention on the cigar and Bryant solitons. 3.1. Solitons and Killing vector fields on surfaces. If (g, X) is a soliton structure on a surface M2, then X is a conformal vector field. This is simply because ViXj + VjXi = - (R + c:) gij' If (g, 'V f) is a gradient soliton, then J ('V f) is a Killing vector field (where J : T M ~ T M is the complex structure, defined as counterclockwise rotation of tangent vectors by 90°). To see this, we observe that since 'V f is a conformal vector field on a surface,

(CJ('\lf)g)ij

= Vi (Jj'Vkf) + 'Vj (JikVkf) = Jj'Vi'Vkf + Jik'Vj'Vkf = - (

~ + ~) (Jj gik + Jik9jk) =

0,

where the components J! are defined by J (8~i) ~ J! 8~j' That is, J ('V f) is a Killing vector field. In dimension 2 we have the following. LEMMA 1.18. A surface with a Killing vector field is locally a warped product. In particular, a gradient soliton on a surface is locally a warped product.

12

1.

RICCI SOLITONS

PROOF. Let (M2, g) be a Riemannian surface (not necessarily complete) with a Killing vector field K. Let x E M and define a smooth unit speed path'Y: (ro - E, ro + E) -+ M by 'Y (r) = I~~~~I ("( (r)) , 'Y (ro) = x. Define a I-parameter family of smooth paths !3r : (00 - E, 00 + E) -+ M by /3r (0) = K (!3r (0)), !3r (00 ) = 'Y (r), and a I-parameter family of smooth unit speed paths 'Yo : (ro - E, ro + E) -+ M by 'Yo (r) = I~~~~I ("(0 (r)), 'Yo (ro) = !3ro (0). Note that 'Yoo = 'Y. CLAIM. !3r (0) = 'Yo (r). PROOF OF CLAIM. This follows from

J (K) ] [ K, IJ (K)I

1

1

1

2

= IJ (K)I [K, J (K)]- 21J (K)1 3K IJ (K)I J (K) = 0

since [K, J (K)] = 0 and K IJ (K)1 2 = O. Hence (r,O) defines local coordinates on a neighborhood of x. Define f (r) ~ IKI (!3r) , where we are using the fact that IKI is constant on !3r. The metric is given by / J (K)

9 = \ IJ (K)I' =

J (K))

2

/

IJ (K)I dr + 2 \

J (K) ) K, IJ (K)I drdO

+ (K, K) dO

2

dr 2 + f (r)2 d0 2.

o It can be shown on a surface that the integral curves of a Killing vector field J (\7 f) are closed loops. In particular one can prove the following without using the Uniformization Theorem (see [86]).

THEOREM 1.19 (Shrinking surface soliton is spherical). If (M2,g) is a shrinking gradient soliton on a closed surface, then 9 has constant positive curvature. More generally, if (M2, g) is a shrinking gradient soliton on a closed, orient able 2-dimensional orbifold with isolated singularities, then 9 is rotationally symmetric with positive curvature. When the 2-orbifold is bad, i.e., not covered by a smooth surface, a unique Ricci soliton exists, which is a nontrivial shrinking gradient Ricci soliton (see [180] and [372]). Topologically, a closed, bad 2-orbifold has either one or two singular points [343]. REMARK 1.20. There do not exist complete shrinking Ricci solitons on noncompact surfaces.

3.2. Warped products. Many known examples of gradient Ricci solitons have been constructed in the form of warped products. In particular, we consider a metric on the product Mn+1 = I x Nn of the form (1.37)

3.

WARPED PRODUCTS AND 2-DIMENSIONAL SOLITONS

13

where r is the standard coordinate on an interval I c JR, 9 is a given metric on an n-dimensional manifold N, and w(r) > 0 is the warping function, which scales distances along the N-factors in the product. Several well-known metrics may be expressed as warped products, at least on an open subset. When N is the unit circle S1, with 9 = d(P for the standard coordinate e, then w(r) = r, w(r) = sinr, w(r) = sinhr give, respectively, the Euclidean plane, the round sphere (with Gauss curvature +1), and the hyperbolic plane (with Gauss curvature -1) with the origin omitted. When N = sn with the standard metric 9 = gcan of constant curvature +1, then w(r) = r gives the standard (flat) metric on JRn+1 with the origin omitted. In each of these cases, the metric extends smoothly across the origin. When N = sn, Lemma 2.10 on p. 29 in Volume One (Le., Lemma A.2 of this volume) gives sufficient conditions for smoothly closing off the Riemannian manifold M, by a point at one (or both) ends of the interval I. Namely, assuming we wish to close off as r ~ 0, we need limr-+o w(r) = 0 and limr-+o w'(r) = 1 (the prime denotes the derivative with respect to r). For constructing Ricci solitons, it is convenient that 9 be a sufficiently 'nice' metric on N; for the rest of this section, we will assume that 9 is an Einstein metric, with Rc(g) = pg. An easy calculation in moving frames gives the following (see Proposition 9.106 on p. 266 of Besse [27]). LEMMA 1.21 (Ricci tensor and Hessian of warped product). If 9 is an Einstein metric on a manifold Nn, with Einstein constant p, then the Ricci tensor of the warped product metric (1.37) is given b'lf (1.38)

w"

Rc(g) = -n-dr2 W

+ (p -

ww" - (n - 1)(w')2) g.

Furthermore, if f is any function of the radial coordinate r, then the Hessian of f with respect to 9 is given by (1.39)

"V"Vf

=

j"(r)dr2 +ww'f'g.

J:

For example, if f(r) = w(t)dt, then "V "V f = w'(r)g. Conversely, by an argument of Cheeger and Colding, warped products may essentially be characterized by the existence of a function whose Hessian is some function times the metric; see §1 in [71] for details. 3.3. Constructing the cigar and other 2-dimensional solitons. For the sake of starting with something familiar, before we study the Bryant soliton, we will briefly repeat the construction of the cigar metric, which is discussed at greater length in Chapter 2 of Volume One. Suppose we wish to construct a complete, steady, rotationally symmetric gradient soliton metric on JR 2. Such a metric will be a warped product (1.37), 5In the case where (N, g) is the unit n-sphere, this follows from (1.58).

1. RICCI SOLITONS

14

where N = Sl, and it is natural to assume that Rc(g) + 'V 2 f = 0 for a radial function f(r). Using (1.38), (1.39) with p = 0 and n = 1 gives (1.40) Integrating

wi" - w" = 0 = w(w' f' - w").

wi" - w' f' = 0 gives

f'

(1.41)

=

for some constant a, whereupon w' f' (1.42)

2aw

- w" integrates to

w' - aw 2 = b.

Using the closure conditions w(O) = 0, w'(O) = 1, we get b = 1. Integrating, we obtain a smooth odd function w(r) whose type depends on the sign of a: • for a = 0, w(r) = r, giving the fiat metric; 1 • for a = a 2 , w(r) = -tan(ar); a 1 2 • for a = -a , w(r) = -tanh(ar). a The third case is the cigar soliton (see [180], [373]) (1.43)

gcig =

dr

212

+ 2" tanh a

2

(ar)dO ,

where by (1.41) we see that the potential function may be taken to be f(r) = -2Iogcosh(ar). The Gauss curvature is K = 2a 2 sech2 (ar) > O. In the second case, the metric (see [125]) (1.44)

1 2 2 = dr 2 + 2" tan (ar)dfJ ,

gxpd

a o < r < 7r/ (2a) , which we call the exploding soliton, is not complete, since tan( ar) ---. 00 at a finite distance away from the origin. The potential function is f (r ) = - 2 log cos (ar) and the Gauss curvature is K = -2a 2 sec 2 (ar) < O. The above steady gradient solitons are defined on topological disks. By taking b = -1, we have the following steady solitons on the punctured disk. For a = -a 2 , taking w(r) =

!a coth(ar) yields

9 = dr 2

1

2 (ar)d0 2 , + 2"coth a

which has potential f(r) = -2logsinh(ar) and K = -2a 2 csch2 (ar) < O. For a = a 2 , taking w(r) =

!a cot(ar) yields

9 = dr 2

1

2 (ar)d0 2 , + 2"cot a

which has potential f (r) = 2 log sin( ar) and curvature K = 2a 2 cot( ar) > O. Both of the above metrics are incomplete because of their behavior near r = O.

3.

WARPED PRODUCTS AND 2-DIMENSIONAL SOLITONS

Likewise, taking b

15

= 0, we have the ODE w' = aw 2. If a = -0: 2, then

w(r) = 0:2;+C' f(r) = -2 log (0:2r+C) and K=

-(r+C:-2)2'

EXERCISE 1.22. Show that if w is a solution to (1.42), i.e., w' - aw 2 = b, then v ~ 1/w satisfies v' +bv 2 = -a. Related to the above exercise, in Section 9 of this chapter we shall see Buscher duality exhibited in the above solitons. EXERCISE 1.23. Determine all of the solutions of the rotationally symmetric steady gradient Ricci soliton equation on a surface. 3.4. A metric transformation on surfaces related to circle actions. It is interesting to search for duality transformations, besides Buscher duality, for Ricci solitons. One transformation for rotationally symmetric metrics on surfaces is the following, which is related to the study of collapsing sequences of solutions of the Ricci flow on 3-manifolds. Given a rotationally symmetric metric 9 = dr 2 + w (r)2 d(P on a surface M2, we may define another metric on M2 by

9 ~ dr 2 +

w (r 0:2w (r)

r

+1

d0 2.

A more general transformation is given by Cheeger [69J. The inverse transformation is g=dr 2 + w(r)2 d0 2 . 1-0:2w(r)2' which is defined as long as 0 < w (r) < 1/0:. EXERCISE 1.24. Consider the SI action on (M2,g) x SI (1/0:), where SI (p) ~ lR/271pZ, defined by ¢(r,O,"1)~(r,O+¢,"1+¢),

Show that

¢ESI.

9 is the quotient metric on [(M2,g) x SI (1/0:)] /SI.

SOLUTION. See Proposition 2 of [103J. We now consider some examples. The inverse transformation of the cigar, ~ . 2( o:r )dO 2, 9 cig = dr 2 + 21smh 0: is the hyperbolic metric of constant curvature K == -0: 2. The transformation of the exploding soliton, gxpd = A

dr 2 + 21sm . 2( o:r )d02 ,

0: is the spherical metric of constant curvature K == 0: 2. Since gxpd is only defined for 0 < r < 7r / (20:) , we see that 9xpd is a hemisphere. In summary, we have the following types of examples:

1. RlCCI SOLITONS

16

(1) a constant negative curvature metric on a surface transforming to a steady soliton, (2) an incomplete steady soliton on a surface transforming to an incomplete metric with constant positive curvature. 3.5. Classifying 2-dimensional solitons. PROPOSITION 1.25 (Surface solitons conformal to ]R2). The only complete steady gradient solitons conformal to the standard metric on ]R2 are the cigar and the fiat metric.

1.26. We have not assumed the curvature is bounded or has a sign. Note that since a steady gradient soliton is an ancient solution, by applying the maximum principle to the evolution equation for the scalar curvature, one sees that a complete steady Ricci soliton on a surface with curvature bounded from below is either flat or has positive curvature (see [111]). For a similar result to Proposition 1.25, see Corollary 1.28 below. REMARK

Consider the pullback of the steady gradient soliton metric under the map from the cylinder 51 x ]R to ]R2 defined by PROOF.

x = eU cosv,

y = e U sinv,

where (u, v) are coordinates on 51 x ]R and 51 = ]R/27l"Z. The pullback is a steady gradient soliton on the cylinder, and the vector field X that it is flowing along is conformal. Hence, the complexification6 of X is a holomorphic vector field and is of the form h(w){)I{)w, where w = u + iv and h(w) is a 27l"-periodic entire (analytic) function. Since X is the gradient of a function f, then X can have no closed orbits, and hence any zeros of h must be simple (by virtue of appealing to a local power series expansion). Consequently, critical points of f can only be local maxima and minima, so that indexp (\7 f) = 1 at any critical point p. Since X (51 x]R) = 0, the Poincare-Hopf Theorem, which says that the sum of the indices of \7 f is the Euler characteristic, implies that f has no critical points, hence h has no zeros. The periodicity then implies that h is a constant. Moreover, X must be a constant times 01 ou, since otherwise the corresponding vector field X on ]R2 has orbits which spiral into the origin. Hence we know that X = I up to multiple, and then by the argument of subsection 3.1 of this chapter, J(X) = 0100 is a Killing field, and so by Lemma 1.18 the metric on ]R2 is rotationally symmetric (and a warped product). Then we may appeal to the above calculation of solutions 0 to (1.40).

ro or

6The complexification of a vector field X is defined + iJ (X)). For example, the complexification of r =

~ (X

~

gr

to

xx 88

+

be y

gy

X(l,O)

is

Z

gz

(x :x + y gy + i (Y go; - xgy )) . See the next chapter for a more detailed discussion of

Kahler manifolds (real surfaces are I-complex dimensional Kahler manifolds). Since we are on a surface, the complexification of a conformal vector field is a holomorphic vector field.

4.

CONSTRUCTING THE BRYANT STEADY SOLITON

17

We have the following from Theorem 13 on p. 61 and Theorem 10 on p. 55 of Huber [210] (see also §10 of Li [251]). THEOREM 1.27 (Conformal structure of surface with finite total curvature). If (M2,g) is a complete Riemannian surface with fM K_dl1 < 00, where K_ ~ max {-K,O} , then (M,g) is conformal to a closed Riemannian surface with a finite number of points removed. Furthermore, by the Cohn- Vossen inequality

1M K+dl1 ~ 1M K- dl1 + 27rX (M)
is cigar). If (M2, g) is a complete steady gradient Ricci soliton with positive curvature, then (M, g) is the cigar. This result gives us a classification of complete steady Ricci solitons on surfaces with curvature bounded from below since such solutions are either flat or have positive curvature.

4. Constructing the Bryant steady soliton We may generalize the cigar metric to a rotationally symmetric steady gradient Ricci soliton in higher dimensions on ]Rn+l by setting N = sn, the unit sphere with constant sectional curvature +1. 7 As the following calculations parallel unpublished work of Robert Bryant for n = 2, we will refer to the complete metrics obtained as Bryant solitons. The Bryant soliton is a singularity model for the degenerate neckpinch, a finite time singularity which is expected to form for some (nongeneric) initial data on closed manifolds. (See Section 6 in Chapter 2 of Volume One.) REMARK 1.29. A singularity model is a long existing solution (Le., the time interval of existence has infinite duration) of the Ricci flow obtained from a singular solution (Le., a solution defined on a maximal time interval [0, T)) of the Ricci flow as the limit of dilations about a sequence of points and times approaching the singularity time T. 7For background on Einstein metrics which are warped products over I-dimensional bases, see the notes and commentary at the end of this chapter.

1. RICCI SOLITONS

18

4.1. Setting up the ODE for Bryant solitons. With p = n - 1, substituting (1.38) and (1.39) into the steady gradient soliton condition Rc(g) + \1\1 f = 0 gives the following system of ODE for wand f:

" !" = n~, w

(1.45) Since R !:If -

=

WW ' f'

= ww" -

(n - 1)(1 - (w ' )2).

-!:If for a steady gradient soliton, Proposition 1.15 implies that

1\1 fl2

= C, a constant. On the other hand, tracing (1.39) gives

!:If

f'

=!" + n~. w I

Using this expression for !:If, we obtain w 2!"

Eliminating (1.46)

+ nww' !, -

w 2 (I') 2

= Cw 2.

f" and w" using (1.45) gives a first integral of our ODE system: 2nww' !, + n(n - 1)(1 - (W')2) - w 2(f')2 = Cw 2.

The analysis of solutions to (1.45) is simplified by using variables that are invariant under the symmetries of the system (Le., translating r, translating f, and simultaneously scaling rand w). We choose new dependent variables x and y and independent variable t (not to be confused with time), such that •

X'T

• I -w f' , Y'Tnw

I

W ,

dr

dt~-.

w

From (1.46) we have nx 2 - y2

(1.47)

+ n (n -1) = Cw 2.

Then by (1.45)

dx

2

dt = ww" = ww' !, + (n - 1)(1 - (w') ),

dy dt Thus the

ODE

I

= wy =

I

I

-ww f .

system (1.45) becomes

dx

dy dt = x(y - nx). ' Substituting C = 0 in the first integral (1.47) gives an invariant hyperbola y2 - nx 2 = n(n - 1) for the system (1.48); see Figure 1 on the next page. (1.48)

REMARK

-

dt

1.30. When n = 1, equation (1.48) becomes dx dt = x (x - y),

(1.49)

which implies x

~; = 2x (x -

2

=x -xy+n-1

+y =

dy dt = x(y - x),

const. Assuming, x, y

~

1 as t

~ -00,

we have

1) . The solution of (1.49) with x (t) decreasing is given by 1

x

= 1 + e2t '

1

y

= 2 - 1 + e2t .

4.

CONSTRUCTING THE BRYANT STEADY SOLITON

19

FIGURE 1. Phase portrait for system (1.48), n = 2, drawn using Maple.

Since ftlogw

= x,

we have w

r

=

= (e- 2t + 1)-1/2.

J

wdt

= arcsinh

= In ( et + J 1 + e2t ) (e t ) .

Hence et = sinh r, so that w (r) = (csch 2r the cigar soliton. EXERCISE 1.31. For n and x (t) increasing?

=

Moreover

r 1/2

+1

= tanh r, and we have

1, what is the solution with x, y

~

1 as t

~

-00,

We will establish the existence of a complete steady gradient Ricci soliton on ]Rn+l by doing phase plane analysis on the system (1.48). Note that the stationary solutions of (1.48) satisfy y = nx and x 2 - 1 = 0, so that the stationary points are (x, y) = (1, n) and (x, y) = (-1, -n). These are both

20

1. RICCI SOLITONS

saddle points since the linearization of (1.48) at (1, n) is8 du dt = - (n - 2) u - v, dv

- = -nu+v dt

'

and the negative of this at (-1, -n) Since we want the metric to close up smoothly as r ---t 0, by Lemma A.2, we will need

.9

x

---t

1 and

y

---t

n.

Therefore, we will limit our attention to the two trajectories that emerge from the saddle point (1, n) as t increases. (These, plus the point itself, comprise the unstable manifold of the saddle point; the stable manifold is the hyperbola.) Given either of these trajectories, we can reconstruct functions w, rand f of t by successively integrating dw (1.50) - = x dt, dr = w dt, df = (nx - y)dt. w Furthermore, we can choose constants of integration such that w '\. 0, r '\. 0, and df /dr ---t 0 as t ---t -00. It then follows that f(r) and w(r) smoothly extend to even and odd functions of r, respectively (cf. [217], Proposition 5.2). 4.2. Phase plane analysis of the right-hand trajectory. The linearization of (1.48) at the saddle point shows that the right-hand trajectory emerges from the saddle point into the region where dx / dt > 0 and dy / dt < o. Because trajectories along the boundary of this region only point into the region, x continues to increase and y continues to decrease. Linearization also shows that y - nx is initially negative along the trajectory, and the equation d dx -(y - nx) = x(y - nx) - n dt dt shows that y - nx is negative and monotone decreasing. Because x ~ 1 and y - nx < -C for some positive constant C when t is sufficiently large, 8The linearization of the general system dx dt = f(x,y),

dy dt=g(x,y)

at (x,y) is di dt

= ax (x,y)x+

of

_

of _ oy (x,y)y,

djj dt

= ax (x, y) x + oy (x, y) y.

og

_

og

_

9The eigenvalues of the matrix

are 2 and 1- n, with corresponding eigenvectors (-l,n) and (1,1), respectively.

4. CONSTRUCTING THE BRYANT STEADY SOLITON

21

the differential equation for Y in (1.48) shows that there exists to such that Y :::; 0 for t 2: to. Therefore,

dx 2 ->x +n-1 dt and limt/T x =

+00 for some finite T > to.

PROPOSITION

1.32. The metric associated to this trajectory is incom-

plete. PROOF.

(1.51)

We will show that there exists a constant a E (0,1) such that a x 0 be small enough so that (1.53)

(05 - n)x 2

+ (n -

1)00 < 0 for all x > Xo.

Let (Xl, Yl) be any point on the trajectory such that and choose 0 :::; 00 sufficiently small so that Yl + OXI :::; that this inequality persists for all later values of t.

Xl

o.

> Xo and Yl < 0, Then (1.53) shows D

4.3. Phase plane analysis of the left-hand trajectory. We now turn to the analysis of the other trajectory emerging from saddle point (1, n). By the same reasoning as before, we see that x is decreasing, Y is increasing, and Y - nx is positive along this trajectory. Moreover,

x

---4

0+

and

Y ---4

+00

as t increases. In order to prove that the corresponding metric is complete, we need to know the limiting behavior of wand r. To do this, we introduce

22

1. RICCI SOLITONS

new phase plane coordinates

Y~ y'n(n-l).

x X ~ y'n-, y As x

---t

0+ and y

---t

y

+00, X

---t

0+

and

Y

---t

0+.

(The constants are chosen so that the invariant hyperbola becomes the unit circle in the XY-plane.) With a new independent variable s such that ds ~ y dt, we have

~~ = X 3 -

(1.54)

X

~: = Y

+ ay2,

(X2 - aX) ,

where a ~ 1/ y'n. In these coordinates, the first integral corresponds to a Lyapunov function L ~ X2 + y2 which satisfies dL/ds = 2X2(L -1). Since L < 1 for this trajectory, L is strictly decreasing but dL/ds 2 2L(L - 1). So, the trajectory approaches the origin exponentially as s ---t +00. Since (1.55)

= xdt =

d(logw)

1 dY y'nX -1 Y'

it suffices to find the limiting behavior of X in terms of Y. Note that for X and Y small, (1.54) implies dX 2 r::::: -X+aY ds '

dY

ds

r:::::

as s

---t

-

In particular, we expect -X following. LEMMA

+ ay2

---t

°

-aXY.

+00. Indeed, we have the

1.33.

and

where a =

l/vin.

PROOF. One might try to apply I'Hopital's Rule to find the limit of X /y2. However,

(1.56)

2

tsx _~ (X2X2- - aX1 ) +

.!l.. y2 - y2 ds

1 'nX2 - X

v'·

shows that this is not straightforward, since the limit of X/Y 2 is also involved in the right-hand side. Let g(s) = X and h(s) = y2 and apply the Cauchy mean value formula (see, e.g., [10]):

g'(c)(h(a) - h(b)) = h'(c)(g(a) - g(b))

4. CONSTRUCTING THE BRYANT STEADY SOLITON

23

for some c between a and b. Letting b --t +00 gives g'(c)jh'(c) = g(a)jh(a). Substituting in from (1.56) gives

~ (x~2_-a~) + v'nX ; - X Is=c =

2:

2

Is=a'

Multiplying through by X 2 - aX (evaluated at c) and taking a (which forces c --t +00) give

X . 11m y2 +a

s ...... +oo

--t

+00

= O.

However, to get zero on the right-hand side, we need to first know that Xjy 2 is bounded as s --t +00. We know that Xjy2 2 O. To obtain an upper bound, we use the differential equation :s

(~)

=a -

~ (X2 -

2aX + 1)

.

Since the trajectory approaches the origin in the XY-plane exponentially, there exists a k > 0 such that d (X) ( -X - a ) (e-k s -~ -~ < - ) 1 + ae- ks ~ .

Comparing with the solution of the

ODE

dujds = u(e- ks - 1) + ae- ks

shows that Xjy 2 is bounded above. In particular, for s 2 So

~ (s) ~ u (s) = e-(ke-kS+ s)

(1:

eke-kS+Sae-ksds + e-k

~ (so))

.

The second limit, which gives the next term in the power series expansion of X in terms of Y near the origin, follows by applying the same arguments to the equations

d (X -y4a y2 )

ds

(

2

= -1 - 3X + 4aX

)

(X _y4aY2)

- ~ y2 (X2 -2aX)

and

o EXERCISE

1.34. Verify the second limit.

24

1. RICCI SOLITONS

Now we derive the asymptotic behavior of wand r. We have d X dy(logw+logY) = Y(X-a)'

which by Lemma 1.33 approaches zero as Y ~ O. Hence, wY approaches some positive constant G as Y ~ O. Next, note that dr

dY

w

vn(n - 1)(X2 - aX)

shows that r is monotone increasing as Y decreases to zero, because X - a = a(nx - y)/y is negative along this trajectory. Furthermore, the limiting behavior of X and w implies that there is a constant c such that dr

w

dY

vn(n - l)X (a - X)

-- =

>

G

----===-----

v'n"=1(1 + c)y3

for Y sufficiently close to zero. We can similarly find a constant c' such that dr dY

2(1

+ c')G

-- 0 follows from the fact that w' = x is strictly less than 1. To show that V2 is positive for l' > 0, we will show that w" is negative. From (1.57), it suffices to show that x 2 - xy + n - 1 is negative, or equivalently that X 2 - y'riX + y2 is negative. Linearizing the system (1.48) about the saddle point shows that x 2 - xy + n - 1 is initially negative when the trajectory emerges. Moreover,

.!!..-(X2 _ ds

Vn + y2) = X2(X2 + y2 -

1) + (X2 _ 1)(X2 - ynX

+ y2)

shows that if X 2 - y'ri + y2 were ever equal to zero in the middle of the trajectory, it would have a negative derivative. So, w" is negative for all l' > O. At the origin, all sectional curvatures are equal. The scalar curvature R satisfies R + IV' fl2 = C, and we know C must be positive because

R = n(n -

l)VI

+ 2nv2

lOSee, for example, the solution to Exercise 1.188 in [111].

1. RICCI SOLITONS

26

is positive everywhere else. The curvatures approach zero as r ---t +00, so R has a positive maximum value at some point. Because f' = (y - nx)/w is positive for r > 0, this must occur at the origin. D We end this section by noting that the preceding construction can be generalized to the case where the fiber is a product of an Einstein manifold and a sphere (see [219]). THEOREM 1.38 (Steady solitons on doubly-warped products). Given an Einstein manifold (Nm, 9N) with positive Ricci curvature, there exists a 1parameter family of complete steady gradient soliton metrics on ~n+l x N in the form of doubly-warped productsll,' 9 = dr 2 + v(r)2gcan where

gcan

is the standard metric on

+ w(r)2gN,

sn, n ~ 1.

These solitons have positive Ricci curvature when r > O. However, the sectional curvature along the copies of N can be negative.

5. Rotationally symmetric expanding solitons In Section 4 in Chapter 2 of Volume One, we described the construction of a I-parameter family of rotationally symmetric expanding gradient Ricci solitons on ~2 (see also Gutperle, Headrick, Minwalla, and Schomerus [174]). These solitons are asymptotic to a cone (with any cone angle in the interval (0,27T') possible) and have curvature exponentially decaying as a function of the distance to the origin (see Exercise 4.15 and Corollary 9.60 of [111]). In this section we consider the problem of constructing rotationally symmetric expanding gradient Ricci solitons on ~n+l for n ~ 2. In particular, we generalize the construction from the previous section on the Bryant soliton to show that there exists a I-parameter family of rotationally symmetric complete expanding gradient solitons on ~n+l.

5.1. Setting up the ODE for expanding gradient solitons. To start, we begin with the expanding gradient soliton condition

Rc(g)

+ VV f + Ag =

0,

E

A = "2 > O.

Substituting (1.38) and (1.39) into this equation gives

o=

-n~" dr 2 + (p - ww" - (n - I)(w')2) 9 w

+ (I" dr 2 + ww' fig) + A (dr2 + w 2g) , is the warping function and 9 is the metric on sn with Einstein

where w( r) constant p. Taking p = n - 1 and collecting the components of the metric

llThis construction was inspired by a similar construction for Einstein metrics by Berard-Bergery [24].

5. ROTATIONALLY SYMMETRIC EXPANDING SOLITONS

on jRn+1, we get a system of two second-order of r:

f" + >. = nw" /w,

ww' f'

ODES

for

27

f and w as functions

+ >.w 2 = ww" + (n -

1)((w' )2 - 1).

Again, Proposition 1.15, together with equation (1.25), provides a first integral for this system:

f" + n(w' f' /w) -

(f')2 - 2>.f = C.

We can reduce the order of the system by again introducing variables that are invariant under the symmetries of translating r and translating f:

Y =;=• nw I - w f' .

. I x=;=w,

Because simultaneous scaling of rand w is no longer a symmetry of the system, we must also retain w as a variable, yielding the following firstorder system:

= xw, dx/dt = x 2 - xy + >.w 2 + n -1, dy/dt = x(y - nx) + >.w 2 ,

dw/dt (1.59)

where the parameter t is related to r by dt = w- 1 dr. (Notice that the first integral is not invariant under the translation symmetries, so it does not give an invariant manifold for this system.) Clearly, solutions of the steady soliton system (1.48) are also solutions of (1.59) for which w = O. (Of course, in the steady case, the warping function is not identically zero, but it is recovered by the quadrature d(log w) = x dt.) To get a metric that closes up smoothly at the origin, we need a trajectory that emerges from the singular point P = (0,1, n) as r increases. (We take w,x,y, in that order, as coordinates on the phase space JR.3.) Linearization at P shows that there is a 2-dimensional unstable manifold passing through P, whose tangent space at P is spanned by the vectors (1,0,0) and (0, -1, n)P The trajectory corresponding to the steady soliton lies in this surface and is tangent to (0, -1, n) at P. Of course, we are only 12The linearization of (1.59) at P is diD

_

ill =W, dx

dt

= (2 -

d-y

= -nx- +-y.

dt

n) x - ;ii,

Note that the last two equations appeared in the linearization for the steady soliton equation and the eigenvalues of the matrix

-1) ( 2-n -n 1

are 2 and 1- n, with corresponding eigenvectors (-l,n) and (1,1), respectively.

28

1. RICCI SOLITONS

interested in solutions in the half-space where w is positive. There is a 1parameter family of these trajectories in the unstable manifold, and they are tangent to the vector (I, 0, 0) as they exit from P. (This vector is associated to the positive eigenvalue closest to zero.) Among these is the hyperbolic metric, for which f' = 0, y = nx, and the warping function is w

= Vnj>.sinh (V>.jnr) ,

y

= ncosh (V>.jnr) .

5.2. Analysis of a I-parameter family of trajectories. For the rest of this section, we will consider only the I-parameter family of trajectories that lie in the quadrant of the unstable manifold of P, between the hyperbolic metric and the steady soliton, i.e., those for which wand y - nx are positive for small values of r. Among these is the flat Gaussian soliton on lR.n +1 described in Example 1.9; the corresponding solution of (1.59) is

w = r,

x = 1,

y=n

+ >.r2.

In order to show that the metrics corresponding to our family of trajectories are complete and to study their curvatures, we again introduce rescaled coordinates

w W::§=-,

. vn(n - 1)

x X::§=vn-,

y

y=;=

y

y

We introduce a new independent variable s such that ds system (1.59) becomes

.

= y dt, whereupon

d: = W(X2 _ >'W2), dX ds = X 3

(1.60)

-

X

~~ = Y(X2 -

+ ay2 + >.( vn -

X)W2,

a::§= Ijvn,

aX - >'W2).

In these coordinates, P = (0, a, VI - ( 2 ), and the hyperbolic trajectory has X identically equal to a, approaching the critical point H = (Ij-Jri"X, a, 0) as r --t 00. The trajectories we are considering lie in the unstable manifold of P, and for small r, in the region where X < a and W > o. The flat trajectory satisfies Y = v'11=1 X and X 2 - aX + >.W 2 = O. LEMMA

1.39. These trajectories remain in the region defined by

0< W:S Ij~.

0< X < a, PROOF.

Along the plane where X = a,

dXjds = a 3

-

a

+ ay2 + >'(vn -

a)W2.

Let Q stand for the quantity on the right, which must be negative for r small. The following equation shows that Q remains negative if X < a:

dQjds = aXy2(X - a)

+ >.( vn -

a)(X2 - ( 2)W2 - >.QW 2.

5.

ROTATIONALLY SYMMETRIC EXPANDING SOLITONS

29

This establishes that X < a always along these trajectories, and the equation dX/ds = ay2 + AVnW 2 when X = 0 shows that X remains positive. Let L = X 2 + y2. Then dL/ds

shows that L

~

= 2(L -

1)(X2 - AW2)

+ 2AVnW2(X -

1 is preserved. Finally, when X

a)

< a,

dW/ds ~ W(a 2 - AW 2),

showing that

W cannot exceed a/../'X = 1/vn:>...

D

LEMMA 1.40. As s ~ +00, these trajectories approach the origin in W XY coordinates, and the corresponding metrics are complete. PROOF. Because the right-hand sides of the system (1.60) are polynomial, parameter s is unbounded along these trajectories. Because the trajectories are bounded within the region given by Lemma 1.39, each trajectory must limit to either the critical point H or to the origin 0 as s ~ +00. However, linearization shows that the tangent space of the stable manifold of H intersects the closure of the region given in Lemma 1.39 only in the line through H tangent to the hyperbolic trajectory. Thus, all the trajectories under consideration limit to the origin. To show that the metrics corresponding to these trajectories are complete, note that (1.61 )

J J dr =

W ds =

J

X2

~~W2 .

Along these trajectories, W must eventually become a decreasing function of s. Then it suffices to show that there is a point (Wo, Xo, Yo) along one of our trajectories such that the limit

fWO

!~Je

dW AW2 - X2

is infinity; this follows from AW2 - X2 < AW2.

D

The flat trajectory divides our chosen quadrant of the unstable manifold near S into regions of negative sectional curvature (bordering the hyperbolic trajectory) and regions of positive sectional curvature (bordering the steady soliton trajectory). The following lemma shows that these signs persist; thus, there exist I-parameter families of rotationally symmetric expanding solitons of strictly positive and of strictly negative sectional curvature. PROPOSITION 1.41. Excluding the fiat metric, the sectional curvatures of these metrics are either strictly positive or strictly negative for all r.

30

1. RICCI SOLITONS

PROOF. VI =

In these coordinates, the sectional curvatures are

y2 - (n - 1)X2 1 - (W')2 = -----;---'------:-~;o__ w2 n(n - 1)W2 '

for 2-planes tangent to and perpendicular to, respectively, the orbits of the rotational symmetry. The evolution equations for the numerators are 1d 2 --(Y - (n - l)X 2 ) 2 ds

= (X 2 -,XW 2 -

aX)(Y 2 - (n - l)X 2 )

+ a(n -l)X( _(X2

+ y2 + n,XW2 -

vnX))

and 1d 2 r:::: --(-(X +Y 2 +n,XW 2 -ynX)) 2ds = _(X2 - ,XW2 + ~(aX - 1))(X2

+ y2 + n,XW 2 - vnX) a (Y2 - (n - l)X 2) . + "2X

Observing that the sign of the first nonconstant factor in the second line of each equation is positive, we conclude that the signs VI > 0, V2 > 0 and VI < 0, V2 < 0 are both preserved. 0 We will now obtain the asymptotic behavior of the curvature relative to distance r. PROPOSITION 1.42. Assuming that the sectional curvatures are negative, then they both decay at least quadratically in r, and for large r, the warping function w(r) is bounded between two linear functions of r. PROOF. First, we establish that, as the trajectories approach the origin, the distance r is asymptotic to a constant times l/W. We will do this by establishing a positive lower bound for W 2 / X, by examining the equations

~ ds

(W2) = W 2 (X2 _ ,XW2 + 1- (ay2

X

X

+ ,Xy'nW2))

X'

~

(Y2) = y2 (X2 _ ,XW2 + 1- 2aX _ (ay2 dsX X

+ ,Xy'nW2))

X·

For r sufficiently large, we can bound the nonconstant terms in the parentheses (those that do not involve dividing by X) by a small number c. Then, if we set Y ~ ay2/X and W ~ 'xy'nW 2/X, we have

:s (W

+ Y) ~

(W

+ Y)

(1 - c -

Y-

W) .

Hence, W + Y ~ c for some positive constant c. On the other hand, because sectional curvature VI is negative, y2 / X ~ (n-1)X, and so lims -++ oo Y = o.

5. ROTATIONALLY SYMMETRIC EXPANDING SOLITONS

31

Hence W 2 / X 2: C1 for some positive C1, for r sufficiently large. Thus, the denominator in the integral (1.61) is bounded below, 2

2

AW - X

W4

2

2: AW - (C1)2'

and it follows that r = O(I/W). The orbital sectional curvature satisfies 111

=

y2-(n-l)X2 X2 W2 > --- > --n(n - I)W2 nW2 n(ct)2

and so decays at least as fast as 1/r2. The equation 112

(1.62)

+ (n -

1)111

=-

(X2 - aX + AW2) W2

shows that in order for 112 to also decay to zero, it is necessary that lim X/W 2 s---+oo

(1.63)

=

AVn.

We already know that X/W 2 is bounded above, and we can then apply the Cauchy mean value theorem (as in Lemma 1.33) to the equation 2

.!LX ds

-

dW2 -

~

(X2 - AW 2 + a~

-1) + AVn

-----'-------;::-------:--::-::-,.".---''----

X2 -AW2

ds

to obtain the desired limit. Next, dividing (1.62) by W 2 gives -112 111 W2 = (n - 1) W2

X2

+ W4 +

A - a~ W2 .

In order to show that 112 decays at least as fast as 1/r2, we need only show that the last term is bounded. This follows from the differential equation

:. C-:l) ~(_1+2,\W2 + A(X -

-3X2)

(,,-:l)

X

y2

a) W2 - a 2 W4'

where the boundedness of the terms on the second line follows from the limit (1.63) and the negative curvature condition y2 ~ (n - I)X2. To establish the last assertion, it suffices to show that dw / dr = x = v:n=T X/Y is bounded above and below for r sufficiently large. First, because of (1.62) we can say that -d (W2) = (X2

ds

Y

+ aX -

W2 AW2)-

Because of the limit (1.63), d W2 - l o g - < CW4 ds Y-

W2 < 2X2_.

Y-

Y

32

1.

RICCI SOLITONS

for some constant C > O. Because dr = W ds and W = 0 (1/ r ), then d W2 - l o g - < Cr- 3 dr Y ,

from which it follows that W 2 /Y is bounded above for large r. Thus, there are positive constants Cl, C2 such that

for sufficiently large

T,

and the bounds on X/Y follow.

o

Much more detailed information about the solutions in the positive and negative curvature cases (as well as the steady case) has been obtained by Robert Bryant (in unpublished notes, using different coordinates). In particular, Bryant proved PROPOSITION 1.43 (Bryant). Each positive curvature solution defines a complete, rotationally symmetric expanding soliton, whose sectional curvature decays proportionally to 1/r2. 6. Homogeneous expanding solitons 6.1. Existence. Homogeneous spaces are among the nicest examples of Riemannian manifolds. 13 Einstein metrics are among the nicest examples of Riemannian metrics. It is thus natural to ask if a given homogeneous space M n = G / K admits a G-invariant Einstein metric. If M is closed, the answer is frequently 'yes'. For example, consider the following result of B6hm and Kerr [29].

THEOREM 1.44. Let Mn be a closed, simply-connected homogeneous space. If n < 12, then M admits a homogeneous Einstein metric. On the other hand, Wang and Ziller's example SU(4)/SU(2) of a 12dimensional homogeneous space that admits no homogenous Einstein metric shows that the answer is 'no' in general and that the dimension restriction above is sharp [365]. Every known example of a noncompact, nonfiat homogeneous space that admits an Einstein metric is isomorphic to a solvable Lie group S. (See [198] and [317].) Moreover, for all known examples, the Einstein metric is of standard type. DEFINITION 1.45. A left-invariant metric g on a solvable Lie group S, regarded as an inner product on the Lie algebra.5, is said to be of standard type if the orthogonal complement with respect to g of the derived algebra [.5,.5] forms an abelian subalgebra a of .5. 13See Chapter 7 of [27], for example.

6. HOMOGENEOUS EXPANDING SOLITONS

33

Many noncom pact homogeneous spaces admit no Einstein metrics whatsoever. For instance, there is the following result of Milnor [266, Theorem 2.4]. THEOREM 1.46. If the Lie algebra of a Lie group N is nilpotent but not commutative, then the Ricci curvature of any left-invariant metric on N has mixed sign.

The Ricci soliton structure equation (1.64)

Rc

=

-Ag - £Xg

illustrates the sense in which a Ricci soliton may be regarded as a generalization of an Einstein metric. Therefore, it is natural to look for Ricci solitons on homogeneous spaces, such as nonabelian nilpotent Lie groups, that do not admit any Einstein metric. Existence and uniqueness of Ricci solitons on such spaces has been investigated by Lauret [244]. To describe his results, we must recall some notation from the cohomology of a Lie algebra g, relative to its adjoint representation. (See [232] and [369].) A k-cochain is a skew-symmetric k-linear map 9 x ... x 9 --t g. Denote the vector space of all k-cochains by C k , noting the natural identifications CO = g, C 1 = End(g) = g* ® g, and C 2 = A2 (g*) ® g. The Lie algebra cohomology of 9 relative to its Chevalley complex,

Hk(g) = ker(8 k )jim(8 k -

1 ),

is derived from the coboundary operators 15k : C k

8k(A)(X 1 , .•• ,XHr)

--t

CHI defined by

,,+k (-1)1 [Xj,A(X 1 , ... ,Xj, ... ,XHl)]

=

~

A

l~j~k+1

+ l~i<j~k+1

In particular, one has 80(X)(Y) = -[Y, X] = adx(Y) for all X, Y E g, and

81 (A)(X, Y)

= [X, A(Y)]- [Y, A(X)]- A([X, Y])

for all A E End(g) and X, Y E g. A derivation of 9 is an element ofker(8r). Let g be a left-invariant metric on a simply-connected Lie group G. Regarding the Ricci curvature Rc of g as an endomorphism, Lauret considers the condition

(A E JR),

(1.65) which is easily seen to be equivalent to (1.66)

Rc

= -AI +D

(A E JR, DE ker(8r)).

Notice that equations (1.65) and (1.66) relate the geometry of (G,g) to algebraic data of g. This turns out to be a productive point of view. We now survey some of the results it generates, referring the reader to [244] for the detailed proofs.

34

1.

RICCI SOLITONS

One begins with the observation that conditions (1.64)-(1.66) are equivalent for a nilpotent group. LEMMA 1.47. Let N be a simply-connected nilpotent Lie group with Lie algebra n and a left-invariant metric g. Then there exist A E lR and X E n solving the Ricci soliton structure equation (1.64) if and only if there exist A E lR and a derivation D E ker(8d solving (1.66). DEFINITION 1.48. A standard metric solvable extension of (n, g) is a pair (5, g), where 5 = aEBn is a solvable Lie algebra such that [', ']slnxn = [', ']n and 9 is an inner product of standard type such that [5,5]s = n =a..l and

9Inxn =

g.

Recalling Definition 1.45, note that for 9 to be of standard type is equivalent to a being abelian. The following result relates Ricci soliton structures on nilpotent Lie groups with Einstein metrics of standard type on solvable Lie groups. THEOREM 1.49. Let N be a simply-connected nilpotent Lie group with Lie algebra n and a left-invariant metric g. Then (N, g) admits a Ricci soliton structure if and only if (n, g) admits a standard metric solvable extension (5 = aEBn, g) such that the simply-connected solvable Lie group (8, g) is Einstein. Although its proof is nonconstructive, Theorem 1.49 is a very useful criterion. For example, any generalized Heisenberg group [25] and many other two-step nilpotent Lie groups admit a Ricci soliton structure. On the other hand, if n is characteristically nilpotent, then N admits no such structure. (See [244] for more examples.) If a Ricci soliton structure does exist on N, it is essentially unique. THEOREM 1.50. Let N be a simply-connected nilpotent Lie group with Lie algebra n. If g and g' are both Ricci soliton metrics, then there exist a> 0 and TJ E Aut(n) such that g' = aTJ(g). REMARK 1.51. If one regards a Ricci soliton structure as a 3-tuple (N, g, X) satisfying (1.64), then uniqueness of X must be understood modulo addition of a Killing vector field. (See Example 1.57 below.) Some partial answers are known regarding the existence of Ricci soliton structures on broader classes of Lie groups. For example, if G is semisimpIe, then condition (1.66) implies condition (1.64). However, (1.66) has no solutions other than Einstein metrics: THEOREM 1.52. A left-invariant metric 9 on a simply-connected semisimple Lie group G satisfies (1.66) if and only if D = 0, hence if and only if g is Einstein. It is not known in general whether or not a noncompact, semisimple Lie group admits an Einstein metric.

6. HOMOGENEOUS EXPANDING SOLITONS

35

Lauret also gives a variational characterization of Ricci solitons [244]. Roughly speaking, this says that any simply-connected nilpotent Lie group (N, g) admitting a Ricci soliton structure is a critical point of a functional that measures, in a certain sense, how far (N, g) is from being Einstein. We now make this precise. It will be more convenient to fix an inner product and allow the Lie algebra brackets to vary. (Compare with Examples 1.63 and 1.64 below.) Specifically, fix an n-dimensional inner product space (n, (-,.)) and let A = A2 (n*) ® n denote the space of all skewsymmetric bilinear forms on n. (Compare to C2 ;2 im( 01) defined above.) Let N denote the subspace of all nilpotent elements of A that satisfy the Jacobi identity. Then N may be regarded as the space of all nilpotent Lie brackets on n. To each p, EN, associate the simply-connected Lie group NJ.t and the left-invariant metric gJ.t on NJ.t determined by (-, .). Notice that N is invariant under the natural action of the group GL(n). Thus the GL(n) orbit of p, E N corresponds to all simply-connected homogeneous nilpotent Lie groups N isomorphic to NJ.t, and the O(n) orbit of p, corresponds to all (N, g) isometric to (NJ.t, gJ.t). The fixed metric (.,.) on n induces a metric (-,.) on A defined by

(p"v) =

L (p,(Xi,Xj),Xk) (V(Xi,Xj),Xk)

i,j,k

when {Xl, ... , Xn} is any orthonormal basis of n. Let

Using the Ricci endomorphism, Lauret defines a functional F : Nl

---t

lR by

and proves THEOREM 1.53. Let p, E N l . Then (NJ.t, gJ.t) satisfies (1.64)-(1.66) (i.e. admits a Ricci soliton structure) if and only if p, is a critical point ofF.

To interpret this result, observe that E(p,) = IRc(gJ.t) - *R(gJ.t)II 2 measures the trace-free part of the Ricci endomorphism, i.e., how far gJ.t is from being Einstein. But for p, E N l , one has

Hence a local minimum p, of F in Nl should correspond to a homogeneous space (NJ.t' 9J.t) that is closest to Einstein among all near by (Nv, gv).

36

1. RICCI SOLITONS

6.2. Construction. The first explicit examples of Ricci soliton structures on Lie groups were constructed by Baird and Danielo [15] and independently by Lott [256]. We discuss [15] in this section and [256] in subsection 6.3, below. Baird and Danielo discovered soliton structures on 3-dimensional manifolds by studying semiconformal maps to Riemannian surfaces. Significantly, their constructions give the first known examples of nongradient soliton structures. We will only describe two of their conclusions, referring the reader to [15] for details of the method.

1.54. Let N denote the 3-dimensional Heisenberg group nil3 . Recall that N may be represented as the group of upper-triangular matrices EXAMPLE

under matrix multiplication. It is a general fact 14 that any simply-connected nilpotent Lie group is diffeomorphic to ]Rn. So give ]R3 its standard coordinates (Xl, X2, X3) and define the frame field

a

a

a

a

Fl - 2 F2 = 2 ( - - Xl-), F3 = 2 - . - OXl' OX2 OX3 OX3 It is easy to check that all brackets [Fi' Fj ] vanish except [Fl' F2] = -2F3, hence that (Fl' F 2, F3) is a nil 3 -geometry frame. 15 The connection I-forms may be displayed as VFIFI VFIF2 ( VF2H VF2F2 VF3Fl VF3F2 Using the dual field

define a left-invariant metric on N by 9 = 4 (wI ® wI + w 2 ® w2 + w3 ® w3) . Recalling the standard formula

(R(X, Y)Y, X) =

~ I(ad X)*Y + (ad Y)* XI 2 3

- 41[X, Y]I

2

-

I

((ad X)* X, (ad Y)*Y)

2 ([[X, Y], Y], X)

-

I

2 ([[Y, X], X], Y),

it is straightforward to compute that Rc(g) = -2(w l ® wI) - 2(w 2 ® w2) + 2(w 3 ® w3). l4The exponential map of a connected, simply-connected, nilpotent Lie group is a diffeomorphism. For instance, see [200j. l5See Volume One, Chapter 1, Sections 3-4.

6.

HOMOGENEOUS EXPANDING SOLITONS

37

Define a vector field

X

1 2

1 1 -X2F2 - (-XIX2 2 2

= --xIFI -

+ x3)F3.

It is then easy to see that

= £Xg + 3g,

-2 Rc(g)

hence that (N, g, X) is a Ricci soliton structure. REMARK 1.55. Because the I-form metrically dual to X is not closed, it follows that X =1= grad f for any soliton potential function f. REMARK 1.56. Compact locally homogeneous manifolds with nil3 geometry occur as mapping tori of Y A

:

T2

--t

T2 induced by A =

(~ ~)

E

8L(2, Z) with k =1= 0. The left-invariant metric 9 is compatible with any compact quotient, but the soliton structure is never compatible with compactification. Indeed, the scalar curvature of 9 is R = -1/2, while every compact Ricci soliton of nonpositive scalar curvature is Einstein. EXAMPLE 1.57. Let S denote the simply-connected 3-dimensional solvable Lie group sol3 = lR > 0. In particular,

n] (47rr)-n/2 e-f dJ.L.

Under the system of equations

a

(1.72)

fJt gij

= -2Rij ,

(1.73)

af 2 n at=-f).f+lV'fl-R+2r,

(1.74)

dr =-1 dt '

we shall show in (6.17) the following. THEOREM 1.73 (Entropy monotonicity). If (g (t), f (t), r(t)), r (t) is a solution of (1.72)-(1.74) on a closed manifold Mn, then

> 0,

d

dt W(g(t), f(t), r(t)) =

1M 2r I~j + V'iV'jf -

;; 12 (47rr)-n/2 e- f dJ.L

~ 0.

ft

Note that the right-hand side (RHS) vanishes, i.e., W(g(t), f(t), r(t)) = 0, if and only if 9 (t) is a shrinking gradient Ricci soliton. The above monotonicity formulas beautifully display the utility of considering Ricci solitons. REMARK 1.74. Assuming that (M n , 9 (t)) is a shrinking gradient Ricci soliton in canonical form (1.13) with>' = -1, we have Rc (g (t)) where ~;

+ V'g(t)V'g(t) f

1

(t) - 2r g (t) = 0,

= -1, and hence (1.14) implies f satisfies (1.73).

Finally we consider the expander entropy of Feldman, Ilmanen, and one of the authors [143]. Define the functional W+ (g, f+, r)

~ 1M

Under the system (1. 75) (1.76) (1.77) we have the following.

[r (R + IV' f+12) - f+

+ n]

(47rr)-n/2 e- f +dJ.L'

1. RICCI SOLITONS

46

THEOREM 1.75 (Expander entropy monotonicity formula). For a solution (g (t), f+ (t), T(t)), T (t) > 0, of (1.75)-(1.77) on a closed manifold Mn,

d

dt W+(g(t), f +(t), T(t)) =

1M 2T I~j + '\h'Vjf+ + ;; 12 (47rT)-n/2 e- f +dj.L ~ O.

Here ftW+(g(t),J+(t),T(t)) = 0 if and only if g(t) is an expanding gradient Ricci soliton. Recall that from Proposition 1.7

8;j ~ Rij

(1.78)

+ \//vjf + 2ETgij ~ 0,

where T (t) ~ c:t+ 1 > 0 and ~ denotes an equality which holds for a gradient Ricci soliton (Mn,g(t),f(t),E) in canonical form. EXERCISE

1. 76. Show that

(1) (1.79)

(2) (1.80)

Moreover, if f has a critical point in space, then C (t) is independent of t.

9. Buscher duality transformation of warped product solitons In this section we describe an interesting duality transformation for solutions of the modified Ricci flow on certain warped products which in particular take gradient Ricci solitons to gradient Ricci solitons. The consideration of these warped products with tori of potentially infinite dimensions leads to Perelman's energy functional. 9.1. A metric duality transformation. Let (Mn,g) be a Riemannian manifold and let (Pq, h) be a flat manifold such as a torus or Euclidean space. Given a function A : M ---t (0,00), consider the warped product manifold (M, g) XA (P, h) which is the Riemannian manifold (M x P,g + Ah). The Buscher duality transformation (see [38]' [39]) takes the metric to the metric Ug~g+A-lh.

(This is a special case of T -d uality in string theory.) Although its definition is simple, the transformation has some surprising properties. Let {xi} ~=1

9. BUSCHER DUALITY TRANSFORMATION

47

and {ya}~=l be local coordinates on M and 'P, respectively, such that ha{3 ~

h

(a~o., ~ ) = fJa{3.

We then have

P9ij = 9ij, P9a(3

=

AfJa{3,

and the rest of the components are zero, where p

.p(8 8) 9 8xi '8xj ,

P

9ij =;=

.p(8 8) 9 8ya' 8y{3 ,

9a{3 =;=

and similarly for ~ 9. The explicit formulas below are from Haagensen [175] (see also (1.38) or §J in Chapter 9 of Besse [27] for curvature formulas for warped products). LEMMA

1.77. The Christoffel symbols of p9 are pr~. ~J

= r~.

~J'

p {3 _ {31 . ria - fJa2V'~logA,

A

.

=

prij

p .

= -fJa {3 '2 V'~ log A,

pr~{3

=

r~{3

pr~j

= 0,

and likewise, the Christoffel symbols of the dual metric ~9 are

~rfj

= rfj ,

~r{3 = -fJ{3a ~2 V' . log A , ~a

~

. r~{3

~

1·

= fJ a {3 2A V'~ log A, ~rl'{3 a = ~riaJ. = ~r~. = 0 • ~J

LEMMA

1.78. The Ricci tensor of p9 is 9iven by

~ [~logA+ ~ lV'logAI 2] fJa{3,

PRa{3

=-

PRai

= 0,

P~j

= Rij - ~ V'i V' j log A - ~ V'i log A V' j log A,

and the Ricci tensor of ~9 is

~ Ra{3 = - 2~ ( -~ log A + ~ IV' log A12) fJa{3, ~Raz. -- 0 , M

~j

M

= ~j +

q q 2V'iV'j log A - '4V'i log AV'j log A.

48

1.

RICCI SOLITONS

LEMMA 1.79. The scalar curvature of P9 and ~ 9 are given by PR =

R -

q~logA _

~ R = R + q~ log A EXERCISE 1.80. Show that if A P

Rij =

~j

q (q: 1) IVlogA1 2 ,

- q (q 4+ 1) IV log AI2 .

~ exp (-~f) (i.e., log A

+ ViVjf -

1 q

q + 1 IV f1 2 , q

~ R = R - 2~f -

q + 1 IV fl2 . q

R

-~f), then

-Vi/Vjf,

+ 2~f -

PR =

=

Hence lim R P = R

q-+oo

+ 2~f -

IV fl2 ,

are Perelman's modified Ricci tensor and scalar curvature (see also (5.18) and (5.19)). As a consequence of the above exercise, if h has unit volume, we then have

lMxP Reg) dJ.L. g = 1M (R + 2~f - q; 1 1v f12) e- f dJ.Lg = 1M (R + q ~ 1 IV fl2 ) e- f dJ.Lg.

Taking q ~

00,

this limits to F (g, f) defined in (1.68).

9.2. Buscher duality. For a warped product solution of the modified Ricci flow of the above type we have the following. THEOREM 1.81 (Buscher duality). If Pg

(t) = 9 (t)

+ exp (-~ f q

t

E

(t))

t

dyO< Q9 dyO are called dilatons and the transformation from 11 c/> to Uc/> is called the dilaton shift. The Buscher duality transformation takes solutions of the modified Ricci flow with dilaton ~ c/> to solutions of the modified Ricci flow with dilaton Uc/>. LEMMA

1.83 (Buscher duality preserves modified Ricci tensor after dila-

ton shift).

URo:.8 + 2 Uv 0: ttV.8 Uc/> = URo:i + 2 Uv 0: "Vi Uc/>

=

~2

( 11 Ro:.8 + 2 I1V 0: I1V.8 11 c/> ) ,

~ Ro:i + 2 I1V 0: I1Vi 11 c/> = 0,

URij+2 UVi "Vj Uc/>= l1~j+2 I1Vi I1Vj 11c/>. REMARK

1.84. If log A = - ~ f, then

~ Rij + 2 ~Vi

I1Vj 11c/> = Rij + ViVjf -

~VdVjf + 2V iVj ~c/>, q

1 q

tt Rij + 2 UVi UVj Uc/> = Rij - ViVjf - - VdVjf + 2V i Vj Uc/>. Suppose limit as q ~

%t 00,

~ gab = -2 11 Rab so that ~ c/> = O. By Exercise 1.80, taking the the equation for the metric gij (t) on the base manifold is

a

atgij

= -2 (Rij + ViVjf).

Note that d/-l (11 g) = e- f d/-l/\ dyl /\ ... /\ dyq. The effective action of (11 g,11 c/» is (e.g., see Alvarez and Kubyshin [2], equation (26))

iMXP (Reg) + 41v 11c/>12) e- 2b4>d/-l eg) = iM (R + 2b.f - q: 1 IV fl2 + 41v ~ c/>12) e- 2b4>e- f d/-l,

S eg,11 c/» =

and similarly for the dual pair (Ug,U c/» • (Note that by taking c/>~ = 0, we have S (l1g, 0) = fMxp R (l1g) d/-lb g.) Buscher duality preserves the effective action:

50

1. RICCI SOLITONS

9.3. Examples. The Buscher dual of a rotationally symmetric soliton solution on a surface is the soliton For example,

dr 2 + tanh 2 r d0 2

dr 2 + coth 2 r d0 2

dr 2 + tan2 r d0 2

dr 2 + cot 2 r d0 2

dr 2 + r 2d0 2

dr 2 + r- 2d0 2

dr 2 + d0 2

dr 2 + d0 2

Let (Nm, h) be an Einstein metric with Rc = £g, c E R Consider the doubly-warped product metric on R2 X Nm: ~g ~ ds 2 + F (s)2 d0 2

-+ G (s)2 h,

where s is the radial coordinate, 0 is the coordinate on the circle, and F and G are positive functions. The gradient Ricci soliton equation ~ Rab + 2 ~Va ~VbI = 0 becomes (see equation (1) in [219]) the following system for the triple (F, G, J): (1.83)

G"

F"

-21" = -C - F'

(1.84)

F" = -n F'G' - l' F',

(1.85)

C =

G

G"

c G2 - (n - 1)

(G')2 G

-

F'G' G' FG - l' G .

The Buscher dual metric is 1 ~g = ds 2 + --2d02

F(s)

+ G (s)2 h.

We know that ~g is a steady gradient Ricci soliton if ~ 9 is, and so we leave it to the reader to verify that indeed the triple (*, G, 1 - log F) is a solution to (1.83)-(1.85) if (F, G, J) is. 10. Summary of results and open problems on Ricci solitons

In this section we collect some known results and open problems about the properties and classification of gradient Ricci solitons. Besides the discussion earlier in this chapter, one may consult [111] for some of the proofs.

10.

SUMMARY

OF

RESULTS AND OPEN PROBLEMS ON RICCI SOLITONS

51

10.1. Gradient Ricci solitons on surfaces. The following is a compendium of known results about complete 2-dimensional gradient Ricci solitons with bounded curvature (shrinkers, steadies, and expanders). In proving them, one may use the fact that if (M 2 ,g) admits a nontrivial Killing vector field X which vanishes at some point 0 E E, then (M,g) is rotationally symmetric. (1) A shrinker has constant positive curvature. In particular, the underlying surface is compact. (2) A steady is either flat or the cigar. (3) A compact expander has constant negative curvature (if X (M) < 0, then there are no nonzero conformal Killing vector fields). (4) An expander with positive curvature is rotationally symmetric and unique up to homothety (see [241] and [111]). In part (4), one can apply the arguments of subsection 3.1 of this chapter and the following result (see [286] and [111]). THEOREM 1.85. If (Mn,g) is a gradient Ricci soliton on a noncompact manifold with Rij ~ ERgij for some E > 0, where R ~ 0, then R decays exponentially in distance to a fixed origin. In particular, if (M2, g) is an expanding gradient Ricci soliton on a surface, then R decays exponentially. A complete proof, using a more direct method, is given in [241]. PROBLEM 1.86. Are there any other expanders on a surface diffeomorphic to ]R2 besides the positively curved rotationally symmetric expander and the hyperbolic disk? PROBLEM 1.87. Do all complete 2-dimensional gradient Ricci solitons have bounded curvature? Are all complete 2-dimensional Ricci solitons gradient? Note that in dimension 3 there are complete homogeneous expanding solitons which are not gradient; see Section 5 of this chapter.

10.2. Gradient Ricci solitons on 3-manifolds. In dimension 3 we have the following results. This subsection is abbreviated since we pose some more problems in the next subsection for dimensions at least 3. (1) (Perelman) Any nonflat shrinker with bounded nonnegative sectional curvature is isometric to either a quotient of the 3-sphere or a quotient of S2 x R In particular, any shrinker with bounded positive sectional curvature is isometric to a shrinking solution with constant positive sectional curvature. (2) There exists a rotationally symmetric steady with positive sectional curvature, namely the Bryant soliton. (3) There exists a rotationally symmetric expander with positive sectional curvature.

52

1.

RICCI SOLITONS

PROBLEM 1.88. Are there any 3-dimensional steady gradient solitons besides a flat solution, the Bryant soliton, and a quotient of the product of the cigar and 1R? This is equivalent to asking if a steady gradient Ricci soliton with n = 3 and sect (g (t)) > 0 is isometric to a Bryant soliton.

10.3. Gradient Ricci solitons in higher dimensions. Here we assume n ~ 3. Note that any expanding or steady Ricci soliton on a closed n-dimensional manifold is Einstein. PROBLEM 1.89. Is a shrinking gradient Ricci soliton with n Rm (g (t)) > 0 compact?

~

4 and

PROBLEM 1.90. Is a compact shrinking gradient Ricci soliton, n ~ 4, and Rm (g (t)) > 0 isometric to a shrinking constant positive sectional curvature solution?16 PROBLEM 1.91. Are there n-dimensional steadies with positive curvature operator besides the Bryant soliton? PROBLEM 1.92. Are there any n-dimensional expanders with positive curvature operator besides the rotationally symmetric one? PROBLEM 1.93. Does there exist an expander with n ~ 3 and positively pinched Ricci curvature, i.e., Rij ~ ERgij for some E > 0, where R> O? By Theorem 1.85, such an expander has R decaying exponentially.

11. Notes and commentary Section 1. The first occurrence of the notion of a Ricci soliton in the literature is in Friedan [145], where non-Einstein Ricci solitons are called quasi-Einstein metrics. See Besse [27] for a comprehensive treatment of Einstein manifolds. The Gaussian soliton first appeared in §2.1 of Perelman [297]. In the case of surfaces we have encountered Ricci solitons in Chapter 5 of Volume One. There, solitons motivated several aspects of Hamilton's original proof of convergence of the Ricci flow for surfaces with R (go) > 0, including the scalar curvature, Harnack, and entropy estimates, as well as the estimate which shows the metric approaches a soliton. See in Volume One, Corollary 5.17 on p. 115, Proposition 5.57 on p. 145, Proposition 5.39 on p. 134, and Corollary 5.35 on p. 130. Expanding solitons are of interest because they constitute borderline cases for Hamilton's Harnack inequality. (Expanding solitons also model the formation of certain Type III singularities.) In fact, as we mentioned in Section 2 of this chapter and will discuss in more detail in Part II, the consideration of quantities which vanish on solitons led to the discovery of the Harnack quadratic. Steady solitons, where the isometry class of the metric is independent of t E (-00,00), occur as limits of Type II singularities. 16T he answer to this problem is 'yes' and follows from the recent work of B6hm and Wilking [30] (see Part II of this volume).

11.

NOTES AND COMMENTARY

53

Section 4. Einstein metrics are special cases of Ricci solitons. In general, if a warped product metric 9 is Einstein, then the fiber ('P, g) is Einstein. If the base is I-dimensional, then provided w (r) is not constant, the metric 9 = dr 2 + w (r)2 9 satisfies Rc (g) = )..g if and only if w' (r)2

+ ~w (r)2 =

_P_, n n-I where Rc (9) = pg and n = dim 'P. Without loss of generality (e.g., up to scaling and diffeomorphism), we have the following cases:

I

p =

~~i I).. = R~) I w (r) I

9

- (n - 1)

-n

0

-n

eT

including hyperbolic cusp

n-I

-n

sinhr

including hyperbolic space

n-I

0

r

Ricci flat cone

n-I

n

smr

including sphere

0

0

1

Ricci flat product

coshr including hyperbolic with two ends

As a consequence, we have the following (see Theorem 9.110 on p. 268 of [27]). THEOREM 1.94 (Einstein warped products over I-dimensional base). If a warped product (M, g) over a I-dimensional base, with the dimension of the fiber at least 2, is a complete Einstein manifold, then either (1) 9 is a Ricci flat product, (2) M is topologically a cone on P and the fiber is Einstein with positive scalar curvature, or (3) the base is JR, the fiber is Einstein with nonpositive scalar curvature, and 9 has negative scalar curvature.

CHAPTER 2

Kahler-Ricci Flow and Kahler-Ricci Solitons Symmetry, as wide or narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection. - Hermann Weyl

The Kahler-Ricci flow is simply an abbreviation for the Ricci flow on Kahler manifolds. In this chapter we first review some basic definitions and properties for Kahler manifolds and survey some of the fundamental results on the existence of Kahler-Einstein metrics. Then we discuss elementary properties of the Kahler-Ricci flow and state some of the fundamental longtime existence and convergence results. We also give a survey of KahlerRicci solitons. Some other highlights of this chapter are an exposition of tensor calculations in holomorphic coordinates, the proof of the long-time existence and convergence of the Kahler-Ricci flow on Kahler manifolds with Cl < 0, construction of the Koiso solitons and other U(n)-invariant solitons, proofs of differential Harnack estimates under the assumption of nonnegative holomorphic bisectional curvature, and a survey of uniformization-type results for complete noncompact Kahler manifolds with positive bisectional curvature. We assume the reader either has some knowledge of Kahler geometry or will read other references on Kahler geometry along with this chapter, so we make no attempt to be completely self-contained. l The latter chapters do not depend on this chapter and the reader interested only in Riemannian Ricci flow may skip this chapter. 1. Introduction to Kahler manifolds

In this section we introduce the basic concepts of complex and Kahler manifolds including the point of view of having an almost complex structure on an even-dimensional Riemannian manifold satisfying natural properties. Let M be a real 2n-dimensional differentiable manifold. A system of holomorphic coordinates on M is a collection {Zi : Ui ---t Zi (Ui) c en} , where {Ui} is a cover of M and Zi are homeomorphisms such that the maps

Zi

0

zjl : Zj (Ui n Uj ) ---t Zi (Ui n Uj )

lSee the notes and commentary at the end of this chapter for some references on complex manifolds and Kahler geometry. 55

56

2. KAHLER-RICCI FLOW

are holomorphic (complex analytic), and hence biholomorphic, whenever

Ui n Uj i= 0. Two systems of holomorphic coordinates {Ui' Zi} and {Vj, W j} on M are equivalent if whenever Ui n Vj i= 0, Wj 0 Zi- 1 :

Zi (Ui

n Vj)

----t

Wj

(Ui n Vj)

is a biholomorphism. A complex structure on a differentiable manifold M is an equivalence class of systems of holomorphic coordinates. A complex manifold is simply a differentiable manifold with a complex structure. A complex manifold has a natural real analytic structure and is orientable (using holomorphic coordinates {zO: ~ xO: + Ryo:} an orientation is

:=1 '

{/xr, i!;r, ... ,

determined by requiring that the frame a~n , a~n } be positively oriented). In this chapter and only in this chapter we shall use Mn to denote a complex manifold M of complex dimension n (half of the real . ) ,.I.e., n:::;=. d'Ime MId' · d ImenSlon = 2' Im]R M . A real sub manifold N of a complex manifold M n is a complex submanifold if for every pEN there exist holomorphic coordinates {ZO:}:= 1 in a neighborhood U of p such that

N

nU = {q E U : zk+

1

(q) = ... = zn (q) =

o} .

Clearly a complex submanifold is itself a complex manifold. In the above definition, k is the complex dimension of N. Some elementary examples of complex manifolds are (1) complex Euclidean space C n , (2) complex projective space cpn, (3) complex submanifold of cpm = algebraic manifold = zero set of a finite number of homogeneous polynomials, (4) complex torus C n Ir, where r is a lattice. A complex structure defines at each point p E M a map J : T Mp ----t T Mp by J = d (z-l 0 R 0 z) , where z are holomorphic coordinates defined in a neighborhood of p (this definition is independent of the choice of z). Since R 0 R = - iden, it is easy to see J2 = - idTM . In general, given an even-dimensional differentiable manifold M, an automorphism J : T M ----t T M is called an almost complex structure if J2 = - idTM . (A manifold with an almost complex structure is called an almost complex manifold.) Hence a complex structure induces an almost complex structure. We say that an almost complex structure is integrable if there exists a complex structure which induces the almost complex structure. Given an almost complex manifold (M, J), the Nijenhuis tensor is defined by

N J (X, Y) ~ [JX, JYJ - J [JX, YJ - J [X, JYJ - [X, YJ for X, YET M. By the Newlander-Nirenberg Theorem, a necessary and sufficient condition that an almost complex structure be integrable is that the Nijenhuis tensor vanish.

1. INTRODUCTION TO KAHLER MANIFOLDS

57

It is interesting that 8 6 admits an almost complex structure. In fact 8 2 and 8 6 are the only even-dimensional spheres which admit almost complex structures (see §8 of Chern [95] for example). The following is a longstanding unsolved question. 2.1 (Existence of complex structure on 8 6 ). Does 8 6 admit a complex structure? More generally, one may ask which closed evendimensional manifolds admit almost complex structures but not complex structures. PROBLEM

Let (M, J) be an almost complex manifold. A vector field V is an infinitesimal automorphism of the almost complex structure if the Lie derivative of J with respect to V is zero, i.e., (2.1)

LVJ

= O.

Note that (LVJ) (W)

= LV (JW) - J (LVW) = [V, JW]- J ([V, W]),

so (2.1) is equivalent to J ([V, W]) = [V, JW] for any vector field W. There are various equivalent ways to define a Kahler manifold. We say that a Riemannian manifold (M,g) with an almost complex structure J : T M ~ T M is a Kahler manifold if the metric 9 is J-invariant (or Hermitian): 9 (JX, JY) = 9 (X, Y)

and J is parallel: \7 J = 0

or equivalently, \7 x (JY) = J (\7 x Y)

for all X, Y, where \7 is the Riemannian covariant derivative. 2 The metric 9 is called a Kahler metric. Note that almost complex structures which yield Kahler manifolds are necessarily integrable. Indeed, by the Newlander-Nirenberg Theorem, we only need to check that the Nijenhuis tensor vanishes for a Kahler manifold: NJ (X, Y)

= \7 JxJY -

\7 JyJX - J (\7 JXY - \7yJX)

- J (\7 x JY - \7 JY X) - \7 x Y

=-

[J (\7 JX Y) - \7 JxJY]

+ [J(\7yJX) -

+ \7 Y X

+ [J (\7 JY X)

- \7 JyJ X]

\7 y J(JX)]- [J(\7xJY) - \7xJ(JY)]

=0.

A complex manifold (M, J) is called a Kahler manifold if it admits a Kahler metric. 2Since 0

= (V xJ) (Y) = V x

(JY) - J (V x Y).

58

2. KAHLER-RICCI FLOW

Given an almost complex manifold (M, J), the complexified tangent bundle is TeM ~ T M ®lR C. For each p E M we may extend the almost complex structure J to a complex linear map

Jc : TeMp

~

TeMp.

Since (Jc)2 = - idTcM p ' the eigenvalues of Jc are Rand define the holomorphic tangent bundle by

Tl,o M ~ {V

E

R.

We

TeM : Je (V) = HV}

and the anti-holomorphic tangent bundle by

TO,l M ~ {V E TeM : Je (V) = -HV} . This gives us a decomposition

TeM = Tl,o M EB TO,l M. A vector in Tl,o M is called type (1,0) and a vector in TO,l M is called type (0,1) . Alternatively, we can obtain the above decomposition by decomposing a (real) tangent vector VET M as V = Vl,o + VO,l, where

Vl,o ~

'12 (V -

HJV)

E

Tl,o M,

VO,l ~

'12 (V + HJV)

E

TO,l M.

I.e.,

Tl,oM = {Vl,o: V

E

TM},

TO,lM = {VO,l : V

E

TM}.

Since TeM is the complexification of a real vector space, the complex conjugate (bar operation), X + A Y ~ X - AY, where X, YET M, is defined on TeM. Given Z E TeM, the real part of Z is defined by Re (Z)

~ ~ (Z + 2) .

Note that if VET M, then 2 Re (Vl,O)

= 2 Re (VO,l) = V.

The almost complex structure satisfies

Je (if) = Je (V),

V

E

TeM,

and we have Tl,oM = TO,lM and TO,lM = Tl,oM. The complexified cotangent bundle is TcM = (TeM)* = T* M ®lR C, which decomposes into

Tc M = Al,oM EBAO,lM, where Al,oM ~ (Tl,oM)* and AO,lM ~ (TO,lM)*.

1. INTRODUCTION TO KAHLER MANIFOLDS

59

A covariant tensor of type (p, q) is a section of ®p,q M ~ (®P A1,0 M) ® (®q AO,lM).3 The complex conjugate extends to the tensor bundles and

Q9p,qM=Q9q,PM. We say that a differential (p + q)-form "1 is of type (p, q) if it is a section of the vector bundle AP,qM ~ (APT1,OM) A (AqTO,lM) c Ap+qTcM. We denote the space of (p, q)-forms by Op,q (M). We say that a (p,p)-tensor (or form) "1 is real if ij = "1. When we are considering complex manifolds, it is most natural to carry out calculations in holomorphic coordinates. Let {zC 0 (respectively, Ke (Z)

~ 0) for all Z

{a}.

T 1,o M By (2.8), positive (respectively, nonnegative) Riemannian sectional curvature implies positive (respectively, nonnegative) holomorphic sectional curvature. Given Z,W E T 1,oM let X ~ Re(Z) and U ~ Re(W). The (holomorphic) bisectional curvature in the directions (Z, W) is defined

{a},

E

66

2. KAHLER-RICCI FLOW

to be J(,

(Z W) == Rm(X, JX,JU,U) = Rme (Z,Z, W, w)

e,

.

IZI 21WI 2

IXI21U12

Clearly we have Ke (Z, Z) = Ke (Z). We say that the bisectional curvature is positive (respectively, nonnegative) if Ke (Z, W) > 0 (respectively, Ke (Z, W) 2: 0) for all Z, WE Tl,O M Clearly positive (respectively, nonnegative) bisectional sectional curvature implies positive (respectively, nonnegative) holomorphic sectional curvature. We also have the following (see Proposition 1 on p. 32 of [270] for example).

{o} .

LEMMA 2.16. Po.sitive (respectively, nonnegative) Riemannian sectional curvature. implies positive (respectively, nonnegative) bisectional sectional curvature.

Similarly to the notation for the components of the complex Riemann curvature tensor, we define

V o.R{3"y8TJ

~ ((Vc) {}~'" Rm e) (0~{3' 0~'Y ' 0~8 ' O~TJ )

.

The second Bianchi identity says that

(2.9) since - ) + ( V {} Rme )(0 - 0 - 0 - 0) ( V {} Rme )( - 0 - 0 - 0 0 {}zO oz{3' oi'Y' oz8' oiTJ {}z'Y ozo.' oz{3' oz8' oiTJ

+ (V a;P {} Rme) (o~z'Y , -00 zo. , 00ZU

0

o~zTJ )

=0

(Le., V o.R{3"y8ii + V (3R;yo.8ii + V"yRo.{38ii = 0, where V"yRo.{38ii = 0). Taking the trace, we have

(2.10) Note that Vo.R{3;y = ~R{3;y - r~{3R8"y' The volume form is d/1g ~ ~wn, where w n ~ w 1\ n = dime M, which in local coordinates is wn = n! ( R) n det(go.,B)dz l

1\

di l

1\ ... 1\

... 1\

dz n 1\ din.

The volume is

(2.11)

Vol (M) = Volg (M) =

~ {

n·JM

The total scalar curvature may be written as (2.12)

w (n times) and

wn .

2.

CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION

Let V i3 (2.13)

~

67

V 8 / 8z i3' The Laplacian acting on tensors is given by

1 ~ ~ "2l~/3 (VOtVi3

+ Vi3VOt)

1 ="2 (VOtVa

+ VaVOt).

NOTATION 2.17. At this point we have begun to use the extended Einstein summation convention where not only pairs of repeated indices consisting of an upper index and a lower index are summed but also pairs of repeated lower indices consisting of a barred index and an unbarred index are summed. For example, aOta ~ L:~,/3=1 gOti3aOti3' Formulas which are expressed using this convention hold in the literal sense in unitary holomorphic coordinates at a point p. Usually we use the extended Einstein summation convention in lieu of computing in unitary coordinates at a point. In this sense, the formulas in (2.7) and throughout this chapter hold in arbitrary holomorphic coordinates. EXERCISE 2.18. Show that the Laplacian defined above in (2.13) is onehalf of the Riemannian Laplacian. Acting on functions, the Laplacian is A

u =

g

Oti3n v

n 01 V

i3 = g

Oti3

since, acting on functions, VOl V i3 = 8z~~z~ is a real-valued function on M, then (2.14)

{j2

8z Ot 8z/3 '

= V i3V

01'

We also note that if ¢

~ Igrad¢1 2 = gOti3vOt¢Vi3¢ ~ IV¢1 2 ,

where the LHS is the Riemannian gradient. Likewise, (2.15)

1 2 = 1V OtVi3¢ 12 "2IHess¢1

+ IV

Ot

V/3¢1 2,

where the LHS is the Riemannian Hessian. In our calculation of evolution equations under the Kahler-Ricci flow we shall often use the following. LEMMA 2.19 (Kahler commutator formulas). On a Kahler manifold we have the following commutator formulas for covariant differentiation acting on tensors of type (1,0), (0,1), and (1,1) , respectively: (2.16) (2.17) (2.18)

V 01 V i3a-y - V i3 V Ota-y

= - R~i3-yao,

= -R~i3;ybiS = R~a}iS' VOl V i3 a-yiS - V i3 V Ota-yiS = - R Ot i3-yf/aT/iS + R Oti3T/iS a-yf/' VOl V i3 b;y - V i3V Otb;y

Analogous formulas hold for higher degree tensors and forms (see Exercise 2. 22).

REMARK 2.20. Note that ROti3T/iS = -ROt i3iST/' which exhibits the consistency of the formula above with the analogous formula in Riemannian geometry. We have also used the extended Einstein summation convention.

2. KAHLER-RICCI FLOW

68

Using the fact that the Christoffel symbols are zero unless all of the indices are unbarred or all are barred, we compute PROOF.

and

Va V /Ja'Y = OaO/Ja'Y - r~'Y0/Jao,

r~'Yao) .

V 13 Vaa'Y = 013 (Oaa'Y Hence, using (2.4), we have

Va V /Ja'Y - V13 V aa'Y = o/Jr~'Yao = - R~/J'Yao, which is (2.16). Equation (2.17) is the complex conjugate of (2.16). If a is a (1, I)-tensor, then we compute V /Ja'Y1, = 0/Ja'Y1, - r~1,a'Ye,

Vaa'Y1,

r~'YaTJ1,

= Oaa'Y1, -

and

Va V/Ja'Y1, = Oa (o/Ja'Y1, V 13 Vaa'Y1,

r~1,a'Ye)

-

r~'Y (o/JaTJ1, - r~1,aTJe)

,

= 013 (Oaa'Y1, - r~'YaTJ1,) - r~1, (Oaa'Ye - r~'YaTJe) .

Hence, after some cancellations, we obtain in holomorphic coordinates {za} at a point p where it is unitary,

Va V /Ja'Y1, - V13 Vaa'Y1, = -Oar~1,a'Ye + 0/Jr~'YaTJ1, = Ra/Je1,a'Ye - Ra/J'YfiaTJ1,'

o LEMMA 2.21 (Va and V (3 commute). We have [Va, V{3] = 0 acting on tensors of any type. That is, if a is a (p, q)-tensor, then

Va V{3a'Yl "''Y1'1,l ... 1,q - V{3 V aa'Yl "''Y1'1,l ...1,q = O. PROOF.

We have

V{3a 'Yl"''Y1'Ol'''Oq - - -- o{3a'Yl···'Y1'Ol···Oq - - -

r TJ a - {3'Yj 'Yl"''Yj-1TJ'Yj+l''''Y1'dl'''Oq '

It is easiest to compute the second covariant derivative in normal holomorphic coordinates centered at any given point p E M, where r~{3 (p) = O. In these coordinates we have at p,

Va V{3a'Yl ···'Y1'1,1···1,q = Oa (0{3a'Y1···'Y1'1,l ... 1,q -

- a o{3a -

a

- - -

'Y1"''Y1'd1'''dq

r~'Yj a'Yl "''Yj-1 TJ'Yj+1 "''Y1'1,1 ...1,q)

(a r a

TJ ) {3'Yj

a'Y1"''Yj-1 TJ'Yj+1"''Y1'01'''Oq' - -

2. CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION

69

Hence V' a V'(3a,Yl' "'Yp81 .. ·8q

-

V' (3 V' aa'Yl .. ''Yp81 ···8q

-- (8(3r'1/a'Yj - {)a r'1/(3'Yj ) a'Yl···'Yj-l'1/'Yj+1···'Yp,h···8q· - -

Now from (2.3) we have at p,

8(3r~'Yj

-

8ar~'Yj

= g'1/5.. (8(38ag'Yj5.. - 8a8(3g'Yj5..) = 0,

o

and the lemma follows.

EXERCISE 2.22. Compute the commutator formula for the operator [V' a, V',a] ~ V' a V',a - V',aV' a acting on (p, q)-tensors and forms. SOLUTION TO EXERCISE 2.22. If a is a (p, q)-tensor, then V' a V' ,aa'Yl"''Yp81 ... 8q

-

V',a V' aa'Yl ''''Yp81 •.. 8q

p

L R a,a'Yifj a'Yl···'Yi-l'1/'YHl···'Yp8 ···8

=-

1

q

i=l

q

+L

Ra,a'1/8j a'Yl···'Yp81 ···8j _ 1 fj8j+l···8q·

j=l

The commutator [~, V',a] , acting on functions, is ~ V',af

1

= "2 (V' a V'ii + V'ii V' a) V',af =

1

1

"2 V' a V',a V'iif + "2 V',a V' ii V' af 1

= "2R'Y,aV'''ri + V',a~f.

(2.19)

LEMMA 2.23 (Kahler V' and ~ commutator). The commutator [V' a, ~l ,

acting on (0, I)-forms, is given by (2.20)

1

1

V'a~a,a - ~V'aa,a = "2V'a R8,aa8 - "2Ra8V'8 a,a + R a;Y8,aV''Y a8'

PROOF. We compute

+ V';y V' 'Y) a,a - (V''Y V';y + V';YV''Y) V' aa,a = V''Y (V' a V';y - V';y V' a) a,a + (V' a V';y - V';y V' a) V''Ya,a = V''Y (R a;Y8,aa8) - R a;Y'Y8V'8 a,a + R a;Y8,aV''Y a8 = V' aR8,aa8 - Ra8 V' 8a,a + 2R a;Y8,a V' 'Y a8'

2 (V' a~a,a - ~ V' aa,a) = V' a (V' 'Y V';y

where we used [V'a'V''Yl

= 0 and V''YRa;Y8,a = V'aR8,a from (2.9).

0

REMARK 2.24. By taking the complex conjugate of the lemma above,

we have (2.21)

2.

70

KAHLER-RICCI FLOW

Since the Kahler-Ricci flow is a heat-type equation for Kahler metrics, some evolution equations we shall derive later in this chapter use the following commutator formula. If a is a time-dependent (0, I)-form, then under the Kahler-Ricci flow,

[:t

-~, \/0:] a{J = ~\/ o:R8{Ja8 - ~RO:8\/8a{J + RO:18{J\/"(a8'

Another basic formula is the commutator of the heat operator and the Hessian. First, by (2.19) and (2.20), we have for any function f on M,

\/ 0: \/ {J~f

= ~L \/0: \/ {Jf ~ ~ \/ 0: \/{Jf

(2.22)

+ Ro:{J81 \/"( \/8f -

1 1 2 R o:8\/8\/{Jf - 2R"({J\/ 0: \/1f,

where ~L is called the (complex) Lichnerowicz Laplacian. If function also of time, then (2.22) tells us \/ 0: \/ {J (:t -

(2.23)

~)

f = (%t -

~L )

f is a

\/0:\/{Jf.

Note that since \/0: \/ {J = 8o:8f3 acting on functions, we have when acting on functions.

3t (\/0:\/(J) = 0

REMARK 2.25. Formula (2.23) has the following Riemannian analogue. For a time-dependent function f and under the Ricci flow, we have

(2.24) where

\/i\/j (%t ~L,

~)

f = (%t -

~L) \/i\/jf,

defined by ~LVij ~ ~Vij

+ 2Rkijf.Vkf. -

~kVjk

- RjkVik,

is the (Riemannian) Lichnerowicz Laplacian acting on symmetric 2-tensors (see Lemma 2.33 on p. 110 of [111]). In fact, (2.24) is a special case of (2.23). 3. Existence of Kahler-Einstein metrics In this section we discuss what is known about the existence and uniqueness of Kahler-Einstein metrics, which are canonical (constant Ricci curvature) metrics on Kahler manifolds. The Kahler-Einstein equation is elliptic whereas the Kahler-Ricci flow, discussed beginning in the next section, may be considered as its parabolic analogue. First recall the following 88-Lemma, which is a consequence of the Hodge decomposition theorem. LEMMA 2.26 (d-exact real (1, I)-form is 88 of real-valued function). Let M be a closed Kahler manifold. Ifw is an exact real (1, I)-form, then there exists a real-valued function 'IjJ such that Ff.88'IjJ = w. That is, n

82 8zo:8zf3 'IjJ

= wo:{J,

3.

EXISTENCE OF KAHLER-EINSTEIN METRICS

where W = HWo./3dzo. 1\ dz/3 and wo./3

71

= W/3ii'

PROOF. This is a standard result in the theory of Kahler manifolds; see the book by Zheng [383] for example. 0 REMARK 2.27. More generally, we have the following. Let b be a (p, q)form, where p, q > O. If b is d-closed and either d-, 8-, or 8-exact, then there exists a (p - 1, q - I)-form

.w for some >. E R If M n admits a Kahler-Einstein metric g, then

(We have>. = ~, where the (complex) scalar curvature R is constant.) Therefore, a necessary condition for the existence of a Kahler-Einstein metric on M is that its first Chern class have a sign. By having a sign we mean that ct (M) = 0, < 0, or > 0 if there exists a real (1, I)-form in the first Chern class which is zero, negative definite, or positive definite, respectively. COROLLARY 2.29 (Cl = 0: existence of Kahler Ricci flat metrics). If (Mn, go) is a closed Kahler manifold with Cl (M) = 0, then there exists a Kahler metric 9 with [w] = [wo] such that Rc (g) == O. Kahler Ricci flat metrics are called Calabi-Yau metrics and Kahler manifolds with Cl (M) = 0 are called Calabi-Yau manifolds. Another consequence of Theorem 2.28 is that if Cl (M) < 0 (respectively, Cl (M) > 0), then in each Kahler class there exists a metric with negative (respectively, positive) Ricci curvature. When Cl (M) < 0, we have the following result about the existence of Kahler-Einstein metrics conjectured by Calabi and proved by Aubin [11,12] and Yau [378, 379]. Calabi proved that such a Kahler-Einstein metric is unique if it exists [43]. THEOREM 2.30 (Cl < 0 Calabi conjecture: R < 0 Kahler-Einstein metrics). If Mn is a closed complex manifold with ct (M) < 0, then there exists a Kahler-Einstein metric 9 on M, which is unique up to homothety (scaling), with negative scalar curvature. A consequence of Theorem 2.30 is the following Chern number inequality:

~

(-It 2 (n + 1) cl(M)n-2 c2 (M). n (See Yau [378] and also Corollary 9.6 on p. 226 of [383] for an exposition.) (-It cl(Mt

REMARK 2.31. Another consequence of Theorem 2.30 is that if (M2, g) is a closed Kahler surface homotopically equivalent to CP2, then M2 is biholomorphic to Cp2 (see Yau [378]; earlier related work in all dimensions was done by Hirzebruch and Kodaira [204]). If Cl (M) > 0, however, there are obstructions to the existence of KahlerEinstein metrics. An example is the Futaki invariant (see [147]). On a closed manifold Mn with Cl > 0 (Le., a Fano manifold), fix a Kahler metric 9 such that [w] is a positive real multiple of cl(M). (This is the so-called

3. EXISTENCE OF KAHLER-EINSTEIN METRICS

73

canonical case.) By scaling the metric, we may assume [w] = cl(M). By Lemma 2.26 there exists a smooth function f : M --+ IR such that p-

27l'W

=

Raar

(One can make f unique by the normalization fM e- f d/-t = 1.) Let 1}(M) denote the space of (real) holomorphic vector fields on M. The Futaki functional F[wJ : 1}(M) --+ C is defined by

F[wJ (V)

~ 1M V (I) d/-t = 1M (V, \7 f)

d/-t.

Futaki [147] showed that F[wJ is well defined, i.e., that it depends only on the homology class [w]. It is then clear that if M admits a Kahler-Einstein metric, then F[wJ vanishes. However, Tian has shown that 1}(M) = 0 (which implies F[wJ = 0) does not imply there exists a Kahler-Einstein metric. The following uniqueness result in the Cl > 0 case was proved by Bando and Mabuchi [21]. THEOREM 2.32 (Uniqueness of Kahler-Einstein metrics). Let (Mn,g) be a closed Kahler manifold with Cl (M) > O. The Kahler-Einstein metric (with positive scalar curvature), if it exists, is unique up to scaling and the pull back by a biholomorphism of M.

The following result was proved by Andreotti and Frankel [144] for n = 2, Mabuchi [260] for n = 3, and Mori [271] and Siu and Yau [336] in all dimensions; Mori proved a more general algebraic-geometric result. Work on characterizing cpn was done by Kobayashi and Ochiai [237]. THEOREM 2.33 (Frankel Conjecture). If(Mn, g) is a closed Kahler manifold with positive bisectional curvature, then M n is biholomorphic to cpn.

In fact, if the bisectional curvature is nonnegative everywhere and positive at some point, then M is biholomorphic to cpn. When n = 2 and Cl (M) > 0, we have the following (see Tian [345]). 2.34 (Cl > 0 surfaces: R > 0 Kahler-Einstein metrics). If M2 is a closed complex surface with Cl (M) > 0 and the Lie algebra of the automorphism group is reductive, then there exists a Kahler-Einstein metric g on M with positive scalar curvature. THEOREM

2.35. Note that such surfaces are biholomorphic to CP2 blown up at p points, where 3 ~ p ~ 8. On the other hand, for n ~ 2, cpn blown up at 1 or 2 points does not admit a Kahler-Einstein metric (see p. 156 of Lichnerowicz [254] and Yau [376]). See Section 7 of this chapter for the existence of Kahler-Ricci solitons on CP2 blown up at 1 or 2 points. REMARK

There are a number of additional works related to stability and the existence of Kahler-Einstein metrics with Cl > O. Notably Aubin [14], Siu [335]' Nadel [281]' Tian [346], Donaldson [129], [130]' and Phong and

74

2. KAHLER-RICCI FLOW

Sturm [304]. The existence of Kahler-Einstein metrics on complete noncompact manifolds with Cl < 0 and Cl = 0 has been well studied (see Cheng and Yau [94] and Tian and Yau [349], [350] for example). For work on the existence of singular Kahler-Einstein metrics on certain classes of closed Kahler manifolds where Cl does not have a sign, see Tsuji [360] (for some further recent work see Cascini and La Nave [60] and Song and Tian [337]). Kahler-Einstein metrics are closely related to Kahler-Ricci solitons and hence the Kahler-Ricci flow which will be discussed next in this chapter.

4. Introduction to the Kahler-Ricci flow In this section we introduce the Kahler-Ricci flow system and its equivalent formulation as a single parabolic Monge-Ampere equation. We discuss some basic estimates which may be proved using the maximum principle. 4.1. The Kahler-Ricci flow equation. Let (Mn, J) be a closed manifold with a fixed almost complex structure. Given a Riemannian metric g, we may define a 2-tensor w by w (X, Y) ~ 9 (JX, Y). Recall that when 9 is Hermitian, w is antisymmetric (Le., defines a 2-form) and w is called the Kahler form. If a solution 9 (t) to the Ricci flow ~gij = -2~j is Hermitian at some time t, then w satisfies the equation gtW = -2p at that time, where p = p (t) is the Ricci form of 9 (t) . Hence

~ (dw) = d (~w) at at

= -2dp = 0

whenever 9 (t) is Kahler (we can define even when 9 is not Hermitian). This suggests that if 9 (0) is Kahler, then under the Ricci flow 9 (t) is Kahler for all t 2: O. Consider the Kahler-Ricci flow equation (2.26)

a

at gai3

= -Rai3

for a 1-parameter family of Kahler metrics with respect to J, which is obtained from the Ricci flow by dropping the factor of 2. Now we derive the parabolic complex Monge-Ampere equation to which the Kahler-Ricci flow is equivalent. For a complete initial Kahler metric with bounded curvature, we will use this scalar equation to prove the short-time existence of a solution to the initial-value problem for the Kahler-Ricci flow. On a closed manifold we will use the scalar equation to prove that the Kahler property of an initial metric is preserved under the Ricci flow and to prove the long-time existence of solutions to the Kahler-Ricci flow. By (2.26), gt [w] = - [p (t)] = - [p (0)] , so that the Kahler class of the metric at time t evolves linearly,

[w (t)] = [w (0)] - t [p (0)] , and the real (1, 1)-forms w (t) - w (0) + tp (0) are exact for t 2: O. Let g~i3 ~ gai3 (0). Using Lemma 2.26, for each t there exists a real-valued function

4.

INTRODUCTION TO THE KAHLER-RICCI FLOW

75

'-P (t) defined on all of M such that o

-

0

-

ga:fi (t) = ga:!3 + tOa:o,a log det g')'6 + oa:o,a'-P (t) .

(2.27)

By (2.6) we have

Hence, by differentiating (2.27), we obtain

Oa: 8,a (:t '-P) = - Ra:fi - oa: 8,a log det g~6 _

det

(g~6 + to')'8olog det g~iI + o')'8o'-P (t) )

= oa:o,a log

dO.

etg')'6

Hence we conclude that the Kahler-Ricci flow equation on a closed manifold is equivalent to the following parabolic (scalar) complex MongeAmpere equation: (2.28)

o'-P

~ = log

vt

det

(g~6 + to-y8o log det g~iI + o')'8o'-P (t») d

0

etg')'6

+ Cl (t)

for some function of time Cl (t) . By standard parabolic theory, given any Coo initial function '-Po on a complete Kahler manifold with bounded bisectional curvature, there exists a unique solution '-P (t) to (2.28) with '-P (O) = '-Po, defined on some positive time interval 0 ~ t ~ E. We also have the following. LEMMA 2.36 (The Kahler property is preserved under the Ricci flow). If (M n , J, go) is a closed Kahler manifold, then there exists a solution to the Kahler-Ricci flow 9 (t), 0 ~ t ~ E, for some E > 0 with 9 (0) = go. Furthermore 9 (2t) is a solution of the (Riemannian) Ricci flow. Also any solution 9 (t) of the (Riemannian) Ricci flow with 9 (t) = go must be Kahler (preserving the compatibility with the almost complex structure). PROOF. Given go, we can find a solution '-P (t), 0 ~ t ~ E, of (2.28) with (t) == O. From the derivation of (2.28), we know that 9 (t) defined by (2.27) is a solution of (2.26). Hence 9 (2t) is a solution of the Ricci flow. The last statement follows from the uniqueness of the initial-value problem for the Ricci flow. D Cl

REMARK 2.37. From the derivation of (2.28) it is clear that if we have a bounded C 4-solution '-P (t) for some Cl (t) on any complex manifold (regardless of completeness and compactness), then we get a C 2 -solution 9 (t) defined by (2.27) to the Kahler-Ricci flow.

2. KAHLER-RICCI FLOW

76

4.2. The normalized Kahler-Ricci flow equation. Let (Mn, J, go) be a closed Kahler manifold. Now we make the basic assumption (corresponding to the canonical case), holding for the rest of this section and the next section, that the first Chern class is a real multiple of the Kahler class, i.e., that

[pol

=

c[wol

for some c E JR. Note that this is possible only if the first Chern class has a sign, i.e., is negative definite, zero, or positive definite. Comparing (2.11) and (2.12), we find that c = ~, where r ~ 1M Rodj.Lga/ Volga (M) is the average (complex) scalar curvature, so that

r 1 -2 [wol = -2 [pol = 7rn

7r

Cl

(M) .

So r depends only on the cohomology class [wol, n, and The normalized Kahler-Ricci flow is

o

otga~ = -Ra~

(2.29)

r

+ ;:;,ga~'

Cl

(M).

for t E [0, T).

The solution of (2.29) can be converted to the solution (2.26) by scaling the metric and reparametrizing time, and vice versa (see Section 9.1 in Chapter 6 of Volume One or subsection 9.1 below). Hence, from Lemma 2.36, we know that the initial-value problem for (2.29) with 9 (0) = go has a solution for a short time. By a derivation similar to that of (2.28), we get the following parabolic (scalar) complex Monge-Ampere equation, corresponding to (2.29) with ga~ (t) = g~~ + Oa8f3'P (t) ,

(2.30)

o'P ot

det

(g~J + O-y 80'P )

= log

detg O-

r +;:;,'P - 10 + Cl (t)

'"(0

for some function of time

Cl

(t). Here 10 is defined by Ra~ (go) - ~g~~ =

Oa8f310; this is possible because [-po

+ ~wo] = O.

4.3. Basic evolution equations. Let 9 (t) be a solution of either the Kahler-Ricci flow or the normalized Kahler-Ricci flow. We define the potential function 1 = 1 (t) by (2.31) This equation is solvable since [- p + ~w] = 0 and by Lemma 2.26. Note that 1 is determined up to an additive constant. Taking the trace of (2.31), we have (2.32)

R-r='~.f.

4.

INTRODUCTION TO THE KAHLER-RICCI FLOW

77

Differentiating (2.3), we find that for both the Kahler-Ricci flow and the normalized Kahler-Ricci flow, the Christoffel symbols evolve by

8 r"{ at Ot{3 -

(2 .33)

"{8'r7 R -

-9

VOl.

{30'

The volume form and scalar curvature evolve according to the following. LEMMA 2.38 (Evolution of df-L and R for normalized flow). Under the normalized Kahler-Ricci flow (2.29),

8 8t df-L = (r - R) df-L and

-8R = flR + 1R 01.{3- 12 - -nr R. 8t

(2.34)

In particular, since fM (r - R) df-L = 0, the normalized Kahler-Ricci flow preserves the volume. PROOF. We first compute, using (2.29), that

8

(2.35)

88

8t log det 9"{8 = 9 "{ 8t 9"{8 = r - R.

Hence

8 8t df-L = (r - R) df-L.

The evolution of the Ricci tensor is (2.36)

8

- (88t

8t ROt/3 = -8Ot8{3

log det 9"{8

)8Ot8{3R. =

From this and

8R _ 01./3 8

at -

9

_

8

_

_

at ROt{3 - at 9Ot{3' ROt{3

D

we easily derive (2.34).

EXERCISE 2.39 (Evolution of R for unnormalized flow). Show that under the Kahler-Ricci flow %t 9Ot/3 = -ROt/3' we have %tdf-L = -Rdf-L, %tROt/3 = 8Ot8{3R, and

(2.37)

8 R -_ at

9

01./3 at 8 ROt{3_ + 1R {3_12 -_ flR Ot

+ 1R Ot{3_1 2 .

EXERCISE 2.40 (Total and average scalar curvature evolution). Show that if M n is closed, then under the Kahler-Ricci flow ft9Ot/3 = -ROt/3'

! 1M

Rdf-L =

1M (I ROt/312 - R2) df-L,

and hence we have

~: = (1M df-L) 1M (I ROt/312 -1

R2) df-L + r2.

78

2.

KAHLER-RICCI FLOW

REMARK 2.41 (I-dimensional normalized Kahler-Ricci flow). Recall the following facts, due to Hamilton [180], about the normalized Ricci flow on Riemannian surfaces £g = (r - R) g. Throughout this remark, Rand r denote the Riemannian scalar curvature and its average, respectively. Note that 9 (t) ~ 9 (!t) is a solution of the complex I-dimensional normalized Kahler-Ricci flow. The potential function f, defined by (2.32) and normalized suitably by an additive constant, satisfies (see Lemma 5.12 on p. 113 of Volume One) of (2.38) ot = b.f + r f. By the maximum principle, this implies If I ~ Ce Tt • The gradient quantity H ~ R - r + 1\7 fl2 satisfies (see Proposition 5.16 on p. 114 of Volume One) oH (2.39) at = b.H - 21MI2 + rH, where M ~ \7\7 f - !b.f· g. By the maximum principle, we have (see Corollary 5.17 on p. 115 of Volume One)

_CeTt ~ R - r ~ H ~ CeTt . This gives the exponential decay of IR - rl when r < O. The norm squared of the tensor M evolves by (see Corollary 5.35 on p. 130 of Volume One)

:t

(2.40)

IMI2 = b.IMI2 - 21\7 MI2 - 2R IMI2 .

Generalizing the I-dimensional formula (2.38) to higher dimensions, the potential satisfies a linear-type equation. (Strictly speaking, the equation is not linear since the Laplacian is with respect to the evolving metric.) LEMMA 2.42 (The potential f satisfies a linear-type equation). Under the normalized Kahler-Ricci flow on a closed manifold Mn, the potential function f, defined by (2.31) and normalized by an additive constant, satisfies (2.41 )

PROOF. From (2.31) we compute On[)(3

(it) =

:t (on[)(3f)

= on o-(3R -

=

! (Rn~

r0

-

~gn~)

- (

r) .

-:;;, otgn~ = on o (3 b.f + -:; , f

Since M is closed, it follows that (2.42)

~~ =

b.f + ~f + c(t)

for some function c (t) and the lemma follows from the fact that we have the freedom of adding a time-dependent constant in our choice of f (x, t). 0

4.

INTRODUCTION TO THE KAHLER-RICCI FLOW

79

COROLLARY 2.43 (Estimate for f). If Mn is closed, then, for the function f given by Lemma 2.42, we have If I :S Ge;;t.

(2.43)

This is the first hint that the Kahler-Ricci flow in the case where Cl (M) < o is the easiest and that the case where Cl (M) > 0 is the hardest. We now compute for f in Lemma 2.42, for the normalized Kahler-Ricci flow, that (2.44)

%t

IV' c.d1 2= illV'afl 2 - IV'a V' ,8f1 2 - IV'a V' ,6f1 2 + ~ IV'afl 2 .

Define

h ~ ilf + IV' afl 2 = R - r Similarly to (2.39), we have

+ IV'afl 2 .

LEMMA 2.44 (Ricci soliton gradient quantity evolution). For the normalized Kahler-Ricci flow on a closed manifold Mn,

~~

(2.45)

= ilh - IV'a V' ,8f1 2 + ~h.

PROOF. We compute

{) (R at (2.46)

r) =

il (R -

r)

+ IRa,6 12 -

=

il (R -

r)

+ lV' a V',6fl + -ilf + -n n

=

il (R -

r)

+ IV'a V' ,6f1 2 + :cn (R -

rR -:;;, 2

2r

r2

r --R n

r),

where we used (2.31) and (2.32). Equation (2.45) follows from summing this equation with (2.44). 0 COROLLARY 2.45 (Estimate for R). rt

rt

-Ge n :SR-r:SGe n ,

(2.47)

lV'fI2:s Ge;;t.

(2.48)

PROOF. By (2.46), the lower bound for R - r follows from

(!

-il) (R-r)

~ ~(R-r).

To get the upper bound for R-r, we observe that by the maximum principle, we have rt R - r :S h:S Gen. This also implies (2.48) since IV' fl2 = h - (R - r) :S h + Ge;;t. 0 REMARK 2.46 (Exponential decay when Cl < 0). When Cl (M) < 0, so that r < 0, (2.47) says that R approaches its average exponentially fast. This suggests that the Kahler-Ricci flow converges to a Kahler-Einstein metric. Indeed, this is Theorem 2.50 below.

2. KAHLER-RICCI FLOW

80

Here is an interesting equation due to Hamilton. LEMMA 2.47 (Ricci soliton vanisher evolution equation). For the potential function f in Lemma 2.42, we have

(2.49)

%t IV' 0 V' /3f12 =

~ IV' 0 V' /3f12 -

IV' "I V' 0 V' /3f12 - lV'i V' 0 V' /3f12

- 2 RoihJ V' a V'i fV' /3 V' 8f. PROOF.

By (2.42) and the commutator formulas, we have

%t (V'oV'rd)

=

V' 0 V'/3

= V'0V'/3

(~{) - (%tr~/3) V''Yf (~f + ~f) + V'oRtJiV''Yf.

On the other hand, for any function V'oV'/3~f =

f,

V'oV'/3V''YV'if = V''YV'oV' i V'/3f

= V' /v i V' 0 V' /3f - V'"I (Roi!3JV' 8f) =

1

'2 (V' "I V'i + V'i V' "I) V' 0 V' /3f -

V' oR/3JV' 8f - ROi/3JV' "I V'/d

1

- '2 (ROi V'"I V' /3f + RtJi V' 0 V'"If) since Hence

a

at (V' 0 V' /3f) = ~ V' 0 V' /3f

r

+ -;;, V' 0 V' /3f -

Roi!3JV' "I V' 8f

1

- '2 (ROi V' "I V' /3f + R/3i V' 0 V'"If)

o

and one easily derives (2.49) from this.

2.48. Find geometric applications of (2.49) in the study of the Kahler-Ricci flow. PROBLEM

EXERCISE

2.49. Show that when dime M = 1,

1V'0V'/3fI2 =

~ lV'iV'jf - ~~f9ijI2

and

Show also that (2.49) generalizes (2.40). 5. Existence and convergence of the Kahler-Ricci flow In this section we present some of the proofs of the basic global existence and convergence results for the Kahler-Ricci flow due to H.-D. Cao [46].

5.

EXISTENCE AND CONVERGENCE

81

5.1. Cao's existence and convergence theorem. When the first Chern class has a definite sign, either negative, zero, or positive, Cao proved that the normalized Kahler-Ricci flow exists for all time. Let KRF and NKRF denote the Kahler-Ricci flow and the normalized Kahler-Ricci flow, respectively.

THEOREM 2.50 (NKRF: C1 0 global existence). Let (Mn,go) be a closed Kahler manifold with

(1) either C1 (M) < 0, C1 (M) = 0, or C1 (M) > 0, and (2) r;;- [wo] = 21l" C1 (M). Then there exists a unique solution g(t) of the normalized Kahler-Ricci flow defined for all t E [0,00) with g(O) = go. When the first Chern class is nonpositive, Cao proved that the normalized Kahler-Ricci flow converges to a Kahler-Einstein metric in the same

Kahler class as the initial metric. THEOREM 2.51 (KRF: C1 SO convergence). Let 9 (t) be a solution of the normalized Kahler-Ricci flow, as in Theorem 2.50, with ct (M) S o. Then 9 (t) converges exponentially fast in every Ck-norm to the unique KahlerEinstein metric goo in the Kahler class [wo]. Note that the initial-value problem for the normalized Kahler-Ricci flow equation

o

r

otga{J = -Ra{J + ;,ga{J' ga{J (0) = g~{J' is equivalent to the following parabolic Monge-Ampere equation for the metric potential function '0 ~ Y, we have PROOF.

(2.71)

( -a - uA) 1og Y < at

Roii"/'Y (0)

-

Y

>'0 + -r < C 1 ~ L.." -1 + -r >."/ n ,,/=1 >."/ n'

where Cl is a constant depending only on a lower bound of the bisectional curvatures of g(O). To control the bad terms on the RHS above, we consider equation (2.54) . (2.72)

(ata) - cp = - f A

Acp = -

1 L ~' n

f - n+

,,/=1

"/

where the second equality follows from (2.60). Consider the modified quantity w = logY - (C1 + l)cp. Combining (2.71) and (2.72), we have (2.73)

(ata) w 0, then C 2 depends on time). By the maximum principle, (2.73) implies that w can be bounded above on M x [0, T) by a constant C depending on T.

88

2.

KAHLER-RICCI FLOW

To get a bound for w independent of T when r ~ 0, we need to work harder. When r ~ 0, we shall use the term - E"Y >..\ to dominate (from below) a function of w which approaches -00 as w ~ 00. Using equation (2.50), f = - %tc.p, and (2.61), we have

f (x, t) - f(x, 0)

det 9cr[J(X, t) (0) , et 9cr[J x,

r

+ -c.p(x, t) = -log d n

and hence

y e!(x,t)- !(x,O)+~ 0 such that the Kahler metric gaiJ g~iJ + cp aj3 satisfies

~

(2.78) By the fact that the bounded function f satisfies t:::.f = R - r and that we have the scalar curvature bound (2.47), standard Lq theory (which only requires the uniform boundedness of the coefficient matrix (gaiJ) from above and below) implies that Ilfllco(M) + IIVV flILq(M) is bounded and hence by (2.77), Ilhllco(M) + IIVVhIILq(M) is bounded (uniformly in t when Cl (M) ::; 0) for any q < ;:0 (independent of t). Note that h is globally defined whereas h is only locally defined. However, for compactly contained open subsets U of a holomorphic coordinate chart of go, Ilhllco(u)+ IIVVhIlLq(u) and Ilhllco(u) + IIVVhIILq(U) are equivalent. Let B(R) denote the Euclidean ball of radius R centered at the origin in en. Since M is compact, there exists a finite collection of open sets {Uk}~~l and normal holomorphic coordinates Zk = {zn:=l defined on Uk (independent of t) such that No

B (3RkO) C Zk (Uk)

and

U z;;l (B (RkO)) = M, k=l

where RkO > O. Hence it suffices to prove the C2,a- estimate for cp in each open set z;;l (B (RkO)) assuming that (2.79)

Ilhllco(uk)

+ IIVVhIlLq(uk)

::; C

< 00.

From now on we work in a fixed coordinate chart. More precisely, we use Zk to push forward our discussion to the Euclidean ball B (3RkO)' For

90

2. KAHLER-RICCI FLOW

simplicity we drop the indices k in our notation below. Since 9 is Kahler, we may write 9Ct /3 locally as the complex (Hermitian) Hessian u Ct /3 of a function u. We shall show that the second derivative of u has bounded Holder norm. Now the equation reads locally as log det (u Ct /3) = h.

(2.80)

It is convenient to write log det as a function F(p), where p lies in the domain of positive definite Hermitian symmetric matrices. An important property that we shall make full use of is that F(p) = log det(p) is a concave function of p, a fact which can be easily checked. Taking the derivative of (2.80), we have

8F

- 8- uCt/3-y = h-y, PCt{3 82F

8

-8 - UCt/3-yUJ.Lii,,(

8F

+ -PCt(3 8_uCt/3-y"( = h-y"(

PCt{3 PJ.LV for each 'Y = 1, ... ,n. By the concavity of F we have (2.81)

On the other hand recall that 8F

--=U

a{3-

8Pa/3

=9

a{3-

,

where (uCt/3) is the inverse of (u a/3) = (9a/3).5 Therefore we can rewrite (2.81) as ~uw ~

(2.82)

h-y"(,

where

u-y"( and ~u denotes the Laplacian with respect to the metric 9Ct /3' For any R ~ Ro, let w

(2.83)

M(s)

~

= sup wand

m(s)

=

B(sR)

inf w. B(sR)

Also define the oscillation function:

w(sR)

~

M(s) - m(s).

The following weak form of the Harnack inequality, which holds in general for linear elliptic operators of divergence form, plays a crucial role in our estimate. One can find the proof of this result in various papers and 5T his is not different from the real version of this formula used in Volume One, which is the variation formula -logdetA = A -l)ij & Q &AQ

a

for any invertible matrix A ij .

(

a

5.

EXISTENCE AND CONVERGENCE

91

books on PDE, such as Moser [276], Morrey [274], Gilbarg and Trudinger [155], Han and Lin [195] (e.g., see Theorem 4.15 on p. 83 of [195]). THEOREM 2.61 (Harnack inequality). Let function such that

U

:

B(3Ro)

---+

lR be a C 2

1

Ao (tSa.a) ~ (ua.a) ~ Ao (tSa.a) forsomeA o E [1,00). Suppose that a nonnegative function v E W 2,2(B(3Ro)) and 9 E LQ(B(3Ro)), for some q > m/2, satisfy ~uv ~

9

in the weak sense in B(3Ro), where ~u denotes the Laplacian with respect to the metric ua.a. Then for any 0 < (j ~ T < 1 and 0 < p < m~2' there exists a constant C = C(p, q, 2n, Ao, (j, T) < 00 such that for any p ~ 2Ro, 1

(2.84)

(~r v (y)P dY) P ~ C ( inf v + i-rg-lIgIlLq(B(P») . pm JB(7'p) B(Op)

REMARK 2.62. The reason for why we can apply this theorem to (ua.a) = (ga.a) =

(g~.a+CPa.a)

is that the C 2 -estimate for cP yields (2.78). Note that

~u

(M(2) - w)

~

-h"(i and -h"(i E Lq for all q

~

00 (e.g.,

< 00), so we may apply the above theorem to M(2) -w (for example, with m = 2n, p = 2R,6 and (j = T = ~, so that (jp = Tp = R) to obtain for any q> nand 0 < p < that there exists C = C(p, q, n, A) < 00 such that q

n:1

1

r (M(2) - w (Y))PdY) P (R~ JB(R) n

~ C (M(2) -

(2.85)

M(1)

+ R 2 (q;n) Ilh"(iIILq(B(2R»)

since infB(R) (-w) = M(1). On the other hand, the concavity of F implies

F(Ui](X)) ~ F(Ui](Y))

8F

+ 8p .., (Ui] (y)) (Ui](X) -

Ui](Y))'

~J

Namely we have (2.86) The following linear algebra fact enables us to estimate w(R). 6Note that

R::; Ro.

92

2.

KAHLER-RICCI FLOW

LEMMA 2.63 (Linear algebra). There exist unitary vectors r}, . .. , rN E en with the property that, as Hermitian symmetric positive definite matrices, N

(giJ(y)) = Lav(y)rv®rv .

(2.87)

v=1

Here av(y) E lR and lA ~ av(y) ~ AA for some constant A > O. Moreover we may assume that the first n vectors r}, ... , rn form a unitary basis of en . REMARK 2.64. Let {ei}~=1 denote the standard basis for en and write rv ~ l:~=1 (rv)i ei for each 1/. By (2.87) we mean that N

(gil(y)) = L av(y) (rv)i (rv)1. v=1

EXERCISE 2.65. Prove the above linear algebra lemma. Define

Wv ~ Hess(u) (rv, rv) = uiJ(rvMrvk Now we let Mv(s) and mv(s) denote the quantities M(s) and m(s) defined by (2.83) using Wv instead of w. Then (2.86) implies N

L av(y)(wv(y) - wv(x)) ~ h(y) - h(x).

(2.88)

v=1

Choosing x E B(2R) to be a point where wdx) implies

= ml(2), this in particular

al(y)(wl(y) - Wl(X)) ~ h(y) - h(x) + L av(Y)(wv(x) - wv(y)) v~2

and hence

WI(y) - ml(2)

~ C(A, A) (RIIV'hllco + L(Mv(2) - wv(Y))) , v~2

where we used al(y) ~ Thus

lA' and av(y) ~ AA and wv(x) ~ Mv(2) for 1

{ (WI (Y) - ml(2))PdY) p (R; n iB(R)

~ C(A, A)RIIV'hllco 1

(2.89)

+ C(A, A) L (R;n v~2

( iB(R)

(Mv(2) - wv(y))PdY) P

1/

~ 2.

93

5. EXISTENCE AND CONVERGENCE

On the other hand, applying (2.85) to bound the V-norm of Mv(2) - WV , we have for each /J 2: 2, 1

r

(R; n JB(R) (Mv(2) - Wv (Y))PdY) (2.90)

:::; C ( Mv(2) - Mv(1)

P

+ R 2(q;n) II Hess(h) (rv, rv )IILq(B(2R))) .

Combining (2.89) and (2.90), we have

(2.91)

r (WI (y) - m1(2))PdY) (R; JB(R)

1

P

n

:::; C

(max (Mv(2) - Mv(l)) + RllVhllco + R 2(q;n) IIVVhIILq) . v:=::2

Now let

w(sR)

~ twv(SR) ~ tv=l (sup Wv v=l B(sR)

inf wv)

B(sR)

N

= L:(Mv(s) -mv(s)).

v=l We then have 1

r (WI(Y) - ml(2))p) P (R;n JB(R) (2.92)

:::; C (W(2R) - w(R)

+ RllVhllco + R 2(q;n) IIVVhIlLq)

On the other hand, from (2.85) we also have 1

r

(R; n JB(R) (Ml(2) - WI (y))p) P (2.93)

:::; C (W(2R) - w(R)

+ R 2(q;n) IIVVhIlLq)

.

.

94

2. KAHLER-RICCI FLOW

Putting these together, we obtain 1

wl(2R) = (

s; (

V01~(R) LIR) (Ml(2) - ml(2))p)'

V01~(R) LIR/W1 (Y) - ml(2))p

r 1

1

+ ( V01~(R)

LIR) (Ml(2) - (Y))P) , WI

~ C (W(2R) -

w(R)

+ RIIVhlico + R 2(q;n) IIVVhIILq) .

Since there is nothing special about the index 1, summing the corresponding upper bounds for w/I(2R) implies w(2R)

~ C (W(2R) -

with a different constant C

w(R)

+ RllVhllco + R 2 (q;n) IIVVhIILq)

< 00. We conclude the following.

LEMMA 2.66 (Oscillation estimate). There exists 6 < 1 (i. e., 6 = 1 such that lor any R ~ Ro we have on B(3Ro), (2.94)

w(R) ~ 6· w(2R) 2(q-n)

+ RllVhllco + R

2(q-n)

q

b)

_

IIVVhIILq.

_

Now since R q IIVVhIlLq(U) and IIVhllco(u) are bounded by (2.79), the Holder continuity of VVu on B(Ro) can be derived from (2.94) by a standard argument; see Moser [276], or Corollary 4.18 on p. 91 and Lemma 4.19 on p. 92 of Han and Lin [195], for example. Finally, the Holder continuity of VVu is equivalent to the Holder continuity of VV'P' 5.4. Proof of Theorem 2.51. Finally, we give the proof of Theorem 2.51, i.e., the proof of the convergence of the normalized Kahler-Ricci flow in the case where Cl < O. Assume, without loss of generality, that r = -n, which can be achieved by scaling the initial metric go. Notice that we have that I = - ~ satisfies

a

atl = b.g(t)1 - I and I/(x, t)1 ~ Ce- t and (2.48).) That is,

IVII (x, t)

~

Ce- t . (The last inequality is by

for some Cl < 00. 2 Now b.g(t) = gCt/3 az~az{3 and gCt{J = g~{J

+ 'PCt{J'

So the C2,Ct-estimate for

'P implies a CCt-estimate for the coefficients gCt{J. Thus we may apply the

6.

SURVEY OF SOME RESULTS FOR THE KAHLER-RICCI FLOW

95

parabolic Schauder estimate (e.g., Theorem 5 on p. 64 of Friedman [146]) to obtain Ilfll c 2 ,(M) ~ C2e- t for some C 2 < 00. Iterating the Schauder estimate, we have IlfII C 2m '(M) ~ Cme- t for some constants Cm < 00 and all mEN. This implies the estimate ~ C2 e- t and hence implies the exponential convergence of I ~ II C 2m ,(M)

'P (', t)

---t 'Poo (.) in Coo as t ---t 00 for some smooth function 'Poo. This proves that the normalized Kahler-Ricci flow converges in Coo to a Kahler-Einstein metric with negative scalar curvature. Theorem 2.51 is proved.

6. Survey of some results for the Kahler-Ricci How 6.1. Closed Kahler manifolds with nonnegative bisectional curvature. Using the short-time existence of the Kahler-Ricci flow, the result of Mori, Siu and Yau (Theorem 2.33) was generalized by Bando [19] when n = 3 and Mok [269] for n ~ 4.7 Mok also used techniques from algebraic geometry. THEOREM 2.67 (Kahler manifolds with nonnegative bisectional curvature). If (Mn,g) is a closed Kahler manifold with nonnegative bisectional

g)

curvature, then its universal cover (Mn, is isometrically biholomorphic to the product of complex Euclidean space, compact irreducible Hermitian symmetric spaces of rank at least 2, and complex projective spaces with Kahler metrics of nonnegative bisectional curvature. REMARK 2.68. Note that the above classification is up to isometry. Any complex projective space admits a metric with constant holomorphic sectional curvature (Le., the Fubini-Study metric). The proof of the theorem above uses the following result, proved by Bando for n = 3 and Mok for n > 4. We discuss this result further in Section 8 below. THEOREM 2.69 (KRF: nonnegative bisectional curvature is preserved). If(Mn,g(O)) is a closed Kahler manifold with nonnegative bisectional curvature, then the solution g (t) to the Kahler-Ricci flow has nonnegative bisectional curvature for all t ~ O. If in addition g (0) has positive Ricci curvature at one point, then g (t) has positive holomorphic sectional curvature and positive Ricci curvature for all t > O. Using the existence of a Kahler-Einstein metric, the convergence in the case of positive bisectional curvature was settled by Chen and Tian [87],

[88]. 7The case of nonnegative curvature operator was considered by eao and one of the authors [50].

96

2.

KAHLER-RICCI FLOW

THEOREM 2.70 (KRF: compact positive bisectional curvature). Suppose (Mn, 9 (0)) is a closed Kahler manifold with nonnegative bisectional curvature everywhere and positive bisectional curvature at a point. Then the solution 9 (t) to the normalized Kahler-Ricci flow, which has positive bisectional curvature for all t > 0, converges exponentially fast to the Fubini-Study metric of constant holomorphic sectional curvature on cpn. REMARK 2.71. Without using the existence of a Kahler-Einstein metric, Cao, Chen, and Zhu [49J proved a uniform curvature estimate (see Theorem 2.92). 6.2. Uniformization of noncom pact Kahler manifolds with nonnegative bisectional curvature. In this subsection we recall Yau's fundamental conjecture on the uniformization of complete noncompact Kahler manifolds with nonnegative bisectional curvature. CONJECTURE 2.72 (Noncompact Kahler uniformization Kc (V, W) > 0). If (Mn, g (0)) is a complete noncompact Kahler manifold with positive bisectional curvature, then M is biholomorphic to C n . Using the Kahler-Ricci flow on noncompact manifolds, Chau and Tam [65J proved the following result, which affirms Yau's conjecture in the case of bounded curvature and maximum volume growth. THEOREM 2.73 (KRF: noncompact positive bisectional curvature). If n (M , g (0)) is a complete noncompact Kahler manifold with bounded positive bisectional curvature and maximum volume growth, then M is biholomorphic to cn. There have been a number of works on the Kahler-Ricci flow on noncompact manifolds with positive bisectional curvature. For example, the reader may consult Shi [331], Tam and one of the authors [290], [292J, and Chen and Zhu [79], [80J. 6.3. Limiting behavior of the Kahler-Ricci flow on closed manifolds. There are also the following results about the limiting behavior of the Kahler-Ricci flow due to Sesum [323]. THEOREM 2.74. If (Mn, 9 (t)) , t E [0,00), is a solution to the KahlerRicci flow on a closed manifold with uniformly bounded Ricci curvature, then for any sequence ti -+ 00 there exists a subsequence such that (M, g (t + ti)) converges to (M~,goo (t)), where goo (t) is a solution to the Kahler-Ricci flow. The convergence is outside a set of real codimension 4. When n = 2, Sesum improved the above result to the following. THEOREM 2.75. If (M 2 ,g(t)) , t E [0,00), is a solution to the KahlerRicci flow on a closed manifold with uniformly bounded Ricci curvature, then for any sequence ti -+ 00 there exists a subsequence such that (M, 9 (t + ti)) converges to (M ~, goo (t)) , where goo (t) is a K ahler-Ricci soliton. The convergence is outside a finite number of points.

7. EXAMPLES OF KAHLER-RICCI SOLITONS

97

For some other recent work on the Kahler-Ricci flow the reader is referred to Phong-Sturm [305], [306], Chen [84J, Chen and Li [85], Song and Tian [337], Cascini and La Nave [60]' Tian and Zhang [351]' and [234J. 7. Examples of Kahler-Ricci solitons In this section, we provide a brief and regrettably incomplete sampling of some results on Kahler-Ricci solitons. These special solutions model singularities of the Kahler-Ricci flow. A Kahler-Ricci soliton is a Kahler manifold (Mn, g, J) such that the soliton structure equation

(2.95)

1

Rc+,\g + 2LXg

=

a

holds for some constant ,\ E IR and some real vector field X which is an infinitesimal automorphism (2.1) of the complex structure J. Note that X is an infinitesimal automorphism if and only if its (1, a)-part is holomorphic: a = '\7 o.X~ = a~'" Xf3. One imposes this requirement for the following reason. As we saw in Lemma 2.36, a solution of Ricci flow that starts with a Kahler metric on a complex manifold remains Kahler with respect to the same complex structure. On the other hand, if 'Pt is any family of diffeomorphisms of M, then each pullback 'Pt (g) is Kahler with respect to the complex structure 'Pt(J). Now consider the evolving metric h(t) := (1 + ,\t)'Ptg, where 'Pt is the family of diffeomorphisms generated by 2(1~At) X. If X is an infinitesimal automorphism of the complex structure, then 'Pt (J) = J, which implies that h( t) remains Kahler with respect to the same complex structure. Furthermore, using (2.95), it is easy to see that h solves the Kahler-Ricci flow:

gth = - Rc(h). One may also define a Kahler-Ricci soliton to be a Kahler manifold (M n , g, J) together with a constant ,\ E IR and a real vector field X satisfying the complex soliton equation (2.96) Equation (2.96) is equivalent to the conjunction of equation (2.95) and the statement that X is holomorphic. Notice that if we restrict our attention to gradient solitons (so that X is the gradient of a real-valued function), then (2.96) is equivalent to (2.95) without any extra hypotheses. (See §2.2 of [142J for the detailed argument.) 7.1. Existence and uniqueness. Any Kahler metric satisfying (2.96) with X = a is Kahler-Einstein. In this sense, Kahler-Einstein metrics may be regarded as trivial Kahler-Ricci solitons. 8 So if no Kahler-Einstein metric exists, a natural replacement is a Kahler-Ricci soliton. In fact, existence of a Kahler-Einstein metric and a nontrivial gradient Kahler-Ricci soliton 80f course, there is nothing 'trivial' about Kii.hl.er-Einstein metrics!

98

2.

KAHLER-RICCI FLOW

are mutually exclusive: applying the Futaki functional F[wJ to the holomorphic vector field X = grad f, one gets

Moreover, Kahler-Ricci solitons on a compact Kahler manifold (Mn, J) are unique up to holomorphic automorphisms. (See Tian and Zhu [352, 353, 354J.) Specifically, we have the following theorem. THEOREM 2.76 (Uniqueness of Kahler-Ricci solitons). Let (Mn, J) be a compact Kahler manifold. If metrics 9 and 9' on M satisfy (2.95) with respect to holomorphic vector fields X and X', respectively, then there is an element CT in the identity component of the holomorphic automorphism group such that 9 = CT*g' and X = (CT-I)*X'. For recent results on uniqueness and other properties of noncom pact Kahler-Ricci solitons, see [63, 64J, [35J and [78J. One might ask, therefore, whether there exists either a Kahler-Einstein metric or else a Kahler-Ricci soliton on every compact Kahler manifold Mn with CI (M) > O. The answer is yes if n ~ 2. A compact complex surface with CI > 0 is p2#k p2 for some k E {O, 1, ... , 8}. (Here and below, pn = cpn is complex projective space.) A Kahler-Einstein metric exists for k = 0 and 3 ~ k ~ 8. (See Theorem 2.34.) In the remaining cases k = 1,2, there is a (non-Einstein) Kahler-Ricci soliton. (See [239], [366], [47], and Section 7.2 below.) In higher dimensions, however, the answer is no. There exist 3-dimensional compact complex manifolds that admit no Kahler-Einstein metric and no holomorphic vector fields, hence no KahlerRicci soliton structure. (See [346, §7J as well as [215, 216J and [278J.) More generally, Tian and Zhu have exhibited a holomorphic invariant that generalizes the Futaki invariant and acts as an obstruction to the existence of a Kahler-Ricci soliton metric [354J on a compact complex manifold (Mn, J). 7.2. The Koiso solitons. As was noted in Proposition 1.13 or Proposition A.32, all compact steady or expanding solitons are Einstein. This is not true for shrinking solitons. The first examples of nontrivial (Le. nonEinstein) compact shrinking solitons were discovered by Koiso [239J and independently by Cao [47J. These are Kahler metrics on certain k-twisted projective-line bundles pI ~ Fk -* pn-I first described by Calabi [44J. We will discuss their construction in considerable detail, because it serves as a prototype for later examples. We begin with Calabi's bundle construction. pn-I is covered by n charts (CPa: Ua ~ cn-I), whereUa = {[XI, ... ,XnJ E pn-I: Xa =F O} Xa-l Xa±l ~) • (lXT . an d CPa.. [Xl"",Xn J f---t (~ , ••• , - - , , ••• , vve wn't e Xa InXa: Xo; Xo: Xo. a stead of x here and in the next paragraph in order to simplify some formulas below.) In particular, one may define complex projective space by

7. EXAMPLES OF KAHLER-RICCI SOLITONS

99

lpm-l = (Il~=ICPo(Uo))/~, where, for example,

CPI (UI ) :3 (Zl, ... , Zn-l )

~

1 Z2 Zn-l) ( -, -, ... , - E CP2 (U2). Zl Zl Zl

Given kEN, we formally identify pI = e u {oo} and define the k-twisted bundle :Fk = (Il~=l (Uo X pI)) / "', where Uo x pI :3 ([Xl, .. . , Xnl;~) '" ([YI, .. . , Ynl; 1]) E U(3 X pI if and only if [Xl' ... ' xnl = [YI, ... , Ynl and 1] = (~)k~ for each Q. Equivalently, one may define where, for example, 1 Z2 Zn-l k I (-, -, ... , - ; Zl () E CP2(U2) x p . Zl Zl Zl Notice that 80 = {[Xl, ... ,xnl; o} and 8 00 = {[Xl, ... , xnl; oo} are two global sections of Fk. The key to constructing Kahler-Ricci solitons on F'J: (as well as examples on other topologies to be considered below) will be to find a Kahler potential on en \ {o} satisfying certain symmetries and boundary conditions. To see = Fk\(80 U 8 00 ) and define 'IjJ : en\{O} - t so that why this is so, let

CPI(UI ) x p

I

:3 (Zl, ... , Zn-l;

()

~

Fr:

Fr:

'IjJ: (XI, ... ,Xn ) f---+ ([xI, ... ,xnl;x~) if Xo =1= O. It is easy to see that ([Xl, ... , xnl; X~) '" ([Xl, ... , xnl; x~) whenever Xo =1= 0 and x(3 =1= 0, hence that 'IjJ is well defined. The map 'IjJ is clearly surjective. If 'IjJ(XI, ... ,xn) = 'IjJ(YI, ... ,Yn), where, say Xo =1= 0 and Y(3 =1= 0, then

(Xl, ... , Xa-l , Xa+l , ... , Xn; X~) ~ (YI , ... , Y(3-1 , Y(3+1 , ... , Yn; Y~). Xa Xa Xa Xa Y(3 Y(3 Y(3 Y(3 The equivalence relation ~ then implies that Y~ = x~, hence that Y(3 = (}x(3 for some k-th root of unity (). Because [Xl, ... , xnl = [YI, ... , Yn], it follows that Y'Y = (}x'Y for all T = 1, ... , n, hence that 'IjJ is a k-to-one map. Therefore, a Kahler potential P on e n \ {O} will induce a well-defined Kahler metric on provided that 8ap((}xI, . .. , (}x n ) = 8ap(XI, ... , xn). With these considerations in mind, our method will be to construct a suitable Kahler potential P : en \ {O} - t IR whose asymptotics as Izi - t 0 and Izi - t 00 ensure that the induced metric extends smoothly to 8 0 and 8 00 • If we are interested in shrinking solitons, what properties should P possess? Let's suppose that P determines a Kahler metric g. As above, we take the Kahler and Ricci forms to be w = Agai3 dz o 1\ dz(3 and p = A Rai3 dz a 1\ dz(3, respectively. Then (locally) we have

Fr:

82

g-a(3 - 8za8z(3 P

2. KAHLER-RICCI FLOW

100

and

82

= - 8z Ci 8z!3 log det g,

RCi[J

exactly as in (2.6). If Q denotes the soliton potential function with gradient vector field X, then equation (2.96) reduces to

82

8zCi 8z!3 (log det 9 - Q - AP) = O. Because we are interested in shrinking solitons, we may assume that A < O. Then (modifying the Kahler potential P by an element in the kernel of V'V if necessary) we may assume that Q = log det 9 - AP. This will give us a soliton provided X = grad Q is holomorphic, that is, provided that

o = ~X!3 =~ ( !3'Y~Q) . 8zCi 8zCi 9 8z'Y Substituting Q = log det 9 - AP, we obtain a single fourth-order equation for the scalar function P, namely (2.97)

8~Ci

[gf3'Y

8~'Y (log det 9 -

AP)]

= O.

To proceed, we adopt the Ansatz that the potential P is invariant under the natural U(n) action in the sense that it is a function of r = log I:~=1 IzCi l2 alone. In this case, setting cP = Pr, we have (2.98)

gCi[J

= e- rcp8Ci!3 + e- 2r (CPr - cp)ZCi Z!3,

so our P will be a Kahler potential if and only if cP and CPr are everywhere positive. Now (following [47] and [142]) we can write (2.97) as the fourthorder ODE n n n Prrrr - 2P;rr T + nrrrr - ( n - 1) P;r p2 + 1\'( rrrrrr - p2rr ) = O. rr r We shall see that only two of the four arbitrary constants in its solution are geometrically significant. Q = ~ZCi will be holomorTo simplify (2.99), notice that XCi = gCi[J~~ z rrr phic if and only if Qr = p,Prr for some p, E lR. Since 9 will be Kahler-Einstein if X = 0, we may assume that p, i= O. Substituting Q = log det 9 - AP, one then obtains

( ) 2.99

(logCPr)r + (n -l)(logCP)r - P,CPr - Acp - n

= 0,

which is a second-order equation for cP, hence a third-order equation for P. (This integration can also be accomplished by standard ODE techniques.) Because CPr > 0 everywhere, one may regard r as a function of cP and hence F(cp). One finds (remarkably) that F satisfies a linear may write CPr equation

n-l

F' + (-cP

-

p,)F - (n+ Acp)

= 0,

7.

EXAMPLES OF KAHLER-RICCI SOLITONS

101

whose solution is

r.pr = r.pl-neJ.l.'P(1/ + >..In + nIn-l), where

1/

is another arbitrary constant and In

r.pr = I/r.p

I-n

= J r.pne-J.l.'P dr.p. This leads to

J.I.'P >.. >.. + /1. ~ n! j"J+1-n e - -r.p - l+n L..J -., /1. 'I-' , /1.

/1.

j=O

J.

which is a separable first-order equation for r.p. The third and fourth arbitrary constants arise when one finds the implicit solution r( r.p) and then integrates r.p to obtain P. These are geometrically insignificant, because they disappear in (2.97); the two geometric degrees of freedom are the parameters /1. i= 0 and 1/. In summary, what we have thus far accomplished is to construct a potential function for a U(n)-invariant Kahler-Ricci soliton metric (2.98) on Cn\{O}, hence a possibly incomplete Kahler-Ricci soliton metric on What remains is to choose /1. and 1/ in order to get a complete metric on FJ:. Our choices of the two parameters will be determined by the two boundary conditions as r -4 ±oo. Since r.pr > 0, we may define a < b E [0,00] by a = limr---+_oor.p(r) and b = limr-+oo r.p(r). If a > 0, one may write P(r) = ar + p(e Qr ) in a neighborhood of Izl = 0, with p smooth at zero, p(O) = 0, p'(O) > O. Similarly, if b < 00, one may write P(r) = br + q(e-f3r ) in a neighborhood of Izl = 00, with q smooth at zero, q(O) = 0, q'(O) > O. One then takes advantage of the following observation.

Jr.

LEMMA 2.77 (Calabi). Assume that k > 0 and a, bE (0,00). (1) When a = k, the potential P(r) = ar + p(e Qr ) induces a smooth Kahler metric on a neighborhood of So in FJ:. Any pI in So has area a7r. (2) When (3 = k, the potential P(r) = br + q(e- f3r ) induces a smooth Kahler metric on a neighborhood of Soo in FJ:. Any pI in Soo has area b7r. For the proof, see [44] or [142, Lemma 4.2]. With more work, one finds that it is possible to satisfy both boundary conditions by appropriate choices of /1. and 1/. (See [47] or adapt the arguments in [142, §4.1].) Normalizing by fixing>.. = -1, these choices yield a = n - k and b = n + k. Since one needs a > 0, one obtains a unique gradient shrinking Kahler-Ricci soliton on FJ: for each k = 1, ... , n - 1. These are the Koiso solitons. 7.3. Other U(n)-invariant solitons. The construction we have described in Section 7.2 above has natural generalizations allowing the discovery of other explicit Kahler-Ricci soliton examples. Namely, one searches the 2-dimensional (/1., 1/) parameter space of U (n)- invariant Kahler potentials

102

2. KAHLER-RICCI FLOW

P(r) on en\{O} for those whose behavior at the boundary izl = 0 implies that 1_: the metric is completed by adding a smooth point at r = -00; 2_: the metric is completed by adding an orbifold point at r = -00; 3_: the metric is completed by adding a Ipm-l at r = -00; or 4_: the metric is complete as r - t -00; and whose behavior at the boundary Izl = 00 implies that 1+: the metric is completed by adding a smooth point at r = +00; 2+: the metric is completed by adding an orbifold point at r = +00; 3+: the metric is completed by adding a pn-l at r = +00; or 4+: the metric is complete as r - t +00. Of course, not all combinations of these alternatives are globally compatible. For example, it is easy to see that the growth condition cpr > 0 prohibits completing the metric by adding a Ipm-l at Izl = 0 and a smooth point at Izl = 00. Nonetheless, this has been a productive line of research. In the remainder of this section, we will survey some of its results. It is possible to add a smooth point at Izl = 0 and to construct a unique steady Kahler-Ricci soliton on en that is complete as Izl - t 00. In complex dimension n = 1, this is just the cigar soliton discovered by Hamilton and discussed in Chapter 2 of Volume One; the examples in higher dimensions are due to Cao [47]. These solitons have the following asymptotic behavior: in the sphere s2n-1 at metric distance r » 0 from Izl = 0, the Hopf fibers U(1) . z have diameter 0(1), while the pn-l direction has diameter O(Vr). Accordingly, one calls this cigar-paraboloid behavior. It is also possible to add a smooth point at Izl = 0 and to construct expanding Kahler-Ricci solitons on en that are complete as Izl - t 00. There is in fact a 1-parameter family (en,go)o>o of such examples in each dimension, due to Cao [48]. (It is a heuristic principle that expanding solitons are easier to find than their shrinking cousins. For these examples, satisfying the boundary condition at Izl = 0 reduces the parameter space by one dimension, but completion as Izl - t 00 comes for free.) Each soliton (en, go) is asymptotic as Iz I - t 00 to the Kahler cone (en \ {O}, go), where the metric go is induced by the Kahler potential P(r) = eOr jB. For each k = 2,3, ... , the authors of [142] add an orbifold point at Izl = 0 and a Ipm-l at Izl = 00 to construct a unique shrinking Kahler-Ricci soliton on an orbifold, which is called Yk' The compact orbifold Yk may be regarded as Ipm jZk branched over the origin and the pn-l at infinity. The orbifold singularity at the origin is modeled on en jZk. For each dimension n ~ 2 and k = 1, ... , n -1, the authors of [142] add a k-twisted pn-l at Izl = 0 and construct a unique shrinking Kahler-Ricci soliton metric that is complete as Izl - t 00. The resulting soliton has the topology of the complex line bundle e ~ L"!:..k - pn-l characterized by (CI' [E]) = -k, where CI is the first Chern class of the bundle and E ~ pI is a positively-oriented generator of H2(pn-l; Z). (For example, the total

8.

KAHLER-RICCI FLOW WITH NONNEGATIVE BISECTIONAL CURVATURE

103

space of L"!:..I is simply Cn blown up at the origin.) As Izl --t 00, the soliton metric is asymptotic to a Kahler cone (C n\{0},90)/Zk, where 0 = O(n,k). For each dimension n 2: 2, Cao [47] adds an n-twisted IF-I at Izl = 0 and constructs a unique complete steady Kahler-Ricci soliton on the total space of the bundle C '---t L"!:..n - pn-I. The metric exhibits cigar-paraboloid behavior at infinity. For each dimension n 2: 2 and k = n + 1, ... , the authors of [142] add a pn-I at Izl = 0 and construct a I-parameter family of complete expanding Kahler-Ricci solitons on C '---t L"!:..k - IF-I. The solutions are parameterized by 0 > 0, where (cn\{0},90)/Zk is the asymptotic Kahler cone at infinity.

8. Kahler-Ricci flow with nonnegative bisectional curvature The study of the Kahler-Ricci flow of Kahler metrics with nonnegative bisectional curvature is somewhat analogous to the study of the Riemannian Ricci flow of metrics with nonnegative curvature operator. One aim is to uniformize Kahler metrics with nonnegative bisectional curvature in both the compact and noncompact setting. In particular, one would like to flow such metrics to canonical metrics, or to infer the existence of canonical metrics from the long-time behavior of the flow. One would also like to deduce properties of the underlying complex structure of the Kahler manifold, and when possible, classify the manifold up to biholomorphism.

8.1. Nonnegative bisectional curvature is preserved. Consider the Kahler-Ricci flow 9a-'!3 = -RafJ. We shall prove that the Kahler-Ricci flow preserves the nonnegativity of the bisectional curvature. As in the real case, the key is Hamilton's weak maximum principle for tensors. This result was proved first by Bando for n S 3 and by Mok in any dimension. (See Theorem 2.69 above.) The result was also extended to the complete noncompact case by W.-X. Shi under the additional assumption of the bisectional curvature being bounded. We say that a Kahler metric has quasi-positive Ricci curvature if the Ricci curvature is nonnegative everywhere and positive at some point.

gt

THEOREM 2.78 (Nonnegative bisectional curvature preserved). The nonnegativity of the bisectional curvature is preserved under the Kahler-Ricci flow on closed Kahler manifolds. Moreover, if the initial metric also has quasi-positive Ricci curvature, then both the Ricci curvature and the holomorphic sectional curvature are positive for metrics at positive time. The basic computation in the proof of the above result is the following.

104

2.

KAHLER-RICCI FLOW

PROPOSITION 2.79 (Evolution equation for the curvature). Under the Kahler-Ricci flow, (2.100)

(:t - ~)

Ro.iJrJ

= Ro.jiIlJ R J.1.i3'""(V -

Ro.ji'""(vR J.1.i3I1J

+ Ro.i3l1ji R J.1.v'""(J

1

- "2 (Ro.ji R J.1.i3'""(J + RJ.1.i3 R o.ji'""(J + ~jiRo.i3J.1.J + RJ.1.J R o.i3'""(ji) . REMARK 2.80. The Riemannian analogue of this formula is given by Lemma 6.15 on p. 179 of [108].

In the proof of the proposition we find it convenient to use a formula relating ordinary derivatives and covariant derivatives at the center of normal holomorphic coordinates. LEMMA 2.81 (Relation between ordinary and covariant derivatives). If 'fJ is a closed (1, I)-form, then, at the center of normal holomorphic coordi-

nates, we have

(2.101)

{p

\7 i3 \7 0. 'fJ'""(J

=

{)zo.{)z{3 'fJ'""(J + 'fJ)..J R o.i3'""()'" {)2

(2.102)

\7 0. \7 i3'fJ'""(J

=

{)zo.{)z{3 'fJ'""(J + 'fJ'""().,Ro.i3)..J.

PROOF. We compute that at the center of normal holomorphic coord inates, \7i3 \7 0. 'fJ'""(J = {)i3 \1 0. 'fJ'""(J - r~o \7 0. 'fJ,""(e

= {)i3 ({)o.'fJ'""(J - r;'""('fJgJ) = {)i3{)o.'fJ'""(J - {)i3 r ;'""('fJgJ {)2

{)zo.{)z{3 'fJ'""(J

+ R~i3'""('fJgJ'

where we used (2.4) in the last line; this proves (2.101). Note that (2.102) 0 is just the conjugate of (2.101). Now we give the PROOF OF PROPOSITION 2.79. We compute the evolution equation for Ro.i3'""(J at any point x and time t using normal holomorphic coordinates {zo.} centered at x with respect to 9 (t) . In such coordinates, a;;~ (x, t) = Recall from (2.5) that R ____ {)2go.i3 o.{3'""(O {) z'""( {)ZO

+ 9

pu {)go.u {)gpi3 {)z'""( {) ZO .

o.

8. KAHLER-RICCI FLOW WITH NONNEGATIVE BISECTIONAL CURVATURE 105

This implies

8

8 (8) 8tga~

= 'V"I 'V &Ra~ -

(2.103)

8 2

2

8t Raih & = - 8z"l8z0

=

8Z"l8zoRa~

Ra>..R"I&>"~'

where we used (2.102). Since (2.103) is tensorial, it holds in any holomorphic coordinate system. We wish to compare the above formula with t:.Ra~"I& ~

1

2" ('V /1- 'V jl + 'V jl 'V /1-) Ra~"I&'

To this end we compute (apply the second Bianchi identity (2.9) and commute covariant derivatives (Exercise 2.22)) 'V"I'VJRa~ = 'V"I'V&Ra~/1-jl = 'V"I'VjlRa~/1-&

= 'V jl 'V /1-Ra~"I& - ~jl/1-iJRa~v&

~jlaiJRv~/1-&

+ R"Ijlv~RaiJ/1-&

+ R"Ijlv&Ra~/1-iJ

and

'V jl 'V /1-Ra~"I& = 'V /1- 'V jlRa~"I& + R/1-jlaiJRv~"I& - R/1-jlv~RaiJ"I&

+ R/1-jl"liJRa~v& - R/1-jlv&Ra~"IiJ = 'V /1- 'V jlRa~"I& + RaiJRv~"I& + R"IiJRa~vJ - Rv&Ra~"IiJ'

Rv~RaiJ"I&

Combining the formulas above yields

8 8t Ra~"I& = 'V jl 'V /1-Ra~"I&

- ~jlaiJRv~/1-&

+ R"Ijlv~RaiJ/1-&

+ R"IjlvJRa~/1-iJ - Ra>..R"I&>"~ = t:.Ra~"I& - ~jlaiJRv~/1-& + R"Ijlv~RaiJ/1-& + R"IjlvJRa~/1-iJ - ~iJRa~v&

1

- 2" (RaiJRv~"IJ + Rv~RaiJ"(& + ~iJRa~v& + Rv&Ra~"IiJ)

,

o

and the proposition follows.

As a consequence of the proposition we have the following evolution equations for the bisectional curvature and the Ricci tensor. COROLLARY

(! -

2.82 (Bisectional curvature evolution).

t:. ) RaOt"l'Y =

t

(IRajlv'Y12 - IR ajl"liJl2

+ RaOtvjlR/1-iJ"I'Y)

/1-,v=l n

(2.104)

- L Re (RajlR/1-Ot"l'Y + ~jlRaOt/1-'Y) . /1-=1

Here Re(A) = ~(A +..4) denotes the real part of a complex number A.

2. KAHLER-RICCI FLOW

106

PROOF.

Indeed, substituting j3

= a and 8 = l' in (2.100), we have

(%t - A ) Rcxory'Y = RCXilv'YRJ.l.c,'Yv - RCXil'YvRJ.l.c,v'Y

+ Rcxc,vilRJ.l.v'Y'Y

1

- .2 (RcxilRJ.l.c,'Y'Y + RJ.l.c,Rcxil'Y'Y + ~ilRcxc,J.I.'Y + RJ.I.'YRcxc,'Yil) . o COROLLARY 2.83 (Ricci tensor evolution). The Ricci tensor satisfies the Lichnerowicz heat equation:

a

at RcxfJ = ARcxfJ + RcxfJ'YiS R 6'Y -

(2.105)

Rcx'YR'YfJ

= ALRcxfJ·

REMARK 2.84. More generally, we say that a real (1, I)-tensor hcxfJ satisfies the Lichnerowicz heat equation if a 1 1 athcxfJ = ALhcxfJ ~ AhcxfJ + RcxfJ'YiS h6'Y - .2 Rcx'Yh'YfJ - .2 R 'YfJ hcx'Y'

See also (2.22). PROOF.

Summing (2.100) over 'Y = 8 from 1 to n, we have

(%t - A ) Rcxi3 =

t

(%t - A ) Rcxi3'Y'Y

+ R6'YRcxfJ'YiS

k=l

= RCXilV'YRJ.l.fJ'Yv -

RCXil'YvRJ.l.fJv'Y

+ Rcxi3vilRp.v + RcxfJ'YiSR6'Y

1

- .2 (RCXil R p.i3 + Rp.i3 Rcxil + ~ilRcxfJp.'Y + Rp.'YRcxfJ'Yil) = RcxfJ'YiS R 6'Y - R cxil RJ.l.i3'

o

after cancelling terms to get the last equality.

REMARK 2.85. Equation (2.105) may also be derived from (2.36), (2.10) and commuting covariant derivatives. In particular, 1 ARcxi3 = .2 (V'Y V'Y + V'Y v\) Rcxi3

1

=

1

.2 V'Y V fJRcx'Y + .2 V'Y V cxR'YfJ 1

1

= V cx V fJR - .2 R'YfJcxiS R 6'Y + .2 R6fJ R cxiS 1

(2.106) That is,

1

+ .2 RcxiS R 6i3 - .2 R'Ycxj36 R 'YiS = V cx V i3 R - RcxfJ'YiSR6'Y + RcxiSR6fJ. V cx V fJR = ALRcxfJ·

Based on the evolution equation (2.104) and Hamilton's maximum principle for tensors (see Chapter 4 of Volume One or Part II of this volume) we present the following.

8.

KAHLER-RICCI FLOW WITH NONNEGATIVE BISECTIONAL CURVATURE

107

PROOF OF THEOREM 2.78. (I) We first prove that the nonnegativity of the bisectional curvature is preserved under the flow. Analogous to Theorem 4.6 on p. 97 of Volume One, by the Kahler version of the maximum principle for tensors (see Proposition 1 in §4 of [19]), we need to show that the quadratic on the RHS of (2.104), i.e., (2.107)

QOdi'Y"Y

~

n

L (IRctjLlI"Y1 p.,I1=l

2 -IRctjL")'iiI 2

+ RctiilljLRp.ii")'''Y)

n

- L Re (RctjLRp.ii")'''Y + ~jLRctiiP."Y) , p.=1

satisfies the null eigenvector assumption. That is, we assume Rctii,,),"Y = 0 for some a and 'Y at some point x, and we shall prove that (2.108) at x. First observe that since Rctii'Y"Y = 0 at x and the bisectional curvatures are nonnegative, we have at x, n

n

L RctjLRp.ii")'''Y = p.=1 L ~jLRctiip."Y = o.

p.=1

By (2.107), in order to prove (2.108) at x, it suffices to show that n

n

L RctiilljLRp.ii,,),"Y ~ p.,I1=l L (I R ctjL")'iiI p.,I1=l

(2.109)

2

-IRctjLlI"Y1 2 ) .

We shall prove (2.109) below, but first we show how the positivity of the Ricci tensor and holomorphic sectional curvatures follow from the quasipositivity of the Ricci curvature at t = O. (2) Recall that the Ricci tensor Rct/3 satisfies the Lichnerowicz heat equation, so that by taking a = {3 in (2.105), we have

a

at R ctii =

(2.110)

b.Rctii

+ Rctii'Y8 R 8"Y -

Rct"Y~ii'

Since the nonnegativity of the bisectional curvature is preserved, by applying Hamilton's strong maximum principle for tensors to (2.110) (see Theorem A.53 and also Part II of this volume), the Ricci tensor becomes positive for all positive time. Now suppose there exists a space-time point (xo, to), with to > 0, at which some holomorphic sectional curvature R ctiictii is zero. Since the holomorphic sectional curvature is nonnegative everywhere, by (2.104), we have at (xo, to),

o~

(:t - b.)

R ctiictii

=

t

p.,I1=l

(2 1R ctiilljL 12 - IRctjLctiil 2) ,

2. KAHLER-RICCI FLOW

108

where we used I::=1 Re (Rap.R/-,aaa + Rap.Raa/-,a) other hand, by (2.109) with Q = I, we have n

=

a at (xo, to). On the

n

L

L

IR aavp.12 ~

IR ap.ailI 2 .

/-"v=l

Hence we conclude Raavp. = RaP.ail = a at (xo, to) for all j.L, v. This in turn implies Raa = a, which is a contradiction. (1) continued. We now verify (2.109). Consider the following Hermitian symmetric form defined by the bisectional curvatures: (-a a +sX'-a a +sX'-a a +sY'-a a +sY ) ~a Q(X,y,s)~Rme za

~

~

~

for X, Y E r 1,0 M and s E R At a point where the bisectional curvatures are nonnegative and Raa'Yi = a, we have Q(X, Y, s) ~ a and Q(X, Y, a) = a for all X, Y E 1,0 M and s E R Therefore the second variation at s = a is nonnegative:

r

d2

a:::; ds 2

1

= Rme

_

8=0

Q(X, Y, s)

( -a8) + X,X, az'Y' az'Y

+ 2 Re ( Rme

( X,

Rme

a~a ' Y, a~'Y )

(a8-) Y, Y aza' aza'

+ Rme ( X,

a~a' a~'Y' Y) )

.

In terms of a unitary (1, a)-frame {ei}i=l' we may write this as Ri3'YiXi Xj

+ 2 Re ( RiajiXiyj + Ria'Y3XiYj) + Raai3 yiyj ~ a,

where X ~ I:~1 Xiei and Y ~

I:j=l yjej. By Lemma 2.86 below, we have

n

n

L Ri3'YiRaaj'i ~ L (l~ajiI2 -I Ria'Y31 2) . i,j=l i,j=l The claimed inequality (2.109) follows and this completes the proof of Theorem 2.78. D LEMMA 2.86. Let Q(X, Y) be a Hermitian symmetric quadratic form

defined by Q(X, Y) = Ai3Xi Xj

+ 2 Re ( BijXiyj + Di3XiYj) + Gi] yiyj.

If Q is semi-positive definite, then n

L

i,j=l

Ai3 Gj'i

~

n

L

i,j=l

(I B ij12 -I Di312).

9.

MATRIX DIFFERENTIAL HARNACK ESTIMATE

109

For the proof of this lemma, which is elementary in nature, we refer the reader to Mok [269]. 9. Matrix differential Harnack estimate for the Kahler-Ricci flow In this and the next section we discuss various differential Harnack estimates for the Kahler-Ricci flow and their geometric applications. Differential Harnack estimates for the Riemannian Ricci flow will be discussed in Part II. For Kahler-Ricci flow, a fundamental result is H.-D. Cao's differential Harnack estimate for solutions with nonnegative bisectional curvature (see [46]). Define (2.111) a J RaP Z {X)aP ~ at RaP + RaiRyP + '\l'YRaPX'Y + '\l iRaPX'Y + RaP'YJX'Y X + -tfor any {l,O)-vector X

= X'Y 8~"" and where Xi ~ X'Y.

THEOREM 2.87 (Kahler matrix Harnack estimate). If (Mn,g{t)) is a complete solution to the Kahler-Ricci flow with bounded nonnegative bisectional curvature, then (2.112) for any {I, O)-vector X.

This result may be considered as the space-time analogue of Theorem 2.78. We shall also see a similar analogy for the Riemannian Ricci flow in Part II, where Hamilton's matrix differential Harnack estimate will appear as the space-time analogue of the result that nonnegative curvature operator is preserved under the Ricci flow. 9.1. Trace differential Harnack estimate for the Kahler-Ricci flow. Taking the trace of the estimate (2.112) leads to the so-called trace differential Harnack estimate, after applying the second Bianchi identity. COROLLARY 2.88 (The Kahler trace differential Harnack estimate). Let (Mn,g{t)) be a complete solution to the Kahler-Ricci flow with bounded nonnegative bisectional curvature. Then (2.113) PROOF. By (2.112) we have O ~ 9apZa(3- -- 9ap

R ata R a(3- + 1R a(3_1 2 + ~v 'Y RX'Y + ~v i RXi + R'YfJ-X'YXJ + t'

and (2.113) follows from (2.37).

0

2. KAHLER-RICCI FLOW

110

> 0, the {l,O)-vector minimizing the LHS of (2.113) is X'Y = I - (Rc- )'YP\7 pR, where (Rc-ItP R'YP = t5~. Hence, ifRc > 0, then (2.113) When Rc

is equivalent to

aR at

+ Rt

_ (Rc- I )'Y5 \7 R\7-R > O. 'Y

r

Since R'Y5 ~ Rg'Y5' we have - (Rc- I 5 ~ {}

(2.114)

at log (tR) - 1\7 log (tR)1

2

6-

_~g'Y5, {}

and hence

1

= at log R + t

- 1\7 log RI 2 ~ o.

Without assuming Rc > 0, we still obtain (2.114) by taking X'Y = -~\7'YR in (2.113) and using RaP ~ R9ap . COROLLARY 2.89

(Integrated form of Kahler trace Harnack estimate).

If (Mn,g(t)) is a complete solution to the Kahler-Ricci flow with bounded nonnegative bisectional curvature, then for any Xl, X2 E M and 0 < II < t2, we have R (X2' t2) > -tl e _l~ ----'----'4 R(XI,tl)-t2 ' where ~ = ~ (Xl, tl; X2, t2) ~ inf'Y fttl2 Ii' (t)I;(t) dt, and the infimum is taken over all paths 'Y : [tl, t2J ~ M with 'Y (tl) = Xl and 'Y (t2) = x2· By the fundamental theorem of calculus and (2.114), we have for any'Y : [tl, t2J ~ M with 'Y (tI) = Xl and 'Y (t2) = X2, PROOF.

log

t2R (X2' t2) lt2 d R( ) = -d [log tR b (t) , t) J dt tl

XI. tl

t

tl

= 1lt2 [(:tlogtR) ('Y(t),t) + (\7logtR,i'(t))g(t)] dt

~ lt2

[1\7l0gtRI;(t)

tl

+ (\7l0gtR,i'(t))g(t)]

dt

~ -~ lt21i' (t)I;(t) dt.

tl The corollary follows from taking the infimum over all 'Y.

o

For the normalized Kahler-Ricci flow, we have the following. COROLLARY 2.90 (Integrated trace Harnack for normalized Kahler-Ricci flow). If (Mn,g(t)) is a solution to the normalized Kahler-Ricci flow on a closed manifold with nonnegative bisectional curvature, then for any Xl, X2 E M and 0 < tl < t2,

(2.115)

where

~

is as above and r

~

0 is the average (complex) scalar curvature.

> eii-tl-l R EMARK 2 .91 . Note t h at l-e-ii-tl l-e-nrt 2 - ~. en 2-1

111

9. MATRIX DIFFERENTIAL HARNACK ESTIMATE

Before we prove the corollary, we first recall how to go from the KahlerRicci flow to the normalized Kahler-Ricci flow on closed manifolds. Let 9 (t) be a solution to ~gQJ3 = -ROti3 and let 9 (t) ~ 'IjJ (t) 9 (t) , where 'IjJ (t) is to be defined below. We compute

_.

{)

{)t9Oti3 = -'ljJROti3 since ROti3 we have

+ 'ljJgOti3

= ROti3' Hence if we define a new time parameter i by -Nt = i %t'

{) -

{)igOti3

= -

~-

R

Oti3 + 'ljJ2 gOti3'

In particular, to obtain the normalized Kahler-Ricci flow, where mains in the same Kahler class, we set ~ (t)

f

'IjJ (t)2

n

9 (t) re-

,

where f is the average scalar curvature of 9 (t), which is independent of time since 9 (t) stays in the same Kahler class. Thus we take 'IjJ (t) ~ (1 - ~t) -1 . Since di = 'ljJdt, we may take i ~ -~ log (1- ~t) . That is, t

(1 - e-~l) .

=~

2.90. Let 9 (i) be a solution of the normalized Kahler-Ricci flow %t 9Ot i3 = -ROti3 + ~9Oti3' Then PROOF OF COROLLARY

is a solution of the Kahler-Ricci flow and we have the estimate ~~;;::;~ >

~e-~A. This implies

where

Li (Xl, i I ; X2, i2) = inflt21 ddT' "f

since

tl

t

(t)1

2

g(t)

dt

= inf "f

ft = 'IjJ-9t, 9 (t) = 'IjJ-I 9 (i) , and dt = 'IjJ-Idi.

(21

itl

d~ (i) 12 dt

ii(i)

di

o

Since 9 (t) has nonnegative bisectional curvature, and in particular it has nonnegative Ricci curvature, under the normalized Kahler-Ricci flow we have itgOti3 ~ ~gOti3' which implies 9 (t) ~ e~(t-tl)g (tI) for t ~ ti' Hence (2.115) implies

112

2.

KAHLER-RICCI FLOW

by taking 'Y (t) to be a minimal geodesic, with respect to 9 (tt), joining to X2 with speed Ii' (t)l g(t1) = ce-f;(t-tt}, where

Xl

r(

t»)-l dg(h) (XI,X2). c=r;, 1-e- nret 2-1

which is the estimate we would have obtained from (2.115) by using e;;;(t-h) ~ e;;; (t2 -t1) and taking 'Y (t) to be a minimal geodesic joining Xl to X2 with speed I'Y· (t)1 g(h) -- d9 (t1)(X1,X2) t2-t1 .

9.2. Application of the trace estimate. A beautiful application of the trace differential Harnack estimate and Perelman's no local collapsing theorem is the following uniform bound for the curvatures of a solution of the normalized Kahler-Ricci flow on a closed manifold with nonnegative bisectional curvature. This proof is due to Cao, Chen, and Zhu [49] and gives a simple proof of an estimate of Chen and Tian [87], [88], who proved convergence of the Ricci flow for the normalized Kahler-Ricci flow on closed manifolds with positive bisectional curvature (Theorem 2.70). THEOREM 2.92 (NKRF: Kc (V, W) ;:::: 0 curvature estimate). If(Mn,go) is a closed Kiihler manifold with !f:: [wo] = 27rCl (M) and nonnegative bisectional curvature, then the solution g (t) to the normalized K iihler-Ricci flow, with g (0) = go, has uniformly bounded curvature for all time. PROOF. Without loss of generality we may assume the solution is nonflat and the average scalar curvature r is equal to n, independent of time. Given any time t > 1, there exists a point y E M such that R (y, t + 1) = n. By (2.116) we have for any X E M

n R(x,t)

=

R(y,t+1) 1-e- t (d;(t)(X,y)) R(x,t) ;:::: 1- e-(t+1) exp -4(1- e- l )

Since 11~e(t:1) ~ ~=:=~ = 1 + e- l for t ;:::: 1, we conclude for any (2.117)

.

X

E M,

9. MATRIX DIFFERENTIAL HARNACK ESTIMATE

113

In particular, if x E B t (y, 1) , then

R (x, t) :S n (1

+ e- 1) exp

(4 (1 ~

e- 1))

.

Hence, since g (t) has nonnegative bisectional curvature, the curvature of g (t) is bounded9 in Bg(t) (y, 1). By Perelman's no local collapsing theorem, which holds for a solution to the normalized Kahler-Ricci flow since its statement is scale-invariant and since the corresponding solution to the Kahler-Ricci flow must blow up in finite time, we conclude that there exists a constant K, > 0 depending only on the initial metric such that Volg(t) (Bg(t) (y, 1)) 2:

K,.

In particular, K, > 0 is independent of t > 1 and the choice of y E M such that R (y, t + 1) = n. We may obtain a uniform diameter bound for g (t) by Yau's argument. In particular, let x E M be a point with d (x, y) = d 2: 2. Since Rc 2: 0, by the Bishop-Gromov relative volume comparison theorem, we have (2.118) Volg(t) (Bg(t) (x, d + 1)) - Volg(t) (Bg(t) (x, d - 1)) (d + It - (d - It Volg(t) (Bg(t)(x,d-l)) :S (d-lt

C(n)

< - -d - . Since Bg(t) (y, 1) c Bg(t) (x, d + 1) \Bg(t) (x, d - 1) and Bg(t) (x, d - 1) C Bg(t) (y, 2d - 1) , by (2.118), we have Volg(t) (M) 2: Volg(t) (Bg(t) (y, 2d - 1))

2: Volg(t) (Bg(t) (x,d -1)) 2:

Volg(t) (Bg(t) (y, 1)) C (n) d.

Taking d = diamg(t) (M) , we have

. C (n) Volg(t) (M) C (n) Volg(t) (M) dlamg(t) (M) < --~--=::...:~--:- < . - Volg(t) (Bg(t) (y, 1)) K, Since under the normalized flow, Volg(t) (M) is constant, we obtain a uniform upper bound C for the diameter of g (t). Hence (2.117) implies

R (x, t) :S n (1 + e-1 ) exp ( 4 (1 ~2e- 1 )) which is our desired uniform estimate for R.

,

o

9We actually only need an upper bound on the scalar curvature for Perelman's no local collapsing theorem (Theorem 6.74).

114

2. KAHLER-RICCI FLOW

9.3. Proof of the matrix Harnack estimate. In this subsection we prove Theorem 2.87. Let

The following computation is the one corresponding to Proposition 2.79. PROPOSITION 2.93 (Evolution of the Harnack quantity Zo.(3)' Suppose that a vector field X satisfies

--

1

'\lJX'Y = '\l oXy = R'YJ + i 9'Y J, '\l oX'Y = '\lJXy = 0, and

Then Zo.13 = Z (X)o.13 defined by (2.111) satisfies the evolution equation:

(:t - Do) Zo.13 = Ro.13'YJZ,yo - ~ (Ro.1Z'Y13 + R'Y13 Zo.1) (2.119)

+ po.J'YPo131 -

Po.1JP13'Yo -

2

i Z o.13 '

The proposition follows from Lemmas 2.94 and 2.95 below. Assuming the proposition, we now prove the differential Harnack estimate, i.e., Proposition 2.93. Applying Zo.13 to a (1, O)-vector field W, we have the following general formula:

(:t - Do) (Zo.13W o. w13 ) = ( (:t - Do) Zo.(3) Wo.W13 + Zo.13 ((:t - Do) (wo.W13)) + '\lTZo.13'\lf (Wo.W13) + '\l fZo.13'\lT (Wo.W13)

,

where W13 ~ W.B. Thus, if we have a null vector W of Zo.13 at a point (xo, to) and if we extend W locally in space and time so that at (xo, to) (2.120) (2.121)

W 0:,1--'?i = Wa ,fJ = 0, (J

(~ at - Do) Wo. = 0'

9. MATRIX DIFFERENTIAL HARNACK ESTIMATE

115

where Wa = 9,6aW,6, then (2.119) implies

(:t - Ll) (ZQ~WaW,6) = (RQ~1'8Zc5'Y) Wa W,6

+ (PQ8I'P£~'Y - PQ8'YPc5~I') Wa W,6

1

2

- 2 (RQ'YZI'~ + ZQ'YRI'~) Wa W,6 - tZQ~WaW,6 = RQ~1'8Zc5'YWaW,6 + IMI'812 -IMI'c512. Here and Now we use the facts that ZQ~ ~ 0 (at least on all of M x [0, toD and W is a null vector of ZQ~. An algebraic fact, similar to Lemma 2.86 and using a second variation computation similar to that in the proof of Theorem 2.78, shows that

RQ~1'8Zc5'YWaW,6 ~ IMI'812 -IMI'c512. Thus at a point where W satisfies (2.120)-(2.121), we have

Hence, by the maximum principle, on a closed manifold we have ZQ~ ~ 0 on all of space and time. In the complete noncom pact case, one can adapt the proof in Part II of this volume of Hamilton's matrix Harnack estimate for complete solutions to the Ricci flow with nonnegative curvature operator to this Kahler setting without significant modifications. Now we give the two lemmas which are needed to complete the proof of Proposition 2.93. LEMMA

2.94.

(:t - Ll) (LlRQ~ + RQ~1'8R;yc5) 1

1

= 2LlRQpRp~ + 2RQpLlRp~ + 2RQP'I'Rp~,'Y 1 + 2Rc5'Y (\lJ\lI'RQ~

+ \l1'\l8RQ~)

1

-

Ll (RQ'YRI'~)

+ 2(\lJ\lI'RQ~+ \ll' \l8RQ~-RQPI'8Rp~-RQpRp~1'8)Rc5'Y (2.122)

+ 2RQ~1'8Rc5pRP'Y

a

+ RQ~1'8 at Rc5'Y.

2. KAHLER-RICCI FLOW

116

PROOF.

Let hOtij be a real (1, I)-tensor. Using (2.33), we compute

%t ('V s'V-yhOtij) =

(2.123)

'VS'V -y (%t hOtij) - 'V s ( (%t

r~Ot ) hpij) -

(%t

r~/3 ) 'V -yhOtp

= 'VS'V-y (%thOtij)

+ 'VS (gPu'V-yROtuhpij) + gUP'VSRiju'V-yhOtP

= 'VS'V-y (%thOtij)

+ 'VS'V-yROtphpij

+ 'V-yROtp'VShpij + 'VSRpij'V-yhOtp. Taking the complex conjugate of (2.123), we have

:t

(2.124)

('V-Y'VShOtij) = 'V-y'VS (%thOtij)

+ 'V-Y'VSRpijhOtp

+ 'VSRpij'V-yhOtp + 'V-yROtp'VShpij' Next we compute, using (2.123), (2.124) and tracing, that

a at

a [g-Ys ('Vs'V-yhOtij+'V-y'VshOtij) ] (~hOtij) = '12 at 1

=

-

a

'2 g-Yo at

('Vs'V-yhOtij

+ 'V-y'VshOtij)

1

+ '2 Ro;y ('V S'V -yhOtij + 'V -y 'V ShOtij) =

~ (:t hOtij ) + ~Ro;y ('Vs'V-yhOtij + 'V-y'VshOtij) 1

+ '2 ('V;y 'V -yROtphpij + 'V -yROtp'V ;yhpij + 'V ;yRpij'V -yhOtp) 1

+ '2 ('V-y'V;yRpijhOtp + 'V;yRpij'V-yhOtp + 'V-yROtp'V;yhpij). In particular, simplifying and taking hOtij to be the Ricci tensor, we have

(! - ~) (~ROtij) = ~ (ROtij-ySRo;y) - ~ (ROt;yR-yij) 1

+ '2 Ro;y ('V S'V -y ROtij + 'V -y 'V SROtij ) 1

+ '2 ('V;y 'V -yROtpRpij + 'V -y 'V ;yRpijROtp) + 'V -yROtp 'V ;yRpij + 'V ;yRpij 'V -yROtp, On the other hand, using (2.103), we compute

a

at (ROtij-ysR;yo)

= ('V -y 'V SROtij - ROtpRpij-yS) R;yo

+ 2ROtij-ySRopRp;y.

a

+ ROtij-yS at R;yo

9. MATRIX DIFFERENTIAL HARNACK ESTIMATE

117

Hence

(:t -

Ll ) ( LlRai3 + Rai3-y;SR-yo)

1

= 2 (\71 \7 -y R apRpi3 + \7 -y \71 Rpi3 Rap ) + 2\7 -yRap \7 ;yRpi3 1

+ 2RO;Y (\7;S\7-y R ai3 + \7-y\7;SRai3) +

(\7 -y \7 ;SRai3 - Rap Rpi3-Y;S ) R-yo

Ll (Ra;yR-Yi3)

a

+ Rai3-Y;S at R-yo

+ 2Rai3-y;SRopRP"Y. Finally we obtain (2.122) from the commutator equations: 1

2 (\7-y\7;SRai3 -

\7-y\7;SRai3) =

1

-2 (R-y;SapRpi3 -

R-y;Spi3Rap)

and \7;y \7 -yRap - \7 -y \7 ;yRap

= - (R-y-yauRup + R-y-ypuRau) = RauRup - RupRau = 0,

which imply

o LEMMA

2.95.

(:t -

Ll ) (\7 -y R ai3 X -Y)

1

2 (Rap \7-yRpi3 + \7-y RapRpi3 + \7 pRai3R-yp) X-Y + (\7 pRauR-Yi3up - Rau-yp \7 pRui3) X-Y + \7-y (Rai3puRup) X-Y

= -

(2.125)

+ \7-yRai3

PROOF.

(:t - Ll) X-Y - \7;S\7-yRai3\7oX-Y - \7o\7-yRai3\7;SX-Y.

We compute using (2.105)

(:t - Ll)

\7 -y R ai3 = \7 -y

(:t - Ll) Rai3 +

(\7 -yLl - Ll \7 -y) Rai3

- (!r~a) Rpi3 = \7-y (Rai3p;SRop - RapRpi3) + gpif\7-yRau Rpi3 (2.126)

1

+ 2 (-Rui3\7 -yRau + Rau \7-y R ui3) - R-ypau \7 pR ui3 + R-ypui3 \7 pRau -

1

2R-yu \7 uRai3'

2. KAHLER-RICCI FLOW

118

Here we also used (2.33) and the following general identity: 1 (V'-yDo. - Do. V' -y) haj3 = "2 V' p (V' -y V' p - V' pV' -y) haj3 1

+ "2 (V'-y V' p 1

V' pV' -y) V' phaj3

R-ypau h qj3

+ R-Y{iuj3 h au)

+ "2 (- R-yu V' qh a j3 -

R-ypau V' phqj3

= "2 V' p ( 1

(2.127)

+ R-ypqj3 V' phau )

1

= "2 (-V'-y R au hqj3 + V' -yRqj3hau) - R-ypau V' phqj3

1

+ R-ypqj3 V' phau - "2 R-yu V' qhaj3'

where we used the second Bianchi identity (2.9). Simplifying (2.126), we have

(%t - Do.) V'-y R aj3 = - ~ (Rqj3 V'-yRau + Rau V'-yRqj3 + R-yu V'qRaj3) + R-ypqj3 V' pRau -

R-ypau V' pRqj3

+ V' -y (Raj3pSR8p)

.

Equation (2.125) now follows from this and the general formula

(%t - Do.) (V'-y R aj3X-Y) = [ (%t - Do.) (V'-y R aj3) ] X-y + V'-y R aj3 (%t - Do.) X-y - V'SV'-y R aj3V'8 X -Y - V'8V'-y R aj3V'SX-Y.

o EXERCISE

2.96. Prove Proposition 2.93 using Lemmas 2.94 and 2.95.

10. Linear and interpolated differential Harnack estimates In this section we consider a differential Harnack estimate related to the estimate of H.-D. Cao considered in the previous section. This estimate has applications in the study of the geometry and function theory of noncom pact Kahler manifolds with nonnegative bisectional curvature. Let (M n , 9 (t)), t E [0, T), be a complete noncompact solution of the Kahler-Ricci flow with bounded nonnegative bisectional curvature. By Shi's theorem, given an initial metric which is complete with bounded nonnegative bisectional curvature, such a solution exists, at least for some short time T > 0, with

IV' Rm (x, t) I :s

c

t 1/ 2

for some C < 00. Analogous to the Riemannian case (see Remark 2.25 in this chapter or Theorem 10.46 on p. 415 of [111]), we consider a solution to the linearized Kahler-Ricci flow. That is, we let haj3 be a Hermitian symmetric (1, I)-tensor satisfying the Kahler-Lichnerowicz Laplacian heat equation:

10.

LINEAR AND INTERPOLATED DIFFERENTIAL HARNACK ESTIMATES

119

(2.128)

A bound for reasonable solutions is given by the following result (see Lemma 1.2 and Proposition 1.1 in Ni and Tam [290]). PROPOSITION 2.97 (Exponential bound for h Ot i3)' Suppose a solution hOti3 of (2.128) satisfies, for some constants A and B, the following inequalities:

IhOti3 (x, 0) I ::; eA (1+ r o(x))

(2.129) and (2.130)

Then there exists a constant C
O. The analogue of the linear trace differential Harnack estimate for the Riemannian Ricci flow, Theorem A.57, is as follows (see Theorem 1.2 on p. 633 of Ni and Tam [290]). The Kahler linear trace differential Harnack quadratic is defined by (compare with (A.27))

Z (h, V)

~ ~gOti3 (V'13 div (h)Ot + V'

01

div (h)i3)

+ ROti3h/3a.

+ gOti3 ( div (h)Ot Vi3 + div (h)i3 VOl) + hOti3 V/3 Va. + ~, where V is a vector field of type (1,0), H ~ gOti3 hOti3' and (2.131)

div (h)Ot ~ g'Yi3V''YhOti3'

THEOREM 2.98 (Kahler linear trace differential Harnack estimate). Suppose that (Mn,g(t)), t E [O,T), is a complete solution of the Kahler-Ricci flow with bounded nonnegative bisectional curvature and (hOti3) 20 is a solution of the Kahler-Lichnerowicz Laplacian heat equation (2.128) satisfying (2.129) and (2.130). Then Z (h, V) 20 on M x [0, T) for any vector field V of type (1,0) . The proof of this theorem requires a number of calculations which we state and prove. In these calculations the theme is to derive a heat-type equation for each of the quantities under consideration. We then need to combine terms in a good way so that we obtain a supersolution to the heat equation. The way this is accomplished, as in the Riemannian case, is to look for terms which vanish on gradient Kahler-Ricci solitons.

2. KAHLER-RICCI FLOW

120 LEMMA

2.99. We have the following equations and their complex conju-

gates: (1) (2.132)

(:t - ~)

div (h)a

(:t - ~)

(ga J3 \7J3 div (h)a)

= R/LiJ \7//ha{.L + h{.L// \7 aR/LiJ -

~RaiJ div (h)//,

(2) (2.133)

= R/La \7 {.L div (h)a + \7 aR/LiJ \7//ha{.L

+ \7 aR/LiJ \7 ah{.L// + R/LJ\7 a \7//ha{.L + h{.L// \7 a \7 aR/LiJ. PROOF.

Using (2.128) and (2.131), we compute

(! - ~) \7-y haJ3 = \7 (:t - ~)haJ3 + (\7-y~ -y

= \7 -y ( RaJ3e8hU 1

-

~ \7-y) haJ3 -

(:t r~a )

hliJ3

~ (RaeheJ3 + ReJ3hae) ) 1

- "2 \7 -y R a8 hliJ3 + "2 \7 -yRliJ3 h a8 - R-Yfia8\7.,.,hliJ3

+ R-yfiliJ3\7.,.,ha8 -

1 "2R-Y8\7lihaJ3

+ \7 -yRa8hliJ3'

since

2 (\7 -y~

-

~ \7 -y) haJ3

= \7-y (\7.,., \7 fi + \7 fi \7.,.,) haJ3 - (\7.,.,\7 fi + \7 fi \7.,.,) \7 -yhaJ3 = \7.,., (\7 -y \7 fi - \7 fi \7 -y) haJ3 + (\7 -y \7 fi - \7 fi \7 -y) \7.,.,haJ3

+ R-YfiliJ3 ha8 ) R-Yfia8\7.,.,hliJ3 + R-YfiliJ3\7.,.,ha8

= \7.,., ( - R-yfia8 hliJ3

- R-Yfi.,.,8\7li h aJ3 = -\7-yRa8hliJ3 + \7 -yRliJ3ha8 - 2R-Yfia8\7.,.,h liJ3 + 2R-YfiliJ3\7.,.,ha8 - R-y8\7lihaJ3· Simplifying, we have

(:t - ~)

\7-y h aJ3

= \7-y R aJ3e8 hlie - R-Yfia8\7.,.,hliJ3 + R-YfiliJ3\7.,.,ha8 1

+ RaJ3e8\7 -yhlie - "2 (Rae\7 -y heJ3 + ReJ3\7-yhae + R-Y8\7lihaJ3) . Since div (h) a

= g-YJ3\7-yhaJ3' taking the trace and cancelling terms, we have

(:t - ~)

div(h)a = g-yJ3

(:t - ~)

\7-y h aJ3 + R/3'Y\7-y h aJ3

= \7 a Re8 h lie + Rlifi \7.,.,ha8 -

which is (2.132).

1

"2 Rae\7 {3heJ3'

10. LINEAR AND INTERPOLATED DIFFERENTIAL HARNACK ESTIMATES

121

Next we verify (2.133). Using (2.132) and (2.21), we compute

(:t - ~) = (:t - ~)

(V 13 div (h)a)

V 13

div (h)a

= V 13 ( Rf..liJ V vhajl

+ (V t3~ -

+ hjlv VaRf..liJ -

~ V13) div (h)a

~RaiJ div (h)v)

+ ~V t3RJa div (h)8 - ~Rt38VJ div (h) a + Rt3,Ja Vi div (h)8 = V t3Rf..liJ V vhajl + V t3hjlV VaRf..liJ + Rf..liJ V 13 V vha'i + hjlV V 13 V aRf..liJ -

~ RaiJ V13 div (h) v

~Rt38 VJ div (h)a + Rt3,Ja Vi div (h)8·

Tracing and cancelling terms, we have

(:t - ~) (:t - ~)

(gat3V 13 div (h)a)

= gat3

(V 13 div (h)a)

+ R,Ba V 13 div (h) a

= V aRf..liJ V vhajl + V ahjlV V aRf..liJ + R,Ba V 13 div (h) a + Rf..liJ V a V vhail

+ hilv Va VaRf..liJ, D

which is (2.133).

EXERCISE 2.100. Write down the formulas for the complex conjugate equations to (2.132) and (2.133).

Now let for c

>0

and

hat3 ~ hat3 + cgat3· Since Z is of the form Z = A+BaVa +BaVa + hat3 Va V,B and hat3 2: cgat3 > 0, at each (x, t) we have that Z attains its minimum for some V. By taking the first variation, we immediately see that (2.134) Differentiating this, we have (2.135) (2.136)

+ V ,Bhai V, + hai V,B V, = 0, V 13 div (h)a + V t3hai V, + hai V 13 V, = o.

V,B div (h)a

122

2. KAHLER-RICCI FLOW

In each of the above instances, we also have the complex conjugate equations; we leave it to the reader as an exercise to write these down. Recall that Cao's Kahler matrix differential Harnack quadratic is (2.111): Za~

=

fl.Ra~

Ra~

+ Ra~-y8Ry6 + V-yRa~V-y + V-yRa~V-y + Ra~-y8V-yV6 + -t-·

From (2.132) and (2.133) and their conjugate equations, while substituting in (2.134), (2.135), (2.136) and their conjugate equations, we obtain

(~ at -fl.) Z = Z aJJf-Ih-a aJJ + h -yu:«v (2.137)

1( - t -h-y8 (V6 V-y

a

V--y - R a-y- - ~g t a-y-)

(v-v" a

u

R-r au - ~g-r\ t au)

+ V-yV6) + 2R6-yh-Y8 + 2 (H t+ en) + 2eR)

+ h-Y8 V Q V-y Va V6 .

From (2.134) and the trace of (2.136), we have ~ 11Z = Ra~hQ/3 - 2ha~VQV/3 - 2h/3QVaV~

Substituting this into the

RHS

+

H

+ en t

+eR.

of (2.137) yields

Z aJJf-Ih-a+h --~g Oi.JJ -yu:«v a V--R -y a-y t a-y-)

(V-V"-R-r-~g-r) a u au t au

>0 - .

Hence we have for the minimizer V satisfying (2.134)

(:t - fl.) (t z) 2: 2

O.

By applying the maximum principle (see pp. 639-640 of [290] for details), we may conclude that t 2 Z 2: 0 for all t > O. We have the following matrix differential Harnack estimate due to one of the authors [287]. THEOREM 2.101 (Matrix interpolated differential Harnack estimate). Let (Mn, 9 (t)) be a complete solution of the e-speed Kahler-Ricci flow

(2.138) where e > 0, with bounded nonnegative bisectional curvature, and let u be a positive solution of the forward conjugate heat equation au (2.139) at = fl.u + eHu. Then for any (1, O)-form V we have

(2.140) Equivalently, f (2.141)

~

log u satisfies fa~

1

+ eRa~ + tga~ 2: O.

10. LINEAR AND INTERPOLATED DIFFERENTIAL HARNACK ESTIMATES

123

PROOF. First we observe that the equivalence of (2.140) and (2.141) follows from the fact that the minimizing {I, O)-form Va for the LHS of (2.140) is equal to -~ and dividing (2.140) by u. We compute

af 2 at = b.f + IY' -til + ER. Using (2.22) and commuting a pair of derivatives, we have

+ IY'_rf12 + ER) = b.LfOt.i3 + EY' a Y'i3 R + f a,rfi3;y + f a;yfi3"Y + Y';yfY'"Yfai3 + Y'"YfY' ;yfOt.i3 + Ra;yoi3Y'JfY'"Yf.

:t fai3 = Y' a Y'i3 (b.f (2.142)

Using the analogue of (2.105) for the E-speed Kahler-Ricci flow,

a

at ROt.i3 = Eb.LR ai3 = EY' a Y' i3 R , we then compute

:t

(fOt.i3

+ EROt.i3 +

t

+ EROt.i3) + fa"Yfi3;Y + fOt.;yfi3"Y + Y';yfY'''Yfai3 + Y'''YfY';yfai3 + Ra;yoi3Y'JfY'''Yf + E2 b.ROt.i3 + E2 ROt.i3"YJRo;y - E2 Ra;yR"Yi3

9ai3) = b.L (JOt.i3

1 E - t29Ot.i3 - Rai3'

t

Hence letting SOt.{3 ~ fai3

+ ERai3 + t9ai3

(! - b.L) SOt.i3

+ E2

=

fa"Yfi3;Y

and using (2.106), we have

( b.Rai3 + Rai3"YJ R O;Y +

:t

Rai3 )

- EY';yfY'''YROt.i3 - EY'''YfY';yROt.i3 + Ra;yoi3Y'JfY'''Yf + Y';yfY'"Y S ai3 + Y'"YfY' ;ySOt.i3

+ ~Sa;y (fi3"Y +~

ER"Yi3 - t9"Yi3 )

(fa;y - ERa;y - t9a;y) Si3"Y'

Now for the E-speed Kahler-Ricci flow, by Cao's Kahler matrix differential Harnack estimate (2.111) with Xa = -~Y'af, we have

1

o ::; b.Rai3 + Rai3"YJ R o;y + Et ROt.i3 1 1 - -Y';yfY'''YROt.i3 - -Y'''YfY';yRai3 E

E

1 + 7. Ra;yoi3 Y' J fY' "Y f. E

124

2.

KAHLER-RICCI FLOW

Hence

(:t -

b.. L ) 8 a(3 2 fa,d(3'Y +

+ \!'Yf\!"(8a(3 + \!"(f\!'Y8a(3

~8a'Y (f(3"( -

+~

cR"((3 - t g"((3)

(fa'Y - cRa'Y - tga'Y) 8(3"(.

The estimate (2.141) follows from an application of the maximum principle; 0 see [287] for details. Tracing (2.140), i.e., multiplying by ga(3 and summing, we have COROLLARY 2.102 (Trace interpolated differential Harnack estimate). Under the hypotheses of Theorem 2.101, nu b..u + cuR + + gaf3 (uaV(3 + u(3Va + uVaV(3) 20,

t

which, by taking Va = - ~ and then dividing the resulting expression by u, implies the equivalent inequality n b..log u + cR + -t > - O.

J

Let N ~ M u log udf..t be the (classical) entropy of u. We have under (2.139), $tdf..t = -cRdf..t (since $t detg"(8 = -cRdetg"(8)' and

~ = 1M (b..u + cRu + (logu) b..u) df..t

1M (b..logu+cR)udf..t 2 -!f 1M udf..t.

=

In other words, (2.143) 11. Notes and commentary Some books containing material on or devoted to complex manifolds and Kahler geometry, in essentially chronological order, are Weil [370], Chern [95], Goldberg [157], Kobayashi and Nomizu [236], Morrow and Kodaira [275], Griffiths and Harris [166]' Aubin [13]' Kodaira [238]' Besse [27], Siu [334]' Mok [270]' Tian [347]' Wells [371], and Zheng [383]. We refer the reader to these books for the proper study of Kahler geometry. For the Ricci flow on real2-dimensional orbifolds, see L.-F. Wu [372] and [112]. For the Ricci flow on noncompact Riemannian surfaces, see Wu [373]'

11. NOTES AND COMMENTARY

125

Daskalopoulos and del Pino [119]' Hsu [206], [207], and Daskalopoulos and Hamilton [120].

CHAPTER 3

The Compactness Theorem for Ricci Flow Although this may seem a paradox, all exact science is dominated by the idea of approximation. - Bertrand Russell

The compactness of solutions to geometric and analytic equations, when it is true, is fundamental in the study of geometric analysis. In this chapter we state and prove Hamilton's compactness theorem for solutions of the Ricci flow assuming Cheeger and Gromov's compactness theorem for Riemannian manifolds with bounded geometry (proved in Chapter 4). In Section 3 of this chapter we also give various versions of the compactness theorem for solutions of the Ricci flow. Throughout this chapter, quantities depending on the metric gk (or gk (t)) will have a subscript k; for instance, 'V'k and Rmk denote the Riemannian connection and Riemannian curvature tensor of gk. Quantities without a subscript depend on the background metric g. Often we suppress the t dependence in our notation where it is understood that the metrics depend on time while being defined on a space-time set. Given a sequence of quantities indexed by {k}, when we talk about a subsequence, most of the time we shall still use the indices {k} although we should use the indices

{jk} .

1. Introduction and statements of the compactness theorems

Given a sequence of solutions (Mr, gk (t)) to the Ricci flow, Hamilton's Cheeger-Gromov-type compactness theorem states that in the presence of injectivity radii and curvature bounds we can take a Coo limit of a subsequence. The role of the compactness theorem in Ricci flow is primarily to understand singularity formation. This is most effective when the compactness theorem is combined with monotonicity formulas and other geometric and analytic techniques, in part because these formulas and techniques enable us to gain more information about the limit and sometimes enable us to classify singularity models. This has been particularly successful in low dimensions. In latter parts of this volume we shall see some examples of 127

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

128

this:

ICompactness Theorem I INo local collapsing I 1

+---

I]\10notonicity I

./

ISingularity analysis I In general, there are three scenarios in which we shall apply the compactness theorem for the Ricci flow. The compactness result may be applied to study solutions (M n , 9 (t)) to the Ricci flow defined on time intervals (0', w) , where w ::; 00 is maximal, i.e., the singularity time. To understand the limiting behavior of the solution 9 (t) as t approaches w, we shall take a sequence of times tk ~ wand consider dilations of the solution 9 (t) about the times tk and a sequence of points Ok E M by defining (3.1)

gk (t) = Kkg (tk

+ KJ:1 t ) ,

where Kk = IRm (Ok, tk)1 is the norm of Rm (g (tk)) at the point Ok. We are interested in determining when there exists a subsequence of pointed solutions to the Ricci flow (M, gk (t) , Ok) which limits to a complete solution (M~, goo (t) ,000 ) . This limit solution reflects some aspects of what the singularity looks like near (Ok, tk). Similarly, when 0' = -00 for solution (M, 9 (t)), which arises when we already have a (first) limit solution of a finite time singularity, we may consider sequences tk ~ -00 and take a second limit, now backward in time. Yet other limits that we shall consider arise from dimension reduction on a limit solution. Here tk remains fixed whereas Ok tends to spatial infinity. Many of the topics in this volume are related to the study of the geometry (and topology) of the limits of these solutions when they exist. 1.1. Definition of convergence. Now we review the definition of Coo_ convergence on compact sets in a smooth manifold Mn. By convergence on a compact set in CP we mean the following.

3.1 (CP-convergence). Let K c M be a compact set and let {gdkEN' goo, and 9 be Riemannian metrics on M. For p E {O} UN we say that gk converges in CP to goo uniformly on K if for every c > 0 there exists ko = ko (c) such that for k ~ ko, DEFINITION

sup sup IV o (gk - goo)lg

< C,

O::;o::;pxEK

where the covariant derivative V is with respect to g. Note that since we are on a compact set, the choice of metric 9 on K does not affect the convergence. For instance, we may choose 9 = goo. In regards to Coo-convergence on manifolds, with the noncompact case in mind, we have the following. We say that a sequence of open sets {UdkEN in a manifold Mn is an exhaustion of M by open sets if for any compact set K c M there exists ko EN such that Uk ::) K for all k ~ ko.

1.

INTRODUCTION; STATEMENTS OF THE COMPACTNESS THEOREMS

129

DEFINITION 3.2 (CC)()-convergence uniformly on compact sets). Suppose {UdkEN is an exhaustion of a smooth manifold Mn by open sets and gk are Riemannian metrics on Uk. We say that (Uk,gk) converges in Coo to (M, goo) uniformly on compact sets in M iffor any compact set K c M and any p > 0 there exists ko = ko (K,p) such that {gdk>k _ 0 converges in CP to goo uniformly on K. In order to look at convergence of manifolds which come from dilations about a singularity, we must ensure that the form of convergence can handle diameters going to infinity. When this happens, a basepoint, or origin, is carried along with the manifold and the Riemannian metric to distinguish what parts of the manifolds in the sequence we are keeping in focus. This allows us to compare spaces that either have diameters going to infinity or are noncompact. DEFINITION 3.3 (Pointed manifolds and solutions). A pointed Riemannian manifold is a 3-tuple (Mn, g, 0), where (M, g) is a Riemannian manifold and 0 E M is a choice of point (called the origin, or basepoint). If the metric g is complete, the 3-tuple is called a complete pointed Riemannian manifold. We say that (Mn, g (t), 0), t E (a,w) , is a pointed solution to the Ricci flow if (M,g(t)) is a solution to the Ricci flow. REMARK 3.4. In [187] Hamilton considered marked Riemannian manifolds (and marked solutions to the Ricci flow), where one is also given a frame F = {e a } at 0 orthonormal with respect to the metric g (0) with o E (a,w). Since for most applications, the choice of frame is not essential, we restrict ourselves to considering pointed Riemannian manifolds in this chapter.

:=1

Convergence of pointed Riemannian manifolds is defined in a way which takes into account the action of basepoint-preserving diffeomorphisms on the space of metrics. DEFINITION 3.5 (COO-convergence of manifolds after diffeomorphisms).

A sequence {(Mk,gk,Ok)}kEN of complete pointed Riemannian manifolds converges to a complete pointed Riemannian manifold (M~, goo, 0 00 ) if there exist

(1) an exhaustion {UkhEN of Moo by open sets with 0 00 E Uk and (2) a sequence of diffeomorphisms q,k : Uk ---t Vk ~ q,k (Uk) C Mk with q,k (0 00 ) = Ok such that (Uk,q,t: [gklvk]) converges in Coo to (Moo,goo) uniformly on compact sets in Moo. We shall also call the above convergence Cheeger-Gromov convergence in Coo. The corresponding definition for sequences of pointed solutions of the Ricci flow is given by the following.

130

3.

THE COMPACTNESS THEOREM FOR RICCI FLOW

DEFINITION 3.6 (COO -convergence of solutions after diffeomorphisms).

A sequence {(Mk' 9k (t) ,0k)}kEN' t E (a, w) , of complete pointed solutions to the Ricci flow converges to a complete pointed solution to the Ricci flow (M~, 900 (t) ,000 ) , t E (a, w) , if there exist (1) an exhaustion {UdkEN of Moo by open sets with 0 00 E Uk, and (2) a sequence of diffeomorphisms O.

Then there exists a subsequence {jkhEN such that {(Mjk' 9jk (t) ,0jk)}kEN converges to a complete pointed solution to the Ricci flow (M~, goo (t) ,000 ), tE (a,w), as k~oo. . Note that the second theorem only supposes bounds on the curvature, not bounds on the derivatives of the curvature. This is because, for the Ricci flow, if the curvature is bounded on (a, w) , then all derivatives of the curvature are bounded at times t > a (see Chapter 7 of Volume One or Theorems A.29 and A.30 of this volume).l In particular all derivatives of the curvature are bounded at time t = 0 and we can apply Theorem 3.9 to {(M k, gk (0), Ok) hEN· In the next section we follow the proofs of Hamilton in [187]. We shall assume Theorem 3.9, which will be proven in Chapter 4. We will show that if there is a subsequence such that (M k,9k (0) ,Ok) converges to a complete limit (M~, 900 (0) ,000 ) , then there is a subsequence (Mk' gk (t) ,Ok) which converges at all times. 1The bounds on the derivatives of Rm get worse as t

-+

Q.

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

132

2. Convergence at all times from convergence at one time In this section we give the proof that the compactness theorem for Ricci flow (Theorem 3.10) follows from the compactness theorem at time t = (Theorem 3.9). This is done by showing that bounds on the metric and covariant/time-derivatives of the metric at time t = extend to bounds on the metric and covariant derivatives of the metric at subsequent times in the presence of bounds on the curvature and covariant derivatives of curvature (for all time). This is shown in subsection 2.1 below. The Arzela-Ascoli theorem is then used to show that these bounds on the covariant/timederivatives of the metric imply that a subsequence converges to a solution of the Ricci flow for all times (in subsection 2.2.2 below).

°

°

2.1. Uniform derivative of metric bounds for all time. In order to extend the convergence at one time to convergence at all times, the following derivative bounds need to be shown. LEMMA 3.11 (Derivative of metric bounds at one time to all times). Let M n be a Riemannian manifold with a background metric g, let K be a compact subset of M, and let gk be a collection of solutions to the Ricci flow defined on neighborhoods of K x [,8, 'l/J], where to E [,8, 'l/J]. Suppose that

(1) the metrics gk (to) are all uniformly equivalent to 9 on K, i. e., for all V E TxM, k, and x E K, C- 1 g (V, V) :::; gk (to) (V, V) :::; Cg (V, V) , where C < 00 is a constant independent of V, k, and x; and (2) the covariant derivatives of the metrics gk (to) with respect to the metric 9 are all uniformly bounded on K, so that

for all k and p :2: 1, where C p < 00 is a sequence of constants independent of k; and (3) the covariant derivatives of the curvature tensors Rmk (t) of the metrics gk (t) are uniformly bounded with respect to the metric gk (t) on K x [,8, 'l/J] :

(3.2)

IV'~ Rmklk :::; C~ for all k and p :2: 0, where C~ is a sequence of constants independent of k.

Then the metrics gk (t) are uniformly equivalent to 9 on K x [,8, 'l/J], e.g., (3.3)

B (t, to)-l g(V, V) :::; gk (t) (V, V) :::; B (t, to) g(V, V),

where

B (t , t a) = C e 2v'n-1Cblt-tol ,

2.

CONVERGENCE AT ALL TIMES FROM CONVERGENCE AT ONE TIME

133

and the time-derivatives and covariant derivatives of the metrics gk (t) with respect to the metric 9 are uniformly bounded on K x [.8, '1/1], i.e., for each (p, q) there is a constant Cp,q independent of k such that

(3.4) 1

8q gd) -'\lP t -< Cp,q 8tq

1-

for all k. REMARK 3.12. Since we often assume bounds on Rm whereas the metric evolves by Rc, we note -In=lIRmlg:::; Rc:::; In=lIRmlg· Since we often interchange 9 and gk norms, we recall the following elementary fact. LEMMA 3.13 (Norms of tensors with respect to equivalent metrics). Suppose that the metrics 9 and h are equivalent:

C-Ig:::; h:::; Cg. Then for any (p, q)-tensor T, we have

(3.5)

ITlh :::; C(p+q)/2I T lg·

PROOF. We can diagonalize 9 and h so that gij = The assumption implies C- I :::; Ai :::; C for all i. Then

bij

and

h ij

=

Aibij.

... h kpl p hidl ... hiqjqT.kl··:kPT~l···~p ITI2h = "h L...J klll tl···tq Jl···Jq

< "

L...J

Tkl··:kPT.kl··:kP. 1l···1q

1l···1q

kl···kpjil···iq

o PROOF OF LEMMA 3.11. For the first part, since

8

at gk (t) (V, V) = -2 Rc k (t) (V, V) and IRck (t) (V, V)I :::; In=lCb9k (t) (V, V), we can estimate the time-derivatives 1

8 I at loggk (t) (V, V) =

We have proved

(3.6)

1-2RC k (t)(V,V)1 ~, gk (t) (V, V) :::; 2vn - 1Co·

134

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

where C ~ 2Jn=lCb. Now we compute

C It 1

-

tol

~

1:

1 1

%t log 9k (t) (V, V) 1 dt

~ 11:1 %t log9k (t) (V, V) dtl -iiog 9k (tl) (V, V) 1 9dto) (V, V) , or equivalently, e-Clt1-toI9k (to) (V, V) ~ 9k (tl) (V, V) ~ eClh-tol9k (to) (V, V). Hence we have C-le-Clt1-toI9 (V, V) ~ 9k (it) (V, V) ~ CeClt1-tol9 (V, V) . This completes the proof of (3.3). For the second part we need to estimate the space- and time-derivatives of 9k (t). We begin with estimating the first-order covariant derivatives of 9k (t) . Note that

8 V' a (9k)bc = 8x a (9k)bc -

d

d

r ab (9k)dc - r ac (9k)bd ,

so if we take the right combination, we see that

(9k)ec (V' a (9k)bc

+ V'b (9k)ac -

V' c (9k)ab)

= 2 (rk)~b - r~b - (9k)ec r~c (9khd - r~b - (9k)ec ric (9k)ad (3.7)

= 2 (rk)~b -

+ (9k)ec r~ (9k)ad + (9k)ec r~c (9k)bd

2r~b'

This implies that

(3.8) From we have

(3.9) Hence the tensors V'9k (t) and rk (t) - r are equivalent. We recall that the derivative of the Christoffel symbols (see, for instance, (6.1) on p. 175 of Volume One) is

:t (rk)~b

= -

(9k)cd [(V'k)a (RCk)bd

+ (V'k)b (RCk)ad -

(V'k)d (Rck)abl·

2. CONVERGENCE AT ALL TIMES FROM CONVERGENCE AT ONE TIME

135

So as tensors, we find that I:t (fk - r)lk ::; 3lVd Rc k)lk ::; Thus

3Jn=1C~ It

1 -

tol

~

1:

1

I:t (fk (t) - r)lk dt

~ 11: ~ Ifk

3Jn=1C~.

1

:t (fk (t) - f) dtl k

(t1) - flk - Ifk (to) - flk .

Hence we have a bound Ifk (t) - flk ::; 3Jn=1C~ It - tol (3.10)

::;

3Jn=1C~ It -

tol

+ Ifk (to) -

flk

+ ~C3/2Cl

using (3.8) and (3.5). Since It - tol ::; '!f; - f3, we have in (3.3): B (t, to) ::; B ('!f;, f3) for all t E [f3, '!f;]. Thus by (3.9) and (3.10), (3.11)

where

C1 ,0 ~ B 3 / 2 ('!f;, f3) (6Jn=lC~ ('!f; - f3) + 3C3 / 2 C 1 ) This proves (3.4) for p = 1 and q = O. Next we prove inductively that for p IVP Rc kl ::; C; IVPgkl

~

+ C;'

.

1, IVPgkl::; Cp,o

and

(where C~, C~', and Cp,o are independent of k). If p and (3.10),

=

1, then using (3.8)

+ Vk RCklk fklk IRcklk + IVk RCklk)

IVRckl (t) ::; B (t, to)3/21(V - Vk) RCk

::; B (t, to)3/2 (If ::; B (t, to)3/2 (

(3Jn=1C~ I'!f; -

If the estimates hold for p < N with N p = N. First we have

~

f31

+ ~C3/2C1) Cb + C~) .

2, then we will prove them for

N

IV N

RCkl

=

LV

N- i

(V - Vk) V1- 1 RCk

N- i

(V - Vk) V1- 1 RCkl

+ Vt' RCk

i=l

N

: ; L IV i=1

+ IVt' RCkl·

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

136

Note that, using (3.7), we can rewrite V' - V'k = r - rk as a sum of terms of the form V'9k. When i = 1, we can bound IV'N-l (V' - V'k)Rckl by a sum of terms of the form IV'N- j 9kllV'j RCkl, 0 ~ j ~ N - 1. When 2 ~ i ~ N, we can bound IV'N-i (V' - V'k) V'1- 1 RCkl by a sum of terms of the form IV'N-i-j+19kllV'jV'~-1 RCkl, 0 ~ j ~ N - i. We can also bound lV'jV'1- 1 RCkl = I((V' - V'k)

+ V'k)j V'1- 1 RCkl

by a sum of terms which are products of lV'i+i-l Rc kl ' 0

~ f ~

j, and

severallV't'9kl, 1 ~ f ~ j. By the assumption of Lemma 3.11, the induction assumption and the equivalence of 1·1 and 1·l k , we get IV'N Rc k 1 ~

C'/v IV'N 9k 1+ C'jJ.

Now we turn to bounding IV'N 9kl. Since 9 does not depend on t,

8

N

N

8t V' 9k = -2V' RCk and 8 1V' N9k 12 8t

8 8 V' N 9k, V' N = 2 \/ 8t 9k) ~ 18t V' N9k 12

= 41V'N RCkl2 + IV'N 9kl 2 ~

+ 1V' N9k 12

(1 + 8 (C'/v )2) IV'N 9kl 2 + 8 (C'jJ) 2 .

Integrating the above differential inequality of 1V' N 9k 12 , we get (compare with (7.47)) IV'N 9kl 2 (t)

~ e(1+8(C~y)(t-to) (IV'N 9kl 2 (to) +

8 (C'jJ)2

).

1 + 8 (C'/v) 2

This implies IV'N 9k (t)1 ~ CN,O, and the induction proof is complete, as well as (3.4) for the q = 0 case. Note that the above proof of bounding 1V' N Rc k 1 can be used to show that IV'PV'%Rckl, IV'PV'%Rkl, and IV'PV'%Rmkl are bounded independent of k. aq aq - 1 When q ~ 1, then atq V' P9dt) = V'P atq-l (-2Rcdt)). Using the evoV'P 9k (t) is lution equation of the curvature Rm k (t), we know that bounded by a sum of terms which are products of

l%t:

1V'PI V'%l Rm k 1(t) ,

1V'PV'% Rc k I,

and

I

1V'PV'%Rk 1.

D

2.

CONVERGENCE AT ALL TIMES FROM CONVERGENCE AT ONE TIME

137

2.2. Convergence at all times from convergence at one time. 2.2.1. The Arzela-Ascoli theorem. With uniform derivative bounds on the metrics in the sequence, the compactness theorem will follow from the Arzela-Ascoli theorem. LEMMA 3.14 (Arzela-Ascoli). Let X be a a-compact, locally compact Hausdorff space. If {fkhEN is an equicontinuous, pointwise bounded sequence of continuous functions fk : X --t lR, then there exists a subsequence which converges uniformly on compact sets to a continuous function foo : X --t R The reader is reminded that a-compact simply means that the space is a countable union of compact sets, and hence any complete Riemannian manifold satisfies the assumption. COROLLARY 3.15 (Metrics with bounded derivatives preconverge). Let (Mn, g) be a Riemannian manifold and let K c Mn be compact. Furthermore, let p be a nonnegative integer. If {gd kEN is a sequence of Riemannian metrics on K such that sup O~Q:~p+1

sup 1V'Q:gkl ::; C
0 such that gk (V, V) 2:: 8g (V, V) for all VET M, then there exists a subsequence {gk} and a Riemannian metric goo on K such that gk converges in CP to goo as k --t 00.

PROOF (SKETCH). We need to show that {(gk)behEN form an equicontinuous family. We use the fact that in a coordinate patch

V' a (gk)be

=

oxoa (9k )be - r dab (gk)de - raed (gk)bd .

Thus if IV'gkl is bounded, then Ia~a (gk)bel is bounded for each a, b, c in each coordinate patch. Hence, by the mean value theorem, the (gk)be form an equicontinuous family in the patch and there is a subsequence which converges to (goo)be. Since K is compact, we may take a finite covering by coordinate patches and a subsequence which converges for each coordinate patch. We have thus constructed a limit metric. Note that the uniform upper and lower bounds on the metrics gk ensure that goo is positive definite. Similarly, we can use the bound on 1V'2gkl to get bounds on second derivatives of the metrics Iax~~xd (9k) be I in each coordinate patch and thus show the first derivatives are an equicontinuous family. Taking a further subsequence, we get convergence in C 1 . Higher derivatives are similar. 0

2.2.2. Proof of the compactness theorem for solutions assuming the compactness theorem for metrics. We will now use Corollary 3.15 together with Lemma 3.11 to find a subsequence which converges and complete the proof of Theorem 3.10. Recall that we have assumed Theorem 3.9 and hence there is a subsequence {(Mk, gk (0), Ok)} which converges to (M~, goo, 0 00 ) .

138

3.

THE COMPACTNESS THEOREM FOR RICCI FLOW

We shall show that there are metrics 900 (t), for t E (a,w), such that 900 (0) = 900 and {(Mk' 9k (t) ,Ok)} converges to (Moo, 900 (t) ,000 ) in Coo. Since {(Mk' 9k (0) ,Ok)} converges to (Moo, 900,000 ) , there are maps 0 independent of k such that Bgk(o)

(Ok,p) x (a,w)

and injgdO) (Ok) ~ /'0,

then there exists a subsequence such that {(Bgk(o) (Ok,P),9k(t),Ok)}kEN converges as k ---t 00 to a pointed solution (B~,900 (t) ,000 ), t E (a,w), in Coo on any compact subset of Boo x (a, w). Furthermore Boo is an open manifold which is complete on the closed ball Bgoo(o) (000 , r) for all r < p. EXERCISE 3.17. Prove Theorem 3.16. A simple consequence of Theorem 3.16 is (see [186]) the following corollary.

COROLLARY 3.18 (Compactness theorem yielding complete limits). Let {(M k,9k(t),Ok)}kEN' t E (a,w) :3 0, be a sequence of complete pointed solutions to the Ricci flow. Suppose for any r > 0 and c > 0 there exist constants Co (r, c) < 00 such that IRm klk ::; Co (r, c)

on

Bgk(o)

(Ok, r) x (a + c, w - c)

3.

EXTENSIONS OF HAMILTON'S COMPACTNESS THEOREM

139

for all kEN. We assume injgk(O) (Ok) ~ "0 for some "0 > O. Then there exists a subsequence {(Mk' gk (t) ,Ok)} which converges to a complete solution to the Ricci flow (M~, goo (t) ,000 ) , t E (a, w) . REMARK 3.19. Note that the limit solution (M~, goo (t), 0 00 ) may not have bounded curvature. Without the injectivity radius estimate, we may use the trick of locally pulling back the solutions by their exponential maps (since the pulled-back solutions satisfy an injectivity radius estimate). We have the following. COROLLARY 3.20 (Local compactness without injectivity radius estimate). Let {(M k,gk (t) , Ok)} kEN' t E (a, w) :3 0, be a sequence of complete solutions to the Ricci flow with IRmklk ~ Co

in Bgk(o) (Ok, p) x (a,w).

Then there exists a subsequence such that {(BTokMk

(a, c) ,(expR(O))* gd t )) ,a} kEN'

where

c~ min {p,7f/JCo} ,

converges to a pointed solution (B~,goo (t) ,000 ) , t E (a,w), on an open manifold which is complete on the closed ball Bgoo(o) (000 , r) for all r < c. REMARK 3.21. There is a similar result for geodesic tubes; see §25 of Hamilton [186].

. 3.2. Compactness for Kahler metrics and solutions. Without much difficulty, the compactness theorems apply to Kahler manifolds and solutions of the Kahler-Ricci flow (see also Cao [48] and Theorem 4.1 on pp. 16-17 of Ruan [314]). THEOREM 3.22 (Compactness for Kahler metrics). Let {(M~n, gk, Ok)} be a sequence of complete pointed Kahler manifolds of complex dimension n. Suppose 1\7~Rmklk ~ Cp on Mk for all p ~ 0 and k, where Cp < 00 is some sequence of constants independent of k, and injgk (Ok) ~ "0 for some constant "0 > O. Then there exists a subsequence {jkhEN such that {(Mjk' gjk' Ojk)}kEN converges to a complete pointed complex n-dimensional Kahler manifold (M~,goo,Ooo) as k ~ 00. See the ensuing proof for the meaning of the convergence of the complex structures Jk ~ J oo . PROOF. Since Kahler manifolds are Riemannian manifolds, we can apply the Cheeger-Gromov Compactness Theorem 3.9 to obtain a pointed limit (M~,goo,Ooo) which is a complete Riemannian manifold. So the only issue is to show that the limit is Kahler. Let Jk denote the complex structure of (Mk' gk) . We have for each kEN,

(1)

Jf = -

idTM k ,

140

3.

THE COMPACTNESS THEOREM FOR RICCI FLOW

(2) (gk 0 Jk) (X, Y) ~ gk (JkX, JkY) = gk (X, Y) for all X, YET Mk, (3) V' kJk = O. Since {(Mk' gk, Ok)} converges to (Moo, goo, 0 00 ) , there are diffeomorphisms ~k : Uk ~ Vk such that ~kgk ~ goo uniformly on compact sets. From (1) we know (~klL 0 Jk 0 (~k)*' as (1, I)-tensors, are uniformly bounded on any compact set K c Moo with respect to the metrics ~kgk' Since the ~kgk are equivalent to goo on K, we conclude that (~klL 0 Jk 0 (~kt, as (1, I)-tensors, are uniformly bounded on any compact set K c Moo with respect to the metric goo. Note that (3) is equivalent to V' 0 and vo > 0 are two constants independent of k. Then either of the following hold. (1) limk--+oodgk(O) (Ok, E k ) > o. In this case there exists a subsequence {(Mkj' gk j (t), OkJ} which converges to a complete pointed orbifold solution of the Ricci flow (M~,goo(t),Ooo) with 1 Rmg""lg"" ~ Co on Moo x (a,w) and Volg<XJ(o) Bgoo(o) (000 , ro) 2:: vo. Furthermore 0 00 is a smooth point in Moo. (2) limk--+oodgk(O) (Ok, E k ) = O. In this case there exists a subsequence {(Mkj,9kj(t),Okj)} such that limj--+oo d(Okj,Ekj) = O. Furthermore if we choose O~j E Ekj with dgk(o) (Okjl O~j) = dgk(o) (Okj' EkJ, then there is a subsequence of { (Mk j , gkj (t),

O~j )}

which converges to a complete pointed

orbifold solution of the Ricci flow (M~, goo(t), 0 00 ) with 1 Rm goo Igoo ~ Co

142

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

on Moo x (o:,w) and Volgoo(o) Bgoo(o) (000 , ro) singular point in Moo.

~

Vo. Furthermore 0 00 is a

Idea of the proofs. Theorem 3.25 can be proved from Theorem 3.24 in the same way as we have proved Theorem 3.10 from Theorem 3.9. On the other hand, Theorem 3.24 can be proved with some modification of the proof of Theorem 3.9 to handle the singularity (see [257]). Note that the Bishop-Gromov volume comparison theorem holds for orbifolds. Fix r > 0; for any k and qk E Mk with dgk(o) (Ok, qk) ::; r, we have Volgk(o) Bgk(o) (qk, ro) ~ VI, where VI is a positive constant independent of k but depending on Vo, r, ro, n, Co. This implies that there exists rl independent of k such that Bgk(o) (qk, rd has the orbifold topological model B n IG (qk) , where B n is the unit ball in Euclidean space centered at the origin and G (qk) cO (n) is a discrete subgroup with rank IG (qk)1 bounded independent of k. The existence of rl implies that we can modify the choice of X'lr in Definition 4.26 and Proposition 4.22 so that the ball Bk ~ B (xl:, Nl< 12) has the orbifold topological model B n IG (xl:) . The key observation in the proof of Theorem 3.24 is that we can choose a subsequence of orbifolds so that the groups G (xl:) and their actions on Bn are independent of k. We can then use the balls Bk, B k, Bk to build the limit orbifold. 4. Applications of Hamilton's compactness theorem

In this section we discuss some applications of Theorems 3.10 and 3.16. We will see more applications of the compactness theorems later in this volume. 4.1. Singularity models. Theorem 3.16 may be applied to study singular, nonsingular, and ancient solutions of the Ricci flow. For example, let (Mn,g(t)) , t E [O,T), where T E (0,00]' be a complete solution to the Ricci flow. Given a sequence of points and times {(Xk' tk)}kEN' let Kk ~ IRm (Xk' tk)l. We say that the sequence {(Xk' tk)} satisfies an injectivity radius estimate if there exists LO > 0 independent of k such that injg(tk) (Xk) ~ LoK;;I/2. Given a complete solution of the Ricci flow, we can obtain a local limit of dilations provided we have an injectivity radius estimate and a local bound on the curvatures after dilations. 3.26 (Existence of singularity models). Let (M n , 9 (t)), t E (0:, w), be a complete solution to the Ricci flow. Given a sequence of points and times {(Xk' tk)}kEN' let Kk ~ IRm (Xk' tk)1 > 0 and COROLLARY

gk (t) ~ Kkg (tk

+ K;;lt) .

Suppose that the sequence {(Xk' tk)} satisfies an injectivity radius estimate, i.e., injg(tk) (Xk) ~ LoK;;I/2,forsomeLo > 0, and suppose thatO:k,wk,O:oo,w oo

~ 0 with O:k ~ 0: > 0, Wk ~ Woo, and [tk - ~,tk +~] 00

C

(o:,w) are such

4.

APPLICATIONS OF HAMILTON'S COMPACTNESS THEOREM

that there exist positive constants p ::; (3.12) sup

00

and C
0,

IRc(gk) - !R (gk) gkl R (gk)

IRc (goo) - !R (goo) goo I

~--~~~----~---t~------~------~

R (goo)

(we have swept under the rug the fact that in Cheeger-Gromov convergence one must pull back by appropriate diffeomorphisms from an exhaustion of Moo to M; we leave it to the reader to justify the arguments in this proof).

4.

APPLICATIONS OF HAMILTON'S COMPACTNESS THEOREM

145

On the other hand, CR;;8 R (9k)-8 ----+ o· R (goo)-8 = O. Hence we conclude that Rc(goo) = kR (goo) goo on the subset of Moo where R (goo) > O. On the other hand, the contracted second Bianchi identity implies R (goo) = const in any connected subset of the set where R (goo) > O. Hence we conclude R (goo) == 1 on all of Moo, so that Rc (goo) == kgoo on Moo. 0 4.3. Ricci flow on closed surfaces with X > O. Now we give various proofs, which are variations on a theme, of the following consequence of Theorem 5.77 on p. 156 of Volume One. THEOREM 3.31. If (M 2 ,go) is a closed Riemannian surface with positive Euler characteristic, then a smooth solution 9 (t) of the Ricci flow with 9 (0) = go exists on a maximal time interval [0, T) with T < 00. Moreover, there exists a sequence {(Xk,tk)} with tk ----+ T such that gk (t) ~ Rk9 (tk + R;;1t) , with Rk = R(Xk' tk), converges to a solution (M2, goo (t)) with constant positive curvature. First we recall the ideas of some proofs from Volume One. The first proof relies on the monotonicity of the quantity lV'iV'jf - ~LlfgijI2. PROOF #1. In this proof, which is the original proof of Hamilton, we actually recall the exponential convergence (not just sequential convergence) in Coo of the normalized flow. Consider the normalized flow %t g = (r - R) 9 on the maximal time interval [0, T). In the rest of this proof we abuse natation by using 9 (t) , t E [0, T), to stand for the normalized solution rather than the unnormalized solution in the statement of the theorem. One can prove that T = 00. If R (g (to)) > 0 for some to < 00, then we may combine the entropy and Harnack (or Bernstein-Bando-Shi derivative) estimates to show that there exist c > 0 and C < 00 such that O o. 0 The next proof uses Hamilton's isoperimetric estimate. PROOF #IIA. Suppose that M2 is diffeomorphic to the 2-sphere. Given an embedded loop, separating M into two connected components Ml and M2, the isoperimetric ratio of, is defined by CH (r)

~ L (r)2 (Area~M1) + Area ~M2))

and the isoperimetric constant of (M, g) is CH (M,g) ~ infCH (r) ~ 47r. 'Y

Then (see Theorem 5.88 on p. 162 of Volume One) under the Ricci flow

d dt CH (M, 9 (t)) ~ 0, so that CH (M,g(t)) ~ CH (M,go)

> O.

On the other hand, in the presence of a curvature bound, the isoperimetric constant bounds the injectivity radius by inj (M,g)

~ (4~ax CH (M,9)) 1/2,

where Kmax ~ maxM K (K is the Gauss curvature). Hence we may dilate about a sequence {(Xk' tk)} approaching the singularity of the unnormalized flow 9 (t), as in the proof of Theorem 3.30, and apply Hamilton's compactness theorem to obtain a limit solution (M~, goo (t)) . This limit solution is a complete ancient solution with bounded positive curvature. In the case of a Type IIa singularity, by choosing {(Xk' tk)} suitably, the limit is an eternal solution (attaining the supremum of R in space-time), which must be the cigar soliton

(R2, t~:t!~~ ).However, the isoperimetric estimate is pre-

served in the limit,2 which contradicts the existence of such a limit. Hence the singularity is Type 1. In this case the limit (Moo,goo (t)) is compact (see [186J or Proposition 9.16 of [111]) and has constant entropy, which implies that it is a gradient shrinker and hence is a constant curvature solution. 0 2More precisely, the limit being a cigar soliton implies that limi~oo C H (M2, 9 (ti)) 0, which leads to a contradiction.

=

4.

APPLICATIONS OF HAMILTON'S COMPACTNESS THEOREM

147

REMARK 3.32. In the above proof, we could have replaced Hamilton's isoperimetric estimate by Perelman's no local collapsing theorem in Chapter 6, which enables the application of the compactness theorem and at the same time rules out the formation of the cigar soliton singularity model. Now we give a proof that Type I limits are round 2-spheres using Perelman's entropy (see Chapter 6 for properties used in the proof below; the reader may wish to come back to this part after reading that chapter). PROOF #IIB, USING PERELMAN'S ENTROPY FOR LAST STEP. For the last step in the above proof, we may use Perelman's entropy instead of Hamilton's entropy. This has the advantage that the entropy is defined for solutions with curvature changing sign, so that we may apply it to the original solution 9 (t), t E [0, T), rather than the limit solution. We assume that 9 (t) forms a Type I singularity. Let W(g (t), f (t), T (t)) denote the entropy with T ~ T - t, which is defined for T E (0, T]. Taking f to be the constant h (t) ~ -log vOl~7)(M)' so that it satisfies the constraint

f M (47rT)-l e- h dJ.1 = 1, we see that J.1 (g (t) ,T (t)) ::; W(g,

47rT

h, T) ::; TRmllJ{ (t) - log Volg(t) (M) - 2.

In particular, by the long-time existence theorem for the Ricci flow on the 2-sphere, we have Volg(t) (M) = 87r (T - t). Hence we have an upper bound for the J.1-invariant: J.1 (g (t) , T (t)) ::; C - 2 + log 2,

where we have used the Type I assumption (T - t) RmllJ{ (t) ::; C. In particular, by the monotonicity of J.1, 1£ J.1 (g (t) , T (t)) ~ 0, the limit J.1T ~ lim J.1 (g (t) , T (t) ) t-+T

exists. Dilate the solution about (Xk' tk) with gk (t) = Rkg (tk + R"k1t) as in the proof of Theorem 3.30. By the scaling property of J.1 we have J.1 (gk (t) ,Rk (T - tk) - t)

= J.1 (g (tk + R"k1t) ,T - tk - R"k 1t ) .

Thus for each t E (-00, woo), the (maximal) time interval of existence of the limit solution (M~,goo (t)) , J.1T

== lim J.1 (gdt) ,RdT - tk) - t) = J.1 (goo (t) ,woo - t) . k-+oo

(We may assume Rk (T - tk) ~ Woo converges. Here we also used the continuity of J.1 (g, T) in 9 and the fact that the convergence, after the pull-back by diffeomorphisms, of gk (t) to goo (t) is globally pointwise in Coo since M~ ~ M is compact.) Now the theorem follows from the result that a solution having constant J.1 is a gradient shrinker. D

148

3. THE COMPACTNESS THEOREM FOR RICCI FLOW

5. Notes and commentary

Some basic references for compactness theorems for Riemannian metrics are Cheeger [70], Greene and Wu [165], Gromov [169], and Peters [300]. A survey of compactness theorems in Riemannian geometry has been given in Petersen [301]. The compactness theorem for Ricci flow in this chapter was proven by Hamilton in [187] and was used to classify singularities and nonsingular solutions in [186], [190] and [297]. Cheeger-Gromov theory was also directly used to study the Ricci flow in Carfora and Marzuoli [57]. Further compactness theorems on the Ricci flow which extend Hamilton's results can be found in [257] and [156]. It should be noted that there has been much work to ensure the injectivity radius bound for dilations of singularities, most notably by Hamilton [186] and Perelman [297]. Additional work on injectivity radius estimates has been done by Wu [372] and by the authors of [112] in the case of 2-dimensional orbifolds and [109] for sequences of solutions with almost nonnegative curvature operator.

CHAPTER 4

Proof of the Compactness Theorem We think in generalities, but we live in details. - Alfred North Whitehead There is no royal road to geometry. - Euclid

1. Outline of the proof

We now prove the compactness Theorem 3.9. This is a fundamental result in Riemannian geometry and does not require the Ricci flow. The compactness theorem is in the spirit of Cheeger [70] and Gromov [169] (see also Greene and Wu [165]' Peters [300], and the book [37]). We follow the proof for pointed sequences converging in Coo given by Hamilton [187]; as Hamilton notes there, things are easier because we can assume bounds on all covariant derivatives of the curvature. Theorem 3.9 will be proved in several steps. It is outlined as follows. STEP A: Construct a sequence of coverings of each manifold Mk which we can compare to each other. The covers should consist of balls BI: C Mk with a number of properties, most notably: • they are diffeomorphic to Euclidean balls, and for each fixed O! they have the same radii for all sufficiently large k, • they are numbered sequentially in O! starting from balls centered at the origin to balls with centers further and further away from the origin, • if we take smaller radii (BI: c BI:), they are disjoint, and if we take larger radii (BI: C BI: c BI:), they contain their neighbors, • we can bound the number of balls intersecting a given ball (the most is I (n, Co), where Co is the curvature bound), and • we can bound the number of these balls that it takes to cover a large ball in Mk, independent of k for k large (it takes fewer than A (r) balls to cover B (Ok, r) if k 2': K (r)). The specifics of this are contained in Lemma 4.18. This process is carried out in Section 3 of this chapter. STEP B: Use our nice covering to construct maps Ffd : BI: ~ M£. We do this by taking the inverse of the exponential map on the ball BI: C Mk, identifying the tangent space of Mk at the center of the ball BI: with Euclidean space and with the tangent space of M£ at the center of B't and 149

4.

150

PROOF OF THE COMPACTNESS THEOREM

then mapping by the exponential map to Me. We do this for each ball. This is done in subsection 4.1 below. STEP C: Use nonlinear averaging to glue together the maps Ffj to obtain maps Fke : B (Ok, 2k) --+ Me which take Ok to Oe (done in subsection 4.2 below). By taking subsequences, we can ensure that compositions of Fk,k+1 are approximate isometries which are getting closer to isometries as k goes to 00. STEP D: We form the limit manifold M~ as the direct limit of the directed system {Fk,k+1 : B (Ok, 2k) --+ B (Ok+1, 2k+l)} . The coordinates of B (Ok, 2k) then form coordinates for the limit Moo, i.e., for each coordinate Hk : EO: --+ B (Ok, 2k) there is a coordinate for the limit manifold defined as H~,k = h 0 Hk : EO: --+ Moo, where Ik is the inclusion of B (Ok, 2k) into Moo. Furthermore, for each coordinate EO: of B (Ok, 2k) there are Riemannian metrics 9k,e which are obtained by the pullbacks F"kege. Since Fke are approximate isometries, the sequence is equicontinuous and thus by the Arzela-Ascoli theorem we can find a convergent subsequence and get local metrics g~ k' It is not hard to see that these metrics form a Riemannian metric 900' on Moo via the coordinate charts H~ k . We can then show that the limit metric is complete and (Moo, 900' 0 00 ) s~tisfies the theorem, where 0 00 is the equivalence class of the base points in the direct limit. This is done in subsection 4.4 below. At many points in this construction we will take a subsequence; to simplify notation, at each stage the sequence will be re-indexed to continue to be k. 2. Approximate isometries, compactness of maps, and direct limits In this section we shall introduce some basic concepts which are essential to the construction of the limit manifold (M~, 900,000 ) . 2.1. Approximate isometries. In the following, the notation ITlg means the length at a point of the tensor T with respect to the Riemannian metric 9. DEFINITION 4.1 (Approximate isometry). For any 0 < c < 1 and p E NU {O}, a smooth map {[> : (M n ,9) --+ (Nn, h) is an (c,p)-pre-approximate isometry if sup I{[>*h - 91g ~ c, xEM

sup sup 1\7~ ({[>*h) l:'So::'SpxEM

I

~ c.

g

An (c,p)-pre-approximate isometry is an (c,p)-approximate isometry if it is a diffeomorphism and

I

sup ({[>-1)* g - hi xEN

h

~ c,

2.

APPROXIMATE ISOMETRIES, COMPACTNESS OF MAPS, DIRECT LIMITS 151

i.e., -1 :

(N, h)

~ (M,

g) is also an (E,p)-pre-approximate isometry.

Note the condition I(-1)* 9 - hlh ~ 10 is equivalent to Ig - *hlcp*h ~ 10 and IVI: [(-1)* g] Ih ~ 10 is equivalent to IV~*hglcp*h ~ E. Another way to express the condition sUPXEM 1*h - gig ~ 10 is

j)

oaob 12 _ ([ ik Oa] ob k) (OC [ .e Od] 1hab oxi oxj - gij 9 habg oxi oxj - I j oxk hcdi' oxf. - I k =

I(dd which converges to a map cI>oo. By symmetry the same argument applies to {cI>;;I}; hence cI>oo is invertible. Taking the derivatives of both sides of (4.4), we get that

8 8 2 (cI>k)a8(cI>k),B h 8xc (gk)ab = 8xc8xa 8xb (k)a,B

+

8(cI>k)a8 2 (cI>k),B h 8xa 8xc8xb ( k)a,B

+ 8 (cI>kt 8 (cI>k),B 8 (cI>k)!' ~ (hk) 8xa

8xb

8x c 8y!'

a,B .

From this equation we can express ~;;;2: as a polynomial function of 1 a (gk) ab' ay"l a (h k) a,B an d ---axaa(d' usmg . (gk) ab' (gk-1)ab ,(h) k a,B' (h-k )a,B ' axe symmetry in the usual way (see §5 of [187] for the explicit formula). Thus

I~;;;2: I can

be bounded. By differentiating the formula for ~;!'a~: and using induction, we can bound all higher derivatives of (cI>kt . This implies the corollary. 0 2.3. Review of direct limits. Let {(Ab h)}kEN be a sequence of topological spaces and open embeddings: II A 2 ---t h A 1 ---t

• • • ---t

A k ---t Ik A k+ 1

---t • •• •

Consider the compositions

hi

~

1£-1 01£-2 0 ... 0

defined for k ::; €, where hk

~

fk+l

id Ak : Ak

---t

0

fk : Ak

---t

Ai

A k , the identity map. Clearly

I£m 0 hi = hm for all k ::; € ::; m. That is, ({ AkhEN , {hih~i) is a directed system of topological spaces (see Definition 15.1 in [159]). We will use II to denote disjoint union. DEFINITION 4.12 (Direct limit). The direct limit is

limAk = (IIkAk)/ "', ~

2.

APPROXIMATE ISOMETRIES, COMPACTNESS OF MAPS, DIRECT LIMITS

157

where x y if x E Ak and y E Ae for some k, fEN and either Ae (x) = Y (if k ~ f) or !ek (y) = x (if f ~ k). The relation", is an equivalence relation. The topology on limAk is the quotient topology. "oJ

Note that direct limits can be defined for more general directed systems, but this is sufficient for our needs. Let l-e : Ae ."', and 205e 20cC >. "', respectively.

4. THE LIMIT MANIFOLD

(M~,goo)

167

J:

To obtain the local convergence of f3 as k -+ 00, we take some further subsequences. Since the J:f3 are Riemannian isometries between llk ~ (H;:r 9k and ~ for each k and since the derivatives of the metrics are bounded, we have that for each pair a, f3 ~ A (r) such that B;: n B~ i- 0 for all k 2 K (r), there exists a subsequence such that the J:f3 converge to a limit transition map J~ : ECt -+ ef3 in Coo uniformly on compact sets (by Corollary 4.11). In fact, J:;! is a f3 is a restriction of Riemannian isometry between g~ and gf!. Since J:f3, we also have that J:f3 converges to a map J:;!. We then diagonalize the sequences so that J:f3 converges for every a and {3. Notice that since J~Ct 0 f3 = idf3 : Ef3 -+ ef3, the identity embedding, we must have that

J:

J:

J!Ct 0

J~ = idf3 .

4.2. Constructing approximate isometries Fke;r of large balls in Mk into Me. We can now construct approximate isometries Fke;r between the ball B (Ok, r) C M k and an open set in Mf. for sufficiently large k and f. (We shall use some results about the center of mass given in Section 5 of this chapter.) The following is the main result of this subsection. PROPOSITION 4.33 (Existence of an approximate isometry on a large ball). For every r > 0, c > 0, and p > 0 there exists ko = ko (r,c,p) > 0 such that for k, f > ko there is a diffeomorphism

Fke;r : B (Ok, r)

-+

Fke;r (B (Ok, r))

c

Me

which is an (c,p)-approximate isometry.

Let FfJ. : B;:

-+

Be

(from a ball in Mk to a ball in Mf.) be defined by

Ct . H eO Ct (HkCt )-1 . F kf.7 Roughly speaking, we construct the desired map Fkf.;r by averaging the local maps Ffj for a ~ A (r). Notice that in terms of the (inverse) coordinate systems (Ef3,Hf) and (e f3 where f3 satisfies B~nBk i- 0, the map

,iif),

Ffj corresponds to the map

Ct .. Ef3 F kf.,f3

-+

E-f3

between Euclidean balls defined by Ffj,f3

~ =

(4.10)

(iif) (iif)

H~

-1 0

Ffj

-1 0

He (H;:r 1

0

0

0

Hf

168

4.

PROOF OF THE COMPACTNESS THEOREM

Hence we have the following local property which is a key step to Proposition 4.33. PROPOSITION 4.34 (Local maps converge to the identity in a sense). If a and j3 are such that Bk n Be =f. 0 for k sufficiently large, then the maps F[d,{3 : E{3 ~ jj{3 converge to the identity (inclusion) map id{3 as k, £ ~ 00. N ow we proceed to average the local maps to construct a map on a large ball. To apply Proposition 4.53 on averaging maps, we need to construct a partition of unity subordinate to the covering {Bk} ok (x); by the choice of balls in Lemma 4.18 we can apply Proposition 4.53 and conclude the existence of Fkf;1' when k, f are large enough. The map Fkf;1' is smooth with all its derivatives IV7P Fkf;1'1 bounded by constants Cp +1 independent of k. From the construction of the weights c/>k we have Fke;1' (Ok) = 0(. Note by Proposition 4.53 and the definition of the center of mass, Fke;1' satisfies (i) Fkf;1' (x) is the minimizer y E Me of A(1')

Ix (y)

=

L c/>k (x) d~f (y, Fke (x)),

a=O

(ii) Fkf;1' (x) is the solution y

E Me of

.4(1')

L ¢k (x) exp;1 Ffj (x) = 0,

(4.12)

a=O

where the exponential map is with respect to ge. With respect to the coordinates (E!3, equation (4.12) can be written in E(3 as

He) ,

L a::;.4(1')

c/>k

0

He (X) exp-1

(j

Fkf;roHk (X)

Ffj

0

He (X) = O.

4.

170

PROOF OF THE COMPACTNESS THEOREM

Define the local versions of Fkejr by

G~ejr ~ We may pull back to

L

o 0, E > 0, and pEN U {O} there exists ko = ko (r, E,p) such that for

(3:::;A(r),

!Vp (G~ejr -

idj3)! :::;

E

for all k, P ~ ko, where V and 1·1 are the covariant derivative and norm with respect to the Euclidean metric on E/3.

As a corollary we have the following. COROLLARY 4.36 (Fkejr is a local diffeomorphism). There exists ko = ko (r) such that if k,P ~ ko, then FkejrlB!3 is a diffeomorphism for each k (3:::;A(r). PROOF. If G~e-r is sufficiently close to the identity map, then it must be injective since its derivative is nonsingular. 0 Now we turn to proving that given (c,p) and r, for k, P large enough, Fke,r is an (E,p)-pre-approximate isometry. First we have the following general result. LEMMA 4.37 (Limit of almost-identity pullbacks). Let cPk : U ~ U C jRn be diffeomorphisms, let id : U ~ U be the identity map, and let {hk} kEN and hoo be Riemannian metrics on U. Suppose hk and hoo are uniformly equivalent to the Euclidean metric for all kEN and their derivatives (covariant derivatives with respect to the Euclidean metric) are uniformly bounded. If cPk ~ id and hk ~ hoo in Coo uniformly on compact sets, then for every E > 0, pEN, and compact set K c U, there exists ko = ko (C, p, K) such that if k ~ ko, then

4.

THE LIMIT MANIFOLD

(M~,goo)

171

where 1·1 is the Euclidean norm and \7 is the Euclidean covariant derivative (i.e., partial derivative). PROOF. Let x = let 0 and p > 0 there exists K, = K, (c, p) such that

1\7~k (Fkl;r9l - 9k) I gk ~ c for all q ~ p if k, R. ;:::: isometry.

K,

(c, p) . Hence Fkl,r is an (c, p) -pre-approximate

PROOF. We work in a coordinate chart

(E/1, He) . By Proposition 4.35,

for any c > 0 there exists ko = ko (r, c) such that l\7q ( G~l;r - id/1) I < c if k,R.;:::: ko. By Proposition 4.32, the metrics = * 9k are uniformly equiv-

(He)

ge

E/1.

alent in the Coo-norm to the Euclidean metric on Thus it suffices to estimate the partial derivatives of Fkl -r9l - 9k using the Euclidean metric. Since

G~h ----+ id/1 and 9~ ----+ 9~,

r

we

~ay use Lemma 4.37 to conclude that

(G~l;r ~~ ----+ 9~ in the Coo-Euclidean norm as k, R. -

timates now follow from the fact that 9~ as k ----+ 00.

----+

00.

The desired es-

9~ in the Coo-Euclidean norm D

172

4.

PROOF OF THE COMPACTNESS THEOREM

Next we turn to prove that Fkf.;r is a diffeomorphism. Since Ff.k is the inverse of Fre, by Proposition 4.34, for each (3 we have Fre,(3 ~ id(3 and FCk ,(3 ~ id(3 when k, £ ~ 00. Then by a simple argument using Proposition 4.54 we conclude that Ff.k.roFkC,r and Fkf.,roFf.k,r both approach the identity map when k, £ ~ 00. It follows that Fkf.,r is invertible. Now it follows from the inverse function theorem and Proposition 4.35 that Fke~r is an (c',p)-pre-approximate isometry. Hence, given (c,p) and r, there is a ko such that Fkf;r is an (c, p )-approximate isometry for k, £ ~ ko. Proposition 4.33 is proved. 4.3. The directed system. We are now in a position to construct a directed system whose direct limit will give us the limit manifold (M~, goo). We first show that, after passing to a subsequence, the existence of approximate isometries whose compositions are also approximate isometries, as close as we like to isometries. PROPOSITION 4.39 (Metrics are almost isometric on large balls). There

exists a subsequence

{(Mk' gk) } J

there exists jo

J

JEN

such that for any c > 0 and pEN

= jo (c,p) EN such that if j > jo, then there exist maps Wj: B (Ok j ,2j ) ~ B (OkJ+1'2j+l)

with

Wj (Ok j ) = OkJ+l such that for any £ E N the composition map Wj,f.

~ Wj+C-I 0 ' "oWj+1 oW j: (B (Ok j ,2 j ) ,gkj ) ~

(B (Ok jH,2 jH ) ,gkjH )

is an (into) (c,p)-approximate isometry. PROOF. We may assume that Cj is increasing as j increases and that

Co ~ 1. We shall inductively define the subsequence

{(M k., gk) } J

J

JEN

. It

j)-

is sufficient to construct a sequence {Wj }jEN such that Wj is a ( C;12- j , approximate isometry. In this case we can use Corollary 4.8 to see that Wr,f. is a (Cr ~r~;-l C i- 12- i , r )-approximate isometry. In fact, since Cj is increasing in j, we have

r+f.-l 00 00 1 2- i < C ~ C:- 12- i < ~ 2- i = 21- r Cr ~ C:~ t r~ t -~ , i=r i=r i=r which implies that

(2 l - r , r )-approximate isometry. We also have by Wo (B (Oka' 1)) C B (Ok!' (1 + COl) 1/2) C B (Ok!, 2)

wr,f

Proposition 4.4 that since

is a

4.

THE LIMIT MANIFOLD

(M~,goo)

173

Similarly, we have

again since C r 2: 1. Hence, given c > 0 and p > 0, we can take jo ::;:: max (1 -log2 E',p) . For j = 0, make ko large enough so that Fko£; 1 is a (Co 1 , 0) -approximate isometry for any f 2: ko (we can do this by Proposition 4.33). By induction, suppose we can do this up to k r . We then make kr+1 large enough so that Fkr+1£;2r+1 is a (C;~12-(r+1), r + I)-approximate isometry for any f 2: k r+1' Now choose w1• ::;:: Fkrkr+l;2r. 0 Let us re-index the subsequence taken in the previous proposition so that it is once again indexed by k and the index of W coincides with the index of M. We may now take the final subsequence to get metrics on B (Ok,2 k ) C Mk which will become the limit metric. Since Wj,£ are approximate isometries, we may consider the sequence {W J* £gj+£}OO ,

£=0

of Rie-

mannian metrics on B (OJ, 2j) . Since Wj,e are (c, p )-approximate isometries independent of f, the metrics W;,egj+£ are uniformly bounded together with its derivatives and so there is a subsequence in f so that they converge to a limit metric gj,oo on B (OJ, 2j) . We can use this argument diagonally to get the following proposition. PROPOSITION 4.40 (Existence of almost-isometric limiting metrics on large balls). There exist a subsequence {kj } ~1 and Riemannian metrics gkj,oo on B (Ok j , 2kj) such that for every c > 0 and p 2: 0 there exists jo = jo (E',p) such that

for all r :S p and f 2: 0 if j 2: jo.

PROOF. This essentially follows from Lemma 4.37 again.

o

We again re-index, replacing Wkj with Wj, which equals Wkj+l 0 . . . 0 Wkj+2 0 Wkj+l 0 Wk j in the old notation, so that we have a sequence of maps Wj : B (OJ, 2j) ---t B (0}+1, 2}+1) (note that if j corresponds to k j , then we have shrunk the ball of radius 2kj to the ball of radius 2j ). We note that Wj : (B (OJ,2 j ) ,gj,oo) ---t (B (0}+1,2 j +1) ,g}+l,oo) is an isometry since

Iw;gj+1,oo - gj,ool :S Iw; (gj+1,oo - W;+I'" W;+£_lgj+e) 1 + IW;W;+1 ... W;+£_lgj+£ - gj,ool and both terms on the RHS go to zero as f

---t

00.

174

4.

PROOF OF THE COMPACTNESS THEOREM

4.4. Construction of the limit. We are now ready to construct the limit manifold (M~, 900) . Topologically, we take the direct limit

M~ ~ ~B (Ok,2 k ), where the directed system comes from the maps Wk. Note that since Wk are approximate isometries, they must be open embeddings. Hence Moo is a Hausdorff space by Lemma 4.17. We recall the embeddings h : B (Ok,2 k ) --+ Moo defined in (4.5). The coordinate maps HI: : gl< --+ B (Ok, 2k) induce coordinate maps H~,k ~ h 0 HI: : E Ol --+ Moo. Note that the transition maps

f3 )-1 (Hoo,k

0

Ol Hoo,k+r = (f3)-1 h 0 Hk

(3)-1

= (Hi

(4.13)

0

0

h+r 0 H kOl+r

Wk,r

0

HI:

are Coo diffeomorphisms (when the domain and range are suitably restricted) and hence they induce a Coo structure on Moo. Furthermore, since Wk,r are isometries between 9k,00 and 9k+r,00, we easily see that the transition maps are isometries and there exists a metric 900 on Moo such that I k900

= 9k,00·

We now show that {(M k,9k, Ok)} converges to (M~, 900, 0 00 ) . Given a compact set K c Moo, it must be contained in h [B (Ok,2 k )] for some k > 0 and hence must also be contained in Ii [B (Oi, 2i)] for all f. ~ k. We now claim that for every p there exists ko = ko (K,p) such that for any € > 0 sup xEK

for all a get

~

Iv~oo

p, k

~

IVOl (900 -

(1;;1)* 9k) I

0 is sufficiently small for later purposes). Since y is not in the cut locus of x, there exists a smooth family of unique minimal geodesics 'Ys : [0,1] --t M, such that 'Ys (0) = x and 'Ys (1) = a (s) for s E (-E, E) . Then 'Yo = 'Y is the minimal geodesic joining x and y. Define (7 : (-E, E) X [0,1] --t M by (7

(s, r) ~ 'Ys (r)

= expx

[r exp,;1 a (s)]

= eXPa(s)

((1 - r) exp~ts) x) ,

176

4.

PROOF OF THE COMPACTNESS THEOREM

where we are considering the curves "Is both in terms of geodesics from x and geodesics from a (s) . Note that

8(J 8(J . 8s (s,O) = 0, 8s (s, 1) = a (s), 8(J -1 81' (s, 1) = - eXPa:(s) x. The second derivative of

f

has the following expression.

LEMMA 4.44 (Hessian of the distance squared function). The Hessian of

f is given by

where J (1') is the Jacobi field along the geodesic between x and y parametrized on l' E [0,1] such that J (0) = 0 and J (1) = Y E TyM. PROOF. Given Y E TyM, define "Is and (J as above. Let J s (1') ~ ~~ (s, 1') be the Jacobi field along "Is; then J s (0) = 0 and J s (1) = a (s). Using 81T( -1 x, we comput e 81' s, 1) = - eXPa:(s) \7yexp;1 x

= - \7{)/8s ~~ (S,1')1

(s,1')={O,I)

= - (\78/81'JO) (1).

o The following is essentially the Hessian comparison theorem. LEMMA 4.45 (Hessian comparison). If the sectional curvatU1'e of (M n , g) is bounded above by K, then there exists a constant C = C (K) > 0 such

that for any y E B (4.14)

(x, (2VK)) 1r /

(Hess!) (Y, Y)

not in the cut locus we have

= -g (\7y exp;1 x, Y)

~ C 1Y12,

Y E TyM.

PROOF. By Lemma 4.44 we need to estimate

g((\78/8r J) (l),J(l)) Note that we can write Y and c E R Then

d id ="21 dr [g(J(r),J(r))] 1'=1 = IJI dr IJI 1'=1' 1

= y-L +ci' (1) , where Y -L J (1') = J-L (1')

is perpendicular to 'Y (1)

+ C1''Y (1'),

where J-L (1') is a Jacobi field satisfying J-L (0) = 0, J-L (1) = y-L and J-L (1') ..l 'Y (1'). Now it is clear that we only need to estimate IJI IJII1'=1 assuming that Y is orthogonal to 'Y (1) .

t.

177

5. CENTER OF MASS AND NONLINEAR AVERAGES

We compute using the Jacobi equation

~ IJI = !£ dr2

dr

(9 (\7 IJI

a/ar J , J))

(\7 a/arJ, J)2 9 (\7 alar \7 a/arJ, J) j\7a/arJj2 =IJI 3 + IJI + IJI 9

9 (\7 a/ ar J,J)2

IJI

3

9(R(J,i)i,J)

IJI

-

j\7 alar Jj2

IJI

+

'

so

::2 IJI + K bl 21J1 = IJI- 1 (IJ121i1 2K

- 9 (R (J, i) i, J))

+ 1J1-3 (IJI2j\7 a/arJj2 ~

where we used J (r) ..1

-y (r)

9

(\7 alar J, J)2)

0, to conclude that

IJ1 21-y12 K -

9 (R (J,

-y) -y, J)

~

O. The corresponding ODE for cp (r) is

cp" + K

(Iii = d (x, y)

1-y12 cp = 0

is a constant since 'Y is a geodesic) and has solutions

cp (r) = cp (0) cS K1 i'12 (r) where

+ cpt (0) sn Khl 2 (r),

fi sin ( JK,r ) { sn/i; (r) = r ~ sinh (J-K,r)

if if if

K,

> 0,

K,

= 0,

K,

< 0,

and COS

cs/i; (r)

=

{

(JK,r)

1 cosh (J-K,r)

if if if

K,

> 0,

K,

= 0,

K,

< O.

The functions sn/i; (r) and cs/i; (r) are the solutions to cp" + K,cp = 0 with sn/i; (0) = 0, sn~ (0) = 1, cS/i; (0) = 1, and cs~ (0) = O. Note that J /IJI is a unit vector (and has a limit as r ----t 0) and that

\7 alarJ (0) is well defined. From

i. IJI

= 9 (\7 a/arJ,

ir

I~I)

we know that the

limit limr-+o+ IJI is also well defined. Now we compare IJI (r) with the solution cp (r) which satisfies cp (0) = IJI (0) = 0 and cpt (0) = limr-+o+ i.IJI. Note that

cp (r) = cpt (0) snKhl 2 (r)

4. PROOF OF THE COMPACTNESS THEOREM

178

and that ¢ (r) is nonnegative for r E [0,1] when K ~ 0 or when K > Ii'I ~ 1r. Assuming Ii'I < 7r if > 0, we compute

JK

JK

K

0 and

(IJI' ¢ - IJI ¢')' = IJI" ¢ - IJI ¢" = (::2 IJI + K 1i'12IJI) ¢ 2: 0 for all r E [0, 1] . Integrating this from 0 to r gives us

IJI' (r) ¢ (r) -IJ (r)1 ¢' (r) 2: IJI' (0) ¢ (0) + IJ (0)1 ¢' (0) = 0, that is, for r E (0,1]' IJI' (r) 2: IJ (r)1 :

(4.15)

(~}.

Hence

IJI.!!:.-IJII dr

provided

2:

IJI 2(1) ¢' (1) ¢ (1)

r=1

JK 11'1 < 7r when K> O. This proves 1

-g (Vy exp; x, Y) CS K1

d

= IJId IJI r

I

2:

r=1

. 12 (1)

Note that"! is positive either when K sn 2 (1) KhI

JKIi'ol

cSK bl 2

snKIi'12 ~

(1)

2

(1) WI .

0 or when K > 0 and

D

O. COROLLARY 4.46 (Local convexity of the distance squared function). Suppose the sectional curvatures of (Mn, g) are bounded above by K. Then

the function f (y) ~ !d2 (x, y) is convex for any y E in the cut locus of x. PROOF.

This follows directly from (4.14).

B(x, 7r/ (2JK))

not

D

We also have COROLLARY 4.47 (Convexity of small enough balls). Suppose the sectional curvatures of (Mn, g) are bounded above by K. Then the ball B (0, r)

is convex if r

~ min {inj 0, 7r /

(2JK) }.

PROOF. Suppose x, y E B (0, r) . Let 'Y (t) be the constant speed minimal geodesic between x and y. We simply need to show that d ("( (t), 0) < r for every t. Consider the function f (z) = !d (0, z)2 . By Corollary 4.46, we have that

5.

CENTER OF MASS AND NONLINEAR AVERAGES

which implies that the maximum of f

h' (t))

179

occurs at the endpoints. Hence

d(O,'Y(t)):S max{d(O,x),d(O,y)} < r.

o The next lemma will be used in proving the smooth dependence of the center of mass in the next subsection. PROPOSITION 4.48 (On the derivatives of exp-1). Let (Mn, g) be a Riemannian manifold such that all derivatives of the curvature are bounded: l\7 t Rml :S Gt

for f

°

= 0,1,2, ....

There is a constant c (n) > such that for any p EM and x, y E B (p, r1), where r1 :S min inj (p) , c/ JCo} , if x is not in the cut locus of y, then (i) we have

{t

(4.16)

1\7~1 \7;; exp;l xl

:S Ct1+t2+1

for f1' f2

°

= 0,1,2, ... ,

where Ct = Ct (n, inj (p) ,f, Go, . .. , Gt) > are constants independent of x and y, and \7 y and \7 x are the covariant derivatives with respect to y and x, respectively; (ii) when x, y ~ p* E B (p, r1), we have (4.17)

(\7 x exp;l x : TxM

~ TyM) ~ (id : Tp.M ~ Tp.M) ,

(\7y exp;l x: TyM ~ TyM) ~ (-id: Tp.M ~ Tp.M) , where we use parallel translation to identify TxM and TyM with

Tp.M and to define the convergences above. PROOF. (i) Let w = {w k } be normal coordinates on B (p, r1). By Proposition 4.32 (Corollary 4.12 in [187]) we have in the coordinates w, (4.18)

~ (8ij ) :S (gij) :S 2 (8ij )

and

I(:;r:tgijl

:S Clol'

where a is a multi-index. In particular the Christoffel symbols (4.19)

rfj

satisfy

ij

80 k I I (8w)Or :S Cjol+1'

N ow we consider the exponential map exp : T M ~ M with expy z = x for Z E T M using the coordinates w; here we abuse notation in that x = (xk) stands for both a point in M and its coordinates in the coordinate system w (and the same for y). Define f (r, y, z) , :S r :S 1, by d2 fk k dfi dfj (4.20) dr 2 + r (f) dr dr = 0,

°

ij

fk (O,y,z) = yk, dfk k dr (O,y,z)=z.

4. PROOF OF THE COMPACTNESS THEOREM

180

Uk

Then (1.y,z)) theorem to

(Xk) =

expy z. We will apply the implicit function

F(x,y,z) ~ f(l,y,z)-x to prove that z = is a smooth function of (x, y) and that expy 1 x has the required derivative estimates. Consider the boundary value problem for the first-order ODE expy 1 x

d:Ifk _ hk dr ,

d~k +

rr

j

(1) hihj = 0,

fk (O,y,z) = yk, hk (O,y,z)

= zk.

From f(r,y,z) E B(p,rt) , Ih(r,y,z)l g = Izlg(p) = d(y,x), and !(Iij)::::; (9ij), we have Ifk(r,y,z)1 ::::; V2rl and Ihk(r,y,z)1 ::::; V2d(y,x). From the smooth dependence property for ODE (see Theorem 4.1 on p. 100 of Hartman [196]), fk (r, y, z) is a smooth function of (r, y, z) . Using the proof of Theorem 3.1 on p. 95 of [196] and (4.19), it follows from an induction argument on the order of derivatives that for r E [0, 1] ,

8 0 + 13

(4.21)

(8y)Ct (8z)f3 fk (r, y, z) ::::; CICtI+If3I+l'

8Ct +f3

(8yt (8z)f3 hk (r,y,z) ::::; CICtI+lf3l+1' Actually the proof of Theorem 3.1 on p. 95 of [196] implies the estimate above for Inl + 1,81 = 1. Let "'fy,z be the geodesic with "'fy,z (0) = y and (fr "'fy,z) (0) = z. Let Jy,z,z (r) denote the Jacobi field along the geodesic "'fy,z with Jy,z,z (0) = 0 and (:1rJy,z,z) (0) = Z E TyM. Then the covariant partial derivative in the direction is

z

Dzf (1, y, z) (i)

= :sf (1, y, z + Si)ls=o = Jy,z,z (1).

To show Dzf (1, y, z) : TyM ---- Texpy zM is invertible, we prove that there is a constant Co > 0 such that IJy,z,z (1)1 ~ Co Iii. This follows from the Rauch comparison theorem; here we give a proof using (4.15). As in the proof of Lemma 4.45, it suffices to prove that IJy,z,z (1)1 ~ Co Iii for those which

z

are orthogonal to z in TyM. From (4.15), we have (iJY;i:r)l)' r = I-I z sncolzl2 (r ) . S'mce I'Imr-->O IJy,z,z(r)1 ¢(r)

A. ( )

'I'

IJy,z,zl (1) ~ ¢ (1)

~ 0, where

1 we h ave IJy,z,z(r)1 > 1 an d ¢(r) _

----,

= Iii sncolzl2 (1)

~ Co

Iii·

181

5. CENTER OF MASS AND NONLINEAR AVERAGES

Using JzJ ::; 2c/VCo, we can choose Co ~ sn4c2 (1). We have proved (4.22) Now we can apply the implicit function theorem to F (x, y, z) -x. From Dyf

(1, y, z)

~

f (1, y, z)

{)z

+ Dzf (1, y, z) {)y = 0,

Dzf

(1, y, z)

~:

- id = 0,

we can take higher-order derivatives of the equations above to get formulas c 8"+ lj z ' f h . 1 d" 8",+{3 fk (1 lor (8y)"'(8xi III terms 0 t e partIa envatIves 0 f (8y)"'(8z)i3 ,y, z ) and (Dzf

(1, y, z))-l . From (4.21) and (4.22) we can estimate

{)Ot+!3z ({)yt ({)x)!3

_

3r for all q E B (p, r). Then there exists a unique minimizer cm {q1, ... , qk} of 1> in M. Furthermore we have cm {ql, .. . , qk} E B (p, 2r) and

uniformly in J-L 1, ... , J-Lk· PROOF. It is clear that for any q E M \ B (p, 2r) , we have 1> (q) > 1> (p) . Hence the minimizer of 1> exists and must be contained in B (p, 2r) . Note that if q E B (p, 2r), then q E B (qi, 3r). Since B (qi, 3r) C B( qi, 7r / (2VK)), by Lemma 4.43, the functions q 1---+ ~d2 (q, qd are strictly convex in B (p, 2r) . Since the weights J-Li are nonnegative and J-Ll +- . '+J-Lk > 0,1> is strictly convex in B (p, 2r) . Hence the minimizer must be unique. To see the last statement, we apply the first part of the statement to B (q*, r *) in place of B (p, r), where r * is small. We get that when ql,···,qk E B(q*,r*), we have cm(JL1, ... ,JLkdql, ... ,qk} E B(q*,2r*) for all

0

(J-Ll, ... ,J-Lk). By Lemma 4.43 we have k

grad1>(q)

=-

LJ-Liexp;lqi. i=l

The minimizer occurs at a point q where the gradient of 1> is zero, so that k

L J-Li exp;l qi = O. i=l The following proposition tells us about the derivatives of the center of mass.

5. CENTER OF MASS AND NONLINEAR AVERAGES

183

4.51 (Dependence of cm on weights and points). Suppose (Mn, g) is a Riemannian manifold such that all of the derivatives of the curvature are bounded: PROPOSITION

IV'I!

Rml

"'.5:.

Cl

for.e

= 0,1,2, ... .

Let iiI, . .. , lik be nonnegative weights with iiI + ... + lik > 0. There exists a constant c(n) E (O,~) such that for any p E M, if inj (q) > 3r for all q E B (p, r) , where r

< ~. Then we have the following.

(i) (Bounds on the derivatives of cm) The unique center of mass cm(J.t1, ... ,J.tk)

{ql,"" qkl

is a smooth function of ql, ... , qk E B (p, r) and iiI, ... , lik' The V'~V'~ -covariant derivatives of cm(J.t1 ,... ,J.tk) {ql, ... , qk} , with respect to ql, ... ,qk and iiI, ... , lik, satisfy

IV'~V'~ cm(J.t1, ... ,J.tk) {ql," . , qk}1

(4.24)

"'.5:.

0lal+I13I+1'

B B ) where V' q = (V' q1' ... , V' qk) and V' J.t = ( BJ.t1'···' BJ.tk and qal+I13I+1 are constants depending on n, inj (p), lal+I.BI, and Co,· .. ,qal+lf3I+1' (ii) Forql, ... ,qk E B(p,r) such thatqI, ... ,qk ---+ q* E B(p,r) (i.e., the points tend to each other), we have (a) (change in a weight has negligible effect on cm)

IV' J.ti cm(J.t1, ... ,J.tk) {ql, ... ,qkll

---+

0,

(b) (effect of the change in a point on cm) ( V'qi CmeJ.t1, ... ,J.tk)

---+ (z:/Li . j=1 J.t]

{ql, ... , qk} : Tqi M

id : Tq.M

---+ Tcm(

1'1,···,l'k

){q1, ... ,qk}M)

---+ Tq.M) ,

(c) (effect of the change in a weight and point on cm)

aa

( V' qj lij ---+

(L

k1

cm(J.tl,. .. ,J.tk)

.

i=1 J.t,

{ql, . .. , qkl : Tq;M

id : Tq.M

---+ Tcmeu

u ){q1 .... ,qd M )

r1>-"' 'rk

---+ Tq.M) .

The convergences above are defined using parallel translation to identify Tqi M with Tq. M. PROOF.

(i) We apply the implicit function theorem to the family of

maps defined by k

Gq1, ... ,qk'~1. ... ,J.tk

(q) ~

L Iii exp;1 qi· i=1

184

4. PROOF OF THE COMPACTNESS THEOREM

By the previous lemma, cmC/Ll, ... ,/Lk) {ql, ... ,qk} is the unique solution of the equation

G (q) ==. G ql,···,qk./Ll,···,/Lk (q) Consider the partial derivative

= 0.

k

\1qG = LJLi\1qexp;l qi: TqM

---7

TqM.

i=l

By Lemma 4.45, \1 qG is positive definite with smallest eigenvalue being bounded from below by a constant depending only on Co and JLl,·.·, JLk. It follows from the implicit function theorem that the unique solution cmC/Ll, ... ,/Lk) {ql, ... , qd is continuous in ql, ... , qk and JL 1, ... , JLk· To see that the \1~\1~-covariant derivatives of cmC/Ll, ... ,/Lk) {ql, ... , qk} are bounded, we compute the other partial derivatives of G:

Hence (4.25)

(\1qi G , 8~j

G) + \1qG· (\1q; cm {ql, ... ,qk}, \1/Lj cm {ql, ... , qd)

= 0,

where q = cm {ql, ... , qd. Thus

(\1 qi cm {qI, ... , qk}, \1/Lj cm {ql, ... , qk})

= - (\1 qG)-l ( \1q;G, 8~j

G) .

This and Proposition 4.48(i) implies (4.24) when lal + 1,81 = l. To bound the higher derivatives of cm { ql, ... , qk} , we argue inductively on the order of the derivative lad + 1,81. We take the appropriate derivatives of (4.25) of order lal + 1,81 - 1 with respect to ql,···, qk and JLl,···, JLk so that \1 qG . \1~\1~ cmC/Ll, ... ,/Lk) {ql, ... ,qd appears in the resulting equality. Then

\1~\1~ cmC/Ll ,... ,/Lk) {ql, ... , qk} can be expressed in terms of \1~~ exp;l qi with i l ::; lal + I,BI , \1~~\1 q exp;;l qi with £2::; lal + 1,81, \1~l\1~lcmC/Lh .. '/Lkdql, ... ,qd with lall + 1,811 ::; lal + 1,81- 1, and (\1qG)-l. Now it is easy to see from Proposition 4.48(i) that 1\1~\1~ cmC/Ll, ... ,/Lk) {ql, ... , qk}1 are bounded by constants (\~I+I,BI+l depending on n, inj (p), lal + 1,81 , and Co,· .. , qal+I,BI+1· (ii) When ql, ... , qk ---7 q*, by Lemma 4.50, we have a~; G ---7 O. By Proposition 4.48(ii) we have \1 q;G

---7

JLi id and \1 qG

---7

-

(I:~=l JLi) id.

5.

CENTER OF MASS AND NONLINEAR AVERAGES

185

This proves the first two convergences. Next we estimate 8 V qi -8 cm(J.!l,- .. ,J.!k) {ql,' .. , qd . J.,Li Since we have V qi 8~i G

= V qi eXPql qi

8

--+

id, by taking V qj-derivative of

8

-8 G + VqG· -8 cm{ql, ... ,qd

J.,Lj J.,Lj in (4.25) and taking the limit, we get

=

0

(t

J.,Li) V qj 88. cm(J.!l, ... ,J.!k) {qI, .. . , qk} = O. i=1 J.,LJ This proves the third convergence. id -

D

REMARK 4.52. (i) Note that in Euclidean space ]Rn, we have the following formula for the center of mass 1 cm(J.!l, ... ,J.!k) {qI, ... ,qk} = (J.,Llql + ... + J.,Lkqk). J.,Ll + ... + J.,Lk It is clear that there are many derivatives of the form

V~V~ cm(J.!l, ... ,J.!k) {ql, ... , qd , where lal + 1,81 ~ 2, whose lengths do not approach 0 as ql, ... ,qk --+ q* E ]Rn. (ii) Suppose h is another metric on M. From the proof of Proposition 4.51 and Remark 4.49(ii), it is not difficult to see that, as a function of (J.,Ll, ... , J.,Lk, ql, ... ,qk), the center of mass map cm9(J.!I,···,J.!k ) {ql, ... , qd is close to cm h(J.!l,···,J.!k ) {ql, ... , qk} on any compact set in Coo when g is very close to h on any compact set in Coo. We can use the center of mass to average maps. We have the following. PROPOSITION 4.53 (Averaging maps). Let (Nn,h) and (Mn,g) be Riemannwn manifolds such that all derivatives of the curvature of M a're bounded: e ~ Ce for £ = 0,1,2, ....

IV Rml

Let J.,Li (x) , i = 1, ... , k, be a finite sequence of smooth nonnegative functions on N with compact support in Ui C N and with bounded derivatives. Let W be an open set with closure W C UiJ.,Li l (0,00). Let Fi : Ui --+ M be a finite sequence of smooth functions with bounded derivatives. Suppose that, such that for any j with for any Xo E W, there exist io and ro E (0,

6Jco)

Xo E J-ljl (0,00) we have Fj (xo) E B (Fio (xo) , !ro) and inj (q) > 3ro for all q E B (Fio (xo) , ro). Then there is a function F : W --+ M defined uniquely by minimizing L:~=l J-li (x) d2 (F (x), Fi (x)) . In particular, F (x) satisfies

!

k

(4.26)

LJ.,Li (x)exP'Ftx) Fi (x) = O. i=l

186

4.

PROOF OF THE COMPACTNESS THEOREM

Furthermore F (x) is smooth and its derivatives Vi F (x) are bounded by constants depending on the bounds for IVii l1i (x) I with fi ::; f, the bounds for IVii Fi (x)1 with fi ::; f, and n, f, and Co,···, CHI'

PROOF. Fix Xo E W. By assumption, there exists a small neighborhood Vxo C W of xo, io, ro, and jl,'" ,jm such that Fjk (x) E B (Fio (xo) , ro) for k = 1, ... , m and all x E Vxo and such that I1j (x) = 0 for all j i= jl, ... ,jm and x E Vxo' By Lemma 4.50, for x E V we can define Fxo (x) on Vxo to be cm(". ""11 (xl

". ) {FJ' 1 (x), ... ,FJ· (x)} 7n

""W]7n

which is the composition of cm(. .) {qjl' ... , qjm} and I1jk = I1jk (x) , J.LJI,···,J.L]m qjk = Fjk (x). From the uniqueness of the center of mass it is easy to see that for any two points Xl, X2 E W, FXl (x) = FX2 (x) on VX1 n VX2 ' Hence this defines F : W - 4 M. We now prove the second part of the lemma. Let Xo E W; then, on VXo ' F (x) is the composition of cm(". .) {qjl'" . ,qjrrJ and I1jk = I1jk (x) , ""JI "'·,J.LJm qjk = Fjk (x). It follows from the chain rule and Proposition 4.51(i) that F (x) is smooth on Vxo and that F (x) on Vxo has the required derivative 0 bounds. We also used the following convergence property. PROPOSITION 4.54 (Average by cm of maps limiting to id limits to id). Let Bl C B2 be two open subsets ofJRn and let gk, where kEN, be a family of Riemannian metrics on B 2. Assume all derivatives of the curvatures of gk are uniformly bounded and gk - 4 goo on any compact set in B2 in Coo. Suppose Ft: : Bl - 4 B2, for a = 1, ... ,A, are sequences of smooth maps such that Ft: - 4 id uniformly on compact sets in C l for each a as k - 4 00. Let 11k be partitions of unities for each k on B l . For any compact set K C Bl we can define Fk : K - 4 B2 for k sufficiently large by letting Fk (x) be the center of mass of Ft: (x) with weight 11k (x) with respect to metric gk., i.e., Fk (x) is defined by A

L 11k (x) eXPF~(X) Ft: (x) ~ 0, a=l

where the exponential map eXPFk(X) is with respect to gk. Then Fk converges to id as k - 4 00 uniformly on any compact set in Bl in C l .

PROOF. Because some of IV~V~ cm(J.Ll, ... ,J.Lk) {ql,"" qk}l, lal + 1/31 ~ 2, do not approach 0 as ql, . .. ,qk - 4 q*, we will not prove this proposition using the composition employed in the proof of Proposition 4.53. By Remark 4.52(ii), cm9(k l() A()) {F1 (x), ... , Ft (x)} can be made arbitrarily close J.Lk x ,···,J.Lk X

to

cm9(ool() A()) J.Lk x ,···,J.Lk x

{F1 (x), ... , Ft (x)}

on any compact set x EKe Bl in

Coo when we choose k large enough. On the other hand, fix Xo E Bl and let in (B2' goo) . It follows from Remark

w be normal coordinates centered at Xo

6. NOTES AND COMMENTARY

187

4.49(ii) that by choosing k large and x to be in a very small neighborhood of Xo, we can make cm9(ool() A()) {Ff (x), ... , Ft" (x)} arbitrarily close /l-k x ,···,/l-k x

in the C 1-topology to the Euclidean center of mass

1() 1 A() (llk(x)Ff(X)+"'+Il:(x)Ft"(x)) , Ilk x + ... + Ilk x where we have identified the point FI: (x) E Bl with its coordinates in the coordimite chart w. Since Ilk (x) + ... + 11: (x) == 1, we have

1() 1 A() (llk(x)Ff(X)+"'+Il:(x)Ft"(x))-x Ilk x + ... + Ilk X

= Ilk (x) (Ff (x) - x) + ... + 11: (x) (Ft" (x) - x), which clearly converges to 0 on any compact set within the coordinate chart Now the proposition is proved. 0

w in C 1 when k ---+ 00.

6. Notes and commentary For some additional references on compactness theorems not cited in the previous chapter, see Cheeger and Gromov [73], [74], Gao [152]' Yang [374], [375], Anderson [4] and Anderson and Cheeger [6].

CHAPTER 5

Energy, Monotonicity, and Breathers Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things. - Sir Isaac Newton The most beautiful thing we can experience is the mysterious. It is the source of all true art and science. - Albert Einstein

Much of the 'classical' study of the Ricci flow is based on the maximum principle. In large part, this is the point of view we have taken in Volume One. As we have seen in Section 8 in Chapter 5 of Volume One, a notable exception to this is Hamilton's entropy estimate, which holds for closed surfaces with positive curvature. 1 Even in this case, the time-derivative of the entropy is the space integral of Hamilton's trace Harnack quantity, which satisfies a partial differential inequality amenable to the maximum principle. 2 Indeed, this fact is the basis for Hamilton's original proof by contradiction of the entropy estimate which uses the global in time existence of the Ricci flow on surfaces. 3 Originally, Hamilton's entropy was a crucial component of the proofs for the convergence of the Ricci flow on surfaces and the classification of ancient solutions on surfaces. Via dimension reduction, the latter result has applications to singularity analysis in Hamilton's program on 3-manifolds. An interesting direction is that of finding monotonicity formulas for integrals of local geometric quantities. Beautiful recent examples of this are Perelman's energy and entropy estimates in all dimensions. We briefly touched upon these estimates in Section 8 of Chapter 1 (Theorems 1.72 and 1.73) to motivate the study of gradient Ricci solitons. Perelman's energy is the time-derivative of a classical entropy ((5.64) in Section 4 below). Observe how the resulting calculation in Perelman's proof of the upper bound for the maximum time interval of existence of the gradient flow (Proposition 5.34) is reminiscent of Hamilton's proof of his entropy formula. In fact this upper bound says that a modified classical entropy is increasing (see (5.67)). Monotonicity formulas usually have geometric applications. In particular, Perelman proved that any breather on a closed manifold is a Ricci soliton of the same type. This statement includes the shrinking case which remained open until his work; previously, we have seen the proofs of the ISee [108], Proposition 5.44, for the case of curvature changing sign. 2See (5.70). 3See Theorem 5.38. 189

190

5. ENERGY, MONOTONICITY, AND BREATHERS

expanding and steady cases in Proposition 1.13. To prove the nonexistence of nontrivial breathers, Perelman needed to do a separate study of each type of breather. However, in each case, the method is the same: introduce a new functional, study its properties, and apply them to the proof that there are no nontrivial breathers of each type. All such functionals have three basic characteristics: • they are nondecreasing along systems of equations including the Ricci flow, • they are invariant under diffeomorphisms and/or homotheties, • their critical points are gradient Ricci solitons (of a different type in each case). Moreover, Perelman's functionals are successive modifications of his initial functional :F and are motivated by the consideration of gradient Ricci solitons of each type. So it is important to study the cases of the proofs successively in order to see how the evolutions of the functionals are used and how to modify the functionals gradually to define the entropy functional, which is the key to proving the shrinking case and where the proof follows essentially the same steps as the other two cases but uses the new functional. In this chapter, we shall discuss in detail the energy functional, its geometric applications and its relation with classical entropy; in the next chapter we study Perelman's entropy and some of its geometric applications. The style of this chapter is that of filling in the details of §§1-2 of Perelman [291] in the hopes of aiding the reader in their perusal of [291]. Throughout this chapter Mn is a closed n-manifold. 1. Energy, its first variation, and the gradient flow

The Ricci flow is not a gradient flow of a functional on the space 9J1et of smooth metrics on a manifold Mn with respect to the standard L2-inner product. 4 On the other hand, variational methods have played major roles in geometric analysis, partial differential equations, and mathematical physics. It was unusual that the Ricci flow, a natural geometric partial differential equation, should appear to be an exception to this. Perelman's introduction of the :F functional (defined below) solved the important question of whether the Ricci flow can be seen as a gradient flow. More precisely, as we shall see in this and the following section, the Ricci flow is a gradient-like flow; it is a gradient flow when we enlarge the system. The key to solving the question above is to look for functionals whose critical points are Ricci solitons, that is, fixed points of the Ricci flow modulo diffeomorphisms and homotheties (so that the ambient space in which we consider Ricci flow is 9J1et/Diff x ~+ instead of 9J1et). This is consistent with the point of view we adopted in Chapter 1 on Ricci solitons. 4An exception is when n Kahler-Ricci flow.

=

2 (see Appendix B of [111]), and more generally, for the

1.

ENERGY, ITS FIRST VARIATION, AND THE GRADIENT FLOW

191

1.1. The energy functional F. Let COO(M) denote the set of all smooth functions on a closed manifold Mn. We define the energy functional F : met x COO(M) ~ lR by (5.1) Note, in addition to the metric, the introduction of a function f. This embeds the space of metrics in a larger space. We shall sometimes follow the physics literature and call f the dilaton. Since ~ (e- f ) = (-~f + l~fI2) e- f , we see from fM ~ (e- f ) dJ.L = 0 that (5.2) So we have two other expressions for the energy: (5.3)

1M (R + ~f)e- f dJ.L = 1M (R + 2~f - I~ fI2)e- f dJ.L.

F(g, f) =

(5.4)

The second way of expressing the energy is motivated by the pointwise formula (5.43) in subsection 2.3.2 below. LEMMA 5.1 (Elementary properties of F). (1) Dirichlet-type energy. The geometric aspect of F is reflected by F (g, 0) = fM RdJ.L being the total scalar curvature and the function theory aspect of F is reflected by expressing it as

where w = e- f /2 \ which is a Dirichlet energy with a potential term. (2) Diffeomorphism mvariance. For any diffeomorphism 'P of M, we have F('P*g,f

0

'P) =F(g,f).

(3) Scaling. For any c > 0 and b \.

F (c 2 g, f + b)

=

cn - 2 e- b F (g, f).

EXERCISE 5.2. Prove the properties for the energy in the lemma above.

1.2. The first variation of F. We use the symbol 8 to denote the variation of a tensor. We shall denote the variations of the metric and dilaton as 8g = v E Coo (T* M @sT*M) and 8f = h E Coo (M), and we

5. ENERGY, MONOTONICITY, AND BREATHERS

192

define V ~ gij Vij. Routine calculations give

k 1 kl 8vrij (g) ="2 g (\1iVjl

(5.6)

p

(5.7)

8vrpj

(5.8)

8(v,h) (e-fdl-l)

+ \1jVil -

\1IVij) ,

1

= "2\1jV, =

(~ -h)e-fdl-l.

We calculate the last one, for example,

= -e-fh dl-l + e- f ~gijV'ij dl-l =

8(v,h) (e- f dl-l)

(5.9)

(~ -

h) e- f dl-l.

LEMMA 5.3 (First variation of F). Then the first variation of F can be expressed as

(5.10)

8(v,h)F(g,f)

1M Vij(Rij + \1i\1j f)e- f dl-l + 1M (~ -h) (2D.J-I\1JI +R)e- fd l-l, =-

2

where 8(v.h)F (g, f) denotes the variation oj F at (g, f) in the direction (v, h), i.e., 8(v,h)F(g'f) PROOF.

~

:

s

I

F(g+sv,J+sh).

8=0

Recall (Vl-p. 92), i.e., P !:l·r R tJ·· -- R Ppij -_!:l t Up r Pij - U pj

q rP + r ijq r ppq - r pj iq'

so that

8~j = \1p

(8rfj) - \1i (8r:j) .

Since \1 i\1j = o/lj - rfjOk as an operator acting on functions, we have

8 (\1 i\1j f)

(8rfj) \1pJ.

= \1 i\1j (8f) -

Hence, using (5.7),

8(~j + \1 i \1 j f) =

\1 p

(8rfj) - (8rfj) \1 PJ + \1i

= ef \1 p

(e -f 8rfj) + \1 \1 i

j

(\1 j

(8 f) - 8r:j)

(h - ~) .

We then compute

(5.11)

8 [(~j

+ \1i\1jf) e- f dl-l]

+e-f\1i\1j(h-~) + \1i\1j f) e- f (~ - h)

= [\1p(e- f 8rfj ) + (~j

1dl-l.

1.

ENERGY, ITS FIRST VARIATION, AND THE GRADIENT FLOW

193

So using (5.8),

8 [(R+~J)e-fdl-l]

= gij 8 [(Rij + 'Vi'VjJ) e- f dl-l] - 8gij . (Rij + 'Vi'VjJ) e- f dl-l

=

['Vp(e-fgij8rfj)+e-f~(h- ~)+(R+~J)e-f(~ -h)]dl-l

- Vij . (Rij

+ 'Vi'VjJ) e- f dl-l.

Note that 8rfj is a tensor and we do not need an explicit formula for it in the rest of the proof. By the Divergence Theorem, we have

8(v,h)F(g,J)

1M 8 [(R+~J)e-fdl-l] = 1M (-~ (e- f ) + (R + ~J) e- f ) (~ - 1M Vij . (Rij + 'Vi'VjJ) e- f dl-l, =

h) dl-l

o

from which the lemma follows. REMARK 5.4. By (5.11), the variation of (~j vergence when h = ~:

+ 'Vi'VjJ) e- f dl-l

is a di-

8 [( ~j + 'Vi 'V jJ) e- f dl-l] = 'V p (e -f 8rfj) dl-l. Note also the factor ~ - h in front of the second term in the RHS of (5.10). The significance of when this factor vanishes will be seen in subsection 1.4 below. By (5.8) we have LEMMA 5.5. Define the measure

dm ~ e- f dl-l. If the variations of 9 and f keep the measure dm fixed. that is, 8(v,h) (dm) = 0, then (5.12)

V=2h.

As a consequence of Lemma 5.3, we have COROLLARY 5.6 (Measure-preserving first variation of F). For variations (v,h) with 8(v,h) (e-fdl-l) = 0, we have

(5.13)

8(v,h)F (g, J) = -

1M Vij (~j + 'Vi'VjJ) e- f dl-l.

Notice in formula (5.10) for 8(v,h)F (g, J) the occurrence of the terms

(5.14)

RiJ ~ (Rcm)ij ~ ~j

(5.15)

Rm ~ R

+ 'Vi'Vjf,

+ 2~f -I'V fl2 .

5. ENERGY, MONOTONICITY, AND BREATHERS

194

The first quantity vanishes on steady gradient solitons flowing along V f, whereas the second appeared in (5.4).5 We call RiJ and R m the modified Ricci curvature and modified scalar curvature, respectively; they are natural quantities from the perspective of the Ricci flow. We can rewrite

F(g,f)

= fM9ijRiJe-fdP, = fM Rme-fdp,

and when V = 2h. 1.3. The modified Ricci and scalar curvatures. In this subsection we digress by showing RiJ and Rm are natural quantities. Consider a closed Riemannian manifold (Mn, g) and a metric 9 = e- ~ f 9 conformal to g. Let ~j = Rc (g)ij' ~j = Rc (g)ij' fl = R (g), and R = R (g) . The Ricci and scalar curvatures are related by (see for example subsection 7.2 of Chapter 1 in [111] or (A.2) and (A.3) in this volume) (5.16)

Rij

= ~j +

(1-~) ViVjf + ~fl.f9ij + n -; 2VifVjf n n n

n -; 21Vfl2 gij' n

'fracing this yields

(5.17)

R = e~f (R + 2 (n - 1) fl.f _ (n - 1) (n - 2) n n2

IV f12)

.

The volume forms are related by dp,g = e- f dp, and the total scalar curvature of 9 is given by

fM Rdp,g

= fM e- n;;:2 f ( R + (n - l~;n - 2) IV f12) dp"

where we integrated by parts, i.e., we used

f e- n;;:2 f fl.fdp, = n - 2 f e- n;;:2 f IV fl2 dp,. 1M n 1M Now consider the Riemannian product (Mn, g) x (Tq, h q) ,where (Tq, h q) is a flat unit volume q-dimensional torus. The formulas for the Ricci curvature and scalar curvature of metric e-n!qf (g + h q') are given by (5.16) and (5.17), respectively, where we replace n by n + q. If we take the limit as q ~ 00 while fixing (Mn,g) , then we obtain Perelman's modified Ricci tensor: (5.18) and Perelman's modified scalar curvature: (5.19)

lim R (e - n!qf (g + h q )) = R

q-->oo

+ 2fl.f - IV fl2 ,

5Earlier we also encountered these quantities in Chapter 1.

1.

ENERGY, ITS FIRST VARIATION, AND THE GRADIENT FLOW

195

where we think of Rc (e- n!qf (g + h q)) and R (e - n!qf (g + hq)) as quantities on M since they are independent of the point in Tq. The total scalar curvatures of x Tq, e- n!qf (g + h q)) limit to Perelman's F functional:

(M

+ hq)) df-L

lim {

R (e - n!q f (g

=

{ R (e - n!qf (g

q--+oo lMxTq lim {

q--+oo 1M 1Tq

_ 2 f e ~ (g+hq)

+ hq)) e- 2f df-Lhqdf-Lg

= 1M (R + IV' f12) e- f df-L =F(g,f)· Note that

gijR7] = R+!::..f = R m -!::..f + lV'fI2. There is an analogue of the contracted second Bianchi identity for R7] and Rm. In particular we compute V'i R 7] and 1

2V'j R

m

=

= V'i~j + V'iV'jV'd =

~V'jR + V'j!::..f + Rjk V'kf

1 2 1 1 V'j!::..f - 2V'j lV'fl + 2V'j R = 2V'j R+ V'j!::..f - V'jV'kfV'kf,

which imply (5.20) To understand this formula further, we define

V'*m : Coo (T* M ®s T* M)

---t

Coo (T* M)

by

(V'*ma)j LEMMA

measure dm

~

V'iaij - ajiV'd.

5.7. The operator V'*m is the adjoint of -V' with respect to the = e- f df-L. -

PROOF. For any symmetric 2-tensor aij and I-form bi ,

1M aij (-V'i) bje- f df-L = 1M bjV'i (aije- f ) df-L

= 1M bj (V'iaij - aij V'd) e- f df-L = 1M bj (V'*ma)j e- f df-L.

o Thus (5.20) implies the following, which is the analogue of the contracted second Bianchi identity.

5.

196

ENERGY, MONOTONICITY, AND BREATHERS

LEMMA 5.8 (Modified contracted second Bianchi identity). (5.21)

1.4. The functional pTl and its gradient flow. Unlike F (g, 1) , we can obtain a functional of just the metric 9 by fixing a measure dm on a closed manifold Mn; by a measure we mean a positive n-form on M.6 Define pn : 9J1et --t ~ by (5.22) where (5.23)

j

.

=;=

log

(dJ.L dm ) .

'." REMARK 5.9. The expression (5.23) makes sense because, given a fixed measure dm on Mn, we can define the bijection

COO (A nT* M) --t Coo (M) , W I-t c.p, where c.p is defined so that W = c.pdm (here we have used the fact that AnT;M ~ ~). Thanks to this, it is possible to define the quotient of two n-forms; e.g., if WI = c.pldm and W2 = c.p2dm, where c.p2 > 0, then we set

WI ..:... c.pI W2 c.p2 Without using the notation j, we can write the energy of the metric 9 as

Using the modified Ricci and scalar curvatures, we can rewrite

r

(g)

=

1M gij RiJdm = 1M Rmdm.

REMARK 5.10. Let c.p : M --t M be a diffeomorphism. Note that in general r(c.p*g) 'Ir(g). That is, by fixing the measure dm, we get pn (g) , which breaks the diffeomorphism invariance of F (g, 1) . In subsection 3.1 of this chapter we shall solve this problem by considering a functional>' (g) which is diffeomorphisminvariant. 6For a calculational motivation for fixing the measure, see the notes and commentary at the end of this chapter.

2. MONOTONICITY OF ENERGY FOR THE RICCI FLOW

197

From (5.13) we have 8v :F"'" (g)

(5.24) where

=-

1M Vij (~j + '\l/VjJ) dm,

f is given by (5.23). The L2-inner product on 9J1et, using the metric

9 and the measure dm, is defined by

(aij,bij)m (g)

~ 1M (aij,bij)gdm.

Then by (5.24) we have

'\l:F"'" (g) = where is

f

-(~j

+ '\li'\ljJ),

is given by (5.23). Hence (twice) the positive gradient flow of Fm

(5.25) (5.26) We can also write the above system as (5.27)

~g at .. = ~J

-2

[R-. + '\l.'\l.J log (dJ.l)] dm' ~J

~

We shall call an equation of the form (5.25) by itself, for some function f, a modified Ricci flow. It is clear from taking Vij = -2 (~j + '\li'\ljJ) in (5.13) that we obtain the following. PROPOSITION 5.11 (P evolution under modified Ricci flow). Suppose 9 (t) is a solution of (5.25)-(5.26). Then (5.28)

:t:F"'"(g(t)) = 2

1M I~j + '\li'\ljfI2 e- f dJ.l.

This is Perelman's monotonicity formula for the gradient flow of P . We may rewrite (5.28) as

:t:F"'" =

:t 1M

Rmdm = 2

1M IR iJI2 dm.

Note that for a general measure dm, solutions to the initial-value problem for the gradient flow may not exist even for a short time; however, as we shall see, this will not cause us problems in applications.

2. Monotonicity of energy for the Ricci flow For monotonicity formula (5.28) to be useful, we need a corresponding version for solutions of the Ricci flow. In this section we show that solutions to equations (5.25) and (5.26), if they exist, differ from solutions of the Ricci flow by the pullback by time-dependent diffeomorphisms. Thus ~~is gives a . . .. monotonicity formula for the energy of the Ricci flow.

198

5.

ENERGY, MONOTONICITY, AND BREATHERS

2.1. A coupled system equivalent to the gradient flow of P. There is a coupled system, i.e., (5.29)-(5.30), induced from the gradient flow (5.25)-(5.26) obtained simply by computing the evolution equation for f = log (d/1,j dm) . As we shall see, this coupled system is equivalent to the gradient flow. LEMMA 5.12 (Measure-preserving evolution of f under modified RF). The function f (t) in a solution (g (t) , f (t)) of the gradient flow of P (5.25) and (5.26) satisfies the following equation: af at = -~f - R. PROOF. We calculate a ( -dJ.L ) -a f = -log at at dm

1 i' agij = _g3 i . = _g3_ (~. + 'V-'V-!).

at

2

3

t

3

o Related to the above calculation, we have the following. EXERCISE 5.13. Show that if forms, then

WI

~ log (WI)

(t) and

= %t WI

_

W2

(t) are time-dependent n-

%t W2 ,

at W2 WI w2 where the quotient of two n-forms is defined as in Remark 5.9. Hence we consider the coupled modified Ricci flow

a

(5.29)

at gij

(5.30)

af at

= -2(~j

=

-~f -

+ 'Vi'Vj!), R.

Note that the first equation is a modified Ricci flow equation whereas the second equation is a backward heat equation. LEMMA 5.14. The coupled modified Ricci flow equations (5.29)-(5.30) are equivalent to the gradient flow (5.27). PROOF. If g (t) is a solution to (5.27), then by Lemma 5.12, (g (t), f (t)) , where f = log (dJ.L/dm) , is a solution to the system (5.29)-(5.30). Conversely, if (g (t) , f (t)) is a solution to the system (5.29)-(5.30), then dm ~ e- f dJ.L satisfies

~ (dm) = (- af at at

- R - b. f ) e- f dJ.L

= o·

' that is, 9 (t) is a solution to (5.27) with dm as defined above. Hence, by (5.28), if (g (t), f (t)) is a solution to (5.29)-(5.30), then (5.31)

:t:F(g(t) , f (t))

=2

1M I~j + 'V i'V j fI2 e- f dJ.L.

o

2.

MONOTONICITY OF ENERGY FOR THE RICCI FLOW

199

2.2. Correspondence between solutions of the gradient flow and solutions of the Ricci flow. 2.2.1. Converting a solution of the gradient flow to a solution of Ricci flow. We first show that solutions of the gradient flow, if they exist, give rise to solutions of the Ricci flow with the same initial data (Lemma 5.15). In particular, suppose we have a solution (g (t) (t)) of the flow (5.25) and (5.26) on [0, T]; then we can obtain a solution g(t) of the Ricci flow on [0, T] by modifying 9 (t) by diffeomorphisms generated by the gradient of I (t) .

,I

LEMMA 5.15 (Perelman's coupling for Ricci flow). Let (g(t),/(t)) be a solution of (5.25) and (5.26) on [0, T]. We define a I-parameter family of diffeomorphisms W(t) : M ~ M by

d

-

dt w(t) = V' g(t)i(t) ,

(5.32)

(5.33)

w(o)

= idM .

Then the pullback metric g(t) = w(t)*g(t) and the dilaton f(t) = satisfy the following system:

ag = -2 Rc

( 5.34)

at at

af

(5.35)

= -I)"f

10 w(t)

'

+ IV' fl 2 -

R.

REMARK 5.16. Basically we can see this from the facts that LVI9 = 2V'V' f and Lv f f = IV' fl2 . For the sake of completeness we give the detailed calculations below. PROOF. First note that by Lemma 3.15 of Volume One the system of ODE (5.32)-(5.33) is always solvable. We compute

aa! = (a-) a! + w*

w* (LVgfg)

To obtain the equation for

= -2w* (Rc (g)) = -2Rc (g).

M, we compute

af = a(! w) = alow / (v f' w aw) at at at +\ 'at g 0

0

f)

= (-!il - Ii) = -I)"f - R

0

w+ I(vI) wi; 0

+ IV' fl2 ,

where barring a quantity indicates that it corresponds to 9 (t) .

0

So a solution to the gradient flow (5.25)-(5.26) yields a solution to the Ricci flow-backward heat equation system (5.34)-(5.35). Note that we can first solve the Ricci flow (5.34) forward in time and then solve (5.35) backward in time to get a solution of (5.34)-(5.35); this will be useful in applications.

200

5. ENERGY, MONOTONICITY, AND BREATHERS

2.2.2. Converting a solution of Ricci flow to a solution of the gr'adient flow. Now we show the converse of Lemma 5.15 by reversing the procedure ofthe last subsection. Given a solution 9 (t) of the Ricci flow (5.34) on [0, T], we can construct a solution (g(t), f(t)) of the gradient flow (5.25) and (5.26) on [0, T] by modifying the solution g(t) by diffeomorphisms. In doing so, we also need to solve a backward heat equation with initial data at time T.

tt

5.17. Let g(t) be a solution of the Ricci flow = -2 Rc on [0, T] and let h be a function on M. (i) We can solve the backward heat equation backwards in time LEMMA

of

at

= -~f

f{T) =

2

+ l\7fl -

t

R,

E

[0, T],

h·

(ii) Given a solution f (t) to the equation above, define the I-parameter family of diffeomorphisms cI>(t) : M ---t M by d dt cI>(t)

(5.36)

=

-\7g(t)f(t),

cI>(0) = idM,

which is a system of ODE and hence is solvable on [0, T]. 7 Then the pulledback metrics g(t) = cI>(t)*g(t) and the pulled-back dilaton J(t) = f 0 cI>(t) satisfy (5.29) and (5.30).

(i) Let 7 = T - t. To get the existence of solutions to equation (5.35), we simply set PROOF.

(5.37) and compute that (5.38)

au = ~u-Ru

-

07

'

which is a linear parabolic equation and has a solution on [0, T] with initial data at 7 = O. Indeed, (5.38) follows from

au au of ( 2) 07 = - at = u at = u -~f + 1\7fl - R = ~u -

Ru.

(ii) Let g(t) be a solution of the Ricci flow and let f(t) be a solution of equation (5.35). One can verify that they satisfy (5.29) and (5.30) as in the proof of Lemma 5.15. D 2.2.3. The adjoint heat equation. Let 9 (t) be a solution of Ricci flow and let D ~ %t - ~ be the heat operator acting on functions on M x [0, T] , where M x [0, T] is endowed with the volume form dp,dt. Its adjoint is

(5.39)

D*

==. -

~ at - ~ + R

7Again see Lemma 3.15 of Volume One.

2. MONOTONICITY OF ENERGY FOR THE RICCI FLOW

201

since

faT 1M bOadp,dt = faT 1M b (:t - ~ )

adp,dt

1M [a ( - :t - ~) bdp, - ab %t dP,] dt = faT 1M aO* bdp,dt

= faT

for C 2 functions a and bon M x [0, TJ with compact support in M x (0, T), where we used %t dp, = - Rdp,. By (5.38), if (g (t), f (t)) is a solution to (5.34)-(5.35), then u = e- f satisfies the adjoint heat equation (also known as the conjugate heat equation)

(5.40) It is often better to think in terms of u than in terms of f since u satisfies the adjoint heat equation. In particular, the fundamental solution to the adjoint heat equation is important. 2.3. Monotonicity of F for the Ricci flow. In this subsection we give two proofs of the monotonicity of energy for Ricci flow. In the next section we give an application of this formula to the nonexistence of nontrivial breather solutions. 2.3.1. Deriving the monotonicity of F from the mono tonicity of :pn. By the diffeomorphism invariance of all the quantities under consideration, the monotonicity formula for the gradient flow implies a monotonicity formula for the Ricci flow. This involves a function f (t) obtained by solving the backward heat equation (5.35). LEMMA 5.18 (F energy monotonicity). If (g (t), f(t)) is a solution to (5.34)-(5.35) on a closed manifold Mn, then

(5.41)

:t F (g (t) ,f(t)) = 2

1M IRij + \7i\7j fI2 e- f dp,.

PROOF. Since (g(t),f(t)) is a solution to (5.34)-(5.35), (g(t),](t)) , defined by g(t) ~ cp*(t)g(t) and J(t) = f(t) 0 cp(t), where cp(t) satisfies (5.36), is a solution to (5.29)-(5.30). Now F (g, f) = F (g, l) , so that by (5.31), we have

d d dt F (g (t) ,J (t)) = dt F (9 (t) , f-(t))

= 2 f litj + ViVj]l= e- f djl

1M

=2

9

1M I~j + \7i\7j fI2 e- f dp,.

202

5.

ENERGY, MONOTONICITY, AND BREATHERS

o 2.3.2. Deriving the monotonicity of F from a pointwise estimate. This second approach to the energy monotonicity formula is based on the pointwise formula (5.43), which is a simpler version of the evolution equation for Perelman's backward Harnack quantity (6.22). Let (g (t), f (t)) be a solution to (5.34)-(5.35). Let u = e- f and V ~ (2D..f -

(5.42)

IV fl2 + R)u = Rmu,

where R m is the modified scalar curvature defined by (5.15),8 so that F=

1M Vd/-L.

LEMMA 5.19 (Bochner-type formula for V). If (g (t), f (t)) is a solution to (5.34)-(5.35) and if u = e- f, then we have the pointwise differential equality: (5.43) This calculation, which we carry out below, is in a similar spirit to that of the calculations for the differential Harnack quantities considered in §10 of Chapter 5 in Volume One and Part II of this volume. To obtain (5.41) from the lemma, we compute

:t F (g (t) , f (t))

=

! 1M

V d/-L

1M (:t V - RV) d/-L = 1M 21 R ij + Vi V jfI 2ud/-L.

=

PROOF OF THE LEMMA. Using definition (5.42) and gij %tr~j = 0, a direct calculation shows that

~Rm = ~(2D..f - IV fl2 + R) 8t

8t

(M)

= 4~jViVjf + 2D.. 8t

- 2RijVdVjf - 2V

= 4RijViVjf - D..(2D..f -IV fl2

(M) 8t

.Vf

M + at

+ R) + D..IV fl2 + 2V D..f . V f

2 8R - 2~jVdVjf - 2V(IVfl - R)Vf + at

-

D..R.

From the above we have

(:t + D.) R

m

=

21~j + V i V j fl 2 + 2V R m . V f.

8The above V is not to be confused with our earlier V, which was the trace of the variation v of g.

3.

STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED

203

On the other hand,

av at+~V-RV=

(8t+~Rm a R m ) u+ (au at +~u-Ru ) R m +2'VRm ·'Vu.

Plugging in the equation for

(gt

+~) Rm and using (5.40), we have

The last two terms cancel each other since 'V f lemma.

= - 'Vu/u,

which yields the 0

REMARK 5.20 (Backward heat-type equation for modified scalar curvature). From the proof of the lemma, we have (5.44) Note the similarity to the equation ~~ = ~R + 21Rcl 2 , except now we have a backward heat-type equation.

3. Steady and expanding breather solutions revisited A solution g(t) of the Ricci flow on a manifold Mn is called a Ricci breather if there exist times tl < t2, a constant QI > 0 and a diffeomorphism cp : M -+ M such that

When QI = 1, QI < 1, or QI > 1, we call g(t) a steady, shrinking, or expanding Ricci breather, respectively. Recall that g(t) is a Ricci soliton (or trivial Ricci breather) if for each pair of times tl < t2 there exist QI > 0 and a diffeomorphism cp : M -+ M (QI and cp will in general depend on tl and t2) such that g(t2) = Qlcp*g(tl). Note that if we consider the Ricci flow as a dynamical system on the space of Riemannian metrics modulo diffeomorphisms and homotheties, the Ricci breathers correspond to the periodic orbits whereas the Ricci solitons correspond to the fixed points. Since the Ricci flow is a heat-type equation, we expect that there are no periodic orbits except fixed points. A nice application of the energy monotonicity formula is the nonexistence of nontrivial steady or expanding breather solutions on closed manifolds (§2 of [297]). This was first proved by one of the authors in [218] (see Proposition 1.66 in this volume). In the next chapter we shall see the application of Perelman's entropy formula to prove shrinking breather solutions on closed manifolds are gradient Ricci solitons (§3 of [297]). Hence we confirm the above expectation.

5.

204

ENERGY, MONOTONICITY, AND BREATHERS

3.1. The infimum A of F. Suppose we have a steady breather solution to the Ricci flow with 9 (t2) = rp* 9 (tl) for some tl < t2 and diffeomorphism rp. One drawback of the energy monotonicity formula is that in general the solution f to (5.35) has f (t2) =1= f (td 0 rp, so that in general, F (g (t2)' f (t2)) =1= F (g (tl)' f (td)· By taking the infimum of F among f, we obtain an invariant of the Riemannian metric 9 which avoids this trouble. DEFINITION 5.21 (A-invariant). Given a metric 9 on a closed manifold

M n , we define the functional A : 9J1et --+ lR by (5.45)

Taking w = e- f /2, we have (5.46)

A(g) = inf {9(9, w) :

1M w dp, = 1, w > o} , 2

where, as in (5.5),9 (5.47)

Thus, when we fix 9 and minimize F (g, 1) among f, we are minimizing a Dirichlet-type functional and we get an eigenfunction-type equation for w. Aspects of this point of view are discussed in the next two lemmas. Note that the variation of 9 (g,') is given by

~O(0,h)9 (g, w) =

r 1M

r 1M

(4"'Vw·"'Vh + Rwh) dp, = (-4flw + Rw) hdp" 2 where h = ow. Hence the Euler-Lagrange equation for (note that we dropped the positivity condition on w) A (g)

~ inf { 9 (g, w) : 1M w 2 dp, = 1}

is (5.48)

Lw ~ -4flw

+ Rw = A (g) w.

LEMMA 5.22 (Existence and regularity of minimizer of Q). There exists a unique minimizer Wo (up to a change in sign) of (5.49)

inf { 9 (g, w) :

1M w dp, = 1} . 2

The minimizer Wo is positive and smooth. Moreover, 9In view of Lemma 5.1(1), the monotonicity of:F exhibits a dichotomy, it is analogous to both the monotonicty of the total scalar curvature under its gradient flow, g = - 2 (Rc - %g) , and the monotonicity of the Dirichlet energy under its gradient flow, the backward heat equation ftw = -~w. In this sense, the monotonicity of :F exhibits a beautiful synthesis of geometry and analysis.

it

3. STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED

205

(1) the minimum value ),(g) of9(g,w) is equal to ),1(g), where),1 (g) is the lowest eigenvalue of the elliptic operator -4~ + R, and (2) Wo is the unique positive eigenfunction of (5.50)

-4~wo

+ Rwo =

),1

(g) Wo

with L2 -norm equal to 1. PROOF. To establish the existence of a minimizer Wo of (5.46), one takes a minimizing sequence {Wi}~1 of (5.46) in W 1,2 (M). There then exists a subsequence {Wi}~l which converges to Wo E W 1 ,2 (M) weakly in W 1,2 (M) and strongly in L2 (M) (by the Sobolev embedding theorem). Since

o ~ 1M 1\7 (Wi -

WO)12 dJL

= 1M I\7WiI 2 dJL + 1M l\7wol 2 dJL - 2 1M (\7Wi' \7wo) dJL, weak convergence in W 1,2, we have limi-too fM (\7wi' \7wo) dJL

by the fM l\7wol 2 dJL exists, hence

{ l\7wol 1M

2

dJL

~ li~inf ~-t00

{ l\7wil 1M

2 dJL.

On the other hand, by the strong convergence of {Wi}~1 in L2, we have .lim { Rw;dJL = { RW5 dJL ,

~->001M

1M

{ W5dJL = lim ( w;dJL = 1. 1M ~-t001M Hence Wo is a minimizer of (5.46) in W 1,2 (M) , and Wo is a weak solution to the eigenfunction equation (5.48). By standard regularity theory, Wo E Coo. We also have that any minimizer is either nonnegative or nonpositive, since otherwise ± Iwol is a distinct smooth minimizer which agrees with Wo on an open set, contradicting the unique continuation property of solutions to second-order linear elliptic equations. We now prove Wo is unique up to a sign. Without loss of generality, we may assume below that Wo is nonnegative. Call a minimizer w of 9 with fM w 2dJL = 1 a normalized minimizer. If the nonnegative normalized minimizer is not unique, then there exist two normalized minimizers Wo ~ 0 and WI ~ 0 with fM wow 1dJL = O. Then W2 = awo +bWI is also a normalized minimizer for all a, b E lR such that a2 + b2 = 1. Indeed, since Wo and WI satisfy the linear equation (5.50), so does W2 = awo+bwI, and fM w~dJL = 1. Now it not hard to see that there exist a and b such that W2 changes sign. In particular, if there are points x and y such that WI (x) = CWo (x) and WI (y) = dwo (y), where e =1= d and Wo (x) > 0 < Wo (y), then by choosing a and b with a2 + b2 = 1 such that a + bc and a + bd have opposite signs, we have that W2 (x) = (a + be) Wo (x) and W2 (y) = (a + bd) Wo (y) have opposite signs, which is a contradiction. Hence Wo is unique.

206

5.

ENERGY, MONOTONICITY, AND BREATHERS

Finally we show Wo > O. By the Hopf boundary point lemma (see Lemma 3.4 of Gilbarg and Trudinger [155]), if Wo = 0 somewhere, then there exists a point Xo E an, where 0. = {x EM: Wo (x) > O}, such that an satisfies the interior sphere condition at Xo, so that w (xo) = 0 and l\7w (xo)1 f:. 0, which is a contradiction to Wo ~ O. Finally, properties (1) and (2) follow easily. 0 The existence of a unique positive smooth minimizer Wo of g (g, w) under the constraint iM w 2dji = 1 implies the existence of a unique smooth minimizer fo of F(g, .) under the constraint iM e- f dji = 1. From (5.50) we see the following. LEMMA 5.23 (Euler-Lagrange equation for minimizer of F). The minimizer fo = -2 log Wo of F (g,.) is unique, Coo, and a solution to

,\ (g) = 2/).fo -1\7foI2

(5.51)

+ R.

That is, the modified scalar curvature is a constant, i.e., R m == ,\ (g) . Note that from setting v = 0 in (5.10), for the minimizer f of (5.45), we have

o(o,h)F (g, J)

= - 1M h (2/).f - 1\7 fl2 + R) e- f dji

for all h such that iM he- f djig = O. We can also obtain (5.51) directly from this. We summarize the properties of the functional ,\ on a closed manifold Mn.

(i) (Lower bound for ,\) '\(g) is well defined (i.e., finite) since

r

min R (x)· e- f dji = min R (x) ~ R min . xEM 1M xEM In particular, ,\ (g) ~ Rmin. (ii) (Diffeomorphism invariance) If cp : M --t M is a diffeomorphism, then '\(cp*g) = '\(g).

F(g, J)

~

(iii) (Existence of a smooth minimizer) There exists f E Coo (M) with iM e- f dji = 1 such that '\(g) = F(g, J)' i.e., (5.52)

'\(g)

= 1M (R + 1\7 fI2)e- f dji.

(iv) (Upper bound for ,\) We have

(5.53)

'\(g)

~ Vol ~M) 1M Rdji.

This can be seen by choosing

1M e-fdjig = 1 and '\(g)

f

= log Vol(M), which satisfies

~ 1M(R+ l\7fI 2)e- f dji.

3.

STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED

207

(v) (Scaling) A (cg) =

C- 1A (g).

3.2. The monotonicity of A. Let (Mn,9(t» , t E [O,T], be a solution of the Ricci flow on a closed manifold. In this subsection we discuss some properties related to the continuity and monotonicity of A(g(t». Such properties are key to the proof of the nonexistence of nontrivial expanding or steady breathers. First we show that A(9(t» is a continuous function on [t1' t2]' This is a consequence of the following elementary result (see also Craioveanu, Puta, and Rassias [118] or Chapter XII of Reed and Simon [310]).10 LEMMA 5.24 (Effective estimate for continuous dependence of A on 9). If 91 and 92 are two metrics on M which satisfy 1 1 + c 91 ~ 92 ~ (1

+ c) 91

and

R (91) - c ~ R (92) ~ R (91)

+ c,

thenlI A (g2) - A (gl)

~

((1 + c)~+1 -

(1

+ c)-n/2) (1 + ct/2 (A (gl) -

+ ((1 + 8) max IRg2 -

Rgil

minRgJ

+ 28 max IRg11) (1 + ct/2 ,

where 8 ---t 0 as c ---t 0. 12 In particular, A : 9J1et with respect to the C2-topology.

---t

lR is a continuous function

PROOF. The proof is straightforward but slightly tedious. First note that (1 + c)-n/2 df..L9l ~ df..L92 ~ (1 + ct/ 2 df..Lgl' If w is a positive function on M, then in view of (5.46), we compute (writing a· b - c· d = a (b - d) + (a - c) d)

1M w2df..L9IQ(g2,W) - 1M w2df..L92Q(91 , w) = 4 1M w 2df..L9l (1M lV'wl~2 df..L92 - 1M lV'wl;l df..L9l) + 4 (1M w 2df..L9l - 1M W2df..L92) 1M lV'wl~l df..L9l + 1M w 2df..L9l (1M R92 w 2df..L92 - 1M R9l w 2df..L9l ) + (1M w 2df..L9l - 1M w 2df..L92) 1M Rgl w 2df..L9l , 10T hanks to

[231] for this last reference. lITo denote the dependence on gi, we use the subscript R (gI). 12See the proof for an explicit dependence of 8 on c.

gi

instead of (gi). So R91

=

208

5. ENERGY, MONOTONICITY, AND BREATHERS

so that

1M w 2d/L9l g (92, w) - 1M w 2d/L92 g (91, w) : :; 4((1 + E) ~+1 - 1) 1M w 2d/L9l 1M lV'wl;l d/L9l + 4 (1 - (1 + E)-n/2) 1M w2d/L91 1M lV'wl;l d/L9l + 1M w 2d/L9l 1M w 2 + 11M w 2 ( 1 -

(I

(R92 - R g1 )

~~:: I+ I(~~:: - 1) R9l I) d/L9l

~~::) d/L9l 111M R9l w 2d/L9l I·

(In the above estimates we took into account that R may change sign.) Let fJ ~ max {(I + Et/ 2 - 1,1- (1 + E)-n/2} , so that fJ ----t 0 as E ----t O. Since 1

1-

dJ.L92 dJ.L91

I

< fJ , we have

-

Hence

g (92, w)

g (91, w)

fM w 2d/L92

fM w 2d/L9l

:::; 4 ((1 + E)~+1 _ (1

+ ((1 + fJ) max IRg2

+ E)-n/2)

- Rgli

Taking w to be a minimizer for

fM IV'W~;l d/L9l fM W d/Lg2

+ 2fJmax IRg11) (1 + Et/ 2 .

g (91, .), we have

A (92) - A(91)

:::; 4 ((1

+ E)~+1 _

+ ((1 + fJ) max IRg2

(1

+ E)-n/2)

- R9lI

(1

+ Et/ 2 fM IV'W~;l d/L9l

+ 2fJ max IRgll)

fM W d/L9l (1 + Et/ 2 .

3.

STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED

209

The result now follows from 4 fM

l'Vwl~l d/-Lgl = 9(91, w)

fM w 2d/-L9l ~

fM R9l w2d/-L9l fM w 2d/-L9l fM w 2d/-L9l A(91) - min Rgl .

o The monotonicity of F (9 (t) ,f (t)) under the system (5.34)-(5.35) implies the monotonicity of A (9 (t)) under the Ricci flow. LEMMA 5.25 (A monotonicity). If 9 (t), t E [0, T], is a solution to the

Ricci flow, then d 2 dtA(9(t)) ~ ;;:A 2(9(t)),

and A(9(t)) is nondecreasin9 in t E [0, T]. Here the derivative sense of the lim inf of backward difference quotients.

1t

is in the

REMARK 5.26. See the next subsection for the case where A(9(t)) is not strictly increasing. PROOF. Given to E [0, T], let fo be the minimizer of F (9 (to) , J), so that A (9 (to)) = F (9 (to), f (to))· Solve

a

2

atf=-R-/).f+I'Vfl,

(5.54)

f(to)=fo,

°

backward in time on [0, to]. Then itF (9 (t),j (t)) ~ for all t ~ to. Since the constraint fMe-fd/-L is preserved under (5.54), we have A(9(t)) ~ F(9(t),f(t)) for t ~ to. This, (5.41), and ,X (9 (to)) = F(9(to),f(to)) imply both (5.55)

A (9 (t)) ~ F (9 (t) , f (t)) ~ F (9 (to) , f (to)) = A (9 (to) )

and the following:

! (5.56)

I ~

A(9 (t) ) t=to

I

:t F (9 (t) , f (t)) t=to

= 21M IRij + 'Vi'Vj fI2 e- f d/-Lg(to)

~2

r

.!. (R + /).J)2 e- f d/-Lg(to)

JMn

~ ~ (1M (R + /).J) e- f d/-Lg(to)) 2 =

~A2(9(tO))' n

where f = fo is the minimizer. Hence, from either (5.55) or (5.56), we see that A (9 (t)) is nondecreasing under the Ricci flow. 0 EXERCISE 5.27. Prove (5.56).

5.

210

ENERGY, MONOTONICITY, AND BREATHERS

SOLUTION TO EXERCISE 5.27. We compute 13

~ >. (g (t) ) I dL

~ lim inf >. (g (to)) - >. (g (to - h))

t=to

h . f F (g (to) , fo) - F (g (to - h) , f (to - h))

h--+O+

> 1. -

lmlll

h--+O+

h

'

where fo is the minimizer for F (g (to),·) and f (t) is the solution to (5.54). On the other hand, we conclude by (5.41) that the last expression is equal to 2 fM I~j + "V/vjfoI2 e- fOdpg(to)· 3.3. There are no nontrivial steady breathers. As an application of the monotonicity of the diffeomorphism-invariant functional>' we prove the nonexistence of nontrivial steady breathers.

LEMMA 5.28 (No nontrivial steady breathers on closed manifolds). If (Mn,g(t)) is a solution to the Ricci flow on a closed manifold such that there exist tl < t2 with>' (g (tl)) = >. (g (t2)) , then 9 (t) is a steady gradient Ricci soliton, which must be Ricci flat. In particular, a steady Ricci breather on a closed manifold is Ricci flat. PROOF. Note that if 9 (t) is a steady Ricci breather with g(t2) = t.p*g(tl) for some tr < t2 and diffeomorphism t.p : M - 4 M, then >.(g (t2)) = >.(g (tl)). Hence we only need to prove the first part of the lemma. Suppose that for a solution 9 (t) to the Ricci flow there exist times tl < t2 such that >.(g (t2)) = >.(g (tl)). Let 12 be the minimizer for F at time t2 so that F (g (t2) , h) = >. (g (t2)). Take f (t) to be the solution to the backward heat equation (5.35) on the time interval [tl, t2J with the initial data f (t2) = 12. By the monotonicity formula (5.41) and the definition of >. we have 14 >. (g (tl)) :S F (g (tl), f (tl)) :S F (g (t), f (t)) :S F (g (t2) ,h)

= >. (g (t2))

for all t E [tr, t2J . Since>. (g (tl)) = >. (g (t2)) and>' (g (t)) is monotone, we have F (g (t), f (t)) = >. (g (t)) == const for t E [tr,t2J. Therefore the solution f(t) is the minimizer for F(g(t),·) and ftF (g (t), f (t)) == 0, so by (5.41) we have

1M IRij + "Vi"VjfI2 e- f dp (t) == 0 for all t E [tI, t2J . Thus (5.57)

~j

+ "Vi"Vjf =

0 for t E [tl, t2J.

In particular, g( t) is a steady gradient Ricci soliton flowing along "V f (t) l3Here d1_ denotes the lim inf of backward difference quotients. 14This is the same as (5.55). l5See (1.9), where a gradient soliton is steady if c = o.

.15

3.

STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED

Note by (5.51) that

f satisfies the equation 21),.f -

On the other hand, R

211

1\7 fl2 + R = A (g) .

+ I),.f = 0, so that 1\7 fl2 + R = -A (g).

However, integrating, we have

-A (g) so that A (g)

= 1M (1\7 fl2 + R) e- f df.L = A (g),

=°and I),.f =1\7 fl2 =-R. Note that then 0= 1M (I),.f -

1\7 f12)

e f df.L = -21M

1\7 fl2 e f df.L

implies that f is constant and hence 9 is Ricci flat by (5.57). Alternatively, we could have argued that since I),.f = 1\7 fl2 2:: 0, f is subharmonic and hence constant. D REMARK 5.29. Even when M is noncompact, we have constant for gradient Ricci solitons; see Proposition 1.15.

1\7 fl2 + R

is

3.4. Nonexistence of nontrivial expanding breathers. Recall that A (g) is not scale-invariant, e.g., A (cg) = c- 1 A (g). Thus we define the normalized A-invariant:

.x (g) ~ A (g) . Vol (M)2/n . It is easy to see that .x (cg) = .x (g) for any c > 0, so the invariant .x is potentially useful for expanding and shrinking breathers. We shall prove the monotonicity of .x (g (t)) under Ricci flow when it is nonpositive. For (5.58)

this reason it is most useful for expanding breathers. Recall that by (5.56), we have

(5.59)

:t A (g (t)) 2:: 2 1M

I~j + \7i\7jfI2 e- f df.L,

where -9t A (g (t)) is defined as the lim inf of backward difference quotients. 16 Let V ~ V (t) ~ Volg(t) (M). From (5.59), we compute

!.x (g (t)) =! [A(9(t)). V (t)2/n] = V2/ndA + ~V~-lA dV dt

2:: 2v2/n 1M

n

dt

I~j + \7i\7jfI 2 e- fd f.L+ ~AV~-l 1M (__.R)df.L,

16This also applies to the time derivatives below in this argument.

212

5.

ENERGY, MONOTONICITY, AND BREATHERS

Hence

~ V- 2/ n ~ >.. (g (t)) ~ 1M I~j + \7i\7jf - ~ (R + ~f) gijl2 e- f dp, +

r ~(R+~f)2e-fdp,

JMn

- ~nJrM (R + ~f) e- f dp,· ~V JrM Rdp,. Recall from (5.53) that 1M (R

+ ~f) e- f dp, ::;

I

~dp,.

Assuming >'(t)::; 0, so that IM (R+~f)e-fdp,::; 0, we have

(5.60)

~V-2/n:t>"(g(t))- 1MI~j+\7i\7jf-~(R+Llf)9ijI2 e-fdp,

~ ~ 1M (R + ~f)2 e- f dp, - ~ (1M (R + ~f) e- f dp, ) 2 ~ 0 since IM e- f dp, = 1. Hence LEMMA 5.30. Let 9 (t) be a solution to the Ricci flow on a closed manifold Mn. If at some time t, >.. (t) ::; 0, then

(5.61)

d-

dt>.(g(t))

~ 2V2/ n 1M I~j + \7i\7j f - ~ (R + ~f) gijl2 e- f dp, ~ 0, where V = Volg(t) (M), f (t) is the minimizer for F (g (t),·), and the timederivative is defined as the lim inf of backward difference quotients. By (5.61), if -it>.. (g (t)) = 0, then 9 (t) is a gradient Ricci soliton. This is reminiscent of the fact that under the normalized Ricci flow, the minimum scalar curvature is nondecreasing as long as it is nonpositive, whereas under the un normalized Ricci flow, the minimum scalar curvature is always nondecreasing (see Lemma A.20). However these two facts appear to be quite different in nature. To apply the above monotonicity to the expanding breather case, we need to produce a time to where>.. (g (to)) < O. This is accomplished by looking at the evolution of the volume. Below we also give another proof of Lemma 5.28 using>" (g (t)).

3.

STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED

213

LEMMA 5.31. Expanding or steady breathers on closed manifolds are

Einstein. PROOF. Let (Mn, 9 (t)) be an expanding or steady breather with g(t2) = acp*g(td for some tl < t2 and a 2 1. We have 5. (g (t2)) = 5. (g (tl))' Let V (t) ~ Volg(t) (M) . Since V (t2) 2 V (tt), we have for some to E (tI, t2),

o~

dd t

It=to log V (t) = - f~ ~~J.l (to) ~ -). (g (to)) . to

By Lemma 5.30, if 9 (t) is not a gradient Ricci soliton, then r1t5. (g (to)) > < 0 for some to < to. Now since 5. (g (t)) is increasing whenever it is negative, we have

oand we have 5. (g (to))

5. (g (t2)) = 5. (g (td)

~

5. (g (t~)) < 0,

which implies). (g (t)) ~ ). (g (t2)) < 0 for all t E [tl, t2]' Hence 5. (g (t)) is nondecreasing, which implies 5. (g (t)) is constant. By (5.61), we have 1 n

Rij + "\h\ljf - - (R +!:!..f) gij == 0, and since we are in the equality case of (5.60), we also have

(5.62)

R +!:!..f

=

Cl (t)

= const (depending on time).

That is, we still conclude that 9 (t) is an expanding or steady gradient Ricci soliton. Now let (Mn,g (t)) be an expanding or steady gradient Ricci soliton. Recall 2!:!..f + R - 1"\1 fl2 = C 2 (t) = const. This, combined with (5.62), implies

!:!..f - 1"\1 fl2 = const. Since

1M (!:!..f -

1"\1 f12) e- f dJ.l =

0,

1"\1 fl2 == O. Thus, by the strong maximum principle (or since now 0 = fM (!:!..f - 1"\1 f12) ef dJ.l = -2 fM 1"\1 fl2 ef dJ.l), we conclude that f == const. Hence ~j - ~R9ij == 0 and 9ij is Einstein. (When n = 2, our

we have !:!..f -

conclusion is vacuous.)

0

REMARK 5.32. As a corollary of the above result, we again see that expanding or steady solitons on closed manifolds are Einstein. In the case of shrinking solitons on closed manifolds, using the entropy functional, we shall see in the next chapter that they are necessarily gradient shrinking solitons.

214

5.

ENERGY, MONOTONICITY, AND BREATHERS

Note that on a shrinking breather we have V (t2) < V (tl) for t2 > tl' In particular, it is possible that A (g (t)) > 0 for all t E [tl, t2] (on the other hand, if A (g (to)) < 0 for some to E [tl' t2], then the proof above implies that a shrinking breather is Einstein), which causes difficulty in extending the proof above to the shrinking case; in the next chapter this problem is solved by the introduction of Perelman's entropy. (Note that for an Einstein manifold with R == r = const, under the constraint I e- f dp, = 1 we have

F (g, J) with equality if and only if .x (g) = r Vol (g)2/n > 0.)

=r+ f ==

1M IV' fl2 e- f dp, 2: r

log Vol (g)

=

const. Hence, if r

> 0, then

EXERCISE 5.33 (Behavior of .x on products). Compute.x of spheres and products of spheres. Show that .x (t) of a shrinking S2 x Sl under the Ricci flow approaches 00 as t approaches the singularity time. What happens if we start with S2 x S2, where the S2's have different radii? What is the behavior of .x for the product of Einstein spaces (or Ricci solitons)?

4. Classical entropy and Perelman's energy Define the classical entropy on a closed manifold Mn by

N

(5.63)

~ 1M fe- f dp, = -

1M ulogu dp"

where u ~ e- f . Under the gradient flow (5.29)-(5.30), we have

dN = dt

(5.64)

f afe-fdp,=_ f (R+t1f)e- f dp, 1M at 1M

= -F.

That is, the classical entropy is the anti-derivative of the negative of Perel-

man's energy. In this section we show that, by an upper bound for F, a modification of N is monotone. For comparison, we discuss Hamilton's original proof of surface entropy monotonicity, the entropy formula for Hamilton's surface entropy, the fact that the gradient of Hamilton's surface entropy is the matrix Harnack, and Bakry-Emery's logarithmic Sobolev-type inequality.

4.1. Monotonicity of the classical entropy. The following gives us an upper bound for the time interval of existence of the Ricci flow in terms of 1M dm and the initial value of P. Equivalently, it also implies the monotonicity of the classical entropy (see also [356], pp. 74-75). PROPOSITION 5.34 (Upper bound for F in terms of time to blow up).

Suppose that (g(t), f (t)) is a solution on a closed manifold Mn of the gradient flow for P, (5.25)-(5.26), for t E [0, T). Then we have (5.65)

r(g(O)) ::;

2~ 1M dm,

4. CLASSICAL ENTROPY AND PERELMAN'S ENERGY

215

that is, T
_1_ ( s -2Z)2 , ds 7l"X

(5.69)

where s ~ T~t. From this we conclude that if s02 Z (so) > 0 for some So < 00, then s-2 Z (s) ---+ 00 as s ---+ Sl for some Sl < 00. In other words, if Z (to) > 0 for some to < T, then Z (t) ---+ 00 as t ---+ t1 for some t1 < T. This contradicts our assumption that the solution exists on [0, T). Hence Z (t) ::; 0 for all t and we have proved the following. THEOREM 5.38 (Hamilton's surface entropy monotonicity). For a solution of the Ricci flow on a closed surface with R > 0, we have dN (t) < 0 dt for all t E [0, T).

Note that, from (5.69), we have t ~ (T - t)2 Z (t) is nondecreasing (since X > 0) and hence there is a constant C > 0 such that (T - t)2 Z (t) 2: -C for all t E [0, T). By (5.68), Z= dN dt

=_ { IVRI 2 dA+ { (R-r)2dA,

1M

R

1M

and we have

REMARK 5.39. An inequality of the above type is often referred to as a reverse Poincare inequality. 4.2.2. Entropy formula for Hamilton's surface entropy. Define the potential function f (up to an additive constant) by !:l.f = r - R. In [97J the monotonicity of the entropy was proved by relating its time-derivative to Ricci solitons via an integration by parts using the potential function (Proposition 5.39 in Volume One). In particular, we have

Note that Rij = ~Rgij and r = ~ 1M RdA = (T - t)-l. We have purposely written this formula to more resemble Perelman's formulas (5.41) and (6.17).

218

5.

ENERGY, MONOTONICITY, AND BREATHERS

4.2.3. The gradient of Hamilton's surface entropy is the matrix Harnack quantity. A less well-known fact is that the gradient of Hamilton's entropy in the space of all metrics with the L 2-metric is the matrix Harnack quantity: (5.70)

8v N (g)

=

1M Vij ( -~ log R· gij + ViVj log R -

~R9ij) dA,

where 8g = v (see Lemma 10.23 of [111] and use N (g) - E (g) is a constant). In the space of metrics in a fixed conformal class, the gradient is the trace Harnack quantity. Note that the same relation is true relating the entropy and the trace Harnack quantity for the Gauss curvature flow of convex hypersurfaces in Euclidean space [96]. 4.3. Bakry-Emery's logarithmic Sobolev-type inequality. The proofs of Hamilton's surface entropy formula and Perelman's energy formulas are formally similar to the proof of Bakry and Emery of their logarithmic Sobolev-type inequality [18].

PROPOSITION 5.40. Let (M n , g) be a closed Riemannian manifold with Rc 2:: K for some constant K > O. If u is a positive function on M, then

1M ulogudJ.L::; 2~ 1M u IVlogul 2 dJ.L + log (VOI~M) 1M UdJ.L) 1M udJ.L. PROOF. (See [104] for more details of the computations.) Consider the solution v to the heat equation aa:. = ~v with v (0) = u. The solution v exists for all time and

t~~

V

= Vol ~M)

1M udJ.L.

J

Define E (t) ~ M v log vdJ.L. Then

(5.71)

t~~ E (t) = 1M udJ.L . log ( Vol ~M) 1M UdJ.L) .

We have

~~ = -

1M (Vv, V log v) dJ.L = - 1M v IV log vl

2

dJ.L ::; O.

Note that limt ..... oo ~f (t) = O. Using %t log v = ~logv + IVlogvl2, we compute

ddt2 E2 = 2 1M f v ( IVVlog vl 2

+ Rc (V log v, Vlogv) ) dJ.L.

Using our assumption Rc 2:: K, we find

d2 E dE -d >-2K2 dt . t -

5. NOTES AND COMMENTARY

By limt-+oo ~f (t)

dE

219

= 0 and (5.71), we have

roo d2E

roo dE

= 2Klog ( Vol ~M)

1M UdJ.t) 1M udJ.t -

dt (0) = -

Jo

dt2 (t) dt ~ 2K Jo

dt (t) dt 2KE (0).

Hence

- 1M u IV' log ul 2 dJ.t ~ 2K log ( Vol ~M) 1M UdJ.t) 1M udJ.t - 2K 1M u log udJ.t and the proposition follows.

D

5. Notes and commentary Subsection 1.1. As we remarked earlier, the function 1 is also known as the dilatonj in the physics literature there are numerous references to Perelman's energy functional (see Green, Schwarz, and Witten [162]' Polchinksi [307]' Strominger and Vafa [341] for example), although Perelman is the first to consider it in the context of Ricci flow. The Ricci flow is the 1-loop approximation of the renormalization group flow (see Friedan [145]). Subsection 1.2. For a computational motivation for fixing the measure, see also §4 in Chapter 2 of [111], where Perelman's functional is motivated starting from the total scalar curvature functional. In particular, let 89 = v. The variation of the total scalar curvature is

8

1M RdJ.t = 1M (diV (divv) - ~v - Rc·v + R ~) dJ.t = - 1M (Rc - ~ 9) .vdJ.t.

This says that V' (JM RdJ.t) = - Rc +~ 9, where the gradient is calculated with respect to the standard L2- metric. To try to find a functional F with V'F = - Rc, we want to get rid of the ~9 term. Now this term is due to the variation of dJ.t. So we consider the distorted volume form e- f dJ.t and assume its variation is O. Hence

8

1M Re- f dJ.t = 1M (8R) e- f dJ.t = 1M (div (divv) -

and now we have the extra terms sate for this by considering

8

JM

~v -

Rc ·v) e- f dJ.t

(div (div v) - ~ V) e- f dJ.t. We compen-

1M 1V'112 e- f dJ.t = 1M (8 IV'112) e- f dJ.t = 1M (-v (V'I, V' f) + V'1 . V'V) e- f dJ.t,

220 using

5. ENERGY, MONOTONICITY, AND BREATHERS

¥- =

8F = 8

h ~ 8f. Integrating by parts yields

1M Re- f d/-L + 8 1M 1V'112 e- f d/-L = - 1M Vij (~j + V'iV'jf) e- f d/-L.

Although the Ricci tensor is not strictly elliptic in g, one can ask if the RHS of equation (5.27)

:t

gij = -2

[~j + V'iV'jlog (:~)]

is elliptic in g. The answer is still no. In particular, if 8g

= v,

then

Hence

Since we have 8

(-2

[Rij

+ V'iV'jlog (:~)])

= D.vij - V'iV'kVjk - V'jV'kVik

+

lower-order terms,

where the last pair of terms form a Lie derivative of the metric term. However the second-order operator on the RHS is still not elliptic in v.

CHAPTER 6

Entropy and No Local Collapsing Everything should be made as simple as possible, but not simpler. - Albert Einstein Disorder increases with time because we measure time in the direction in which disorder increases. - Stephen Hawking Close, but no cigar. - Unknown origin

By combining Perelman's energy and the classical entropy in a suitable way, we obtain the entropy functional W, which we shall discuss in this chapter. This is implemented with the introduction of a positive scalefactor T. The advantage which the addition of this scale-factor yields is that from the functional W we can understand aspects of the local geometry of the manifold, e.g., volume ratios of balls with radius on the order of ..ji. Perelman's entropy is also the integral of his Harnack quantity for fundamental solutions of the adjoint heat equation. 1 As such, one can integrate in space the Harnack partial differential inequality to give a proof of the monotonicity formula for W. Note that here Perelman's Harnack quantity is directly related to the entropy whereas in Hamilton's earlier work on surfaces, Hamilton's Harnack quantity is related to the time-derivative of his entropy. 2 This monotonicity formula can be used to prove that shrinking breathers must be shrinking gradient Ricci solitons. More importantly, this monotonicity formula will be fundamental in proving Hamilton's little loop conjecture or what Perelman calls the no local collapsing theorem. In this chapter, we shall discuss in detail the entropy estimates, the two functionals J1 and 1/ associated to W, and their geometric applications. We discuss the logarithmic Sobolev inequality, which is related to the entropy functional. We will also give different versions/proofs of the no local collapsing theorem. In the last part of this chapter we shall discuss some interesting calculations related to entropy. Throughout this chapter Mn denotes a closed n-dimensional manifold. 1. The entropy functional W and its monotonicity

Let (Mn,g(t)) , t E [O,T], be a solution of the Ricci flow on a closed manifold. Note that by the proof of Lemma 5.31, we have that when lSee (6.21). 2Equation (6.21) as compared to (5.70). 221

222

6. ENTROPY AND NO LOCAL COLLAPSING

>.(g(t)) ~ 0 for some t E [tl, t2J, even shrinking breathers, i.e., solutions with 9 (t2) = acp*g (tl) and a < 1, are Einstein solutions (trivial Ricci solitons). In order to handle shrinking breathers when >.(g(t)) > 0 for all t E [tl' t2], we need to generalize the monotonicity formula for the functional F to a monotonicity formula for a functional related to shrinking breathers. This is the monotonicity formula for the entropy W. In this section we introduce the entropy Wand discuss its monotonicity. We also give a unified treatment of energy and entropy in the last subsection. 1.1. The entropy W, its first variation and the gradient flow. 1.1.1. The entropy W. Let 9Jtet denote the space of smooth Riemannian metrics on a closed manifold Mn. We define Perelman's entropy functional W : 9Jtet x COO(M) x jR+ --+ jR by

W(g, 1, T)

(6.1)

(6.2)

~ 1M [T (R + 1\7112) + 1 =

n] (47rT)-n/2 e- f d/-l

1M [T (R + 1\7112) + 1 - n] ud/-l,

where 3 (6.3) This is a modification of the energy functional F (g, j) , which we considered in the last chapter, where we have now introduced the positive parameter T. By (5.1), we have

(6.4)

W(g,j, T) = (47rT)-n/2 (TF (g, j)

+ 1M (f -

n) e- f d/-l)

= (47rT)-n/2 (TF(g,j) +N(f)) - n

(6.5)

1M ud/-l,

where the second equality is obtained using definition (5.63). As we shall see, T plays the dual roles of understanding the geometry of (M, g) at the distance scale -IT and representing a constant minus time for solutions (M,g (t)) of the Ricci flow. 4 The functional W has the following elementary properties. (i) (Scale invariance) W is invariant under the scalings T f--t CT and 9 f--t cg, i.e.,

W(cg, 1, CT) = W(g, 1, T).

(6.6)

(ii) (Diffeomorphism invariance) If cP : M

--+

M is a diffeomorphism,

then

W(g, 1, T) = W(cp*g, cP* 1, T), where cp*g is the pulled-back metric and cp* 1 = 1 0 CPo 3This u is not to be confused with the u = e- f in Chapter 5. 4We may think of T as physically representing temperature (see §5 of [297]).

1.

THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY

1.1.2. The first variation of W. Let 8g 8f = h, and let 87 = (. Since

(6.7)

8 (udp,)

= v E C2(M, T* M

223

® T* M), let

= 8 ((47r7)-n/2 e- f dp,) = ( -;:. (- h + ~)

udp"

where V ~ trg v = gi j vij , the measure (47r7)-n/2e- f dp, preserving variations satisfy

V

n

(6.8)

--(-h+ - =0. 27 2

We find it convenient to write the variation of W so that this quantity is one of the factors. In particular, we have LEMMA 6.1 (Entropy first variation formula). The first variation of W at (g, f, 7) can be expressed as follows: 8W(v,h,() (g, f, 7)

(6.9)

= 1M (- 7Vij + (gij) (~j + \/i\/jf -

(~ -

+ 1M 7

h - ;;) (R

+ 2~f -

2~9ij)udp, 1\ /

fl2

+

f - ; -

1)

udp,.

PROOF. It follows from the first variation formula (5.10) of F with respect to 9 and f (keeping 7 fixed) that 8( v,h,O) (7( 47r7) -n/2 F (g, f) )

=-

1M 7Vij(~j

+ 1M 7

(~ -

+ \/i\/jf)udp,

h)

(2~f -1\/fI2 + R) udp,.

Next we calculate the first variation of the remaining term of W with respect to 9 and f (again keeping 7 fixed), 8(v,h,O) ((47r7)-n/21M (f-n)e-fdP,)

= 1M

[(1+n-f)h+

~(f-n)] udp,.

Now the term from the variation of W with respect to 8(o,o,() (1M [7(R +

= 1M

[( 1-

1\/ f12) + f

7

is

- n] (47r7)-n/2e- f dp, )

~) ((R + l\/fI2) -

;; (f - n)] udp,.

6. ENTROPY AND NO LOCAL COLLAPSING

224

Combining the above three formulas and simplifying a little, we get 8(v,h,() W

(g, f, 7)

= - 1M 7Vij(~j + V/Vjf)udJ.L + 1M 7

(~

+ 1M [h +

-

h) (2b. f - IV fl2 + R + f

(1 - ~) ((R + IV f12) -

~ n) udJ.L

;; (f - n)] udJ.L.

We rewrite the above expression as 8(v,h,() W

(g, f, 7)

= 1M (- 7Vij

+ (gij) (Rij + Vi Vj f)udJ.L

+ 1M7(~ -h-;;)

(2b.f-IVfI2+R+f~n)udJ.L

+ 1M -((R + b.f)udJ.L + 1M + 1M ( 1 -

~( (2b.f -IV fl2 + R) udJ.L

~) ((R + IV fI2)udJ.L + 1M hudJ.L,

which, by combining terms, we further simplify to 8(v,h,() W

(g, f, 7)

= 1M (- 7Vij + (gij) (Rij + ViVjf)udJ.L + 1M7(~ -h- ;;)

(2b.f-IVfI2+R+f~n)udJ.L

+ 1M (n - 1) ( (b.f -IV f12) udJ.L + 1M hudJ.L. Since ( is a constant, (6.9) follows from a rearrangement and the integration by parts identity: fM (b.f REMARK

Rij

-IV f12) e- f dJ.L = O.

0

6.2. Analogous to (5.14) and (5.15), the terms

+ ViVjf -

1 -2 gij 7

and

R + 2b.f -IVfl

2

f-n

+-7

in (6.9) are natural quantities vanishing/constant on shrinking gradient Ricci solitons. 1.1.3. The gradient flow of W. When we require that the variation (v, h, () satisfies V n ( = -1 and '2 - h - 27 ( = 0,

1. THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY

225

i.e., the variation preserves the measure (41l"7)-n/2 e- f dp, on M, we obtain from (6.9) the gradient flow (assuming ~; = -1)

a

at gij = -2 {~j + '\h\1jf) ,

(6.10) (6.11)

af at

(6.12)

~: = -1.

=

-b..f - R

n

+ 27'

LEMMA 6.3 (Entropy monotonicity for gradient flow). If (g{ t), f (t), 7{ t)) is a solution to the system (6.10)-{6.12), then

d dt W{g{t), f{t), 7{t))

=

(6.13) PROOF.

1M 27 I~j + '\h\1jf - 2~9ij12 udp, 2: O.

This follows directly from substituting Vij = -2 {~j h

=

+ '\h\1jf) , n

-b..f - R + 27'

(= -1

into (6.9) and using the facts

~ -h- 2:(=0

and

1M (b..f-I V fI 2)e-f dp,=0. D

1.2. Coupled evolution equations associated to Wand monotonicity of W. 1.2.1. The coupled evolution equations associated to W. As in Chapter 5, there is a system of evolution equations for the triple (g, f, 7) (see (1.3) of [297]) (6.14)

(6.15)

a

at gij = af

-2~j, 2

at = -b..f + IVfl - R+

n

27'

d7 =-1 dt ' whose solution differs from the solution to (6.10)-{6.12) by diffeomorphisms. This leads to the following theorem, which says that (6.16)

d dt W{g{t), f{t), 7{t)) 2: O. Another motivation for studying this system of equations, from considering gradient Ricci solitons, was discussed in Section 8 of Chapter 1.

226

6.

ENTROPY AND NO LOCAL COLLAPSING

THEOREM 6.4 (Entropy monotonicity for Ricci flow). Let (g(t),J(t),7(t)), t E [0, T], be a solution of the modified evolution equations (6.14), (6.15), and (6.16). Then the first variation of W along this solution is given by the following: d dt W(g(t), f(t), 7(t))

(6.17) (6.18)

1M 27 I~j + 'Vi'Vjf - 2~9ij12 . udJ.L = 27 1M I~j ~ ('Vi'Vju - 'Viu:jU + 2~ U9ij ) 12 . udJ.L 2 O. =

-

EXERCISE 6.5. Show that the line of reasoning of deriving Theorem 6.4 from Lemma 6.3 is rigorous.

(6.19)

is exactly the matrix Harnack quantity for a solution U of the backward heat equation. In particular, this same expression for positive solutions of the backward heat equation appeared in Hamilton's derivation of the monotonicity formulae for the harmonic map heat flow, mean curvature flow, as well as the Yang-Mills flow; see [184].5 Equation (6.17) is Perelman's entropy monotonicity formula and implies that W(g(t), f(t), 7(t)) is strictly increasing along a solution of the modified coupled flow except when g(t) is a shrinking gradient Ricci soliton (since 7> 0 in (6.17)), where it must flow along 'Vf and where W(g(t),f(t),7(t)) is constant. This monotonicity is also fundamental in understanding the local geometry of the solution 9 (t) to the Ricci flow as we shall see in the proof of the no local collapsing theorem. The function f allows us to localize and the parameter 7 tells us at what distance scale to localize (yfT) .. Since we do not give the details of how to transform between the systems (6.10)-(6.12) and (6.14)-(6.16), we also compute (6.17) directly through the following exercise. EXERCISE 6.7 (Deriving the formula for ~~.

dJi from ft)· Use the equation for ft to derive

SOLUTION TO EXERCISE 6.7. The effect of the extra term as compared to (5.30) is to add

+~

in (6.11)

-;:. 1M (R+ l'VfI2) e-fdJ.L 5For an exposition of the matrix Harnack estimate asssociated to (6.19), see [104J.

1.

THE ENTROPY FUNCTIONAL

to (5.31), so that we get

dF

dt

W

AND ITS MONOTONICITY

227

r IRj + 'Yi'Yjfl 21 n e- dJ.L - 27 F .

= 2 1M

Similarly, (5.64) becomes

dN = -F +!:. dt

r e- I dJ.L -

271M

!:.N 27

under (6.10)-(6.11). Hence, by (6.5), i.e., W =(47r7)-n/2 (7F(g, f) and iM udJ.L

+N) - n 1M udJ.L

= const, we have

d: = :t [(47r7)-n/2 (7F +N)] = (47r7)-n/2

(!:. (7F +N) _ F + 7 dF + dN) 27

dt

dt

'

so that

(47r7t/2d: = 271M IRj + 'Yi'YjfI2 e- I dJ.L - 2F + ;:. 1M e- I dJ.L

= 271M IRi j + 'Yi'Yjf -

2~9ij12 e- I dJ.L.

1.2.2. Second proof of the monotonicity ofW from a pointwise estimate. Analogously to subsection 2.3.2 of Chapter 5 we again derive (6.17) using a pointwise evolution formula. Let 6 (6.20)

In Part II of this volume we shall see that v is nonpositive when u is a fundamental solution (for this reason v is also called Perelman '8 Harnack quantity). Note that (6.21)

W (g, f, 7)

= 1M vdJ.L.

We shall show the following below. LEMMA 6.8 (Perelman's Harnack quantity satisfies adjoint heat-type equation). Under (6.14)-(6.16) (6.22)

where 0*

O*v

= -27 I~j + 'Yi'Yjf -

2~9ij12 u,

= -It - Ll + R is the adjoint heat operator defined in (5.39).

6This quantity v is not to be confused with v = 8g.

228

6.

ENTROPY AND NO LOCAL COLLAPSING

Theorem 6.4 then follows from

dW dt

= { (~_ R) vd/-L = { (-0* - 1:1) vd/-L

1M

= 2T ~ o.

1M

at

1M I~j + "h'Vjf - 2~9ij12 ud/-L

Before we prove (6.22), we need the following lemma concerning the function u = (47rT)-n/2 e- f as defined in (6.3) (compare with (5.40)). LEMMA 6.9 (u is a solution to the adjoint heat equation). The evolution equation (6.15) of f is equivalent to the following evolution equation of u : (6.23)

O*u

= o.

PROOF. We calculate

o We now show that we can apply the computation in subsection 2.3.2 of Chapter 5 to derive the evolution equation (6.22) for v. PROOF OF (6.22). Let Ibe defined by e- i ~ u. Then 1= f+~ log (47rT) and (g,1) satisfies (5.34) and (5.35). By (5.43), we have that the quantity

V~ (R + 21:11 -IVI12) u = (-21:1 log u -IVlogul 2 + R) u satisfies the equation

Note that

v= [T (R + 21:11 -Iv112) -logu (6.24)

~ log(47rT) -

= TV - (logu + ~ log(47rT) + n) u.

We compute using O*u 0* (ab)

= 0 and the general formula = bO*a + aO*b -

2 (Va, Vb) - Rab

n] u

1. THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY

229

that 0* [(lOgU + ~ log(47f7)

+ n) U] n

= uO* log u + log uO*u - 2 (V log u, Vu) - Ru log u + 27 U n

2

= -Ru-IVlogul U+ 27U, since

0* logu =

~(-

:t -~) + U

IVlogul 2 + Rlogu

= -R + IVlogul 2 + Rlogu. From (6.24), we compute O*v = 70*V + V

+ IVlogul 2 u + Ru -

n

27 U

= -271~j - ViVj logul 2 u

+ (-2~logu -IVlogul 2 + R) u 2

+ IV log ul u + Ru -

n

27 U 2

= -271~j - ViVj logul u

+ 2 (-~logu + R) u -

n 27 u.

o

Completing the square, we obtain (6.22).

1.3. A unified treatment of energy F and entropy W. We finish this section with several exercises. In total, this unifies part of the discussion of shrinking, steady, and expanding gradient Ricci solitons, which correspond to entropy, energy, and expander entropy on a closed manifold Mn (see the definition below), respectively.7 In this subsection we use the following convention: c E JR, and if c i= 0, we take 7 (t) = ct, whereas if c = 0, we take 7 (t) = 1. We only consider all t E JR such that 7 (t) > 0. Let (6.25) Define the c-entropy by

(6.26)

W€(g, f, 7)

~ 1M (7 (R + IV f12) =

c (f - n)) (47f7)-n/2 e- f dj.t

1M Veudj.t,

7Notation: For the most part we use a hat on M to emphasize that we are considering a fixed metric instead of a solution to the Ricci flow.

230

6. ENTROPY AND NO LOCAL COLLAPSING

where u is defined in (6.3). When c < 0, this is Perelman's entropy; when c = 0, this is Perelman's energy; and when c > 0, this is called the expander entropy. 8 The definition of Vc is motivated by the following exercise. EXERCISE 6.10 (Harnack Vc as an integrand for We). Let (Mn,g (t),J (t)) be a gradient Ricci soliton in canonical form. By Proposition 1. 7, the pair (g (t), I (t)) satisfies

(6.27)

Sfj

~ ~j + V/vjl + 2Cr gij

= 0,

where c E R Note that if c ::f. 0, then 9 (t) and I (t) are defined for all t such that r (t) > 0, whereas if c = 0, then 9 (t) and I (t) are defined for all t E (-00, (0) . Show that Vc (g (t), I (t), r (t)) is constant in space. 6.11 (We and the Gaussian soliton). Consider the Gaussian soliton (lRn,!}JE, Ie), where!}JE = L:~=1 (dxi)2 and EXERCISE

Ie (x, t)

~

1

_1~~2

°

Ixl2 -""4t

for t >

°

if c

< 0,

for t E lR if c = 0, for t
0.

Check that

8 2 Ie

c

8 x~'8 x J. + -dij 2r =

°

for all t such that r (t) = ct > 0, and thus S[j == 0. Show Vc = 0, so that We = 0. It is useful to keep this example in mind, which reflects the Euclidean heat kernel, when considering the function theory aspects of the material in this chapter and especially the chapter on Perelman's differential Harnack estimate in the second part of this volume. EXERCISE

6.12 (Basic properties of We). Show that on a closed Rie-

mannian manifold

(Mn, g) ,

(1)

We(g, I, r) = (47f)-n/2 where 9 ~ r- 1 g, (2) for any constant c>

1M (Rg + IVII~ - c (f - n)) e- f djLg,

°

We (cg, I, cr) = We(g, I, r), and (3) for any diffeomorphism cp : M ~

We (cp*g, 1 0 cp, r)

=

M, We(g,

I, r).

8These identifications are true up to constant factors.

1.

THE ENTROPY FUNCTIONAL

AND ITS MONOTONICITY

W

231

EXERCISE 6.13 (First variation formula for We.). Show that, on a closed manifold Mn, if

8g

(6.28)

= v,

8j

= h,

=(

8r

at (g, j, r), then (6.29)

{ ((-rVi j + (gij) Sfj ) 8(v,h,() We;(g, j, r) = 1M + (~ _ h _ (V~ + c) udJ-l,

¥f)

where Sfj is defined by (6.27). In particular, if

then

8(v,h,()We;(g,j,r) =

(6.30)

1M (-rVij + (gij)SfjudJ-l.

SOLUTION TO EXERCISE 6.13. We compute

8v~j = V'p (8rfj)

- V'i

(8r~j)

and

8(v,h) (V'iV'jf) = V'iV'j (8f) - (8rfj) V'pJ. Adding these two formulas together, we get

8(v,h) (Rij

+ V'iV'jf)

= V'p (8rfj) - (8rfj) V'pj + V'i (V'j (8f) - 8r~j) (6.31)

=efV'p(e-f8rfj)+V'iV'j(h-

~).

Tracing this formula, remembering to take the variation of %s r = (imply

lj,

and using

8(v,h,() [r (R + /1f) (47rr)-~ e- f dJ-l] = r (-8gij . (~j

+ V'iV'jf) + gij . 8 (~j + V'iV'jf)) (47rr)-~ e- f dJ-l

+r(R+/1f)(47rr)-~ ((l-~)~-h+ ~)e-fdJ-l =r [

-Vi.J

(~.J + V'iV'·f) e- f + V'P (e- f .£r1!.) J as n

+e- f /1 (h -~)

+ (R+ /1f) e- f (~- h - n22~)

1(4

7rr )-!!:d 2 J-l.

232

6. ENTROPY AND NO LOCAL COLLAPSING

Since We(g, f, 7) = iM (7 (R + /).f) - c (f - n)) (47r7)-n/2 e- f dJ.L, integrating this by parts and applying the divergence theorem, we have

d

ds We(g, f, 7) -Vij (~j

~ LT

+ \1/Vjf) e- f

+ (R + /).f) e- f

(t - h -

~)

n22~)

+t. (e- / ) (h +~ (-ch - c (f - n) (t - h - ~)) e- f - 7Vij (~j

+ \1i\1j f + 2~gij) e- f +7 (R + 2/).f -1\1fI 2) e- f (t - h -~)

(t - h -~) +( (R + /).f) e- f + (~ (/).f -1\1fI2 + ~) e- f -c (f - n -1) e- f

Now, integrating by parts tells us again that the terms on the last line are

1M ( ((R + /).f) + ~ (/).f -1\1 fl2 + ~)) e- f (47r7)-~ dJ.L = 1M ( (R + /).f + ~~) udJ.L. Substituting this into the above formula yields (6.9). In the next two exercises we give a unified proof of the monotonicity formulas for entropy, energy, and expander entropy. EXERCISE 6.14 (Monotonicity of We from the first variation formula). When we require that the variation (v, h, () satisfies ( = c and -h- ~( = o in (6.30) (e.g., preserves the measure (47r7)-n/2 e- f dJ.L), this leads to the following gradient flow:

t

(6.32) (6.33) (6.34)

{)

at gij = {)f

at =

-2 (Rij

+ \1i\1jf) ,

nc -/).f - R - 27'

d7 dt = c.

Show that if (g (t), f (t), 7 (t)) is a solution of the above system, then

1t We (g (t), f (t)

,7

(t)) =

271M ISij 12 udJ.L.

1.

THE ENTROPY FUNCTIONAL

W

AND ITS MONOTONICITY

233

EXERCISE 6.15 (Evolution of Ve and monotonicity of We). Consider the gauge transformed version of (6.32)-(6.34) on a closed manifold Mn:9

8

8t gij = -2~j,

(6.35)

81

8t = -~I

(6.36)

dT dt

(6.37)

2

- R + 1\711 -

nE

2T'

= E.

Let Ve ~ Vcu. Show that if (g (t) ,J (t), T (t)) is a solution of the system above, then (6.38) Also show that this implies

:t

1M IRij + \7i\7jl + ;Tgijl2 ud/-L = 2T1M ISf l ud/-L ~ o.

We (g (t), I (t), T (t)) = 2T

j

2

The next exercise relates the first variation of We to the linear trace Harnack quantity. EXERCISE 6.16 (Variation of We and linear trace Harnack). Show that, for any symmetric 2-tensor W on a closed manifold Mn, we have:

(6.39)

1MWijSfjud/-L = 1M Z (w, \7 f) ud/-L,

where Z is the linear trace Harnack inequality defined in (A.27) (6.40) and W ~ gi j wij . In particular, if c5(v,h,() ((47rT)-n/2 e- f d/-L) c5(v,h,()We (g,I,T)

=

= 0, then

1M Z(W, \7f) ud/-L,

where

Wij ~ -TVij

+ (gij

= -T2c5(v,() (T-1g) .

HINT. Use the identity

1M \7j\7iWije- f d/-L = 1M \7iWij\7jle- f d/-L = 1M Wij (\7d\7 j l - \7i\7j f) e- f d/-L. In the last two exercises in this subsection, we first rewrite We (g, I, T) and then we use the new formula to give a lower bound for We (g, I, T). 9These equations are equivalent to (6.35)-(6.37) after pulling back by diffeomorphisms generated by the vector fields V' f (t) .

6.

234

ENTROPY AND NO LOCAL COLLAPSING

EXERCISE 6.17. Let (6.41) so that w 2 = u. Show that (6.42)

We (g, j, T) = .

r (T (Rw2 + 41\7wI2) (log (w + ~ 10g(47rT) + n) w

1M ~

(6.43)

2)

+E

)

dp,

2

Ke (g, w, T) .

SOLUTION TO EXERCISE 6.17. We obtain (6.42) from substituting the definition of w,

j = -2 log w -

'2n log (47rT) ,

\7w

and

1 2

= --w\7 j

into (6.26). EXERCISE 6.18 (Lower bounds for We). Let (Mn,g) be a closed Riemannian manifold and let Rmin ~ infxEM R (x). Suppose that (g, j, T) satisfies the constraint

1M (47rT)-n/2 e- f dp, = 1. Show that for T > °the following

hold. (1) If E

(6.44)

> 0,

then

We(g, j, T) ~ TRmin - ~ Vol (g)

+E

(i

log (47rT)

+ n) >

-00.

(2) If E < 0, then

We(g, j, T) ~ -2G lEI

+ TRmin + E (i log (47rT) + n) > -00,

where

2T

-'-l€I Vol ...!...

G

()-2/n 9

lEI + 2TGs (M, g)'

and Gs (M,g) is the constant in the Sobolev inequality (6.66). Hence we conclude that when E < 0, for any A < 00, there exists a constant G (g, E, A) < 00 such that

We(g,j,T)

~

-G(g,E,A)

for T E [A-I, A] and j E Goo (M) with fM (47rT)-!j e- f dp, SOLUTION TO EXERCISE 6.18. (1) If expanding case, then (6.44) follows from

(6.45)

E

>

= 1.

0, which corresponds to the

r ulogu dp, ~ -!e Vol (g) >

-00.

1M

(2) If E < 0, which corresponds to the shrinking case, then the logarithmic Sobolev inequality implies that the entropy has a lower bound (see Section 4

235

2. THE FUNCTIONALS Jl. AND v

of this chapter). In particular, by taking a = I~ in (6.65) below, we have for c < 0,

Wc(g, j, r) 2:

1M (4r l\7wl

+ rRmin + c (6.46)

where

2:

2

+ cw 2 10g (w 2) ) dp,

(i log (41T"r) + n)

-21cl C (~~ ,g) + rRmin + c

(i log (41T"r) + n) >

-00,

-2/n 41cl 2r ) _ 2r C ( ~,g - ~ Vol (g) + 2rn2e2Cs (M,g)

is as in Lemma 6.36.

(Mn ,g) ,Jensen's inequality says that if

: M --t M, we have

J-l(g, r) = J-l (<J>* g, r) , v(g) = v(<J>*g). Compare (i) and (ii) with Lemma 5.24. EXERCISE

6.21. Prove properties (i)-(iv) above.

Using the fact that the variation of W : X 8(O,h,O) W

(g, f, r) = -

where h satisfies (6.49),

1M rh (2tl.f + f -

--t

jR

with respect to

(; + 1)

-IV fl2 +

f is

R) udJ-l,

fM hudJ-l = 0, we have the Euler-Lagrange equation of r (2tl.f -

IV fl2 + R) + f

- n

=C

for some constant C. If fT is a minimizer of (6.49) (we will see the existence of fT in the next subsection), then it follows that

J-l (g, r) =

1M [r (2tl.fT -

IV fTI2 + R) + fT

- n] (47rr)-n/2 e- JT dJ-l,

and hence C = J-l (g, r) for a minimizer. Therefore we have the following. I°It is easy to see that I-'(g, r) is semi-continuous in g. This is a consequence of the fact that if a function hex, y) is continuous, then infyEY hex, y) is upper semi-continuous in x.

2. THE FUNCTIONALS

f.L

AND

237

v

LEMMA 6.22 (Euler-Lagrange for minimizer). The Euler-Lagrange equation of (6.49) is (6.51)

For the minimizer fT of (6.49), (6.52)

r (211fT - IV fTI2

+ R) + fT

- n = J.L (g, T) .

Compare (6.52) to the equation 26.f - IV fl2 + R = A (g) for the minimizer f of F (g,.). In terms of w ~ (41rT)-n/4 e- f /2 as in (6.41), a simple computation shows that J.L is the lowest eigenvalue of the nonlinear operator: (6.53)

N (w) ~ -4r6.w + TRw -

(% 10g(41rT) + n) w -

2w logw

= J.L (g, T) w.

2.2. The finiteness of J.L and the existence of a minimizer a consequence of Exercise 6.18 we have the following.

f. As

LEMMA 6.23 (Finiteness of J.L). For any given 9 and T > 0 on a closed manifold Mn,

J.L(g, T) >

(6.54)

-00

is finite. PROOF. Since J.L (Tg, T) = J.L (g, 1), we may assume without loss of generality that r = 1. We need to show that for any metric 9 there exists a constant c = c(g) such that (6.55)

W (g, f, 1) =

1M (R + IV fl2 + f -

n) (41r)-n/2 e- f dJ.L

~c

for any smooth function f on M satisfying (41r) -n/2 1M e- f dJ.L = 1. As in (6.41) with T = 1, let w = (41r)-n/4e- f / 2. By (6.42), the lemma is equivalent to showing that W (g, f, 1)

=

1M (41Vw12 + (R -

~

1i (g,w) ~ c

210gw -

for any w > 0 such that 1M w 2dJ.L = 1. Since R - n ~ infxEM R (x) - n > exists C < 00 such that

-00,

%log (41r) -

n) w 2) dJ.L

it suffices to show that there

1M w 2 10gwdJ.L ::; 2 1M IVwl2 dJ.L + C for all w > 0 with 1M w 2dJ.L = 1. This follows from the logarithmic Sobolev inequality (6.65),11 which we state in the next subsection. D 11With a = 2.

238

6.

ENTROPY AND NO LOCAL COLLAPSING

Next we prove the existence of a smooth minimizer for (6.49); compare the proof below with the proof of Lemma 5.22. LEMMA 6.24 (Existence of a smooth minimizer for W). For any metric g on a closed manifold Mn and T > 0, there exists a smooth minimizer fT of W (g,., T) over X. PROOF. Again assume T = 1. The lemma will follow from showing that there is a smooth positive minimizer WI for 1-l (g, w) under the constraint iM w2dJ.Lg = 1. A smooth minimizer II of W (g,., 1) is then given by II = - 2 log WI - ~ log (471") (see Rothaus [312]). We give a sketch of the proof. Suppose W is such that 1-l (g, w) :S C 1. Then the above considerations imply that since iM w 2dJ.Lg = 1,

C1

~ 1-l (g,w) = 1M (41V'wI2 + (R -

~ 2 1M lV'wl2 dJ.L -

2logw -

~ log (471") -

n) w2) dJ.L

C2,

where we used (6.65) below with a = 1. Hence any minimizing sequence for 1-l (g, .) is bounded in W 1,2 (M). We get a minimizer WI in W 1,2 (M) and by (6.53), WI is a weak solution to

-4~Wl + RWI - 2WllogWl - (~lOg(471") + n) WI = J.L(g, l)Wl. By elliptic regularity theory, we have WI E Coo (see Gilbarg and Trudinger [155] for a general treatise on second-order elliptic POE). Finally, one can prove that WI > 0; see [312] for more details. 0 REMARK 6.25. We also have (6.56)

J.L(g, T)

~ inf {W(g, f, T) : f

E W 1 ,2(M),

1M udJ.L = I} ,

where Coo is replaced by W 1,2 in (6.49) and u is defined in (6.3).

2.3. Monotonicity of J.L. Let (g (t), T (t)) , t E [0, T], be a solution of (6.14) and (6.16) with T (t) > 0. For any to E (0, T], let f (to) be the minimizer of

{W(g (to), f, T (to)) : f E Coo (M) satisfies (g,J, T) E X} and solve (6.15) for f (t) backwards in time on [0, to]. By the monotonicity formula, we have d dt W (g (t) , f (t) , T (t)) ~

°

for t E [0, to]. Note that the integral constraint (6.48) is preserved by the modified coupled equations (6.14)-(6.16). This can be seen from the following calculation: (6.57)

.!!:.- { (471"T)-n/2 e- f dJ.L = { (au dt 1M 1M at

RU) dJ.L = 1M { (-O*u -

~u) dJ.L = 0.

2. THE FUNCTIONALS J.L AND

239

II

Hence we have

(6.58)

I-" (g (t), 7 (t)) ::; W (g (t), f (t), 7 (t)) ::; W (g (to) , f (to) , 7 (to)) = I-" (g (to) , 7 (to) )

for t E [0, to]. The above inequality implies

(6.59)

~lt=tol-"(9(t)'7(t)) 2:

:tlt=to W(g(t),J(t),7(t)) 2: °

in the sense of the lim inf of backward difference quotients. This inequality holds for all to E [0, T] . Actually we have

:t It=to I-" (g (t) , 7 (t) )

2:

1M 27 (to) IRi j (to) + V'iV'jf (to) - 27 ~to)g (to)i I

2

j

x (41l"7 (to))-~e-f(to)dl-"g(to) for the minimizer f (to) of {W(g (to)," 7 (to)) : (g (to)," 7 (to)) EX}. Hence, from either (6.58) or (6.59), we have the following monotonicity formula for

1-". LEMMA 6.26 (I-"-invariant monotonicity). Let (g (t) ,7 (t)) , t E [0, T] , be a solution of (6.14) and (6.16) on a closed manifold Mn with 7 (t) > 0. For all tl ::; t2 ::; T, we have

°: ;

(6.60) In particular, (6.61) for t E [O,T] and r

> 0.

The following exercise continues our discussion in subsection 1.3 of this chapter. Again u ~ (41l"7) - ~ e- f . EXERCISE 6.27 (I-"k ) goo

90011~~cI>k(SUPP(f))'(cI>;1)*goo)

(47rT)-n/2 e- Jocl>;1 df..L(cI>-l)' k goo

-0, which implies

We conclude that

W (Noo , 900' j, T) =

J~ W (Uk, k

[9klvk] , j, T)

= k~oo lim W(Nk ,9k,jo;;!,T) 2:: lim sup f..L (9k, T) . k~oo

To see why the last inequality is true, note that the functions log Ck satisfy Ck - 1 and the constraints

!k

~ j

0

;; 1+

r (47rT)-n/2 e- Jkdf..L9k = 1

iNk

and W (Nk ,9k, j

0

;;1, T) = CkW (Nk,9k, jk, T) - Ck logCk

2:: Ckf..L (9k, T) - Ck log Ck· Since e- J/2 is an arbitrary nonnegative function with compact support and since it satisfies the constraint with respect to 900' we obtain f..L(900,T) 2:: lim sup f..L (9k,T) . k~oo

D

242

6.

ENTROPY AND NO LOCAL COLLAPSING

3. Shrinking breathers are shrinking gradient Ricci solitons Let (Mn,g(t)) , t E [O,T), be a shrinking Ricci breather on a closed manifold with g(t2) = aq,*g(iI), where t2 > tl and a E (0,1) . As discussed at the beginning of Section 1 of this chapter, we only need to show that when >.(g(t)) > for all t E [tl, t2], the shrinking breather is a shrinking Ricci soliton. By Koiso's examples (subsection 7.2 of Chapter 2), such solutions need not be Einstein. In this section we give two variations on the proof that shrinking breathers on closed manifolds are shrinking gradient solitons. The first proof, which also appears in Hsu [208], involves fewer technicalities in that it uses J-L instead of 1/. The first proof also does not use the assumption>. (g(t)) > O. On the other hand, the second proof requires some knowledge of the asymptotic behavior of J-L.

°

3.1. First proof using functional J-L. The following result of Perelman rules out periodic orbits for the Ricci flow in the space of metrics modulo diffeomorphisms and scalings. THEOREM 6.29 (Shrinking breathers are gradient solitons). A shrinking breather for the Ricci flow on a closed manifold must be a gradient shrinking Ricci soliton. FIRST PROOF. Let (M,g(t)) be a shrinking Ricci breather with g(t2) aq,*g(tl), where t2 > tl and a E (0,1). Define T

(

=

t2 - atl t ..... 1 - t, )

...!...

-a

so that ~; = -1,

and T (t2) = aT (tl). By Lemma 6.24, there is a minimizer

12 for

{W(g (t2),f, T (t2)) : f E Coo (M) satisfies (g, f, T) E X}, so that W (g (t2), 12, T (t2)) = J-L (g (t2) , T (t2)). Define f (t) to solve (6.15) on [t}, t2] with f (t2) = h. By the monotonicity formula (6.17) and the definition of J-L, we have J-L

(g (tJ), T (tl)) ::; W (g (tl) , f (tl) , T (tl)) ::; W (g (t), f (t), T (t)) ::; W (g (t2) , 12, T (t2)) = J-L (g (t2) , T (t2))

for all t E [tl, t2]. Since g(tl) = aq,*g(t2) and T (t2) diffeomorphism and scale invariance of J-L, we have J-L

(g (tl) , T (tl)) =

J-L

= aT (tl),

(g (t2) , T (t2)).

This and the fact that W (g (t) ,I (t) , T (t)) is monotone implies W (g (t), f (t), T (t))

= J-L (g (t), T (t)) == const

by the

3.

SHRINKING BREATHERS ARE SHRINKING GRADIENT RICCI SOLITONS

for t E [tt, t2J. Thus W (g (t), f (t), T (t))

It

f (t) == 0,

243

is the minimizer for W (g (t), f (t), T (t)) and so by (6.17), we have

1M I~j + '\h'Vjf for all t E [tl' t2J . We conclude that goo (6.62) ~j + V/vjf - 2~

~; 12 e- JdJl == 0

= 0 for t E [tl' t2J.

Since a breather is a periodic solution of the Ricci flow (modulo diffeomorphisms and homotheties), by the uniqueness of solutions to the Ricci flow on closed manifolds, the behavior of 9 (t) on [tt, t2J determines completely the behavior of 9 (t) on its whole time interval of existence. This is why, from (6.62), which is valid on [tl' t2J , one can deduce that 9 (t) is a breather on [O,T). D

3.2. Asymptotic behavior of Jl and finiteness of //. In the second proof of Theorem 6.29 given below we need the finiteness of //, which in turn depends on A(g(t)) > 0 and the following asymptotic behavior of Jl. We have shown that for each 9 and T > 0, Jl (g, T) is finite. However we have yet to study the behavior of Jl (g, T) as T -+ 00 or T -+ O. Recall that

A(g)

=

Ad-4~ + R) = inf {1M (R + IVfI2)e- JdJl : 1M e- JdJl = I}.

Since Jl and Ware modifications of A and F, we can prove the following. LEMMA 6.30 (Jl

-+ 00

as T -+

00

when A > 0). If A(g)

lim Jl (g, T)

T-+oo

> 0, then

= +00.

REMARK 6.31. The idea of the proof is that when T -+ 00, the F term in the expression (6.4) for W dominates, so if inf F > 0, then inf W -+ 00 as T -+ 00. PROOF. By Lemma 6.24, for any T > 0, there exists a Goo function fT with fM (47rT)-n/2 e-f.r dJl = 1 such that

Jl(g, T) = W(g, fn T) =

1M [T (R + IV fT12) + fT - n] (47rT)-n/2 e-f. dJl. r

We add a constant to fT so that it satisfies the constraint for F (g, .) (instead

1

of W(g,·, T)) and we define ~ fT+~ log (47rT) so that fM e- i dJl = 1. Then by the logarithmic Sobolev inequality (e.g., Corollary 6.38 with b = 1), we have

Jl(g, T)

= 1M [T (R + IV112) + 1 - ~ log (47rT) -

~ 1M (TR + (T -

~ (7 -

1)

1)

n] e- i dJl

IV112) e- i dJl- ~ log (47rT) -

1M (R + IV112) e- i dJl + R

min -

n - Gdg)

~ log (47rT) -

n - Gl (g).

6.

244

Hence, if

7 ~

ENTROPY AND NO LOCAL COLLAPSING

1, we have

/-L(g, 7)

(6.63)

~

(7 - 1) ,\(g) -

Since '\(g) > 0, we have limT->oo /-L (g, 7)

n

2 log 7 -

C2 (g) .

o

= +00.

EXERCISE 6.32. Show that if '\(g) < 0, then limT->oo /-L (g, 7) = -00. In particular, if '\(g) < 0, then v (g) = -00. SOLUTION TO EXERCISE 6.32. Since '\(g) = 1 and

< 0,

there exists

fo

with

fM e- fOd/-L

a,* 1M Define 1,*

fo -

(R + IV' fol 2) e- fOd/-L < 0.

~ log (471'7) so that fM(471'7)-n/2 e- Jd/-L = 1. We have for all

7>0 /-L (g, 7)

50 W (g, 1,7) 50 7 1M

= a7 +

7 ---+

(R + 1V'112) + 1 -

n] (471'7)-n/2 e- Jd/-L

(R + IV' fol2) e-fod/-L + l 1M (471'7)-n/ 2d/-L (471'7 ) -n/2

since xe- x 50 ~ for x When

1M [7

=

e

> 0.

Vol (g) ,

The result follows from a < 0.

0+, we have

LEMMA 6.33 (Behavior of /-L(g,7) for

7

small). Suppose (Mn,g) 'ts a

closed Riemannian manifold. (i) There exists f > such that

°

/-L (g, 7)


0, then v (g) is well defined and finite. Also, there exists 7 > such that v (g) = /-L (g, 7) .

°

3.3. Monotonicity of v and the second proof. The following lemma is the monotonicity property of v(g(t)) along the Ricci flow g(t). LEMMA 6.35 (v-invariant monotonicity). Let (Mn, 9 (t)) , t E [0, T), be a solution to the Ricci flow on a closed manifold. (1) The invariant v(g(t)) is nondecreasing on [0, T), as long as v(g(t)) is well defined and finite.

3.

SHRINKING BREATHERS ARE SHRINKING GRADIENT RICCI SOLITONS

245

°

(2) Furthermore, if A (g (t)) > and ifv(g(t)) is not strictly increasing on some interval, then g(t) is a gradient shrinking Ricci soliton. (3) Ifv(g(to)) = -00 for some to, then v(g(t)) = -00 for all t E [0, to]. PROOF. (1) Given any

°

~

iI < t2 < T, we shall show that

(6.64)

>

Since by assumption, V(g(t2)) such that

-00,

for any E >

°

there exist

12

and T2

W(g(t2), 12, T2) ~ V(g(t2)) + E. Let (f(t), T(t)) , t E [0, t2], be a solution of the backward heat-type equation (6.15) with f(t2) = 12 and T(t2) = T2. By the monotonicity formula (6.17), we have l3

W(g(t2), f(t2), T (t2))

~

W(g(tt}, f(tl), T (tl)),

where equality holds if and only if 1 ~j

°

+ "h'ljf - 2Tgij =

for all t E (tI, t2)'

This implies

V(g(t2))

+ E ~ W(g(t2), f(t2), T (t2))

The result follows since

E

>

°

~

W(g(tl), f(iI), T (tl))

~

V(g(tl))'

is arbitrary.

(2) Suppose V(g(tl)) = V(g(t2)) for some tl < t2' Since A (g (t)) Corollary 6.34, there exist 12 and T2 such that

> 0, by

W(g(t2), 12, T2) = V(g(t2)). In this case, by repeating the argument in (1), we obtain W (g (t), f (t), T (t))

= v(g(t)) == const

for all t E [tl, t2]' As in the proof of Theorem 6.29, we can conclude that g(t) is a gradient shrinking Ricci soliton. (3) If v(g(to)) = -00, then for any N > -00 there exist fa and TO such that W (g (to), fa, TO) ~ N. Let (f(t), T(t)) , t E [0, to], be the solution of (6.15) with f(to) = fa and T(tO) = TO. For all t E [0, to],

v(g(t)) Since N >

-00

~

W (g (t), f (t), T (t))

~

W(g(to), f(to), T (to))

is arbitrary, we conclude v(g(t)) =

-00

~ N.

for all t E [0, to].

D

Using the v-invariant instead of the J.L-invariant, we can give a SECOND PROOF OF THEOREM 6.29. As we stated at the beginning of this section, we only need to consider a shrinking breather g( t) with g( tt} = a 0, t E [tl, t2]' From the elementary properties (iii) and (iv) of J.L and v in subsection 2.1 of this chapter, we have v(g(tt}) =

6.

246

ENTROPY AND NO LOCAL COLLAPSING

1I(g(t2)). Now the theorem follows from Corollary 6.34 and Lemma 6.35(1)(2), which tell us that 1I (g (t)) is monotone and finite and characterizes when 1I (g (t)) is constant. 0 4. Logarithmic Sobolev inequality In this section we give a proof of the logarithmic Sobolev inequality which we have used earlier. The logarithmic Sobolev inequality is related to the usual Sobolev inequality and has the advantage of being dimensionless.

4.1. Logarithmic Sobolev inequality on manifolds. LEMMA 6.36 (Log Sobolev inequality, version 1). Let (Mn, g) be a closed Riemannian manifold. For any a > 0, there exists a constant C (a, g) (given by (6.67)) such that if'P > satisfies iM 'P2dJ.l = 1, then

°

1M 'P210g 'PdJ.l :::; a 1M 1V''P12 dJ.l + C (a, g) .

(6.65)

PROOF. Recall that the Sobolev inequality (see Lemma 2 in [245]) that

if

iM 'P2dJ.l = 1, then (assume n > 2)

(6.66) where V = Voig (M). Note that for en = ~e we have en log x :::; x2/n for all x> 0, so that

1M 'P2+~ dJ.l :::; E1M{ 'P2+~ dJ.l + ~ 1M( 'P2 dJ.l ,

1M

en { 'P210g 'PdJ.l:::; {

E

for any E > 0, since 'P1+~ 'P :::; E'P2( l+~)

+ ~'P2.

By Holder's inequality,

n-2

1M 'P2'P~dJ.l:::; (1M 'Pn2~2dJ.l) n- (1M 'P2dJ.l) Hence, using

iM 'P2dJ.l =

2

TO

1, we have

n-2

en

1M 'P210g 'PdJ.l :::; E (1M 'P n2~2 dJ.l) n- + ~ :::; C (~, g) (1M 1V''P12 dJ.l + v- 2/ n ) + ~. s

Inequality (6.65) follows by choosing (6.67)

C (a,g ) -- aV -2/n

+ an2 e2C4s (M ,g)"

Now we have proved the lemma when n > 2. We leave the n an exercise. EXERCISE 6.37. Prove the above lemma when n

= 2 case as

= 2.

Making the substitution 'P = e-¢>/2 in (6.65), we have the following.

0

4.

LOGARITHMIC SOBOLEV INEQUALITY

247

COROLLARY 6.38 (Log Sobolev inequality, version 2). For any b > 0, there exists a constant C (b, g) such that if a function satisfies fM e-dJ-t = 1, then (6.68)

4.2. Logarithmic Sobolev inequality on Euclidean space. We give a proof of Gross's logarithmic Sobolev inequality on Euclidean space [170]. Although this result will not be used elsewhere in Part I of this volume, we include it here since it is both fundamental and elegant. THEOREM 6.39. For any nonnegative function E W 1,2 (JRn) , we have

Note that the above inequality is scale-invariant, that is, the inequality is preserved under multiplication of by a positive constant. Also, if fRn 2dx = 1, then the inequality says that fn~.n 2 log dx ~ fRn IV12 dx. The following consequence of Gross's logarithmic Sobolev inequality is actually equivalent to it. (We leave the proof of the equivalence to the reader.) COROLLARY 6.40. If fn~.n (47f'r)-n/2 e- f dx (6.69)

kn

(r

In particular, taking r

IV fl2 + f

= 1, then

- n) (47f'r)-n/2 e- f dx 2: O.

= 1/2, we have

(6.70)

provided fRn (27f')-n/2 e- f dx = 1. Moreover, if we can perform an integration by parts, then we may rewrite (6.69) as (6.71)

kn

(r (2D.f

-I VfI 2 ) + f

- n) (47f'r)-n/2 e- f dx 2: O.

REMARK 6.41. Compare the LHS of (6.69) with the entropy (6.1) and compare the integrand on the LHS of (6.71) with Perelman's differential Harnack quantity (6.20). PROOF OF THE COROLLARY. We shall prove just the case where r = 1/2 since the general case follows from making the change of variables x ~ (2r)-1/2 x. Let be defined by f = ~ - 210g, so that e-f = e-lxI2/2. 2

248

6.

and \1 f

ENTROPY AND NO LOCAL COLLAPSING

= x - 2'2/. We compute in

(~I\1 fl2 + f

= 2in where dv

- n) (27r)-n/2 e- f dx

(~ Ixl2 cj>2 -

cj> X . \1 cj> + 1\1 cj>1 2 - cj> 210g cj> - %cj>2) dv,

= (27r)-n/2 e-lxI2/2dx. Now integrating by parts yields

r

J/Rn

_cj> x. \1cj>dv

=~

r

2 J/Rn

ncj>2dv -

~

r

2 J/Rn

Ixl 2 cj> 2dv,

so that we have the identity

Hence (6.72)

in

(~I\1fI2+f-n) (27r)-n/2e-fdx=2in (1\1cj>1 2 -cj>210g cj»dv,

with the constraint

Since log U/Rn cj> 2dv) ity, we have

= 0, by (6.72) and Gross's logarithmic Sobolev inequal-

o EXERCISE 6.42. Show that Gross's logarithmic Sobolev inequality for Euclidean space implies that Euclidean space (JR n , 91E) has nonnegative entropy: W (9IE, f, 7) ~ 0 and

JL (9IE, 7)

= O.

Now we give Beckner and Pearson's proof of Gross's logarithmic Sobolev inequality, which is a consequence of the following [23]. PROPOSITION 6.43. If J/R n 'Ij; (x)2 dx (6.73)

~ log (_2_ 4

r

7ren J/Rn

=

1\1'1j; (x)1 2dX)

1, then

~

r

J/Rn

(log 1'Ij; (x)l) 'Ij; (x)2 dx.

Note that this inequality is scale-invariant. We first show that (6.73) implies Gross's logarithmic Sobolev inequality.

4. LOGARITHMIC SOBOLEV INEQUALITY

249

6.39 FROM THE PROPOSITION. Given f such that

PROOF OF THEOREM

r (27r)-n/2 e- dx = 1,

(6.74)

f

J~n

let 'I/J ~ (27r)-n/4 e-f/2, so that 10g'I/J Then (6.73) implies

~ log (7r~n

(6.75)

~-

In ~ IV'

= -~ - %log (27r)

and J~n 'l/J 2dx

= l.

fl2 e- f (27r)-n/2 dX)

r (L2 + ?!:4 log (27r)) e- f (27r)-n/2 dx,

J~n

so that

~

r lV'fI2e-f (27r)-n/2dx ~ en2 exp{-~nknr fe-f (27r)-n/2 dX}.

2kn

We claim

r

(6.76) en exp {-~ fe- f (27r)-n/2 dX} 2 n J~n which implies the

T

Since J~n (27r)-n/2 e- f dx

?!:exp{~

+f

fl2

(n - f) e- f (27r)-n/2 dx,

= 1,

- n) e- f (27r)-n/2 dx

~ O.

inequality (6.76) is equivalent to

r

(?!:-f)e- f (27r)-n/2 dX } n J~n 2

2

r

J~n

= 1/2 case of (6.69):

In (~IV'

(6.77)

~

~?!:+

r

(?!:-f)e- f (27r)-n/2 dx . 2 J~n 2

If we let du = e- f (27r) -n/2 dx and 9 = ~ - f, then the above inequality becomes

9dU} ~ 1 + ~ gdu. n J~n n J~n 1 + a for all a E R

exp This follows from e a

~

{~

r

r

D

Now we present the PROOF OF PROPOSITION

6.43. By Jensen's inequality, if J~N

IFl r dx =

1, then

(p - r)

r

J~N

(log IFI) IFl r dx =

r

J~N

Slog

log

(IFIP- r ) IFl r dx

(IN IFIP-

r

IFl r dX) = log

for p ~ r > O. The L 2-Sobolev inequality says 2 IIFIILN* S

AN

l

~N

IV'FI 2 dx,

(IN IFIP dX)

6. ENTROPY AND NO LOCAL COLLAPSING

250

2N r(N) )2/N I By Sterl'mg '£ h N * -- N-2 were an d A N -- ( r(N/2) nN(N-2)' s ormu1a, r(N) rv .j2iiNN-!e- N , so that AN rv n;N for N large, where r is the Gamma function. Hence, if fIRN 1F12 dx = 1, then by above two inequalities

log ( AN

IN IV'FI2 dX) ~ ~* log (IN IFI dX) ~ ~ IN (log IF!) 1F12 dx, N•

where we used ~. (N* - 2) = Given f : ]Rn ~ ]R with In~.n by

t.

f 2 dx =

1, let N

= nf and define F : ]RN

~ ]R

i

F (x) ~

IIf (Xk) , k=l

where x

=

(Xl, ... ,

Xi), Xk

E ]Rn for

k

= 1, ... , f.

Now

V'F(x) = (V'f(XI) V'f(Xi)) F (x) f (xI) , ... , f (Xi) . Hence

and

i

since In~.n f (x)2 dx

=

1. Using In~nl F (x)2 dx

=

II fIRnf (Xk)2 dXk = 1, we k=l

have

5. NO FINITE TIME LOCAL COLLAPSING

251

Now

f

JlRnl =

(log IF (x)l)

lnl t IR

IF (x)12 dx

(f (xk)2Iog Ilk (xk)1

k=l

IIf (Xi)2) dX1 ... dxR.

i#k

= f f (log If (x)J) f (x)2 dx.

JlRn

Hence

for all fEN. Recall that AnR. implies log

rv

7r:nR.' which by taking the limit as f

(~ f IV f (x)1 2 dX) 7ren JlRn

-t

00,

2: i f f (x)2Iog If (x)1 dx. n

JlRn

This completes the proof of the proposition.

o

5. No finite time local collapsing: A proof of Hamilton's little loop conjecture In this section we first define the notion of ~-noncollapsed at scale r and show its equivalence to the injectivity radius estimate. We then prove Perelman's celebrated no local collapsing theorem and indicate its equivalence to Hamilton's little loop conjecture. We end this section by showing the existence of singularity models for solutions of the Ricci flow on closed manifolds developing finite time singularities corresponding to sequences of points and times with curvatures comparable to their spatial maximums. Perelman's no local collapsing theorem solves a major stumbling block in Hamilton's program for the Ricci flow on 3-manifolds. In particular, it provides a local injectivity radius estimate which enables one to obtain singularity models when dilating about finite time singular solutions of the Ricci flow on closed manifolds of any dimension. The no local collapsing theorem also rules out the formation of the cigar soliton as a singularity model. 14 The above two consequences of the no local collapsing theorem, together with Hamilton's singularity theory in dimension 3, imply that necks exist in all finite time singular solutions on closed 3-manifolds. This, together with Hamilton's analysis of nonsingular solutions, leads one to hope/expect that 14More precisely, in dimension 3 it rules out singularity models which are quotients of the product of the cigar soliton and the real line.

252

6.

ENTROPY AND NO LOCAL COLLAPSING

Ricci flow with surgery may lead to the resolution of Thurston's geometrization conjecture. Perelman's deeper analysis of 3-dimensional singularity formation greatly strengthens this expectation. I5 5.1. K-noncollapsing and injectivity radius lower bound. 5.1.1. K-noncollapsing on a Riemannian manifold. Let (Mn, complete Riemannian manifold.

g)

be a

DEFINITION 6.44 (K-noncollapsed). Given p E (0,00] and K > 0, we say that the metric 9 is K-noncollapsed below the scale p if for any metric ball B(x,r) with r < p satisfying I Rm(y)1 ::; r- 2 for all y E B(x,r), we have VolB(x,r) > rn -

(6.78)

K.

If 9 is K-noncollapsed below the scale 00, we say that 9 is K-noncollapsed at all scales. Complementarily, we give the following. DEFINITION 6.45 (K-collapsed). We say that 9 is K-collapsed at the scale r at the point x if I Rm(y) I ::; r- 2 for all y E B(x,r) and Vol B(x, r) < K. rn

(6.79)

The metric 9 is said to be K-collapsed at the scale r if there exists x E M such that 9 is K-collapsed at the scale r at the point x. Thus 9 is not K-noncollapsed below the scale p if and only if there exists r < p and x E M such that 9 is K-collapsed at the scale r at the point x. REMARK 6.46.

(1) If Mn is closed and flat, then 9 cannot be K-noncollapsed at all scales since I Rm I = 0 ::; r- 2 for all r and Vol B(x, r) ::; Vol

(M)

so that limr-->oo Vol~~x,r)

= 0 for all x E M.

g)

(2) If (Mn, is a closed Riemannian manifold, then for any p > 0 there exists K > 0 such that 9 is K-noncollapsed below the scale p. We have the following elementary scaling property for K-noncollapsed metrics. LEMMA 6.47 (Scaling property of K-noncollapsed). If a metric 9 is Knoncollapsed below the scale p, then for any a > 0 the metric a 2 g is Knoncollapsed below the scale ap. I5Some aspects of Perelman's singularity theory are discussed in Chapter 8 of Part I and also in Part II of this volume.

5.

NO FINITE TIME LOCAL COLLAPSING

253

PROOF. We leave it as an exercise to trace through the definition of Knoncollapsed and verify that the lemma follows from the scaling properties: Bg(x, r) = B Q 2 g(X, ar), I RmQ 2 g(Y) I = a- 2 1 Rmg(Y)I, and VoI Q 2 g B Q 2 g(X, ar) = anVolgBg(x,r). 0 The next lemma says the property of being K-noncollapsed below the scale p is preserved (stable) under pointed Cheeger-Gromov limits. LEMMA 6.48 (K-noncollapsed preserved under limits). Let {(M k,gk, Ok)} be a sequence of pointed complete Riemannian manifolds. Suppose that there

exist K > 0 and p > 0 so that each (Mk' 9k) is K-noncollapsed below the scale p. Furthermore assume that (Mk' gk, Ok) converges to (M~, goo, 0 in the pointed Cheeger-Gromov C 2-topology. Then the limit (Moo, goo) is K-noncollapsed below the scale p. (0 )

PROOF. This is because the distance function, the curvature, and the volume all converge under the limit. In particular, suppose x E Moo and r < p are such that I Rmgoo (Y)I :::; r- 2 for all Y E Bgoo (x, r). Then for every £ E (O,r), there exists k(£) EN such that I Rmgk(y)1 :::; (r_£)-2 for all y E B9k (x, r - E) and for all k 2: k (E) . Since each gk is K-noncollapsed below the scale p, we have Volgk(Bgk(x, r-£)) 2: K (r - Et for all k 2: k (£). Taking the limit as k ---t 00, we have Volgoo(Bgoo(x,r - E)) 2: K(r - £t. Letting £ ---t 0, we then conclude that Volgoo(Bgoo(x,r)) 2: Kr n as desired. 0 Recall that if Rc 2: 0 on a complete Riemannian manifold (Mn,

g) ,

then for p E M fixed, Vol~~p,r) is a nonincreasing function of r. When M is noncompact, the notion of K-noncollapsed at all scales is closely related to another invariant of the geometry of infinity called the asymptotic volume ratio, which we now define.

g)

DEFINITION 6.49 (Asymptotic volume ratio). Let (Mn, be a complete noncompact Riemannian manifold with nonnegative Ricci curvature. The asymptotic volume ratio is defined as the limit of volume ratios by

(6.80) where

AVR(g) Wn

~

lim VoIB(p,r) < 00, r->oo wnrn

is the volume of the unit ball in

jRn.

We say that (M,g) has

maximum volume growth if AVR(g) > O. REMARK 6.50. The asymptotic volume ratio is independent of the choice of basepoint p EM. EXERCISE 6.51 (AVR> 0 implies noncollapsed). Show that if (Mn,

g)

is a complete noncompact Riemannian manifold with Rc 2: 0 and AVR (g) > 0, then 9 is K-noncollapsed on all scales for K = Wn AVR (g) .

254

6.

ENTROPY AND NO LOCAL COLLAPSING

The next exercise shows that on small enough scales 9 is K-noncollapsed for some K. EXERCISE 6.52. Show that for any Riemannian manifold (Mn,

M, and have

K

< W n , there exists p (x) > 0 such that IRml ~ r- 2

and

g) , x E

for every r E (0, P (x)], we

in B (x, r)

Vol B(x, r)

---'-----'> K. rn SOLUTION TO EXERCISE 6.52. This follows from the facts that lim r2 sup IRml = 0 r-+O

B(x,r)

and lim VoIB(x,r) = rn

Wn.

r-+O

REMARK 6.53. In some sense Exercise 6.52 is a local version of Remark 6.46(2).

5.1.2. K-noncollapsing and injectivity radius lower bound. We now show that K-noncollapsing and a lower bound of the injectivity radius are equivalent. LEMMA 6.54. Let (Mn,g) be a complete Riemannian manifold and fix p

E (0,00]. (i) If the metric 9 is not K-collapsed below the scale p for some K > 0, then there exists a constant 8 = 8 (n, K) which is independent of that for any x E M and r < p, if IRml ~ r- 2 in B (x, r), then inj (x) ~ 8r. (ii) Suppose that for any x E M and r < p with IRml ~ r- 2 in B (x, r) we have inj (x) ~ 8r for some 8 > O. Then there exists a constant K = K (n, 8) , independent of p and g, such that 9 is not K-collapsed below the scale p. p and

9 such

PROOF. (i) Let B (x, r) be a ball satisfying IRml ~ r- 2 in B (x, r) for some r ~ p. Consider the metric r-2g on B (x,r) = B r -2 g (x, 1). Since 9 is not K-collapsed on B (x,r), we have IRmr -2 g I ~ 1 in B r -2 g (x, 1) and Voig B (x, r) n ~ K. r By a result of Cheeger, Gromov, and Taylor (see Theorem A.7), there exists 8 = 8 (n, K) such that injr-2g (x) ~ 8. Hence inj (x) ~ 8r. (ii) Again let B (x, r) be a ball satisfying IRmI ~ r- 2 in B (x, r) for some r ~ p, and consider the metric r-2g on B (x, r) = B r -2 g (x, 1). We have IRmr -2 g l ~ 1 and injr-2g (x) ~ 8. By the Bishop-Gromov volume (or Volr -2 g B r -2 g (x, 1) =

5. NO FINITE TIME LOCAL COLLAPSING

255

Rauch) comparison theorem (comparing (Br-2g (x, 1), r-2g) with the ball of radius 8 in the unit sphere (1)), there exists K = K (n, 8) such that Voig B (x, r) n = Volr -2 g B r -2 g (x, 1) ~ K. r

sn

o 5.1.3. K-noncollapsing in Ricci flow. DEFINITION

6.55. We say that a complete solution (Mn,g(t)) , t

E

[0, T), to the Ricci flow, where T E (0,00], is K-noncollapsed below the

scale p if for every t E [0, T), 9 (t) is K-noncollapsed below the scale p. If M is closed, Tl < 00, and Co ~ sUPMx[O,Td IRml < 00, then, using the metric equivalence e- 2(n-l)Co 9 (0) ::; 9 (t) ::; e2(n-l)Co g (0) for t E [0, T 1 ),16 we see that for every p E (0,00) there exists K = K (n, Co, 9 (0), TI, p) > such that the solution 9 (t) is K-noncollapsed below the scale p. Hence we are interested in K-noncollapsing near T when the solution forms a singularity at time T. We shall see that when T < 00 and M is closed, Perelman's monotonicity of entropy implies that for all p > the solution is K-noncollapsed below the scale p for some K = K(n, 9 (0), T, p) > 0. In §4.1 of [297] Perelman also gave the following.

°

°

DEFINITION 6.56 (Locally collapsing solution). Let (Mn,g(t)) , t E [0, T), be a complete solution to the Ricci flow, where T E (0,00]. The solution 9 (t) is said to be locally collapsing at T if there exists a sequence of points Xk EM, times tk -+ T, and radii rk E (0,00) with rVtk uniformly bounded (from above) such that the balls B9(tk) (Xk' rk) satisfy (1) (curvature bound comparable to the radius of the ball)

IRm [g (tk)]1 ::; r k2

in

B9(tk)

(Xk' rk),

(2) (volume collapse of the ball) . Volg(tk) hm

k-+oo

B9(tk)

rk'

(Xk' rk)

=0.

EXERCISE 6.57. It is interesting to consider solutions to the Ricci flow which are defined on a time interval of the form (0, T) with the curvature becoming unbounded as t -+ 0+. For example, consider an initial metric 90 on a surface which is Coo except for a conical singularity. We expect a smooth solution 9 (t) of the Ricci flow to exist on some time interval (0, T) with 9 (t) -+ 90 as t -+ 0+. It is interesting to ask if solutions on closed manifolds can locally collapse as t -+ 0+. In view of this, for a solution defined on (0, T) , formulate the notion of locally collapsing at time 0.

16For the proof of this metric equivalence, see Corollary 6.50 on p. 204 of Volume One. The argument there is essentially repeated in the proof of the inequalities in (3.3) of this volume.

256

6.

ENTROPY AND NO LOCAL COLLAPSING

5.2. The no local collapsing theorem and its proof. 5.2.1. No local collapsing theorem and little loop conjecture. One of the major breakthroughs in Ricci flow is the following. THEOREM 6.58 (No local collapsing-A). Let g(t), t E [0, T), be a smooth solution to the Ricci flow on a closed manifold Mn. If T < 00, then for such that g(t) is K,any p E (0,00) there exists K, = K,(n,g(O),T,p) > noncollapsed below the scale p for all t E [0, T).

°

We shall prove this theorem in the next subsubsection. Actually Theorem 4.1 of [297] states the result a bit differently. THEOREM 6.59 (No local collapsing-B). If M is closed and 9 (t) is any solution on [0, T) with T < 00, then 9 (t) is not locally collapsing at T. REMARK 6.60. We leave it as an exercise to show that Theorems 6.58 and 6.59 are equivalent.

Hamilton's little loop conjecture says the following (see §15 of [186]). Let (M n , g (t)) , t E [0, T), be a smooth solution to the Ricci flow on a closed manifold. There exists 8 = 8 (n,g (0)) > such that for any point (x, t) EM x [0, T) where

°

1 IRm (g (t))1 ::::; W2

in

Bg(to)

(x, W)

for some W > 0, we have injg(t) (x) ~

8W.

Note that the role of the positive number K, in the definition of K,noncollapsed is similar to the role of 8 in the injectivity radius lower bound which is used in the statement of Hamilton's little loop conjecture. Rephrasing the little loop conjecture (LLC) a little differently, we have the following equivalence between no local collapsing (NLC) at T and the little loop conjecture. LEMMA 6.61 (NLC and LLC are equivalent). Let (Mn, 9 (t)) , t E [0, T), be a smooth complete solution to the Ricci flow where T E (0,00]. The following two statements are equivalent.

°

(i) (Little loop conjecture) For any C > there exists 8 > if (x, t) E M x [0, T) and W E (0, v'ct] satisfy 1 IRm(t)1 ::::; W2

in

Bg(t)

°

such that

(x, W),

then (6.81)

injg(t)

(x) ~

8W.

(ii) (No local collapsing) The solution g(t) is not locally collapsing at T.

5. NO FINITE TIME LOCAL COLLAPSING

257

PROOF. (i) ==> (ii). We prove (ii) by contradiction. Suppose 9 (t) is locally collapsing at T. Then there exists a sequence of times tk / T and a sequence of metric balls B9(tk)(Xk, rk) such that r2 (1) C for some C < 00,

t: : ;

(2) IRm(g (tk))1 ::; r-;;2 in Bg(tk) (Xk' rk), (3) Volg(tk) Bg~k) (Xk' rk) '\. 0 as k ~ 00. rk Hence by (i) we have injg(tk) (Xk) ~ ork for all k, where 0> 0 is independent of k. By Lemma 6.54(ii), the volume collapsing statement (3) above cannot be true, a contradiction. (ii) ==> (i). We also prove (i) by contradiction. If (i) is not true, then there exists C > 0 and a sequence of points and times (Xk' tk) EM x [0, T) and Wk E (0, JCtk] satisfying 1 . IRm (tk)1 ::; W2 In Bg(tk) (Xk' Wk) k

and

injg(tk) (Xk) Wk

'\.

O.

k) Lemma 6.54 (1.).Impl'Ies t h at Volg(tkJ B9(tk)(Xk,W witk '\. 0 as k ~ 00. T hus 9 (t ) IS. locally collapsing at T and we have a contradiction. The lemma is proved.

o It follows from Theorem 6.59 and Lemma 6.61 that Hamilton's little loop conjecture holds for solutions of the Ricci flow on closed manifolds forming finite time singularities. COROLLARY 6.62. Let g(t), t E [0, T), be a smooth solution to the Ricci flow on a closed manifold Mn. If T < 00, then the little loop conjecture holds. That is, for any C > 0 there exists 0 > 0 such that if (x, t) E M x [0, T) and W E (0, JOt] satisfy 1 IRm (t)1 ::; W2 in Bg(t) (x, W),

then we have injg(t) (x)

~

oW.

The little loop conjecture illustrates the essence of no locally collapsing from the injectivity radius perspective. For convenience we give the following DEFINITION 6.63 (Local injectivity radius estimate). We say that a complete solution (Mn,g(t)), t E [O,T), to the Ricci flow satisfies a local injectivity radius estimate if for every p E (0,00) and C < 00, there exists c = c (p, C, 9 (t)) > 0 such that for any (p, t) E M x [0, T) and r E (0, p] which satisfy IRm (., t)1 ::; Cr- 2 in Bg(t) (p, r),

258

6.

ENTROPY AND NO LOCAL COLLAPSING

we have injg(t) (p) 2: cr. Corollary 6.62, i.e., Perelman's no local collapsing theorem, implies that if (Mn,g (t)), t E [0, T), is a solution of the Ricci flow on a closed manifold with T < 00, then g(t), t E [0, T), satisfies a local injectivity radius estimate. 5.2.2. Proof of No Local Collapsing Theorem 6.58. The idea of the proof is that if a metric 9 is K-collapsed at a point x at a distance scale r for K small and r bounded, then W (g, f, r2) is negative and large in magnitude, e.g., on the order of log K, for f concentrated in a ball of radius r centered at x. This contradicts the monotonicity formula for p, (g (t), r (t)) . PROOF OF THEOREM 6.58 ASSUMING PROPOSITION 6.64. We shall say that r is the (space) scale of p, (g, r2); the justification for this terminology occurs below. Since T /2 < T, by the remarks after Definition 6.55, there exists KO = KO (n, 9 (0) , T, p) > 0 such that g(t) is Ko-noncollapsed below the scale p for all t E [0, T /2]. On the other hand, if t E [T /2, T), then for any 0 < r ::; p, we have t + r2 E [T /2, T + p2), and by the monotonicity formula (6.61), we have p, (g (t) , r2) 2: p, (g (0) , t (6.82)

2:

+ r2)

inf TE[T/2,T+p2]

p,(g(O),r)

~

-CI(n,g(O),T,p) >-00

since T < 00. In summary, by the monotonicity formula, since the p,invariant of the initial metric is bounded from below at scales bounded from above and below, the p,-invariant of the solution after a certain amount of time (say T /2) is bounded from below at all bounded scales. The theorem will follow from the important observation that if a Riemannian metric is K-collapsed at some scale r for K small, then its p,-invariant is negative and large in magnitude at the time scale r2 (see Proposition 6.64 below). If x E M, t E [T /2, T), and r E (0, p] are such that Rmg(t) 1 in 2 2 Bg(t) (x, r) , then RCg(t) 2: -Cl (n) r- and Rg(t) ::; Cl (n) r- in Bg(t) (x, r) , where Cl (n) = n (n - 1) . So by (6.82) and (6.83),

I

-CI(n,g(O),T,p)::;p, (g(t),r

2)

::;log

Volg(t) Bg(t) (x, r)

r

n

I ::;

+Cdn,p).

We conclude that Volg(t) Bg(t) (x, r)

rn

> -

(

Kl n, 9

(0)

"

T

)

P

>0

,

where Kl (n, 9 (0), T, p) = e- C1 (n,g(0),T,p)-C2(n,p). The theorem follows with the choice K(n, g(O), T, p) ~ min {KO, Kl}. 0 Now we turn to bounding the p,-invariant from above by volume ratios in a Riemannian manifold. PROPOSITION 6.64 (p, controls volume ratios). Let p E (0,00). There

exists a constant C2 = C2 (n, p) < 00 such that if

(Mn, g)

is a closed

5. NO FINITE TIME LOCAL COLLAPSING

259

Riemannian manifold, pENt and r E (0, pJ are such that Rc ~ and R ~ Cl (n) r- 2 in B (p, r), then ~ 2) J-L ( g,r

(6.83)

~log

Vol B (p, r) r

n

-Cl

(n) r- 2

+C2 (n,p).

That is,

In particular, if for some K > the scale r, then

°

and r E (0, pJ the metric f) is K-collapsed at

J-L (f), r2) ~ log K Proof. As in (6.41) with

T

+ C2 (n, p) .

= r2, define the positive function

W

by

(6.84) From the definition (6.56) of J-L as an infimum of W, we have by rewriting W (f),f,r 2) in terms of W (compare to (6.42)), (6.85)

J-L (f), r2)

~ 1M r2 (41V'w12 + Rw 2) dJ-L + 1M (f -

n) w 2dJ-L

~ J( (f),w,r2) ,

where iM w 2dJ-L = 1 and f = -2 log w- ~ log (47rr2) . While making the convention that f (y) w 2 (y) = when w (y) = 0, we claim that (6.85) holds for nonnegative Lipschitz functions w satisfying iM w 2dJ-L = 1. To see this claim, first by (6.56) we know that (6.85) holds for positive Lipschitz functions w satisfying i M w 2dJ-L = 1. Now given any nonnegative Lipschitz function w satisfying iM w 2dJ-L = 1, define for c E (0,1),

°

we;~Ce;(w+c),

where the constant Ce; is defined by iM w;dJ-L = 1. Clearly lime;->oCe; = 1, We; is a positive Lipschitz function, and hence J-L (f), r2) ~ J( (f), We;, r2) for each c E (0,1). Using lime;->oclogc = and fe; = -2 log We; - ~ log (47rr2) in the definition of K (f),we;,r 2) , we have lime;->oK (f), We;, r2) = K (f),w,r 2). The claim is proved. Now let ¢ : [0,00) --t [0, 1J be a standard cut-off function with ¢ = 1 on [0, 1/2J , ¢ = on [1,00), and WI ~ 3. Assume pENt and r E (0, pJ are such that we have the curvature bounds Rc ~ -Cl (n) r- 2 and R ~ Cl (n) r- 2 in B (p, r) for Cl (n) = n (n - 1) . We make a judicious choice for f so that the RHS of (6.85) reflects the local geometry at p with respect to the metric f). In particular, let

°

°

(6.86)

6. ENTROPY AND NO LOCAL COLLAPSING

260

where dg(x,p) is the distance function and the constant c = c(n,g,x,r) is chosen so that fM w 2df.l = 1. Note that w is a Lipschitz function and definition (6.84) implies (6.87) (We abuse notation and write ¢ (x)

to the volume ratio LEMMA

Vol ~n(p,r)

= ¢ (dg(;,P))

.)

The constant c is related

by the following.

6.65. There exists C3 (n, p)
0 such that 900 (t) is ~-noncollapsed on all scales. PROOF. By Perelman's no local collapsing theorem, we have an injectivity radius estimate at the points Pk with respect to the metrics gk (0). Hence by Hamilton's compactness theorem (Theorem 3.10), there exists a subsequence such that (Mn,gi(t),Pi) converges to a complete ancient solution (M~, goo(t),poo) to the Ricci flow. Since 9 (t) is ~-noncollapsed on the scale JT for all t E [0, T), we have gi(t) is ~-noncollapsed on the scale 18 As usual we denote a subsequence of i still by i rather than ij to simplify our notation.

264

6.

ENTROPY AND NO LOCAL COLLAPSING

v'KiT for all t E [-Kiti,KdT-ti))' Since limi ...... oov'KiT = 00, goo(t) is K-noncollapsed on all scales from Lemma 6.48 (regarding a limit property of sequences of K-noncollapsed solutions). 0 5.3.2. Ruling out the cigar as a finite time singularity model. Finally, we observe that Perelman's no local collapsing theorem implies that the cigar (product with any flat solution like lRn - 2 or a torus Tn-2) cannot be a limit of dilations about a finite time singularity as in Theorem 6.68. This is because the cigar (product with any flat solution) is not K-noncollapsed on all scales for any K > O. An easy way to see this is that the cylinder S1 x lR is a limit of the cigar. Clearly, S1 x IR (product with any flat solution) is not K-noncollapsed on all scales for any K > O. By the property of K-noncollapsed being preserved under limits, this implies the same for the cigar (product with any flat solution). 6. Improved version of no local collapsing and diameter control

In this section we give a proof of Perelman's improvement of his no local collapsing theorem to the case where one assumes, in the ball to be shown to be noncollapsed, only the scalar curvature has an upper bound. We also present the work of Topping [357] on diameter control. We end this section with a variation on the proof of Perelman's no local collapsing theorem. 6.1. Improved version of no local collapsing. We first revisit and revise Proposition 6.64. Let (Mn, be a closed Riemannian manifold and

g)

r > O. Again we shall consider the inequality (6.85) for J.l (g, r2) and the test function w defined by (6.86). It is easy to see that the proof of Lemma 6.65 yields the following, where the estimate now involves Vol B (p, r /2) , which we had previously estimated in terms of Vol B (p, r) under a local Ricci curvature lower bound assumption. LEMMA 6.69 (c and the volume ratio). The constant c in (6.86) satisfies the following bounds: 1 < (4 2)-n/2 -c < 1 VoIB(p,r) 7rr e - VoIB(p,r/2),

(694) . Equivalently,

(6.95)

-~log(47r)+IOg VOIB;~,r/2) ~c~-~IOg(47r)+log VOI~~p,r).

Using the above lemma, we obtain the following (the proof is similar to the proof of Proposition 6.64). PROPOSITION 6.70 (Bounding J.l by the scalar curvature and volume ratio). The J.l-invariant has the following upper bound in terms of local geometric quantities.

For any closed Riemannian manifold (Mn,

g) , point

6.

PE

M,

(696)

.

IMPROVED VERSION OF NLC AND DIAMETER CONTROL

265

and r > 0, we have

J-l g,r

where R+

r

r2 R+dJ-l) VolB(p,r) ( 36 JB(p,r) VolB(p,r) - og rn + + VolB(p,r) VolB(p,r/2)'

(A 2) oo

VolB p,s/2 k Vol B p,s/2k+ 1 )

1'lJ h h VolB{p,s/2k) 3n d VOlB{P,S/2i~ k E 1~ sue t at VolB{p,S/2k+ 1 ) : : ; an VolB(p,s/2 +) 1

Applying (6.97) to B

>

2n

=

3n

&

h . ,t ere eXists

lor a

110

. k ::; '/, < .

(p, s/2 k ) , we get

k)2) VolB(p,s/2k) J.t ( g, ( s/2 ::; log {s/2kt

)) + ( 36 + MR (p,k s/2

n

·3.

6.

IMPROVED VERSION OF NLC AND DIAMETER CONTROL

267

Hence

VoIB(p,s) > (~)n VoIB(p,s/2) Sn 2 (s/2t

> (~)nk VolB (p,s/2 k) -

(s/2k)n

2

2: (~) nk e-3n36eJL(g,s2/22k)e-3nMR(p,s/2k) 2:

(~) nk e-3n36evr(g)e-3nMR(p,r).

o

The theorem again follows in this case.

Now we apply Proposition 6.72 to solutions of the Ricci flow and obtain the following improvement of the no local collapsing theorem. THEOREM 6.74 (No local collapsing theorem improved). Let (Mn, 9 (t)) , t E [0, T), be a solution to the Ricci flow on a closed manifold with T < 00 and let p E (0,00). There exists a constant K, = K, (n, 9 (0) , T, p) > Osuch that if p E M, t E [0, T), and r E (0, p] are such that R ::;

r- 2 in

Bg(t)

(p, r),

then Volg(t) Bg(t) (p, s) >

_....::....0..."----.::....0....:_ _

for all

°< s ::;

sn

-

K,

r.

PROOF. By (6.61) and the definition of Vn we have Vr

(9 (t)) 2: v v'p2+T (9 (0))

for r E (0, p] and t E [0, T). Then the theorem follows from Proposition 6.72. 0 6.2. Diameter control. In this subsection we show how ideas related to the previous subsection can be used to obtain a diameter bound for solutions of the Ricci flow in terms of the L(n-l)/2- norm of the scalar curvature. This result is due to Topping [357] and our presentation essentially follows his ideas. Recall that Proposition 6.72 implies that if (,Mn, is a closed Riemannian manifold, then for any p E ,Mn and

(6.98)

VoIB(p,s) > sn -

°
o Vol~Jp,s) = Wn and lims->oo Vol~;p,s) = 0 since M is closed. Hence for any point p E M, there exists s(p) > 0 such that Vol~~'t~(p))

=

8 (9) and Vol~;p,s) 2': 8 (9) for s E (0, s(p)]. Applying inequality (6.98), we have

MR(p, s(p)) 2': l.

(6.99)

This implies there exists s' (p) ~ s(p) such that

(s'(p))2 VoIB(p, s'(p))

r

>1

R d

JB(p,s'(p))

+ p, -

.

Applying the Holder inequality, we have

VoIB(p,s'(p)) < (s' (p))2 -

~

r

R d

J B(p,s'(p))

+

P, 2

(r

R:21 dP,)

n-l

[VoIB(p,s'(p))]

~=i ,

J B(p,s'(p))

1

so that

VoIB(p,s'(p)) n;-l --:---:-:-":-:------7-':"':" ~ R+ dp,. (s'(p))n-l B(p,s'(p)) We have proved that for every p EM, there exists s' (p) > 0 such that 8 (A) '( ) < VoIB(p, s'(p)). '( ) 9 s P _ (s'(p))n s p =

VoIB(p, s'(p)) 1 < (s'(p)t-

1

n-l

R+2 dp"

B(p,s'(p))

where the first inequality follows from the definition of s(p) and s'(p) ~ s(p). To finish the proof of the theorem, let "f be a minimal geodesic whose length is the diameter of 9). One can show that there exists a countable

(M,

(possibly finite) number of points Pi E "f such that B (pi, s' (pd) are disjoint and cover at least 1/3 of"f (Vitali covering-type theorem). Then

~diam(M'9) ~ L2s'(Pi) ~ i

I

8 2(A) L 9

~ 8 ~9) 1M R:21 dp,.

i

r

JB(Pi,S'(Pi))

R:21 dp,

6.

IMPROVED VERSION OF NLC AND DIAMETER CONTROL

269

The theorem follows from plugging in the definition of 8 (g) .

D

Now we can apply Theorem 6.75 to the Ricci flow and obtain the following. COROLLARY 6.76. Let n ~ 3 and let (Mn, g(t)), t E [0, T), be a solution of the Ricci flow on a closed manifold with T < 00. Assume that A(g(O)) > O. Then there exists C = C(n,g(O)) > 0 such that

(6.100)

diam (M,g(t))

~ C 1M R+ (t) n 2

1

d/-lg(t)·

PROOF. Note that by the monotonicity of the A-invariant we have A(g(t)) ~ A(g(O)) > 0,

and hence the theorem is applicable. Now the corollary follows from ~ l/(g(O)).

1/

(g (t))

D

6.3. A variation on the proof of no local collapsing. In this subsection we give a modified proof that the no local collapsing theorem follows from entropy monotonicity using a local eigenvalue estimate. We also give a heat equation proof of a less sharp form of the global version of this eigenvalue estimate. 6.3.1. Modified proof of no local collapsing theorem. Recall Cheng's sharp upper bound for the first eigenvalue Al of the Laplacian -~ on balls with a lower bound on the Ricci curvature [92].

THEOREM 6.77 (Cheng, local eigenvalue comparison). Let (Mn,g) be a complete Riemannian manifold with Rc(g) ~ -(n - l)g. Then for any

PEM, (6.101) where BlHIn (1) is the open ball of radius 1 in hyperbolic space lHIn of sectional curvature -1. Here Al denotes the first eigenvalue of the Laplacian with the Dirichlet boundary condition.

EXERCISE 6.78. Suppose (Mn,g) is a complete Riemannian manifold with Rc(g) ~ (n - l)Kg, where K ~ O. Given r > 0, determine an upper bound for Al (B(p, r)) in terms of the corresponding model space. We now give the modified proof of no local collapsing using the above eigenvalue estimate and Jensen's inequality. Given the monotonicity of /-l, the first proof we presented relies on inequality (6.83) giving an upper bound for /-l in terms of the volume ratio; it is this inequality for which we give a second proof. Recall from (6.85) that we have for a Riemannian manifold

ENTROPY AND NO LOCAL COLLAPSING

6.

270

J-t(9, r2)

~ 1M r2(41V'wI2 + Rw 2) dJ-t

- 1M (log (w 2) + ~ log(47rr2) + n) w2 dJ-t for all w with fM w 2dJ-t = 1. Using (6.47), we have for any w with supp (w) B(p, r) and fM w2dJ-t = 1 that

c

- 1M log(w 2)w2dJ-t ~ logVolB(p,r). By assumptions Rc (g) ~ - (n - 1) r- 2 and R ~ n (n - 1) r- 2 in B (p, r) ,19 we have for any w with supp (w) c B(p, r) and fM w 2dJ-t = 1,

1M Rw2 dJ-t ~ n (n -

1) .

Let 9 = r- 2g; then Bg(p, 1) = B(p,r) ~ Bg(p,r). By Theorem 6.77 and the Rayleigh principle for eigenvalues, inf

J:

f . lV'wl2 dJ-t-

supp(w)CBg(P,I)

2d M w J-tg

=.AI (Bg(p, 1)) ~ .AI (Bnnn (1)).

9

Hence we have

Therefore log Vol ~ip, r)

~ J-t(g, r2) -

4.AI (Bnnn (1)) - n (n - 2)

+ ~ log(47r).

This provides the needed estimate to replace (6.83).

6.3.2. A heat equation proof of a global eigenvalue estimate. We now recall a global version of Cheng's Theorem 6.77. THEOREM 6.79 (Cheng's eigenvalue estimate, global).

complete noncompact Riemannian manifold with Rc (g) .A 1

(_~) < (n -

4

~

If (Mn,g)

is a

-(n - l)g, then

1)2

It turns out that a weaker version of this estimate, i.e., .AI ~ n(n4-1) , can be proved using the energy/entropy computation of Perelman for the fixed metric case; we give this proof below. First we state a formula which is implicit in [283]. 19We choose these constants for our curvature bounds since they are implied by

_r- 2 :$ sect :$ r- 2 •

6.

IMPROVED VERSION OF NLC AND DIAMETER CONTROL

271

LEMMA 6.80. Let U be a positive solution to the heat equation

(:t-A)U=O on a fixed Riemannian manifold then

(Mn,g). If f = -logu, that is,

U

= e- f ,

(6.102)

ftJ = Af -1V'fI 2 , we calculate ~ 1M IV' fl2u dp, = ~ 1M (AI) udp,

PROOF. (1) Using

= 1M (2Af -IV' f12) Audp, = 21M (AfAu+ V'iV'jfV'dV'ju)dp,. Now integrating by parts yields

1M Af Audp,

= - 1M V' f . V' Audp, = - 1M (V' f·

A V'u - !4jV'dV'ju) dp,

= 1M (V'V' f . V'V' U

-

u!4j V'dV' j I) dp,

= 1M (-u lV' i V'jfI2 - V'iV'jfV'dV'ju - u!4jV'dV'jf) dp,. The lemma immediately follows from combining the above two formulas. (2) Alternatively, one can integrate the following formula to get (6.102) (see (2.1) and Lemma 2.1 in [283]):

(:t - A) (u (2Af -IV' fI2))

= -2u lV'i V'jfI2 - 2u!4jV'dV'jf,

which follows directly from the calculation:

:t (2Af - IV' f12) = 2A (Af - IV' f12) - 2V' f . V' (Af -IV' f12) = A (2Af -1V'fI2) - 2V'f· V' (2Af -1V'fI2)

- A IV' fl2

+ 2V' f

. V' Af

= A (2Af -1V'fI2) - 2V'f· V' (2Af -1V'fI2)

- 21V'i V'jfI2 - 2!4jV'dV'jf.

o Now we can prove the following weaker version of Cheng's Theorem 6.79.

6.

272

ENTROPY AND NO LOCAL COLLAPSING

g)

PROPOSITION 6.81 (Weaker version using the heat equation). If (Mn, is a complete noncompact Riemannian manifold with Rc (g) ~ -(n - l)g, then Al :S n(n4-1) .

SKETCH OF PROOF. Assume that 0 implies that the metric h is Riemannian, i.e., positive-definite. In local coordinates,

(7.12)

hij = hij ,

(7.13)

hOl(3 = 7h Ol(3,

(7.14)

hO~

(7.15)

hiO = hiOi = hOio = O.

-

-

=

N

27

-

+ R, -

Let 1'(s) ~ (x(s), y(s), 7(S)) be a shortest geodesic, with respect to the metric h, between points P ~ (xo, Yo, 0) and q ~ (Xl, YI, 7q ) E .N. Since the fibers SN pinch to a point as 7 ---t 0, it is clear that the geodesic 1'( s) is orthogonal to the fibers SN. (To see this directly, take a sequence of geodesics from Pk ~ (xo, YI, 1/k) to q and pass to the limit as k ---t 00.) Therefore it suffices to consider the manifold jJ' ~ N x (0, T) endowed with the Riemannian metric:

(7.16)

Ii

~ hijdxidxj + (~ + R) d7 2 .

(This metric is dual to the metric considered in [100J.) For convenience, denote x(s) ~ "((s). Now we use s = 7 as the parameter of the curve. Let.:y (7) ~ ~; (7). The length of a path 1 (7) ~ ("((7), 7) , with respect to the metric Ii, is given by the following: 4We shall consider the case where Q = -00 (in which case we define w - Q ~ +00). On the other hand, if w = +00 and Q = -00, we may simply take T = -to However, for the backward Ricci flow we are not as interested in the case where w = +00 and Q > -00.

290

7. THE REDUCED DISTANCE

Lengthji (i)

= fo

Tq

J~

+ R + 11' (r)1 2 dr

r q V[ii. ( . 2) 2;V/1 + 2r N R+ 1,(r)1 dr

= Jo

= fo =

Tq

~ (1 + ~ (R + 11' (r)1

2)

+ a (N- 2 )) dr

fo Tq ~dr + fo Tq J2~ (R + Ii' (r)1

= J2Nrq + v'~N

2)

dr +

fo Tq {fa (N-

fo Tq yT (R + 11' (r)1 2 ) dr + ..j'iTqa (N-

3/ 2 )

dr

3/ 2 ) •

The calculation indicates that as N --+ 00, a shortest geodesic should approach a minimizer of the C-Iength functional defined by

Cb)

~ fo Tq yT (R b (r), r) + 11' (r)I~(T)) dr.

Note that the definition of Cb) only depends on the data of (N, h). EXERCISE 7.4 (Levi-Civita connection of the potentially infinite metric) . Consider the metric h on Nn x (0, T) defined in (7.16) by (7.12), (7.14), and hio = 0 (without the SN factor). The components of the Levi-Civita connection N'fj of h are defined by

N'fj

8 8xa

~ = ~ Nrc ~c aXb

~

ab aX '

c=O

where x O = r. Show that

and

(N = - 2r + R ) ~j, (N )-1 21 ViR , = 2r + R -1

N-O

r ij

N-O riO N-O

roo=

(N-+R )-1 -1 (aR R) 1 -+--. 2r ar r 2r 2

2.

THE C-LENGTH AND THE L-DISTANCE

291

In particular, Nf'!b are independent of N, whereas .

hm

N-O

r··~J

N-+oo .

hm .

'

N-O

riO = 0,

N-+oo

hm

= 0

r oo = - 1-

N-O

N-+oo

27

2.2. The £-length. A natural geometry on space-time (in the sense of lengths, distances and geodesics) is given by the following. DEFINITION 7.5 (£-length). Let (Nn, h (7)),7 E (A, 0), be a solution to the backward Ricci flow h = 2 Rc, and let 'Y : [71, 72J ~ N be a piecewise C 1-path,5 where h,72J c (A,O) and 71 2: O. The £-length of'Y is6

-t

(7.17)

£("()

~ £h ("() ~ 1T2 Vr (R("((7) ,7) + Idd'Y (7)1 T1 h(T) 2

)

d7.

7

Later we shall take

71

= 0 and call 72 = f.

REMARK 7.6. Taking 71 = 0, the subsequent degeneracy introduced by the Vi factor in (7.17) reflects the infinite speed of propagation of the Ricci flow (as a nonlinear heat-type equation for metrics). We also note the formal similarity between R+ 1~ 12 and the quantity R+ 1'\7112 which we considered for gradient Ricci solitons and which also appeared in the definitions of energy and entropy; this seems like more than just a coincidence. The £-length is defined only for paths defined on a subinterval of the time interval where the solution to the backward Ricci flow exists. Note that £ may be negative since the scalar curvature may be negative somewhere. This is in contrast to the energy defined in Section 1 above for a static metric. Often we shall use the following conventions: (7.18) We may rewrite £ as (7.19)

£("() =

r..fi2 (:2 R(f3((]'),(]'2/4) + 1~f3 ((]')1 2 ) d(]'.

J2v'T1

This is especially useful in the case

(]'

71

h(u 2 /4)

= O.

Because of the 1~ 12 term on the RHS of (7.17), £ ("() looks more like an energy than a length. Another way to obtain £, which is related to the 5That is, 'Y

("42) is a C 1function of u.

6 R (f (T), T) is just a notation meaning vature of (N", h (T)) .

Rh(T)

(T (T)), where

Rh(T)

is the scalar -cur-

292

7.

THE REDUCED DISTANCE

above approach of renormalizing the Riemannian length functional, is as follows. We define the space-time graph

l' : h, T2J -t N

h, T2J

x

of the path, by i'(T) ~ (,(T),T), so that ~~ (T) = (~; (T),1). Note that the parameter T, of which, is a function, also serves as time; so it is natural to consider its graph. Define the space-time metric h ~ h + RdT2. In general, this metric is indefinite since R may be negative somewhere. We easily compute

£ (,) Using

0"

= 2y'T,

=

1

72

71

Id-

12

y'T d; (T) h dT.

we may rewrite the £-length as

where O"i ~ 2fo" i = 1,2. That is, £ (,) is the energy of the space-time path l' with respect to the space-time metric h and the new time parameter 0". If 0:: [Tl,T2J-t Nand f3: h,T3J-t N are paths with 0: (T2) = f3(T2), then we define the concatenated path 0: '---' f3 : h, T3J -t N by

We have the following additivity property. LEMMA 7.7 (Additivity of the £-length). (7.20)

£ (0: '---' (3) = £ (0:)

+ £ (f3) .

However, the £-length of a path, is not invariant under reparametrizations of ,. We leave it to the reader to make the easy verification of this fact. The following bound on £ is elementary. LEMMA 7.8 (Lower bound for the £-length). (7.21)

£ (,)

~

-2 (3/2 T2 - Tl3/2) 3

. mf

R.

NX[7I,T21

1

72

This follows directly from £ (,) ~

y'TRinf (T) dT, where Rinf (T)

~

71

infNx{7} R. The Riemannian counterpart of estimate (7.21) is the obvious fact that the length of a path is nonnegative.

292

7.

THE REDUCED DISTANCE

above approach of renormalizing the Riemannian length functional, is as follows. We define the space-time graph

i: h,T2]----t N

x h,T2]

1) .

of the path 'Y by i (T) ~ b (T) , T), so that ~~ (T) = (~~ (T) , Note that the parameter T, of which 'Y is a function, also serves as time; so it is natural to consider its graph. Define the space-time metric h ~ h + RdT2. In general, this metric is indefinite since R may be negative somewhere. We easily compute

[b) Using a

=

=

1 Vi Id-d~

12

T2 Tl

(T) h dT.

2yfi, we may rewrite the [-length as

2,;r;.,

i = 1,2. That is, I:- ('Y) is the energy of the space-time path where ai ~ i with respect to the space-time metric h and the new time parameter a.

T2] ----t Nand {3 : [T2' T3] ----t N are paths with 0: (T2) then we define the concatenated path 0: '-../ {3 : [TI' T3] ----t N by If 0: :

[TI'

=

{3 (T2),

We have the following additivity property. LEMMA 7.7 (Additivity of the [-length). (7.20)

[ (0: '-../ {3)

=

I:- (0:)

+[

({3) .

However, the £-length of a path 'Y is not invariant under reparametrizations of "f. We leave it to the reader to make the easy verification of this fact. The following bound on I:- is elementary. LEMMA 7.8 (Lower bound for the £-length). (7.21)

(3/2 - 3/2) mf .

[ ('Y) ~ -2 T2 3

TI

Nxh,T2l

1

R.

T2

This follows directly from [b) ~

yfiRinf

(T) dT, where

Rinf

(T)

~

T1

infNx{T} R. The Riemannian counterpart of estimate (7.21) is the obvious fact that the length of a path is nonnegative.

2. THE L:-LENGTH AND THE L-DISTANCE

293

2.3. The L-distance function. Just as for the usual length functional (perhaps it is better to compare with the energy functional), one gives the following definition. DEFINITION 7.9 (L-distance). Let (Nn, h (T)), T E (A, 0), be a solution to the backward Ricci flow. Fix a basepoint pEN. For any x E Nand T > 0, define the L-distance by

L (x, T) ~ L~p,o) (x, T) ~ i~f £ h) , where the infimum is taken over all CI-paths 'Y : [0, T] ~ N joining p to x (the graph i joins (p,O) to (x, T)). We call an £-length minimizing path a minimal £-geodesic. We also define

L (x, T) ~ L~,o) (x, T) ~ 2.;TL (x, T) .

(7.22)

Note that the L-distance defined above may be negative. To help the reader have a feeling for the L-distance function, we present some exercises. EXERCISE 7.10 (Scaling properties of £ and L). Let (Nn, h (T)) be a solution to the backward Ricci flow, 'Y : [Tl' T2] ~ N a CI-path, and c > a constant. Show that for the solution it (f) ~ ch (c-lf") and the path 1: [CTl' CT2] ~ N defined by 1 (f) ~ 'Y (c- 1 f) , we have

°

£it (i) = JC£h h)·

Consequently,

L~,o) (q, f) = JCL~p,o) (q, c-lf) . EXERCISE 7.11 (£ and L on Riemannian products). Suppose that we are given a Riemannian product solution (Nrl x N;:2, hI (T) + h2 (T)) to the backward Ricci flow and a C 1-path 'Y = (a, (3) : [Tl' T2] ~ Nl X N 2. Show that

Hence

L~~1~:22,O) (ql, q2, T)

=

L~~l'O) (ql, T) + L~;2'O) (q2, T) .

It is useful to keep in mind Euclidean space as a basic example; more generally we have EXERCISE 7.12 (L-distance for Ricci flat solutions). Let (Nn, h (T) = h o) be a static Ricci flat manifold and let pEN be the basepoint. Show that given any q E Nand f > 0, the £-length of a C 1-path 'Y : [0, f] ~ N from p to q is

£h) =

10

2"fi T

I~; (a 2 /4)

1da, 2

which is the same as (7.2). Hence a minimal £-geodesic 'Y is of the form (7.23)

'Y (T) =

f3 (2.;T) ,

294

7.

THE REDUCED DISTANCE

where (3 : [0, 2v'r] ~ Nn is a minimal constant speed geodesic with respect to ho joining p to q. Thus L ( f) = d (p, q)2 q, 2VT

(7.24)

For reference below, we have L (q, f) ~ d (p, q)2 and £ (q, f) ~ 2~L (q, f)

=

d~~)2 ; the definition of £ will be given again in (7.87). SOLUTION TO EXERCISE 7.12. This exercise is a special case of the discussion in Section 1 above. We also note that d, (7) 12 7 1d

(7.25)

=

g(T)

7

1dd(3 (0') 12 0'

=

2 , IVlg(o)

g(u2/4)

where V ~ limT-+o yT~ (7) = limu-+o ~ (0'). We leave it to the reader to check that for .c-geodesics defined on a subinterval [71,72] c [0, T] , we still have

71~' (7) 12

(7.26)

== const .

g(T)

7

2.4. Elementary properties of L. In this subsection (M n , 9 (7)) ,7 E [0, T], shall denote a complete solution to the backward Ricci flow, and p EM shall be a basepoint. We will assume the curvature bound (7.27)

max

(x,T)EMx[O,Tj

{IRm (x, 7)1, IRc (x, 7)1} ~ Co

< 00.

The curvature bound assumption is written in this way for the convenience of stating later estimates. We prove some elementary CO-estimates for the L-distance and lengths of .c-geodesics, relating them to the Riemannian distance; we shall use these estimates often later. First recall from (3.3) in Lemma 3.11 that for 71 < 72 and x E M, e- 2CO (T2- Tl) 9 (72, x) ~ 9 (71, x) ~ e2CO (T2- TI) 9 (72, x) .

LEMMA 7.13 (.c and Riemannian distance). Let, : [0, 1'] ~ M, l' E

(0, TJ, be a C 1 -path starting at p and ending at q. (i) (Bounding Riemannian distance by .c) For any

d~(o) (p, ,

(7))

7 E

[0,1'] we have

~ 2JTe 2COT ( .c (,) + 2n3Co1'3/2) .

In particular, when M is noncompact, for any l' E (0, T], we have

lim L (q, f)

q-+oo

= q-+oo lim 2VfL (q, f) = +00.

(ii) (Bounding speed at some time by.c) There exists 7* E (0, f) such that

d, 12 7* I-d (7*) 7

g(T.)

Id(3

= -d (0'*)

0'

12 g(T*)

~ 2

nCo _ r;.c (T) + -3- 7, y 7 1

2. THE .c-LENGTH AND THE L-DISTANCE

295

where f3 (cr) ~ ')' (T), cr = 2.jT, and cr* ~ 2..rr;. (iii) (Bounding L by Riemannian distance) For any q E M and L (q, T-)

:::;

e

(p, q) 2-/T

2Co1' d;(1')

f

> 0,

2nCo -3/2

+ -3- T

.

7.14. In each of the estimates above, on the RHS one may think of the first term as the main term and the second term as an error term. Recall by (7.24) that if (Mn, 9 (T) = 90) is Ricci fiat and')' : [0, f] --t M is a minimal .c-geodesic from p to q, then REMARK

d~o (p,,), (T)) = 2y'TL(')'(T) ,T) = 2y'T.c (')'I[O,rl) = ~d~o (p,q) , and for all

T*

E

(0, f) , T*

Id')' (T*)1 dT

2

=

g(r.)

~o (p,q). 4T

Hence for Ricci fiat solutions, .e (')' (T), T) defined in (7.87) is constant (= Ad;o (p, q)) along .c-geodesics. PROOF. (i) Let a- = 2v'T and f3 (a-) ~ ')' (7'). The idea is to first bound the energy of f31 [0,2y'T] . By splitting the formula for .c into two time intervals, we see that

1

2y'T 1 df3 - (0-) 12 dao dOg(q2/4) 2";¥

=.c(')')-

(7.28)

:::; .c (')')

r J2y'T

Idd~(o-)I cr

2

r VfR(')'(7'),f)d7' Jo l'

da--

g(q2/4)

+ 2n3Co f3/2,

since R ~ -nCo. Hence, since 9 (0) :::; e2Cor 9 (7') for 7' E [0, T] , we have

d;(o) (p, ')' (T)) :::; e2Cor ( :::; e2Cor . 2y'T

2 r y'T dd~ (0-) dO-) Jo cr g(q2/4) 1

+ 2~Co f3/2)

(ii) From the proof of (i) we have (take

T

.

= f in (7.28))

2..;¥ 1df3 --=- (a-) 12 dO- :::; -1. c (')') 2..,fi 0 dcr g(q2/4) 2..,fi

-1

2

2y'T 1d~ (a-) 12 r daJo dcr g(q2/4)

:::; 2.J!e2cor ( .c (')')

1

1

nG0 f. +_ 3

296

7. THE REDUCED DISTANCE

By the mean value theorem for integrals, there exists

T*

E (0, f) such that

d;3 12 1 nCo (0'*) ::; 2 ,=C h) + - 3 f. I-d 0' g(T.) V T

(iii) Let 'f} : [0, 2JT] ~ M be a minimal geodesic from p to q with respect to the metric 9 (f) . Then

o 3. The first variation of C-Iength and existence of C-geodesics Now that we have defined the C-length, we may mimic basic Riemannian comparison geometry in the space-time setting for the Ricci flow. We compute the first variation of the 'c-length and find the equation for the critical points of C (the C-geodesic equation). We also compare this equation with the geodesic equation for the space-time graph (with respect to a natural space-time connection) and prove two existence theorems for C-geodesics. 3.1. First variation of the 'c-Iength. Let (Nn, h (T)), T E (A, 0), be a solution to the backward Ricci flow. Consider a variation of the C 2-path , : h, T2J ~ N; that is, let G: h,T2J x (-e,e) ~N be a C2-map such that GI[Tl,T2jX{O}

= ,.

Convention: We say that a variation G (.,.) of a C 2-path , is C 2 if G s) is C 2 in (0', s) .

C:t '

Define

,s

~ GI[Tl,T2jX{S} :

[T1, T2J ~ N for -e < S < e. Let

. aG (T, s) = a,s aT (T)

x (T, s) =;= aT

and Y (T, s)

. as aG (T, s) = a,s as (T)

=;=

be the tangent vector field and variation vector field along tively. The first variation formula for C is given by

,s (T) , respec-

3.

FIRST VARIATION OF .c-LENGTH AND EXISTENCE OF .c-GEODESICS

297

LEMMA 7.15 (L:-First Variation Formula). Given a C 2-Jamily oj curves "fs : h, r2] ~ N, the first variation oj its L:-length is given by

1 d

1

'-

2(8yL:)("fs)~2dsL:("(s)= yrY·X

+

(7.29)

IT2 Tl

lT2 VTY' (~\7R - ~X - \7xX - 2Rc (X)) dr, 2 2r Tl

where the covariant derivative \7 is with respect to h (r) . REMARK 7.16. We use the notation (8y L:) ("(s) since tsL: ("(s), at a given value of s, depends only on "fs and Y along "fs. PROOF. We compute in a similar fashion to the usual first variation formula for length (see [72], p. 4ff for example)

d L: ("fs) = -d d -d

(7.30)

s

S

=

l

lT2 VT (R ("(S (r), r) + 1~ a"f (r) 12) dr ur

T2

h(T)

Tl

VT ((\7 R, Y) + 2 (\7y X, X)) dr ;

Tl

here ( . , . ) = h (r) ( . , . ) denotes the inner product with respect to h (r) . Using [X, Y] = [~~, ~~] = 0 and = 2Rc, we have

:Th

d

(\7yX, X) = (\7x Y ,X) = dr [h(Y,X)]- (Y, \7xX) - 2Rc(Y,X). Hence

1

d 2dsL:("(s) =

lT2 VT (12 (\7R,Y) + dr(Y,X)-(Y,\7x d X )-2Rc(Y,X) ) dr Tl

and integration by parts yields

l T2 VT-dd (Y, X) dr Tl

r

I1T2

= --2

Tl

1 '- (Y, X) dr yr

+ VT (Y, X) I~~ .

The lemma follows from the above two equalities.

D

REMARK 7.17. In comparison, the Riemannian first variation of arc length formula on is

(Mn, g)

(7.31)

where "fu : [0, b]

U~

d {b b du L ("(u) = - Jo (U, \7TT) ds + (U, T)l o ,

~ M is

a I-parameter family of paths, T

:u "fu, and ds is the arc length element.

~ ~ / I~ I'

298

7. THE REDUCED DISTANCE

3.2. The £-geodesic equation. The £-first variation formula leads us to the following. DEFINITION 7.18 (£-geodesic). If 'Y is a critical point of the £-length functional among all C 2 -paths with fixed endpoints, then 'Y is called an £-geodesic.

By the £-first variation formula, 7.19 (£-geodesic equation). Let (Nn, h (T)) , T E (A, 0), be a solution to the backward Ricci flow. A C 2-path 'Y : [Tl' T2J ~ N is an £-geodesic if and only if it satisfies the £-geodesic equation: 1 1 (7.32) V' x X - "2 V'R + 2 Rc (X) + 2T X = 0, COROLLARY

where X (T) ~ ~ (T). For the four terms in (7.32), (1) is the usual term in the geodesic equation, (2) comes from the variation of R in £, (3) comes from :rh, and (4) comes from -IT in £ via integration by parts. In local coordinates, the £-geodesic equation is d2'Yi i d'Y j d'Yk 1 i' i' d'Yk 1 d'Yi (7.33) 0 = dT2 +fjk ("( (T), T) dT dT -"2 h 3V'jR+2h 3 Rjk dT + 2T dT' where 'Yi = xi 0 'Y. We find it convenient to use the notation for the covariant derivative along the curve 'Y. Multiplying (7.32) by T yields

£.

D

T

vT dT (vTX) - 2V'R + 2vT Rc (vTX) = O.

(7.34)

Since the covariant derivative along the curve can be written as D dT V = V'x V , we may write D

vT dT (vTX) = vTV'X (vTX) = V'.;rx (vTX) along 'Y (T) , where the last two terms require extending vector field in a neighborhood of 'Y (T) . Note that

V'.;rx (vTX) = TV'XX

V (T) = -ITX

to a

+ vT (d~ vT) X 1

= TV'XX + "2X, which is different from -ITV' .;rxX because the curve. (7.35)

-IT must be differentiated along Using the convention (7.18) and Z (0') ~ d~~) = -ITX, we get 0'2 V'zZ - SV'R+O'Rc(Z) = O.

3.

FIRST VARIATION OF C-LENGTH AND EXISTENCE OF C-GEODESICS

299

EXAMPLE 7.20 (C-geodesics on Einstein solutions). Let (NO', ho (r)) , r E (0,00), be a 'big bang' Einstein solution to the backward Ricci flow

with RCho (r) = h~~). Then Rho (r) = ~ and the C-geodesic equation (7.32) is (7.36)

\lx X

3

+ 2rX = 0,

so that \l ~ (r3/2 ~)

= 0. Note that since \l is independent of scaling and ho (r) = rho (1), we have \lho(r) = \lho(l) is independent of r. Clearly the constant paths, where X = 0, are C-geodesics. More generally, reparametrize

, and define the path {3 by (3 (p) = , (f (p)) , where (7.37)

= l' (1-1 (r)).

r 3/ 2

Then /3 (p) implies

c:t =

~(f(p))f'(p)

\l/3(p)/3(p) =

=

~~(f(p))f(p)3/2, so that (7.36)

\lr3/2~(r) (r3/2~; (r))

= OJ

i.e., {3 is a constant speed geodesic with respect to ho (1) . Since solutions of (7.37) are given by f (p) = ~( 4 ) , the C-geodesics are of the form Po-P

,(r) = {3

(~-~), JTo VT

defined for r E (0,00), where (3 : (-00,00) geodesic with respect to ho (1) . Note that d, 1 1-(r) dr ho(r)

=VT I'(3

No is a constant speed

(2JTo VT2) 11 ---

-r 3/ 2 ho(l)

const r

That is,

r21~'(r)12 =:const. r ho(r) (Compare with (7.26) for the Ricci flat case.) In particular,

I~r

(r)1 ho(1) =

C:3/~t. In any case, the speed of ~ (r) tends to infinity as r - 0, whereas the speed of for all

~ (r) tends to zero as r -

00. Note

J7iooo I~

ro E (0,00), whereas J;o I~ (r)1 ho(l) dr = 00.

(r)1 ho(l) dr
0 for 0" > 0, we have

d d7 _ log dO" dO"

r80

B

dT

d

~~

27

dO"

= - 2~' Hence, assuming

= da = da = - log ..jT

'

so that

d7 = C..jT dO" for some constant C > O. Since 7 (0) = 0, we conclude 0"2 7 (0") = C 2 4. 3.4. Existence of C-geodesics. Our next order of business is to establish the existence of solutions to the initial-value problem for the C-geodesic equation. In this subsection (M n , g (7)) , 7 E [0, T] , is a complete solution to the backward Ricci flow with curvature bound max {IRml, IRcl} ~ Co < 00 on M x [0, T] . We shall use the following LEMMA 7.24 (Estimate for speed of C-geodesics). Let (M n ,g(7)), 7 E [0, T], be a solution to the backward Ricci flow with bounded sectional curvature. There exists a constant C (n) < 00 depending only on n such that given 0 ~ 71 ~ 72 < T, if'Y : h, 72] ---+ M is an C-geodesic with lim ..jTdd'Y (7) T->Tl 7

=V

E

T-Y(TI)M,

then for any 7 E [71,72] ,

2

71 d'Y (7)1 d7 g(T)

~ e6CoT 1V12 + .

C(n)T -1 mm {T - 72, Co }

(e 6COT - 1) ,

where Co is as in (7.27) and 1V12 ~ 1V1;(Tl) . PROOF. Let O"i ~ 2y1ri. Define f3 : [0"1,0"2] so that

---+

2

(7.43)

lim 1df3 1 a-+al dO" g(a 2 /4)

= 1V12 .

M by f3 (0") = 'Y (0"2/4) ,

304

7. THE REDUCED DISTANCE

Since 72 < T, by (7.27) and the Bernstein-Bando-Shi derivative estimate (Theorem Vl-p. 224), there exists a constant C (n) < 00 such that

IV R (x, 7)1 ~

(7.44)

~ C2

C(n)Co Jmin {T

-

72,

COl}

for all (x, 7) EM x h, 72J. From the .c-geodesic equation (7.35), we compute

.!!..-I d(31 2 deI deI

=

g(u2/4)

(7.45)

09 (d(3 d(3)

OeI

deI' deI

= -eIRc

+ 2 / V rJ1t d(3 \

d(3 d(3) ( deI' deI

d(3) d~ deI' deI

eI 2

d(3)

/

+ 4 \ VR, deI

.

Applying the bounds (7.27) and (7.44) on the curvature and its first derivative, we have

-d Id(31 (7) 12 ) d7 geT)

2nCo ( 723/2 - (271 - 72) 3/2) :S L (q, 72) + -32nCo (3/2 + -371 (7.50)

+2

(2 71 - 72 )3/2)

l

T2 -/1>-1 (7) Id-.2 (7) 12 d7. 2TI-T2 d7 g(q..-l(T))

Recall that f3 (a) is defined by (7.18). By Lemma 7.13(ii), (iii) there exists 7* E (0,72) such that

(7.51) Since 72 :S T - c:, by Shi's derivative estimate we have

IV R (x, 7)1 :S

C(n)Co vmin {T - 72, COl}

~ C2

for any (X,7) EM x [0,72]. From equation (7.46), we have

~ 1 df312


-1 (7) = T2T2 using (7.52) and (7.51)

2

+ 71 :S 7 for 7 E

[271 - 72,72] , we can estimate

7.

308

THE REDUCED DISTANCE

where

C

..!...

3 -;-

2 4Coe+6CoT ( 2COT2 d~(T2) (p, q) e e 4T2

+

2nCo 3 T2

C~T) + 12C6

(using 10" - 0"*1 ::; 2VT). Combining this with (7.50), we obtain 7 L (q, Tl) - L (q, T2)

2nCo (3/2 (2 TI-702 )3/2) 0 such that 'yvl[o,T.] is a minimal L:-geodesic. The next lemma and Rademacher's Theorem (Lemma 7.110) imply that L is differentiable almost everywhere on M x (0, T). LEMMA 7.30 (L is locally Lipschitz). The function L : M x (0, T) - t R is Lipschitz with respect to the metric 9 (T) + dT2 defined on space-time. PROOF. For any 0 < TO < T and qo E M, let E ~ min {f%, T~;o, lo} > O. Then for any Tl < T2 in (TO - E, TO + E) and ql, q2 E Bg(o) (qO, E),

IL (q1, Tl)

- L (q2, T2)1 ::;

IL (ql, Tl) -

L (q2, Tdl

+ IL (q2, Tl) -

L (q2, T2)1·

To prove that L is Lipschitz near (qo, TO), it suffices to prove (1) and (2) below. (1) L (', Td is locally Lipschitz in the space variables uniformly in Tl E (TO - E, TO + E). Let dT denote the distance function with respect to the metric 9 (T) and let 'Y : [0, TIl - t M be a minimal L:-geodesic from p to ql. Let a : h, Tl + do (q1, q2) 1- t M be a minimal geodesic of constant speed 1, with respect to 9 (0) , joining ql to q2. Then 'Y "'-...-/ a :

[0, Tl

+ do (ql, q2)l - t M

4. GRADIENT AND TIME-DERIVATIVE OF THE L-DISTANCE FUNCTION

309

is a piecewise smooth path from p to q2. We estimate, using I~~ (7)1:(T) :::; e2Co T 1dOl (7) 12 = e2Co T that g(O)

dT

L (q2, 71

,

+ do (qI, q2) )

:::; .c (')') +

1

+dO (Ql,q2) Vi (R (Q (7),7)

T1

+ 1ddQ 7

T1

2 (nCo

:::;L(ql,71)+ :::; L (ql, 71)

(7)1

+3 e2CoT )

(

2

)

d7

geT)

3/2 3/2) (71+ dO(ql,q2)) -71

+ C 1do (qI, q2) .

By Lemma 7.28 we have

L (q2, 7t)

+ do (ql, q2)) + C 1do (ql, q2) :::; L (ql, 7t) + C 1 do (ql, q2) :::; L (ql, 71) + C 1d (ql, q2),

:::;

L (q2, 71

Tl

where we have used do (ql, q2) :::; eCoT dTl (ql, q2). By the symmetry between ql and q2 we get

IL (q2, 71)

-

L (ql. 71)1

:::;

Cl dTl (ql. q2) .

(2) L (q, .) is locally Lipschitz in the time variable uniformly in q E Bg(o) (qO, c). For any 71 < 72 in (70 - c, 70 + c), let I : [0, 71l ~ M be a minimal C-geodesic from p to q and let f3 : h, 72l ~ M be the constant path f3 (7) = q. Then I f3 : [0, 72l ~ M is a piecewise smooth path from p to q. Hence 0"""./

L (q, 72)

:::;

.c (f) +.c (f3) = C (')') + 1T2 ViR (q, 7) d7 T1

o < - L (q,71 ) + 2nC 3 ( 723/2

:::; L (q, 71)

+ C1 (72 -

_

3/2) 71

71),

where C 1 depends only on Co and T. Combining this with Lemma 7.28, we obtain where (7.53)

o 7.31. L is differentiable almost everywhere on M x (0, T) and L E Wl~'~ (M x (0, T)). COROLLARY

PROOF.

See Lemmas 7.110 and 7.111.

o

7. THE REDUCED DISTANCE

310

4.2. Gradient of L. We compute the gradient of L via the first variation formula for C. Since L (', r) is not smooth in general, the gradient is defined in the barrier sense as described below. Let 'Y : [0, r] --+ M be a minimal L:-geodesic from p to q so that L (q, r) = C ('Y). For any point x in a small neighborhood U of q and any r E (r - c, r + c) with small c > 0, let 'Yx,r : [0, r] --+ M be a smooth family of paths with 'Yx,r (0) = p, 'Yx,r (r) = x and 'Yq,'f = 'Y. (Recall that our definition of a smooth variation says that

'Yx,r (~2) is a smooth function of (0', x, r) .) Define L : U x (r - c, r

+ c) --+ R

by

L(x,r) =C("tx,r)' Then L (x, r) is a smooth function of (x, r) when r > 0, L (x, r) ~ L (x, r) for all (x, r) E U x (r - c, r + c), and L (q, r) = L (q, r). That is, the function L (.,.) is an upper barrier for L (".) at the point (q, r) . Given a vector Y (r) at q, let q (s) be a smooth path in U with q (0) = q and ~ (0) = Y (r). Consider the smooth I-parameter family of paths 'Ys ~ 'Yq(S),T : [0, r] --+ M. Let Y (r) ~ %s Is=o 'Ys (r) denote the variation vector field along 'Y (r). By (7.29), (7.32), and Y (0) = 0, we have

~ dl s=OL(q(s),r) ~ VL(q,r)·Y(r)= ds = (8 y C)("t) =2v'=r Y(r)·X(r). Hence

v L (q, r) =

2../fX (r) . It follows from Lemma 7.30 that L (.,7) is differentiable a.e. on M. Suppose L(·,r) is differentiable at q. Since L(.,r) is an upper barrier for L(',r) at the point q, it is easy to see that VL(q,r) = VL(q,r) = 2../fX (r). Suppose there is another minimal C-geodesic 'Y' : [0, r] joining p to q. Then we can construct another barrier function L' as above; the same proof will imply V L (q, r) = 2../fX' (r), where X' (r) = ~ (r). Now both 'Y and 'Y' satisfy the same C-geodesic equation and 'Y (r) = 'Y' (f) = q and ~ (f) = ~ (r) = V L (q, r) . By the standard ODE uniqueness theorem, we conclude that 'Y (r) = 'Y' (r) for r E [0, r]. Hence if L (', r) is differentiable at q, then the minimal C-geodesic joining (P,O) to (q, r) is unique. Convention: If the function L (', f) is not differentiable at q, then by writing V L (q, r) = 2..jTX (r),8 we mean that there is a smooth function L satisfying L (x, r) ~ L (x, r) for x E U, L (q, r) L (q, r), and V L (q, r) = 2..jTX (r). We have proved the following. 8Note that X (f) depends on the choice of minimal .c-geodesic, which may not be unique.

4.

GRADIENT AND TIME-DERIVATIVE OF THE

L-DISTANCE

FUNCTION

311

LEMMA 7.32 (Gradient of L formula). The spatial gradient of the Ldistance function is given by

V'L (q, f)

(7.54)

= 2VfX (f),

where X (f) = ~ (f), for any minimal C-geodesic I : [0, f] --t M joining p to q. Furthermore if L (', f) is differentiable at q, the minimal C-geodesic joining (p,O) to (q, f) is unique.

REMARK 7.33. The analogy of (7.54) in Riemannian geometry is as follows. Let dp(x) ~ d(x,p) and suppose dp is smooth at q E

(Mn,g).

Define I : [0, b] --t M to be the unique unit speed minimal geodesic from p to q. By the first variation formula (7.31), for any U E TqM, (V'dp(q),U)

=

d~lu=o L(,u) = h(b),U),

provided the IU : [0, b] --t M satisfy 10 = I, IU (0) = p and U. That is, V'dp (q) = '1' (b).

Iu Iu=o IU (b) =

Taking the norm of (7.54), (7.55) IV'LI2 (q, f) = 4f IX (f)1 2 = -47

R(q, f) + 4f (R (q, f) + IX (f)1 2) .

The reason we rewrite this in a seemingly more complicated way is that both Rand R + IX (f)1 2 are natural quantities. 9 4.3. Time-derivative of L. Next we compute the time-derivative of L. This time we need to choose Ix,r used in subsection 4.2 above a little more carefully. Given (q, f), let I : [0, f] --t M be a minimal C-geodesic from p to q so that L (q, f) = C (,). We first extend I to a smooth curve I : [0, f + oS] --t M for some oS > 0, and then we choose a smooth family of curves Ix,r to satisfy Ix,r (0) = p, I'Y(r),r = II[O,r] and Ix,r (T) = x. Define

(7.56) for (x, T) E U x (f - oS, f + oS) . We compute, using the chain rule and (7.54),

O£( _)_O£(,(T),T) aT q, T aT

=

d~ I

r=T

[£ (, (T) , T)]

- V' £ . X

r=T

=

:7 IT~' [[ v/f (R ("(f), f) + 1~>f)l}fl- 2v/f IX

(7')1'

= Vf (R (, (f), f) + IX (f)1 2) - 2Vf IX (f)1 2 . It follows from Lemma 7.30 that L (q,.) is differentiable a.e. on (0, T). As discussed in subsection 4.2 above, if L (q,') is differentiable at f, then 9As it is the integrand of the L:-Iength,

..fi (R + IXn

is a natural quantity.

7. THE REDUCED DISTANCE

312

~~ (q, f) ~~ (q, f). If L (q,') is not differentiable at 1', then by writing ~~ (q, f) = JT h (f), f) + IX (1')1 2) - 2JT IX (1')1 2, we mean (this is our convention below) that there is a smooth function L (x, I) satisfying L (x, I) ~ L (x, I) for x E U and I E (1' - €, l' + €), L (q, f) = L (q, f) and (q, f) = JT h (f), f) + IX (1')12) -2JT IX (1')1 2 . Now we have proved

(R

fi

(R

7.34 (Time-derivative of L formula). The time-derivative of the L-distance function is given by LEMMA

~~ (q, f) = -# (R (q, f) + IX (1')12) + 2#R (q, f),

(7.57)

where X (f) = ~ (f) , for any minimal C-geodesic 1 : [0,1'] to q.

---+

M joining p

In the case where (Mn,g (I) == go) is Ricci fiat, we have 1 (I) = f3 (2JT) , where f3 : [0,2vT] ---+ M is a constant speed Riemannian geodesic with respect to go. Thus hand L (q, I)

~=

Jr/3 (2JT) and 1/3 (2JT) 1 == ~~. On the other

= d~~2. Hence

aa,L (q,,)_ = - d41'3/2 (q, p) 2 '= = -VI

1

d , _1 2 d, (I) ,

agreeing with (7.57).

5. The second variation formula for C and the Hessian of L Recall that the second variation of arc length formula of a geodesic 1 is (7.58)

2

d 2 1u=o L (,U) du

= fob (1\7"yU1 2- (\7"yU, i)2 - (Rm (U, i) i, U)) ds + (\7u U, i) Ig, where IU : [0, b] ---+ M is parametrized by arc length s and satisfies 10 = 1 and U ~ Iu=o IU' This formula is fundamental to Riemannian geometry for a variety of reasons. For example, on a complete Riemannian manifold, any two points can be joined by a minimal geodesic, in which case 2 PdU 1u=o L (,U) ~ for endpoint-preserving variations. The second variation of arc length can also be used to bound from above the Hessian of the distance function. In this section we consider the analogous second variation formula for C-Iength. The first variation of C-Iength determines the C-geodesic equation and is related to the space-time connection. The second variation formula for Clength at an C-geodesic 1 : [0,1'] ---+ M is related to the space-time curvature and hence Hamilton's matrix Harnack quadratic. In the case of a minimalCgeodesic, it also gives an upper bound for the Hessian of the barrier function

Ju

°

5. THE SECOND VARIATION FORMULA FOR C AND THE HESSIAN OF L

313

t defined in subsection 4.2 of this chapter. At a point q where L (', r) is C2, the second variation formula gives an upper bound for the Hessian of the L-distance function and tracing this estimate yields an estimate for the Laplacian of L. The Hessian upper bound will be very important in discussing the weak solution formulation later. In this section (M n , 9 (7)), 7 E [0, T], will denote a complete solution to the backward Ricci flow satisfying the pointwise curvature bound max {IRml ,IRcl} :S Co < 00 on M x [0, T], and p E M shall denote a basepoint. 5.1. The second variation formula for C. Let r E (0, T) and let ---t M be an C-geodesic from p to q. Let IS : [0, r] ---t M, s E ( -E, E), be a smooth family of paths with 10 (7) = I (7). Recall that our convention about the smoothness of a variation IS of I is that !3s (0') ~ IS ( ~2) is required to be a smooth function of (0', s) . It is easy to see that the nondifferentiability of IS at 7 = causes no trouble in the following calculation. This would also be clear if we use (7.19) to do the calculation. Define ~ (7) ~ X (7, s) and ~ (7) ~ Y (7, s) , so that [X, Y] = 0.10 We I : [0, r]

°

also write Y (7) ~ Y (7, 0). Note that Y (U42, (0', s) . Recall the first variation formula (7.30)

s)

is a smooth function of

which holds for all s E (-E, E). Differentiating this again, we get

(8~C) b) ~ d2~s~IS) Is=o =

Io -IT (Y (Y (R)) + 2 (V'yV'yX,X) + 21V'yXI f

2)

d7.

Now since [X, Y] = 0, (V'yV'yX, X)

= (V'yV'xY, X) =

(R(Y,X) Y, X)

+ (V'xV'yY, X).

Hence (7.59)

(8~C) b) = {f -IT (

Jo

Y (Y (R)) + 2 (R (Y, X) Y, Xl ) d7. +2 (V' x V'y Y, X) + 21V'y XI

10Alternately, given any vector field Y along "I, there exists a family of paths "Is such that ~Is=o = Y. In this case, we extend Y by defining ~ = Y for s E (-c:,c:), so that [X,YJ = O. Technically, X and Y are sections of the bundle G*TM on [O,fJ x (-c:,c:), where G (7, s) ~ "Is (7).

314

7. THE REDUCED DISTANCE

On the other hand, we compute d dT (VyY,X) = (VxVyY,X) + (VyY, VxX)

+ ~~ (VyY,X) + (

(:T V)

y

Y,X).

Now ~ = 2Rc and

((:T V) y y,x)

= 2 (Vy Rc) (Y,X) - (Vx Rc) (Y, Y).

Hence d dT (Vy Y, X)

(7.60)

= (V x Vy Y, X) + (Vy Y, V x X) + 2 Rc (Vy Y, X) + 2 (Vy Rc) (Y, X) - (V x Rc) (Y, Y).

Suppose (7.61)

Y (0)

=0

(this and the fact that ..jTX (T) has a limit as T --t 0 are used to get the third equality below). Then applying (7.60) to (7.59) and integrating by parts, we compute

(&}.c) b) = foT .jT (Y (Y (R)) + 2 (R (Y, X) Y, X) + 21Vy X12) dT +2 =

1 T

o

d~ (VyY,X) - (VyY, VxX) - 2Rc (VyY, X) ) -2 (Vy Rc) (Y, X) + (V x Rc) (Y, Y)

'- (

yT

~

foT .jT (Y (Y (R)) + 2 (R (Y, X) Y, X) + 21VyX12) dT

+2

1 T

o

(VyY, VxX) - 2Rc(VyY,X) ) dT -2 (Vy Rc) (Y, X) + (V x Rc) (Y, Y)

'- (

-

yT

+ 2.jT (Vy Y, X) I~ - foT '=

= 2YT (VyY

X)

+

,

+2

1 T

o

'- (

yT

Jr

(Vy Y, X) dT

1T .jT (Y(Y(R))-VYY.VR 0

)

+2 (R (Y, X) Y,X) + 21VyXI 2

dT

(VyY, [VxX + 2Rc (X) - !VR + 2~X]) ) dT -2 (Vy Rc) (Y, X) + (V x Rc) (Y, Y) -

= 2Vf (VyY, X) +

foT .jT (V~,yR + 2 (R (Y,X) Y, X) + 21VyX12) dT

+ foT .jT (-4 (V y Rc )(Y, X) + 2 (V x Rc )(Y, Y)) dT,

5.

THE SECOND VARIATION FORMULA FOR

I:.

AND THE HESSIAN OF

L

315

where we used the £-geodesic equation (7.32) to get the last equality; in the above, \7~yR ~ Y (Y (R)) - (\7yY) (R) ,

= Hess (R) (Y, Y).

That is, LEMMA 7.35 (£-Second variation - version 1). Let f E (0, T) and let "I : [0, r] ---t M be an £-geodesic from p to q and let Y ~ gs "Is for some smooth variation "Is of "I with Y (0) = O. The second variation of £-length is given by (8~£) ("t) =

+

(7.62)

2v'r (\7yY,X) (f)

i

T

o

r= ( \7~yR + 2 (R (Y, X) Y, X) + 21\7y XI 2 ' -4 (\7y Rc) (Y, X) + 2 (\7 x Rc) (Y, Y)

y 7

REMARK 7.36. Note that by (7.54) we have (8£) ("t)

)

d7.

= 2.jTX (f) . Hence

whose value only depends on Y (7) defined along "I (7). This is analogous to considering (Hess f) (Y, Y) = YY (f) - \7 y Y . \7 f. We now rewrite the £-second variation formula in a better form, which relates to Hamilton's matrix Harnack quadratic, Le., the space-time curvature. ll Since

!

[Rc (Y (7) , Y (7))]

= (:7 RC)

(Y, Y) + (\7 x Rc )(Y, Y) + 2 Rc (\7 x Y, Y) ,

integrating by parts, we have

- foT -IT ( =

:7

RC) (Y, Y) d7

foT -IT (;7 Rc (Y, Y) + (\7 x Rc) (Y, Y) + 2 Rc (\7 x Y, Y) )

d7

- -ITRc (Y, Y)I~.

llSee Section 5 of Chapter 8 for the reason why the space-time curvature is Hamilton's matrix quadratic.

7.

316

Hence (7.62) and Y (0)

THE REDUCED DISTANCE

= 0 imply

1

2 (6}.c) h) -

Vf(V'yY,X)

-

+ JTRc(Y,Y)I~

1T JT ((:r Rc+ 2~ RC) (Y, Y) + ~V'},YR) dr + 1T JT ((R (Y, X) Y, X) -IRc (Y)12) dr + 1T JT (-2 (V'y Rc) (Y, X) + 2 (V' Rc) (Y, Y)) dr + 1T JTIV'x Y + Rc (Y)1 2 dr, =

X

where we used V' x Y = V'y X. Let H (X, Y) denote the matrix Harnack expression

H (X, Y) (7.63)

~ -2 (:r RC) (Y, Y) -

V'},yR + 21Rc (Y)1 2 -

~ Rc (Y, Y)

- 2 (R (Y, X) Y, X) - 4 (V' x Rc) (Y, Y) + 4 (V'y Rc) (Y, X) .

By substituting the definition of H (X, Y) in the above formula, we obtain (6}.c)

h) -

=-1T

(7.64)

2Vf (V'yY,X) (f) + 2VfRc (Y, Y) (f)

JTH(X,Y)dr

+

1T

2JTIV'xY+Rc(Y)12dr.

An even nicer form is LEMMA 7.37 (.c-Second variation - version 2). Let l' E (0, T) and let ---t M be an .c-geodesic. If Y (r) ~ IS (r) , for a smooth variation IS of I, satisfies Y (0) = 0, then

ts

I : [0,1']

(7.65)

(6}.c)

h) -

-1T

JTH (X, Y) dr +

2Vf (V'yY,X) (f) + 2VfRc (Y, Y) (f)

1T

2JT IV' x Y + Rc (Y) -

= IY;1 2

2~ YI2 dr.

PROOF. Since lV'x Y +Rc(Y) -

2~Y12

1 r 2 1 d 2 1 2 = lV'x Y +Rc(Y)1 - 2rdr IYI + 4r21YI ,

1 r

= lV'x Y + Rc(Y)12 - - ((V'x Y , Y) + (Rc (Y), Y)) + -42 1YI2

5. THE SECOND VARIATION FORMULA FOR C AND THE HESSIAN OF L

317

we have

where we integrated by parts. Note that the assumption of the lemma IY (T)12 = 0. The lemma implies that (~2) is smooth in (J and limT->o now follows from (7.64). 0

Y

Jr

We now consider a special case of this formula. As above, let 'Y : [0, f] --+ M be an ,C-geodesic. Fix a vector Y r E T-y(r)M and define a vector field Y (T) along 'Y by solving the following ODE along -"(:

V'x Y = -Rc(Y)

(7.66)

I

+ 2TY'

TE

[0,1'],

Y (f) = Yr. Note that any vector field along 'Y can be considered as a variation vector field. In particular, we may extend Y (T) to Y (T, s) for some smooth variation of 'Y. REMARK

7.38. Equation (7.66) is equivalent to

JrY

which essentially says is parallel with respect to the space-time connection. Note that if 'Ys : [0,1'] --+ M is a I-parameter family of paths such that X = 'Ys and Y = :s 'Ys, then [X, Y] = and (7.66) may be rewritten as

°

:T

( V'

X+ Rc - 2~g) (Y) = 0,

which is reminiscent of the gradient shrinker equation. From (7.66) we compute (7.67)

ddT 1Y12

Solving this

ODE,

=

dd [g (Y, Y)] T

= 2 (V' x Y, Y) + 2 Rc (Y, Y) = ~ 1Y12 .

we have

(7.68) Thus Y (0)

= O. Hence by (7.65) we have the following.

T

7. THE REDUCED DISTANCE

318

LEMMA 7.39 (.c-Second variation under (7.66)). If f E (0, T), YT E T'Y(T)M and Y is a solution to (7.66) with Y (f) = YT, then the second variation of .c-length is given by (8~.c)

(7.69)

=-

h) - 2Vf (\7yY, X) (7) + 2VfRc (Y, Y) (f)

loT .,fTH (X, Y) d7

112

+ IY

5.2. Hessian comparison for L. Corresponding to the .c-second variation formula is an upper bound for the Hessian of the L-distance function, which we derive in this subsection. Given any (q, f), let I : [0,1'] --+ M be a minimal .c-geodesic from p to q so that L (q, f) = .c (,), Fix a vector Y E TqM - { and define the vector field Y (7) along I to be the solution to (7.66) with Y (f) = Y. Let IS : [0,1'] --+ M be a smooth family of curves for s E (-c, c) with

IT}

I

d,s (7) = Y (7) and (\7yY) (f) = 0. ds s=o . Then there exists a small neighborhood U of q, 8 E (0, c], and a smooth family of curves IX,T : [0,7] --+ M for (X,7) E U x (1' - 8, l' + 8) satisfying IX,T (0)

= p,

''Ys(T),T

= IS,

IX,T (7)

and

=x

for s E (-8,8). We define L (x, 7) ~ .c hX,T)' Then L hs (f), f) = .c hs). Since L (', .) is an upper barrier function for the L-distance function L (.,.) at (q, f), we have (Hess(q,T) L) (Y, Y) ::; (Hess(q,T)

when L (', f) is C 2 at q. Since (\7y Y) (f) =

°

L) (Y, Y)

and

dd 2 I .c hs) , s s=o S s=o combining this with Lemma 7.39, we get the Hessian Comparison Theorem for L. ( Hess(q,T)

L) (Y, Y)

2

= dd 2

I L hs (f) ,f) =

2

COROLLARY 7.40 (Inequality for Hessian of L). Given l' E (0, T), q E M, and Y E TqM, let I : [0,1'] --+ M be a minimal .c-geodesic from p to q. The Hessian of the L-distance function L (', f) at q has the upper bound (7.70)

r

(Hess(q,T)L) (Y,Y)::; - Jo .,fTH(X,Y)(7)d7+

IY(1')1 2

.JT

- 2VfRc (Y, Y) (f), where Y (7) is a solution to (7.66) with Y (f) = Y and H is the matrix Harnack expression defined in (7.63). Equality in (7.70) holds when L (', f) is C 2 at q and Y (7) is the variation vector field of a family of minimal .c-geodesics.

5.

THE SECOND VARIATION FORMULA FOR

C

AND THE HESSIAN OF

L

319

If L (., r) is not C 2 at q, the above inequality is understood in the barrier sense; this is our convention below. More precisely, there is a smooth

function

L (., r)

defined near q such that ( Hess(q,T)

equality (7.70) and

L (., r)

i) (Y, Y)

satisfies in-

is an upper barrier function for L (., r) with

L (q, r) = L (q, r) . LEMMA 7.41 (Upper bound for Hessian of L). Fix To E (0, T). Given (0, To], q E M, and Y E TqM, the Hessian of the L-distance function L (., r) at q has the upper bound

rE

where C2 is a constant depending only on n, To, T, Co (Co 2: sUPMx[o,T]IRml is as in (7.27)).

:T

PROOF. From Shi's derivative estimate and the equation for Rc, there is a constant Cl depending on n, T - To, and Co such that Rcl ' IVV RI , IV Rcl , and IVV Rml are all bounded by CIon M x [0, To]. From (7.52) and Lemma 7.13(ii) and (iii), we get

Ifr

e 6CoT I fTX (7 )1 2 + 12C5 C~T e6CoT I.jTX (r)12 < V'* * g(T.)

2

nCo) C T_e6CoT 1 < e 6CoT ( --c, (r) + r + _2

-

2Vf

12C5

3

nCor + __ e2COT < e 6CoT ( d2 -


0 there exists qT EM such that

T

h(T) = L (qn T) - 2nT, and h( T) is a continuous function. If M is closed, then the claim follows from L(x, T) being a continuous function when T > O. When M is not closed, the claim follows from Lemma 7.13(i), which says that limx-+oo L (x, T) = +00. This and the local Lipschitz property of L (x, T) imply the claim. Now we estimate the right lim sup derivative of hat f E (0, T):

d+h(_) ...!...l' T

dT

"7'

lmsup

h(f + s) - h(f) s

s-+O+

. L(qHs, f + s) - L(qr, f) = 1lmsup s-+O+

L(qr, f . :::; 1lmsup

2 n

-

S

+ s) - L(qr, f) s

s-+O+

-

2

n,

where the last inequality follows from the definition of qHs' If L is C 2 at (qr, f), then by using (7.84), we have . L(qr,f+s) -L(qr,f) _ oL( -) 11m sup - ~ qr, T s-+O+

S

uT

:::; 2n - ~L(qr, f) :::; 2n. The reason why ~L(qr, f) 2: 0 is that qr is a minimum and smooth point of L(·,f). We have proved that d;Th(f):::; 0 when L is C 2 at (qr,f). When L is not C 2 at (qr, f).!. let L be a smooth barrier function of L at (qr, f) as in (7.56). Setting L(x, T) ~ 2..jTLix, T), we se~ that qr is a minimum point of the locally defined function L(., f) since L is a barrier function of L from above at (qr, f) and qr is a minimum point of L(., f).

7. THE REDUCED DISTANCE

326

Hence t1t(q'f, f) ~ O. We have

d+h(_)

-d r r

~

l' L(q'f, l' + s) - L(q'f, f) Imsup 8 ..... 0+ s

. L(q'f' l' + s) - L(q'f' f) _ < _ 1Imsup s

8 ..... 0+

aL (q'f,r_) .

~

ur

Then using (7.84), which holds for the barrier function

at

t,

we have

~

or (q'f, f) ~ 2n - t1L(q'f' f) ~ 2n.

We have proved that d;Th (f) ~ 0 when L is not C 2 at (q'f, f). Hence we have proved that d;Th (f) ~ 0 for all l' E (0, T). By the monotonicity principle for Lipschitz functions stated in §3 (Lemma 3.1) of [179], h(r) is nonincreasing. (ii) This follows from (i) and Jim h(f)

T ..... O+

= Tlim min L (q, f) ..... OqEM ~ !im L (p, f) = (dg(o) (p,p))2 T ..... O

=0. D

6.3. The reduced distance function £. To get even better equations than those in Lemma 7.45, we introduce the reduced distance function £. DEFINITION

(7.87)

7.49. The reduced distance £ is defined by

£(x,r) ~

1

1 -

2JT L (x,r) = 4rL(x,r).

Let l' E (0, T) and let "Y : [0,1'] ---+ M be a minimal £-geodesic from p to q and let K = K b, f) be defined as in (7.75). By (7.79), (7.80), and (7.81), we have at (q, f) , (7.88)

a£ =_1_K_~ R of 21'3/2 f + ,

(7.89)

1 £ IV'£I 2 = -R- K +1'3/2 1"

(7.90)

1 n t1£ < - - - K + - -R. -

21'3/2

21'

From these equations (which involve the trace Harnack integral K), (7.86) and Lemma 7.48, we easily deduce the following which do not involve K.

6.

EQUATIONS AND INEQUALITIES SATISFIED

BY L

AND

I.

327

LEMMA 7.50 (Reduced distance - partial differential inequalities). At (q, '1') the reduced distance e(x, 7) satisfies

ae _ ~e + l\lel 2 - R + ~ >0 a7 2T - , 2 e-n 2~e -I\lel + R + -_- :s; 0, 7

(7.91) (7.92)

ae

(7.94)

lim 1'-->0+

(7.95)

e

n

0, rewrite equation (7.106) using the metric 9 (1) instead of 9 (0) . Show that this equivalent form is consistent with (7.108). We follow up on the above exercise by translating time in our Einstein solution so that R (0) --t 00 in (7.103), and correspondingly, R (7) --t 2~' Since we then have d;(O) (p, q) --t 0, we choose to rewrite the formula for £ in terms of d;(T) (p, q) using

d;(O) (p,q) R(r) 1 d;(T) (p, q) = R (0) = 1 + ~ R (0)' In particular, from (7.105), we have £

r (q,)

(1-

=~ 2

tan-l (V27R(0)

V27R(0)

In

In))

R(r)d;(T) (p,q)

1

+----:--J~:::::::::::::;:::::;::::~-::--;:::::::=-r===:::::;::::=;=

In) 2ffnV7R (0)

tan- 1 (V27R (0)

n 2

--t -

as R (0)

--t

00.

EXERCISE 7.70. Let (Mn,g(t)), t E [O,T), be a maximal shrinking Einstein solution of Ricci flow so that R (t) = 2(T~t)' Given any (Xi, ti) E M x (0, T), we define a solution gi (7) ~ 9 (ti - 7) of the backward Ricci flow and i(x."" t.) (x, t) = £9(;x,.,. 0) (x, ti - t) . Check that

+ 4 (T -

t)

From the above remarks, if~i,td (x, t)

J

--t

t;-t T-ti

tan- 1

~ as i

--t

J

00

t;-t . T-ti

if ti

--t

T.

7.2. The £ function on a steady gradient Ricci soliton. Consider a steady gradient Ricci soliton g(7) = O. If 'Y has no conjugate points in the interior, but possibly one at b, then (7.120) still holds although J may not be unique, Furthermore, we have I (W, W) ~ 0 for any W perpendicular to 1 with W (a) = 0 and W (b) = 0, where equality holds if and only if W is a Jacobi field.

8.1. £-Jacobi fields. Let 'Y : [0, f'] ~ M be an £-geodesic, where f' E (0, T), and let X (7) ~ ~ be its tangent vector field. DEFINITION 7.82 (£-Jacobi field). An £-Jacobi field along an £-geodesic 'Y is the variation vector field of a smooth 1-parameter family of £-geodesics 'Ys, s E (-E, E), for some E > 0, all defined on the same time interval as 'Yo = 'Y.

Let X (7, s) ~ ~, Y (7, s) ~ ~,and let Y (7) ~ Y (7, 0) be an £-Jacobi field along "f. Using the £-geodesic equation (7.32), we compute

\7x (\7xY) = \7x (\7yX) = R(X, Y)X

= R (X, Y) X + \7y

+ \7y (\7xX)

(~\7 R -

2 Rc (X) -

2~ X) .

8.

l:-JACOBI FIELDS AND THE .c-EXPONENTIAL MAP

347

Thus we have a linear second-order ODE for the C-Jacobi field Y (7), called the C-J acobi equation: (7.121)

\Ix (\lx Y )

1

= R(X, Y)X + 2\1y (\lR)

- 2 (\lyRc) (X)

1 - 2Rc (\lxY) - -\lxy' 27

Since 7 = 0 is a singular point because of the ~ factor in the last term, we rewrite the equation as D,fox (\I ,foxY)

=7

(\Ix (\lx Y ) +

2~ \lXY)

= R (JTX, Y) JTx

7

+ "2\1y (\lR)

- 2JT (\ly Rc) (JTX) - 2JTRc (\I ,foxY) .

Let Z(O") ~ y'TX(7), where 0" = 2y'T and (3 (0") = 'Y (0"2/4) . Then Z(O") ~~ and we can rewrite the C-Jacobi equation for Y (7) as \I z (\I z Y) = -20" Rc (\I z Y)

(7.122)

=

+ R (Z, Y) Z 0"2

- 20" (\ly Rc) (Z)

+2

\ly (\I R),

where we view Y (0"2/4) as a function of 0". Suppose Z(O) = limT-to y'TX = V E T'""((o)M. We have the following by solving the initial-value problem for (7.122). LEMMA 7.83. Given initial data Yo, Yl E T'""((o)M, there exists a unique solution Y (7) of (7.121) with Y (0) = Yo and (\I zY) (0) = Y1. Since (7.121) is linear, the space of C-Jacobi fields along an C-geodesic 'Y is a finite-dimensional vector space, isomorphic to T'""((o)M x T'""((o)M. REMARK 7.84. If the solution (M n ,9 (7) C-Jacobi equation (7.121) says

= 90) is Ricci fiat, then the

1 \Ix (\lx Y ) = R(X, Y)X - 27 \lxY.

That is, we obtain the Riemannian Jacobi equation for 90, D,fox (\I ,foxY)

= R (JTX, Y)

JTXj

i.e., \I z (\I z Y) = R (Z, Y) Z.

On the other hand, if 9 (7) is Einstein and satisfies Rc = 2~9, then \Ix (\lx Y )

= R (X, Y) X

3

- 27 \lxy'

7. THE REDUCED DISTANCE

348

We now rewrite the £-Jacobi equation in a more natural way in view of the space-time geometry associated to the Ricci flow. Consider the quantity Rc g(r) (Y) . The time-dependent symmetric 2-tensor Rc g(r) is defined on all of M whereas Y is a vector field along the path '"Y ( r) in M. In local coordinates, RCg(r) (y)i = gijRjkyk, so actually we are considering Rc as a (1, I)-tensor. Caveat: When we take the time-derivative of Rc, we consider it as a (2,O)-tensor and then raise an index to get a (1, I)-tensor! By V x [Rc (Y)] (ro) we simply mean the covariant derivative along '"Y (r) of the vector field RCg(ro) (Y h (r))) at r = TO. In this respect the vector field RCg(ro) (Y h (r))) along '"Y (r) should be distinguished from RCg(r) (Y h (r))) , where in the latter case the Ricci tensor depends on time. Combining the equations 14

D;. [Rcg(r) (Y)] = (:rRC) (Y)+ (VxRc)(Y) +Rc(VxY)-2Rc 2 (y) and

Dd~ (V~r)x) =

Vx(VyX) +

= Vx

(:r V)y X

(VyX) + (VyRc) (X)

+ (V x Rc) (Y) - (V Rc) where (V Rc) (~, we have (7.123)

1") (Z) ~ (V Rc) (Z, Y, X), and commuting derivatives,

D.!L (Rc (Y) d.,.

=

(~, 1") ,

+ Vy X)

(:r RC) (Y) + 2 (V x Rc) (Y) + Rc (V x Y) - 2 Rc

+ Vy (VxX) + R(X, Y)X + (Vy Rc) (X) - (VRc)

2

(Y)

(~'1")

,

where we used Vx (VyX)

= Vy (VxX) + R(X, Y)X.

Substituting the £-geodesic equation 1

1

o = V x X - "2 V R + 2 Rc (X) + 2r X in (7.123) and adding this to 1 ) =-Y--VxY 1 1 Dd ( --Y d.,. 2r 2r2 2r ' 14In accordance with the caveat above, :.,. Rc denotes the derivative of Rc as a (2,0)tensor and (:.,. Rc) (y)i ~ gij (t.,. RC)jk yk.

8 . .L:-JACOBI FIELDS AND THE .L:-EXPONENTIAL MAP

349

we have D..!!. (RC(Y) + V'yX dr

~Y) 27

= (:7RC) (Y)+2(V'xRc)(Y)-2Rc 2 (y) +

1

"2 V'y (V'R) - 2 (V'y Rc) (X) - Rc (V'y X)

+ R (X, Y) X +

(V'y Rc) (X) - (V'Rc)

(~, 1)

1 1 + 272 Y - :;:-V'yX.

This may be rewritten as D..!!. (RC (Y) dr

=

+ V' y X

-

~ Y) 27

(aa7 RC) (Y) + ~2 V' y (V'R) - Rc

2

(Y) +

~ Rc (Y) 27

- 2 (V'y Rc) (X) + 2 (V' x Rc) (Y) + R (X, Y) X + (V'Rc)

(1' 1) - (V'Rc) (~, 1)

- Rc (RC(Y) + V'yX -

~y) 27

-~ 7

(RC(Y) + V'yX -

~Y). 27

Define the matrix Harnack expression

J (Y)

::§:: -

(aa7 RC) (Y) - ~2 V' y (V'R) + Rc

2

(Y) -

~ Rc (Y) 27

+ 2 (V'y Rc) (X) - 2 (V' x Rc) (Y) - R (X, Y) X

(1'1) + (V'Rc) (~'1)' so that (note (- (V'Rc) (1'1) + (V'Rc) (~'1)) (Y) = 0) - (V'Rc)

(J(Y),Y) =

~H(X,y).

Thus we have the following. LEMMA

7.85. The £-Jacobi equation is equivalent to

(7.124)

+ Rc +~) (RC (Y) + V' x Y ( D..!!. ~ 7

~ Y) ~

=

-J (Y) ,

where we have replaced V' y X by V' x Y. EXERCISE

7.86. Rewrite the above equation using Uhlenbeck's trick.

350

7. THE REDUCED DISTANCE

8.2. Bounds for L:-Jacohi fields. Let c > 0 and let IS : [0, f] -+ E (-€, €), be a smooth I-parameter family of L:-geodesics. In this subsection we adopt the notation of subsection 8.1 above. Assume Y (0, s) = for s E (-€, €) (for simplicity we may assume IS (0) = I (0) for all s). We shall estimate from above the norms of L:-Jacobi fields Y (7) = Y (7, 0). By the first variation formula for the L:-Iength and the L:-geodesic equation, we have for s E (-€, c),

M, s

o

fJyL: (Ts) = 2Vf (Xs, Ys ) (f). We differentiate this again to get

= 2Vf (\7 x Y, Y) (f) + 2Vf (X, \7y Y) (f) , where we used \7y X = \7 x Y. (fJ~L:) (T)

Now the derivative of the norm squared of the L:-Jacobi field is

d~ 17=1' IYI

2

= :717=1' IY (7)1;(7) = 2 (\7 x Y, Y) (f) = 2Rc (Y, Y)(f)

(7.125)

1

+ v'T (fJ~L:) (T) -

+ 2Rc (Y, Y) (f)

2 (X, \7yY) (f),

which is expressed in terms of the second variation of L:. Let field along I which satisfies the ODE

(\7xY)

(7.126)

(7) = -Rc

(Y (7)) + 2~Y (7),

Y be a

vector

7 E [o,f] ,

Y(f)=Y(f).

(7.127)

(The first equation is the same as (7.66).) As in (7.68),

Iy (7)1

(7.128)

2

=

~ IY (f)1 2 •

In particular, Y (0) = 0 = Y (0) . Now we further assume that the IS are minimal L:-geodesics for each s E (-€, c). Let is : [0, f] -+ M be a I-parameter variation of I with

81

-

-8 is = Y, s s=o

is (f) =

IS (f)

and is (0) = IS (0) ;

this is possible because Y (0) = Y (0) and Y (f) = Y (f). Then L: (is) ~ L: (Ts) for all s, and equality holds at s = O. Hence

(fJ~L:) (T) S (fJ~L:) (T), where equality holds if Y is an L:-Jacobi field. Combining this with (7.125), we get

d~ 17=1' IYI 2 S 2 Rc (Y, Y)(f) + J:r (fJ~L:) (T) -

2 (X, \7yY) (f).

8. 'c-JACOBI FIELDS AND THE ,C-EXPONENTIAL MAP

By (7.69), since (7.126) holds and Y (0)

351

= 5, we have

(8~C) (r) - 2Vf (X, VyY) (f) =Note that

loT VTH (X,Y) dT+ IY~W -2VfRc(Y,Y) (f).

is (f)

= IS

(f) implies

VyY (f) = VyY (f). Hence

7.87 (Differential inequality for length of C-Jacobi field). Let IS : [0, T2J --t M, where T2 E (0, T), S E (-E, E), and E > 0, be a smooth family of minimal C-geodesics with Ys (0) = 5 for S E (-E, E). Then for any LEMMA

l' E

(0, T2J

the C-Jacobi field Y (T)

~I

(7.129)

dT

7=1'

=

¥: Is=o satisfies the estimate

r

IYI 2~ -~ VTH (X,Y) dT+ IY~)12, Vi Jo T

Y satisfies

(7.126) and (7.127) and quantity defined in (7.63).

where

H(X, Y)

is Hamilton's Harnack

Note that the only place where we used an inequality (versus an equality)

(8~C) (r) ~ (8~C) (r) . Hence equality holds in (7.129) if and only if the vector field Y satisfying (7.126) and (7.127) is an C-Jacobi in our derivation is

field. Then

d~ 17=1' 1Y12 = d~ 17=1' lyI 2= IY ~)12 in (7.129), and

(7.130)

J; vrH (X, Y) dT = O. From (7.125) we get

:T 17=1' 1Y12 = 2Rc (Y, Y) (f) + Jr (HessL) (Y, Y) (f) = IY ~)12

Applying Hamilton's matrix Harnack inequality to (7.129), we get LEMMA 7.88 (Estimate for time-derivative oflength of C-Jacobi field). If the solution (Mn, 9 (T)) , T E [0, TJ , to the backward Ricci flow has bounded nonnegative curvature operator and the C-Jacobi field Y (T) along a minimal C-geodesic I : [0, T2J --t M satisfies Y (0) = 5, then for C E (0,1) and l' E (0, min {T2, (1 - c)T}],

d~ 17=1' log IYI 2~ ~ (C£ (r (f), f) + 1), where C = ~. 1fT =

00,

then

352

7. THE REDUCED DISTANCE

7.89. Note that for Euclidean space, .c-Jacobi fields satisfy (and £ is constant along .c-geodesics). REMARK

IY (T)12 = const ·T. In particular, d~ log IY (T)1 2 =

f:

Since 9 (T) has nonnegative curvature operator, Hamilton's matrix inequality holds and we have for any T E [O,fJ, PROOF.

H(X,y) (T) +(~+ T~T) Rc (y,y) (T) ~ O. Since Rc T E

~

Iy (T)1 2

0 and

=

¥IY (1')1 2, from

l'

~

(1 - c) T, we get for

[0,1'],

H(X, Y) (T) ~ - (~+ T ~ T) R(-y(T) ,T) Iy (T)1 T

2

2

T (T _ T) R (, (T) , T) IY (f) I ~

1 2 --=R (, (T), T) IY (1')1 . CT

Then (7.129) implies

~

dd I _1Y12 T T=T

(

2 r JTR (-y (T), T) dT + 1) IY (1')1 CyTJO T

~

~ (~£(-y(f),f)+1) IY~)12, since I is a minimal C-geodesic. Hence d I -d T

1(2-£ (-y (f) ,f) + 1) .

_ log IYI 2 ~ -=-

T=T

T

C

Finally, we leave it as an exercise to check that when T in essence take C = 1 in the inequality above.

=

00,

one can 0

8.3. The .c-exponential map. The .c-exponential map Cexp: TM x [O,T)

--+

M

is defined by Cexp (V, f) ~ Cexpv (f) ~ IV (f), where IV (T) is the .c-geodesic with limT->o JT~~ (T) = V E TpM (and IV (0) = p). Given 1', define the C-exponential map at time l' Cf'exp: TM

--+

M

by Cf'exp (V) ~ IV (f).

353

8. L:-JACOBI FIELDS AND THE .c-EXPONENTIAL MAP

EXAMPLE 7.90 (.L:-exponential map on a Ricci flat solution). To get a feel for the .L:-exponential map, we first consider a Ricci fiat solution (Mn,9 (7) = 90). Here, by (7.23), for V E TpM,

.L: exp (V, f) = exp ( 2v1rV)

,

where exp is the usual exponential map of (M, 90) with basepoint p. Note that for a Ricci flat solution, the .L:-exponential map has the scaling property:

= .L:exp (cv, ;)

.L:exp (V, f)

for any c > O. However this is not true for general solutions of the Ricci flow. The .L:-exponential map at f = 0 is related to expg(O) which is the usual (Riemannian) exponential map with respect to the metric 9 (0) . LEMMA 7.91 (.L:-exponential map as f ---+ 0). Let (Mn, 9 (7)) , 7 E [0, T] , be a complete solution to the backward Ricci fiow with bounded sectional curvature. Given V E TpM, as f ---+ 0, the .L:-exponential map tends to the Riemannian exponential map of 9 (0) in the following sense:

(7.131)

!im .L:exp ( 1"-+0

1G V,

2y 7

f) =

expg(O)

(V) .

From the proof we can see that the convergence in (7.131) can be made into

Coo -convergence. Motivated by the Ricci flat case, we define the path f3 : [0, 1]

PROOF.

---+

M by f3(p) so that f3 (1)

= .L: exp

~ .L:exp ( 2y1G7 V ,p2f)

= "1_1 V (p2f) , 2,j¥

C.i¥ V, f) . (Note that f3 depends on f but we do not

emphasize this in our notation.) We have (7.132)

df3 () d ( ( 2_)) 2 _ d'Y ~ v ( 2-) -d p = -d 'Y_1_V P 7 = p7 d P7 P P 2,j¥ 7

.

Hence the .L:-geodesic equation (7.32) becomes O=V'

1 ~ 2pr dp

(3 ) 1 d (3 ) --V'R+2Rc 1 ( -1_d( ----=2p7 dp 2 2p7 dp

1 1 df3 +--_-_-. 2 2p 7 2p7 dp

Multiplying this by 4p2f2 yields for p E [0,1] , (7.133)

df3 2 2 (df3) V' ~! dp - 2p f V'R + 4pf Rc dp

= O.

The covariant derivative and Ricci tensor are with respect to 9 (p2f) . Since

1

d'Y_1_V lim yT

1"-+0

~,j¥ 7

(7) =

G V, 2y7

354

7. THE REDUCED DISTANCE

we have df3 d'Y~v lim -d (p) = lim 2pf ~..;¥ (p2f) = V p--->O P p--->O r independent of f. Hence, by taking the limit of (7.133) as f "V~~O) fp = 0 and

--t

0, we have

dp

P 1-+ !im £ exp (

1;:; V, p2f) 2yr

r--->O

= !im f3 (p) r--->O

is a constant speed geodesic with respect to g (0) and with initial vector V. Thus when we evaluate it at p = 1, we get that the limit is exp9(0) (V). 0 Next we compute the differential of £fexp. LEMMA

7.92. For f E (0, T), £f exp is differentiable at V and the tan-

gent map

is given by

=

(£fexp). (W)

J(f),

where J(r) is the £-Jacobifield along £exp(V,r) with J(O)

=0

~

and

do-

J(2)

1

4

0'=0

= W.

PROOF. We have a family of £-geodesics £ exp(V for s E (-c, c). By the definition of £-Jacobi field,

J(r)

~

dd 1 s

8=0

1

r

E

[0, fJ,

£exp(V +sW,r)

is an £-Jacobi field. Since £ exp(V + sW, 0)

dd

+ s W, r),

£exp(V + sW,r)

= p, we have

J(O)

= O. From

= V + sW,

0- r=O

where

0-

= 2..j7, we get by taking dd

fa 18=0' J(r)

1

= W.

0- r=O

Note that the tangent map of £1' exp at V is given by D [£1'exp(V)] (W)

=

:SI8=0 £exp(V + sW,f).

The lemma follows. As a simple corollary of the lemma we have

o

8. C-JACOBI FIELDS AND THE C-EXPONENTIAL MAP

355

COROLLARY 7.93. Fix f E (0, T) and consider the £-exponential map £f' exp : TpM - t M. Then V is a critical point of the map £f' exp if and only if there is a nontrivial£-Jacobi field J(7) along £exp(V, 7), 7 E [0, f], such that J(O) = 0 and J(f) = O. PROOF. If V is a critical point of £f' exp, then there exists W such that D [£exp(V, f)] (W) = O. By the lemma, £exp(V + SW,7) is the required £-Jacobi field. On the other hand, if we have a nontrivial £-J acobi field J1 (7) with J 1(0) = 0 and J1(f) = 0, then we define W ~ d~IT=oJ1(7). Since J 1(7) is nontrivial, we have W =1= o. By the uniqueness of solutions of the initial-value problem for £-Jacobi fields, we know that

is Is=o

D [£f'exp(V)] (W)

=

J(f)

=

J1(f)

= o.

o

We see that V is a critical point of £f' expo

The Hopf-Rinow theorem in Riemannian geometry can be generalized for £-geodesics and the £-exponential map. The proof of this result will appear elsewhere. LEMMA 7.94 (£'-Hopf-Rinow). Suppose (Nn, h (7)),7 E [0, TJ, is a solution to the backward Ricci flow satisfying the curvature bound IRm (x, 7)1 ~ Co < 00 for (x, 7) E N x [0, T]. The following are equivalent:

(1) for every 7 E [0, T), the metric h (7) is complete; (2) £exp is defined on all ofTpNx [O,T) forsomepEN; (3) £ exp is defined on all of TpN x [0, T) for all pEN. Moreover,

(4) any of the above statements implies that given any two points p, q and 0 ~ 71 < 72 < T, there is a minimal £-geodesic T : [71,72] - t N joining p and q. 8.4. £-cut locus. We have the following simple lemma which is analogous to the corresponding theorem in Riemannian geometry. LEMMA 7.95 (When £-geodesics stop minimizing). Given V E TpM, there exists 7V E (0, T] such that

£ ( TVI[O,Tl) = L (Tv (7),7)

for all 7 E [0, Tv)

and £, ( TVI[o,Tl)

where TV : [0, T)

-t

> L (Tv (7),7)

for any 7 E (Tv, T),

M is the £-geodesic with limT--+O JT~ (7)

= V.

PROOF. The existence of Tv follows from the additivity property for concatenated paths (7.20). The fact that 7V > 0 follows from Lemma 7.29.

o

7.

356

THE REDUCED DISTANCE

That is, if TV < T, then

is the first time the £-geodesic rV : [0, T) -+ TV = T if and only if rV is minimal. The lemma establishes that either rV is minimal or there exists a first positive time TV past which rV does not minimize. Let r : [0, T) -+ M be an £-geodesic with r (0) = p. We say that a point (, (f) , f) , l' E (0, T), is an £-conjugate point to (p,O) along r ifthere exists a nontrivial £-Jacobi field along r which vanishes at the endpoints (p,O) and (,(1'), f). A point (q, f) is an £-conjugate point to (p, 0) if (q, f) is £-conjugate to (p, 0) along some minimal £-geodesic r (T) , T E [0,1'], from p to q. If r (T) = rV (T) for some V E TpM, then this is equivalent to V being a critical point of the £-exponential map Dr exp (see Corollary 7.93). TV

M stops minimizing. On the other hand,

DEFINITION 7.96. (i) The £-cut locus of (p,O) in the tangent space of space-time is defined by £C(p,O) ~ {(V, TV) : V E TpM} ,

where TV is defined above. Since TV > 0, we have £C(p,O) C TpM x (0, T]. (ii) The £-cut locus of (p, 0) at time l' E (0, T] in the tangent space is defined by (iii) Define n(p,O) (f) ~ {V E TpM : TV > f}. In words, O(p,O) (f) is the open set of tangent vectors at p for which the corresponding £-geodesic minimizes past time f. Note that n(p,O) (f) is not necessarily star-shaped in the sense that if V E n(p,O) (f) , then a V E n(p,O) (7) for any a E (0,1). However n(p,O) (f) is an open subset in TpM and n (T2) en (Td if Tl < T2· Next we define the £-cut locus of the map Dr exp for l' E (0, T) . DEFINITION 7.97. (i) The £-cut locus of (p,O) at time l' is defined by £ Cut(p,O) (f) ~ {£-fexp (V): V E TpM and

TV

= f}.

(ii) We define £ CutZp,O) (f) to be the set of points q E M such that there are at least two different minimal £-geodesics on [0,1'] from p to q. (iii) We define .cCut(p,O)(f) to be the set of points q such that (q,f) is £-conjugate to (p, 0) . EXERCISE 7.98. Show that for l' E (0, T) we have q i £ Cut(p,O) (f) if and only if for every minimal £-geodesic r : [0,1'] -+ M joining p to q, we may extend r as a minimal £-geodesic past time f. There is a characterization of £-cut locus points analogous to the characterization of cut locus points in Riemannian geometry. The first lemma below can be proved using the locally Lipschitz property of the £-distance L, while the second lemma below can be proved via a calculation similar to

8.

C-JACOBI FIELDS AND THE C-EXPONENTIAL MAP

357

the proof of the Riemannian index lemma. We will give the details of the proof elsewhere. LEMMA 7.99 (.L:-cut locus). (i) .L:Cut(p,O)(f) = .L:Cutzp,O)(f) U .L:Cut(p,O)(f) and is closed. (ii) .L:CutZp,O)(f) has measure O. (iii) .L:Cut(p,O)(f) is closed and has measure o. The proof of the above lemma depends on an index lemma for .L:-length. Define the .L:-index form .L: I(Y, W) by

(7.134) .L: I(Y, W)

~

l:

b

l

v'T

1

!V'yV'wR+ (R(Y,X)W,X) + (V' x Y, V' x W) - (V'y Rc)(W, X)

dT.

- (V'w Rc) (Y, X) + (V' x Rc) (Y, W) Note that the second variation of .L:-length (7.62) is related to the .L:-index form by

(6~.L:)

(7.135)

h) =

2v'T(V'yy,X)I~~ +2.L:I(V,V).

On the other hand, the .L:-index form is related to the .L:-Jacobi equation by .L: I(Y, W)

= JT (V' x Y, W) I~~

r

b

[I

V'x (V'x Y ) - Rm(X, Y)X - !V'y (V'R)

- JTa JT \ +2(V'yRc)(X)+2Rc(V'xY)+2~V'xY This can be proved by integrating by parts on the term in (7.134).

)] ,W

dT.

..jT (V' x Y, V' x W)

LEMMA 7.100 (.L:-index lemma). Let'"Y be an .L:-geodesic from (p, Ta) to (q, Tb) such that there are no points .L:-conjugate to (p, Ta) along '"Y. For any piecewise smooth vector field W along'"Y with W(Ta) = 0, let Y be the unique .L:-Jacobi field such that Y(Ta) = W(Ta) = 0 and Y(Tb) = W(Tb). Then .L: I(Y, Y) :::; .L: I(W, W)

and the equality holds if only if Y = W. Here we have used the obvious generalization of the definition of .L:-conjugate point with (p,O) replaced by (p, Ta) . Lemma 7.99 implies the following. COROLLARY 7.101 (Differentiability of L away from .L:-cut locus). Given f E (0, T) and q E M, suppose that there is only one minimal .L:-geodesic '"Y joining (p,O) and (q, f) and suppose that (q, f) is not an .L:-conjugate point of (p, 0), i.e., (q, f) is not an .L:-cut locus point. Then the L-distance L(·,·) and reduced distance £ are C 2 -difJerentiable at (q, f).

7. THE REDUCED DISTANCE

358

PROOF. Suppose limT->o JT~(O) = Vj the hypothesis implies by Lemma 7.92 that there is some E > and some small neighborhood Uv of V E TpM such that the map

°

(C exp, id) : Uv X (1' ~ E, l' + E) ~ M x (1' - E, l' + E) , (C exp, id) (W, r) = (C T exp (W) ,r) is a local diffeomorphism. For each WE Uv and r* E (1'-E, 1'+E), we claim the curve Cexp(W, r), r E [0, r*l, is a minimal C-geodesic. Hence using the local diffeomorphism property, there are an El > 0, a small neighborhood Uq of q, and a family of minimal C-geodesics Iq,T* smoothly depending on the endpoint q E Uq and 'r* E (1' - EI, l' + EI). Now L (q, r*) = C (,q,T.) is a smooth function of (q, r*) , L is differentiable near (q, f). We now prove the claim by contradiction. If the claim is false, then there is a sequence of points (Wi,ri) ~ (V, f) such that Cexp(Wi,r), r E [O,ril, is not a minimal C-geodesic. Let Cexp(Wi' r), r E [0, ril, be a minimal C-geodesic from p to Cexp(Wi, ri). As in the proof of Lemma 7.28, using Lemma 7.13(ii) and the C-geodesic equation, it is easy to show that IWil

g(p,O)

is bounded. Hence there is a subsequence Wi ~ Woo. If Woo =J V, then we get two minimal C-geodesics IV and IWoo joining (p,O) and (q, f), which contradicts the assumption of the lemma. If Woo = V, then (C exp, id) cannot be a local diffeomorphism since (C exp(Wi , rd, ri) The claim is proved and the lemma is proved.

= (C exp(Wi' ri), ri) . 0

As a simple consequence, we have (7.136)

is a diffeomorphism. Note that M\CCut(p,O)(1') is open and dense in M. Now we end this subsection by rewriting the formula for the C-index form in terms of Hamilton's matrix Harnack quadratic. Using

d~ [Rc (Y, W)] = (! RC) (Y, W) + (\7 x Rc) (Y, W) + Rc (\7 x Y, W) + Rc (Y, \7 x W) , we may write (7.134) as

l

CI(Y, W) = -

+

l

d

Tb

Ta

JT dr [Rc (Y, W)] dr

(,t Rc) (Y, W) + Tb

Ta

y'T

+ (R (Y, X) W, X) -

~\7y\7wR - (Rc (Y), Rc (W))

- (\7 y Rc) (W, X)

(\7w Rc) (Y, X) + 2 (\7 x Rc) (Y, W)

+ (Rc (Y) + \7 x Y, Rc (W) + \7 x W)

dr.

8 . .c-JACOBI FIELDS AND THE .c-EXPONENTIAL MAP

359

Rearranging terms and integrating by parts, we express this as follows. Define the symmetric 2-tensor Q by Q (Y, W)

~ (:r RC) (Y, W) + ~ Vy V w R 1 + 2r Rc (Y, W) -

(Rc (Y) ,Rc (W))

.

+ (R (Y, X) W, X) - (Vy Rc) (W, X) (Vw Rc) (Y, X) + 2 (V x Rc) (Y, W)

and

S (Y)

~

Rc (Y)

+ Vx Y -

1 -Y. 2r

Then we have LEMMA 7.102 (C-index form). The C-index form CI(Y, W) can be written as CI(Y, W)

= - ViRc (Y, W)I~: +

+

l

Tb

~

Vi

l

TaTb

ViQ (Y, W) dr

((S(Y)'S(W))+(S(Y)'~))

+ \~, / Y s (W) + (y'\f) ~

Note that, assuming [X, Y]

dr.

= 0 and [X, W] = 0, we have

d~ ((Y'rW )) = ~ (RC + Sym (VX) - 2~g) (Y, W), where Sym (V X)ij ~ ! (ViXj + VjXi) = !CXg. 8.5. C-Jacobian. First we recall the Jacobian in Riemannian geometry. Let (Mn, be a Riemannian manifold, let p EM, and given V E TpM

g)

with IVI = 1, let ')'v : [0, 8V) ~ M be the maximal unit speed minimal geodesic with 'Y (0) = V. Take {Ed?:11 to be an orthonormal frame at p perpendicular to V and define Jacobi fields {Ji (8)} along"YV so that Ji (0) = 0 and (VvJd (0) = Ei. The Jacobian J is defined by J hv (8))

~

Jdet ((Jd 8) , Jj (8)) ),

where ((Ji (8), Jj (8))) is an (n - 1) x (n - 1) matrix. Note that (7.137)

lim J hv (8)) = 1. 8n - 1

s-+O

Let d17sn-l denote the volume form on the unit (n - I)-sphere in TpM, which naturally extends to TpM -

pin

M. We define the

{O}, and let Cut (p) be the cut locus of

(n - I)-form d17 on M\ (Cut (p) U {p}) by d17 ~ (exp;1)* d17sn-l,

360

7. THE REDUCED DISTANCE

where the exponential map is restricted to inside the cut locus in the tangent space. Then the volume form of 9 on M\ (Cut (p) U {p}) is given by dl1g

= J hv (8)) dr Ada.

The volume forms of the geodesic spheres S (p, r) at smooth points, are given by das(p,r)

~ {x

EM: d (x, p)

= r} ,

= J hv (8)) da.

The Jacobian is related to the mean curvatures of the geodesic spheres and the Ricci curvatures of the metric 9 on M by the following formulas:

8

8r logJ

=H

and

~H = -Rc (~,~) -lhl 2 8r 8r 8r < _ Rc (~ ~) _ H2 -

8r'8r

n-l'

where h is the second fundamental form of S (p, r). The Bishop-Gromov volume comparison theorem may be proved this way (see [111] for example). Now we turn to the case of Ricci flow. Let ')'v (7), 7 E [0, T), be an L:-geodesic emanating from p with limT-+o VT'Y (7) = V. Let Jr (7), i 1, ... , n, be L:-Jacobi fields along ')'v with

Jr (0) = 0 and

('VvJr) (0) = EP,

where {En ~=l is an orthonormal basis for TpM with respect to 9 (0). Note that Jr (7) is a smooth function of V and 7 > 0 since g( 7) is smooth. Via the orthonormal basis {EP} ~=l we can identify TpM with IRn. Since D (L:exp(V, 7)) (EP) = Jr (7) (see Lemma 7.92), the Jacobian of the L:exponential map L:Jv (7) E IR (called the L:-Jacobian for short) is the square root of the determinant (computed using the inner products on the tangent spaces from the Riemannian metric 9 (7)) of the basis of L:-J acobi fields: (Jr (7), ... , (7)). That is,

J;:

L:Jv (7) ~ It is clear that L: Jv (7) is a smooth function of (V, 7) when 7 equivalent way of describing L: J v (7) is to define

L: J v (7) dx (V) ~ [( L:T exp(V))* dl1g( T,L:

T

> o. Another

exp(V)) ] '

where dx is the standard Euclidean volume form on (TpM,g(O,p)).

8.

£-JACOBI FIELDS AND THE £-EXPONENTIAL MAP

361

To get a feeling of the £-Jacobian, we calculate an example. EXAMPLE 7.103 (£-Jacobian of Ricci flat solution). Recall the fact that if (M n , 9 (7) = 90) is a Ricci flat solution, then an £-geodesic is of the form "t (7) = (3 (2ft) where (3 ((1) is a constant speed geodesic. Then an £-Jacobi field is of the form JV (7) = K (2ft) where K ((1) is a Riemannian Jacobi field along (3 ((1) with respect to 90. . JV () 2;;; ¥ (2y'T) Hence, by ch oosmg n 7 = V 7 1d /3 (" '-)1 ' d"

£Jv (7)

2yT

9(0)

= 2ftJv (2v'T)

,

where Jv is the Jacobian of the Riemannian exponential map of 90. Since by (7.137), lim Jv ((1) - 1 (1n-1 - ,

u->o+

we have lim £Jv (7) T->O+

= 2n.

7n/2

Note that for Euclidean space R n we have £Jv (7)

= 2n7 n/ 2 .

As suggested by Example 7.103 and (7.131), we now prove the following lemma. LEMMA 7.104 (£-Jacobian as 7 ~ 0). Let (Mn,9 (7)) be a solution of the backward Ricci flow with bounded sectional curvature. We have the following asymptotics for the £-Jacobian at 7 = 0: lim £Jv (7)

(7.138)

T->O+

= 2n.

7 n/ 2

PROOF. Let Ei (7) denote the parallel translation of with respect to 9 (0). Since Jr (0)

= 0 and

E?

(\7 d: Jr) (0) =

along "tv (7)

(\7vJr) (0) =

E?, then by the definition of derivative, . IJr (7) - 2ft E i (7)1 (0) . IJr (7) - (1Ed7) 1 (0) 11m 9 = 11m 9 T->O+ 2ft T->O+ (1

= O.

Hence lim £JV}7) = lim 7- n/ 2

T->O+

7n 2

T->O+

det ((2v'TEi (7), 2v'TEj (7))g(0))

= 2n. D

For the proof of the no local collapsing result via the L-distance in Chapter 8 we need the following properties of the £-Jacobians.

362

7. THE REDUCED DISTANCE

7.105 (Time-derivative of £-Jacobian). Let (M n , g (7)), 7 E [0, T], be a solution of the backward Ricci flow with bounded sectional curvature. Along a minimizing £-geodesic /'v (7) , 7 E [0, Tv), with /'v (0) = p, where Tv is defined in Lemma 7.95, for 0 < f < Tv the £-Jacobian £ J v (7) satisfies PROPOSITION

d ) (f) (-log£Jv d7

(7.139)

~

n --= 27

1 K,

-3

27"2

where K = Kbv, f) is defined by (7.75). Equality in (7.139) holds at the point /,v(f) only if

Rc bV(f), f)

(7.140)

+ (Hess f) bv(f), f)

= g

bV2~)' f).

7.106. The proof of (7.139) is closely modeled on that of the classical Bishop-Gromov volume comparison theorem. Here we follow the derivation using £-Jacobi fields. There are other ways to prove volume comparison such as in Li [246]. REMARK

REMARK

7.107. If we let it

= 2...;T and u = 2ft, then (7.139) says

Compare this with (7.76). PROOF OF PROPOSITION 7.105. Choose an orthonormal basis {Ei (f)} of TW{f)M. Since there is no point on /,v(7) , 0 ~ 7 ~ f, which is £conjugate to (p,O) along /,v, we can extend Ei (f) to an £-Jacobi field Ei (7) along /'v for 7 E [0, f] with Ei (0) = O. Actually for the same reason, we know that both {Jt (7)} E T-yv{r)M and {Ei(7)} E Tw(r)M are linearly independent when 7 E (0, fl. We can write n

Jt (f) =

LA{Ej (f) j=l

for some matrix

(Ai) E GL (n, ~). Then

for all 7 E [0, fJ, since we cannot have a nontrivial £-Jacobi field vanishing at the endpoints 7 = 0, f.

9. WEAK SOLUTION FORMULATION

363

Now we compute that the evolution of the £-Jacobian along I'v is given by

The last inequality is due to (7.129). Here the along I'V satisfying \l X Ei

Ei (f)

where

= Ei (f) and H

Ei (r)

are the vector fields

- ) 1= - Rc ( Ei + 2r Ei,

(x, Ei) (r) is the matrix Harnack quadratic

given by (7.63). By (7.72), we have have

(Ei' Ej) (r)

n

~H (X,Ei)

(r)

=

¥Oij,

and by (7.74), we

= ~H(X) (r).

t=l

So

(d~ log £ J) (f) ~ - 2'1'~/2 loT r 3/2H (X) dr + ; 1

n

= - 2'1' 3 / 2K + 2'1'.

(7.141)

If equality in (7.139) holds, then we have equality in (7.129) for each Y (r) = Ei (r), i = 1,··· ,n. By (7.130), we have ) 2Rc ( Ed'1'),Ed'1')

for each i. Since (7.140).

1 ( ) lEi '(f) + y'f(HessL) Ei (f) ,Ei(T) = 1'

Ei (f) =

r

Ei (f) can be chosen arbitrarily, this implies 0

9. Weak solution formulation The purpose of this section is to prove the integration by parts inequality (7.148) for the reduced distance f and to give the inequalities we proved for f a weak interpretation. We first recall some of the well-known results in real analysis which we shall need. An excellent reference for properties of

7.

364

THE REDUCED DISTANCE

Lipschitz functions and other aspects of real analysis on lRn is the book by Evans and Gariepy [139]. Many of the results in their book easily extend to Riemannian manifolds; when this is the case, we state the extensions without proof. In this section we shall assume that is a complete Riemannian manifold.

(Mn, g)

9.1. Locally Lipschitz functions. Recall the definition of differentiability on Riemannian manifolds. DEFINITION 7.108 (Differentiable function). A function differentiable at p E M if there exists a linear map

f :M~

lR is

Lp: TpM ~ lR such that lim If (expp (X)) - f (p) - Lp (X)I = x-o IXI

o.

When this is the case, by definition we write

If (expp (X)) - f (p) - Lp (X)I

=

0

(IXi)

X ~

as

o.

This implies for every X E TpM that the directional derivative

Dxf ~ lim f (expp (sX)) - f (p) = Lp (X) 8-0

S

exists. REMARK 7.109. Note that differentiability can be defined more generally for differentiable manifolds, but in the definition above we chose to endow the manifold with a Riemannian metric. Now we list three results in Evans and Gariepy's book. The first is Theorem 2 in §3.1.2 on p. 81 of [139]. LEMMA 7.110 (Rademacher's Theorem). Let

(M,g)

be a Riemannian

manifold. If f : M ~ lR is a locally Lipschitz function, then f is differentiable almost everywhere with respect to the Riemannian (Lebesgue) measure. Secondly, Theorem 5 in §4.2.3 on p. 131 of [139]. LEMMA 7.111 (Locally Lipschitz is equivalent to being in Wl~':l Let U be an open set in a Riemannian manifold

(M, g).

locally Lipschitz if and only if f E Wl~~oo (U) . Thirdly, Theorem 2 in §2.4.2 on p. 76 of [139].

Then

f :U ~

lR is

9.

and

WEAK SOLUTION FORMULATION

365

LEMMA 7.112 (Hausdorff dimension of a Lipschitz graph). Let

(Mr, 91)

Ml ~ M2

is locally

(MH\ 92)

be two Riemannian manifolds. If f :

Lipschitz in the sense that for any p E M1, there is an open neighborhood Up of p and a constant Cp such that dg2 (J (qd ,f (q2)) ~ Cpdg1 (ql, q2) for any ql, q2 E Up, then 'Hdim

{(x, f (x)) : x

E

Ml} = n,

where 'Hdim denotes the Hausdorff dimension. In particular, the (n + m)dimensional Riemannian measure vanishes: measMlxM2 {(x,

f (x)) : x E Ml} = O.

Later we shall recall some more basic results, especially about convex functions, as we need them. Now we give a proof that integration by parts holds for locally Lipschitz functions. We say a vector field v on M is locally Lipschitz if for any p E M and local coordinates {xi} in a neighborhood of p, we have for each i that the function vi (x) is locally Lipschitz, where v (x) vi (x) a~i. It is well known that integration by parts holds for Lipschitz functions.

'*'

LEMMA 7.113 (Integration by parts for Lipschitz functions). Let f be a locally Lipschitz function on M and let v be a locally Lipschitz vector field on M. Suppose that at least one of f and v has compact support. Then

1M f div v dJ.Lg = - 1M v . V' f dJ.Lg. Here div v and v . V' f are defined with respect to

9.

PROOF. We prove the lemma in the case where v has compact support; the other case can be proved similarly.15 Rademacher's Theorem says that both derivatives div v and V' f exist almost everywhere. Since v has support in some compact set K, we have lvi, Idiv vi E L oo (K), and

f, IV' fl

E

L~c (M) , so the integrals in the lemma make sense. We can

choose a smooth partition of unity { there exists a C2 function cp (s ) defined on a subinterval (so - 8, So + 8) such that

°

°

f 0 'Y (so) = cp (so)

and

f 0 'Y (s) ::; cp (s )

for all s E (so - 8, So + 8) , and cp" (so) ::; c. Let

cp (s) ~ cp (so) + cp' (so) (s - so) + c (s - sO)2 . cp (so) = cp (so), cp' (so) = cp' (so), and cp" (s) == 2c > c 2: cp" (so).

Note that Hence there exists 81 E (0,8) such that

f 0 'Y (so) = cp (so)

f 0 'Y (s) < cp (s) {so}. We claim that f 0 'Y (s) ::; cp (s) for all and

for s E (so - 81, So + 8d s E [0, a]. By taking c ~ 0, the claim then implies

f 0 'Y (s) ::; f 0 'Y (so) + cp' (so) (s - so) for all so, s E [0, a]. We conclude f 0 'Y (s) is concave on [0, a]. Finally, suppose the claim is false; then there exists SI E [0, a] - {so} such that f 0 'Y (SI) = cp (SI)' Suppose S2 E (so, SI) is a minimum point of f 0 'Y (s) - cp (s) on [so, SI]. Then

f 0 'Y (s) 2: f 0'Y (S2) + [cpl (so) + 2c (S2 - so)] (s - S2) + c (s - S2)2 on [so, SI]. On the other hand, by our hypothesis on f 0 'Y, there exists a C 2 (7.143)

function CP2 (s) defined for s near S2 such that

f 0'Y (s) ::; f 0'Y (S2)

+ cp~ (S2)(S -

S2)

+ ~ (s -

S2)2 ,

7. THE REDUCED DISTANCE

372

which contradicts (7.143). This completes the proof of the claim and part (ii) . (iii) By hypothesis, for every q E B (p, r) there exists a G 2 function 0 and C 2 function ip : B (q, r) -> lR such that ip (q) = f (q), f (x) ::; ip (x) for all x E B (q, r), I'Vip I (q) ::; C I , and

f}.ip (q) ::; k + c.

(ii) A continuous function f : Nt -> lR is said to satisfy b.f ::; k in the weak sense for some function k E Lfoc (Nt) if for any nonnegative C 2 function ip with compact support, we have

(7.144)

1M f b.ipdllg ::; 1M ipkdllg·

374

7.

THE REDUCED DISTANCE

(iii) A continuous function f : M ---t IR is said to satisfy b..f ~ k in the viscosity sense for some continuous function k if for every p E M and any C 2 function r.p : U ---t IR on some neighborhood of p satisfying r.p (p) = f (p), f (x) 2: r.p (x) for all x E U, we have b..r.p (p) ~ k. (iv) A continuous function f : M ---t IR is called a supersolution of b..f ~ k for some continuous function k if for every p E M, any r < inj (p) , and every C 2 function r.p on B (p, r) with b..r.p = k and r.plaB(p,r) = flaB(p,r) we have r.p ~ f on B (p, r) .

i

Now we can prove the following. LEMMA 7.125 (Equivalence of notions of supersolution). Let k : M ---t IR be a continuous function. (i) Let f : M ---t IR be a continuous function with Hess supp (f) ~ Cp < 00 on B (p, ~ inj (p)) for each p EM. Suppose that for each q E B (p, inj (p)) there is a local upper barrier function r.pq for f near

i

q satisfying 1'Vr.pql (q) ~

Cp < 00. We have for any r.p E C~ (M)

1M f b..r.pdp,g ~ 1M b..f . r.pdp,g.

(7.145)

In part~cular if b..f ~ k in the support sense, then f satisfies b..f ~ k in the weak sense. (ii) f satisfying b..f ~ k in the weak sense is equivalent to f satisfying b..f ~ k in the viscosity sense; they are both equivalent to f being a supersolution of b..f ~ k. PROOF. (i) Let 'IjJ be a C2 function on M. Then having b..f ~ k in the support sense is equivalent to b.. (f + 'IjJ) ~ k + b..'IjJ in the support sense, and b..f ~ k in the weak sense is equivalent to b.. (f + 'IjJ) ~ k + b..'IjJ in the weak sense. We use a partition of unity {cPoJ to rewrite r.p = r.pcPa, so that we only need to verify inequality (7.145) and (7.144) when r.p has small compact support, say in B (p, ~r) for some p E M, where r < min { 1,

L

injJp)}

as determined by Lemma 7.122(iii). From Lemma 7.122(iii) and by adding to f another concave C 2 function if necessary, we may assume that f (q) satisfies Hess supp (f) ~ -1 on B (p, r) and that f (x) is a concave function on Bp (r) C TpM in normal coordinates {xi} on B (p, r) . From Lemma 7.117(ii) and Lemma 7.119(ii), there are smooth functions fe (x) such that fe (x)

D2f(x) for x a.e. on Bp

---t

(~r) , (a~:!J~j (x)) ---t 7.119(iii), (a~:!J~j (x)) ~ 0 for

f (x) uniformly on Bp

(~r). By Lemma

all x E Bp (~r) . Hence for any 61 > 0 and any C 2-test function r.p supported in Bp (~r), there exist a sequence ck ---t 0+ and a set W 01 C Bp (~r) with meas (W01 )

~

61 such that (aJ;f (x))

---t

D f (x) and

(::i~:j

(x))

---t

9 WEAK SOLUTION FORMULATION

D2 f (X) uniformly on Bp (~r) \ W81. This in turn implies that b.gfek uniformly on Bp ar) \ W81. We compute using b.fek ::; 0,

375 ---t

Since meas (W81) ::; O. Note by Lemmas 7.59 and 7.60 that we can choose

0 even when M is noncompact.

(M, [})

381

8.

382

APPLICATIONS OF THE REDUCED DISTANCE

EXERCISE 8.1. Show that if (Mn'9) is a complete noncompact Riemannian manifold with RC g ~ - K for some K E ~, then the integral defining if (9, T) converges for all T > 0. In Euclidean space if is the integral of the heat kernel, which is the constant 1. Note also that for any (Mn'9) and p EM, lim if (9, T) = 1

(8.2)

T~O

essentially since manifolds are locally Euclidean. EXERCISE 8.2. Prove (8.2). Let

u (x, T) ~ (41l'T)-n/2 e- d (x,p)2/ 4T , which is a Lipschitz function, and let d (x) ~ d (x, p). We can think of

if (9, T) =

1M u (x, T) dJ.L (x) as a weighted volume centered at p with the

radial weight function u. As T ---+ 0, the weight u concentrates at p and as u diffuses throughout M.

T ---+ 00,

REMARK 8.3. If M is closed, then the upper bound

if (9, T) ~

1M (41l'T)-n/2 dJ.L (x) = (41l'T)-n/2 Vol (9)

implies that limT~oo if (9, T)

= 0.

9)

Now assume that (Mn, is complete with nonnegative Ricci curvature. Since RC g ~ 0, the Bishop-Gromov volume comparison theorem says that the volume ratio r- n Vol B (p, r) is a nonincreasing function of r. It is thus natural to expect that if (9, T) is a nonincreasing function of T since as T increases, the weighting favors larger radii. Indeed we have LEMMA 8.4 (Static reduced volume monotonicity). If (Mn'9) is complete with RC g ~ 0, then

d~ if (9,r) = 1M (:T - ~) udJ.L ~ 0.

(8.3)

In particular, by (8.2),

if (9, T)

~ 1

for all T

> 0.

REMARK 8.5. Clearly this lemma implies limT~oo V (9, T) E [0,1] exists. PROOF. We compute that u is a subsolution, in the weak sense, to the heat equation:

( ~ _ ~) u = u (_~ + ~ + (d~d + IV'dI2) _ ~ IV'dI2) ~

(8.4)

~

~

0,

~

~

~

1. REDUCED VOLUME OF A STATIC METRIC

383

where we used I\i'dl = 1 a.e. and the Laplacian comparison theorem, i.e., dfld::; n - 1. Note that the Laplacian comparison theorem is equivalent to the Bishop-Gromov volume comparison theorem. 0 It is useful to keep the following simple examples in mind when pondering Lipschitz continuous sub- and super-solutions of the heat equation.

8.6 (Heat equation on 8 1 ). Let Ml the standard metric 9 = d0 2 • Consider the function EXAMPLE

f (0, t)

02 t + 2'

=

= 81 =

0 E (-71",71"] and t

~/

(271"Z) with

E R

For each fixed 0, f (0, .) is a smooth (linear) function of time and, for each fixed t, we have f (., t) is Lipschitz on 8 1 and Coo except at 0 = 71". Moreover f is a solution to the heat equation almost everywhere. In particular,

8f 8 2f 8t (0 , t) = 80 2 (0, t) = 1 for all 0 E 8 1

-

{71"} and t

E R On the other hand,

f f(0,t)dO=271">O. ls!

ddt

This is consistent with the fact that f is not a subsolution of the heat equation in the weak sense (as we shall now see, f is a supersolution). Note that

8f 80 (0, t) = 0

for all 0 E (-71",71") and t E ~ (and ~ is undefined for 0 = 71"). In particular, for each t E ~, ~ (., t) has a jump discontinuity at 0 = 71". In the sense of distributions, we have ~ (0, t) = 0 and

82 f

(8.5)

80 2

(.,

t)

= 1 - 271" ·6n ,

where 6n is the Dirac 6-function centered at 0 distributions,

8f

82 f

= 71".

Hence, in the sense of

82 f

at = 802 + 271" . 6n ~ 802.

EXERCISE

8.7. Prove (8.5).

SOLUTION TO EXERCISE

f ls!

8.7. For any C 2 function cp: 8 1

n 02 cp" (0) dO = 02cp' (O)l 2 2 -n

-

f Ocp' (0) dO ls!

f cp(O)dO- Ocp(O)I:n ls! = f cp(0)dO-271"cp(7I"). ls! =

---+ ~

we have

384

8.

APPLICATIONS OF THE REDUCED DISTANCE

That is,

fJ2

80 2

((P) 2

= 1 - 27r . b7r •

The square torus is abo a nice concrete example for which we can compute the static reduced volume explicitly. EXAMPLE 8.8 (Static reduced volume for square torus). Consider the torus M n ~ lRn / (2Zr with the standard fiat metric [} = dxy + ... + dx~. A fundamental domain for the covering lR n -> M is D ~ (-1, l]n. Let p = 0 be the origin so that d (x, p) = Ixl for x E (-1, 1]n = M. The static reduced volume is!

(8.6) using the change of variables a decre1lliing function of T.

x=

2ft. From (8.6) it is clear that V ([}, T) is

1.2. Static reduced volume and volume ratios. We may think of Vas the static manifold analogue of Perelman's reduced volume for the Ricci fiow (see (8.16) defined later in this chapter), which is defined similarly with d2/4T replaced by the reduced distance function .e. The monotonicity formula (8.3) is analogous to Perelman's monotonicity of the reduced volume (8.28). In the Ricci fiat case, V is the same as Perelman's reduced volume (see Exercise 7.12 and (8.16)). Intuitively the static reduced volume V says something about volume ratios r- n Vol B (p, r) at scales r rv ft. Motivated by these elementary yet a posteriori considerations, we now relate V to the volume ratio r- n Vol B (p, r) under the assumption that the Ricci curvatures of [} are nonnegative. be a complete Riemannian manifold with RC g > O. We Let

(Mn, [})

divide the integral (8.7)

V into two parts:

V ([}, T) = f

udJ1 +

JB(p,r)

f.

udJ1.

JM-B(p,r)

For the first term on the RHS, just using the obvious fact that e- r2 /47 ~ 1, we have

f

udJ1

~

(47rT)-n/2 Vol B (p, r) .

JB(p,r)

lSince the torus is (Ricci) flat. the static metric reduced volume is the same Ricci flow reduced volume of Perelman.

88

the

REDUCED VOLUME OF A STATIC METRIC

1

385

Let A (8) denote the volume of the geodesic (n - l)-sphere of radius 8 centered at p. Since RCg 2: 0, we may apply the Bishop-Gromov volume comparison theorem (see (A.8)), which says that for 8 2: r,

8n-l VolB(p,r) n-l A() A() 8::; r n-l::; n n 8 , r r

(8.8)

to estimate the second term on the RHS of (8.7):

{.

JM-B(p.r)

udp, =

/.00 (47fT)-n/2 e- s2 / 4r A (8) d8 . ,.

< n Vol B (p, r) rn =

100 r

(4 7fT )-n/2 e -s2/4r 8 n- 1d8

n -n/2 Vol B (p, r)

-27f

rn

100 r2

e

-'1/

rJ

n-2 2

d rJ,

4r

where we made the change of variables rJ ~ ~~. Hence we have ) (4 )_n/2 VolB (p,r) V-(-g, T::; 7f Tn /2

-_

(8.9)

+ -2n 7f _n/2

4r

Vol B (p, r) (( r2 )n/2

-47fT

rn

J~n

n -n/21°O -'1/ n-2 d ) + -7f e rJ 2 rJ 2 r2

.

4r

This tells us that a lower bound for ratios of balls. Note that

1= {

1

VolB (p,r) 00 -'1/ n-2 d rn 2. e rJ 2 rJ

7f-n/2e-lxI2 dx

V yields

a lower bound for the volume

= nW n7f- n/2

2

roo e-'1/rJ n~2 drJ,

Jo

where Wn is the volume of the unit Euclidean n-ball. Hence LEMMA 8.9 (The static reduced volume is bounded by volume ratios). If

(Mn, g)

is complete w1.th RCg 2: 0, then for all r

V(g,T)::; VolB(p,r) rn

(8.10)

>0

and

T

> 0,

((~)n/2 +~). 47fT

wn

Thus, for r ::; p, the static reduced volume controls the volume ratio in the sense that

VolB(p,r) -IV- (- 2) n 2: Cn g, P , r

(8.11) where have (8.12)

Cn

~

(20r)

n

+ WIn'

Note also that if for some p E Vol B (p, r) n r

::; K,

M and r > 0 we

386

8.

APPLICATIONS

OF

THE REDUCED DISTANCE

then

-( 2) 0< V [}, K,a r

I_an

:::;

K,

2

(47rt Hence if a < 2/n, then assumption (8.12), as

/2 K,

+-. K,

Wn ---t 0, implies

V ([}, K,a r 2)

---t

O. 1.3. The noncompact case and the asymptotic volume ratio. Consider the case where is complete and noncompact with RC g ~ O. It is natural to believe that the limit as 7 ---t 00 of the static reduced volume V ([}, 7) is related to the limit as r ---t 00 of the volume ratios; as we now show, this is indeed the case. Inequality (8.10) implies

(Mn, [})

V ([},7)

lim

:::; inf VoIB(p,r) r>O

T--+oo

where

wnrn

= AVR([}) ,

AVR(9)~ lim VoIB(p,r)

= lim A(s) wnrn s-->oo nwns n- l as in (6.80). Next we show the opposite inequality. Since, by (8.8), A (r) ~ nWn AVR ([}) r n- 1, r--+oo

we have for all

V ([}, 7) =

7

> 0,

1

00

(47r7)-n/2 e- s2 / 4T A (s) ds

~ nWn AVR ([})

1

00

(47r7)-n/2 e- s2 / 4T sn-1ds

= AVR ([}).

Therefore the limit, as 7 tends to infinity, of the static reduced volume is the asymptotic volume ratio.

(Mn,[})

LEMMA 8.10 (Asymptotic limit of V is AVR). If noncompact Riemannian manifold with RC g ~ 0, then lim

T--+oo

is a complete

V (9, 7) = AVR ([}).

REMARK 8.11. When n = 2, (8.9) says for any r > 0 (8.13)

V- (~g,7 ) :::; Area B 2(p, r) (r2 -4 7rr 7

+ e _ 4-.r2 )

•

Note that the function F (x) ~ x + e- x , x ~ 0, is an increasing function and its minimum value of 1 is attained at x = O. In particular, as in (8.10) the upper bound in (8.13) improves as 7 increases (for fixed r > 0).

2. Reduced volume for Ricci flow In this section (Mn, 9 (7)) , 7 E [0, Tj , will be a complete solution to the backward Ricci flow satisfying the curvature bound IRm (x, 7)1 :::; Co < 00 for (X,7) EM x [O,Tj.

2. REDUCED VOLUME FOR RICCI FLOW

387

2.1. Volumes of geodesic spheres in M. We motivate the definition of the reduced volume by computing the volume of geodesic spheres in the potentially infinite-dimensional manifold introduced in subsection 2.1 of Chapter 7. In particular, let P = (xo, Yo, 0) , T E (0, T) , and

(M, 9)

Bg (p, v'2NT) C M ~ M

X

SN

X

(0, T)

denote the ball centered at P with radius v'2NT with respect to the metric:

9 ~ 9ijdxi dx j + T90 r}

C

TpM

is given by Definition 7.96(iii) and satisfies O(p,O) (r2) c O(p,O) (r1) if r1 Recall from (7.136) that the .L:-exponential map restricted to O(r), .L:Texp: O(p,O) (r)

--+

< r2.

M\.L:Cut(p,o)(r)

is a diffeomorphism. If O(p,O) (r) = TpM for some r > 0, then .L: Cut(p,O) (r) = o and M n is diffeomorphic to Euclidean space. Since .L:Cut(p,O)(r) has measure zero in (M,g(r)), we have by the definition of the .L:-Jacobian and (7.136),

V (r) =

j

(47rr)-n/2 exp [-l (q, r)] dJ.Lg(T) (q)

M\C CutCI',O) (T)

(8.18)

=

r

(47rr)-n/2 e -i(')'v(T),T).L:Jv(r)dx(V),

JnCp,Q)M where 'Yv is the .L:-geodesic emanating from p with limT--+o+ ..Jiiv (r) = V and .L: J v (r) is the .L:-J acobian associated to the .L:-geodesic 'Yv. Here dx is the volume form on TpM with respect to the Euclidean metric g(O,p), .L:T exp(V) = 'Yv (r) , and .L: J v (r) dx (V)

= (.L:T exp)* dJ.Lg(c7' exp(V),T)'

2. REDUCED VOLUME FOR RICCI FLOW

391

We will use the convention CJv (7) ~ 0

for 7 ~ TV.

We can then write the reduced volume as

if (7) =

(8.19)

r

(41IT)-n/2 e-f(-Yv(T),T) CJV (7) dx (V).

iTpM

We compute the evolution of f along a minimal C-geodesic 'YV(7) for 7 < TV, where V E TpM. For q = 'YV(7), 7 E [0, TV), the function f(·,·) is smooth in some small neighborhood of (q,7); hence the following derivatives of e at such (q,7) exist. Recall from (7.78) that

o ::;

73/ 2(R + IXI2) (7) = -K + ~L (q,7), where K = K (7) is the trace Harnack integral defined by (7.75). Thus (8.20) Recall equation (7.88):

af

1

K -a7 = -S/2 27

1 - -f+R, 7

and from (7.54) recall that

\If(q,7) = 'Yv (7) = X (7). Hence the derivative of the reduced distance along a minimal C-geodesic is given by af d d7 [fhv(7),7)] = a7 +\If·X 1 f 2 =-K--+R+IXI 3 2 27 /

7

= _~7-3/2K

(8.21)

2

by (8.20). The following lemma can be viewed as an infinitesimal Bishop-Gromov volume comparison result for the Ricci flow geometry. The striking part is that no curvature assumption is needed. 8.16 (Pointwise monotonicity along C-geodesics). Suppose that (Mn, 9 (7)), 7 E [0, T], is a complete solution to the backward Ricci flow with bounded curvature. (i) For any V E TpM and 0 < 7 < TV, LEMMA

(8.22)

d~

[(47r7)-n/2 e-f('YV(T),T) CJ v (7)] ::; 0,

where equality holds if equality in (7.139) holds.

2. REDUCED VOLUME FOR RICCI FLOW

391

We will use the convention .c J v (T)

~

0 for T ;::: TV.

We can then write the reduced volume as

V(T)

(8.19)

= (

(47rT)-n/2 e -£(W(T),T).cJV(T)dx(V).

JTpM

We compute the evolution of f along a minimal .c-geodesic 'IV (T) for T < TV, where V E TpM. For q = 1'V(T), T E [O,TV), the function f(·,·) is smooth in some small neighborhood of (q, T); hence the following derivatives of f at such (q, T) exist. Recall from (7.78) that

o~

T3 / 2 (R + IXI2) (T) where K

= -K + ~L (q, T),

= K (T) is the trace Harnack integral defined by

(7.75). Thus

(8.20) Recall equation (7.88):

and from (7.54) recall that

'Vf (q, T)

= 'Yv (T) = X (T).

Hence the derivative of the reduced distance along a minimal .c-geodesic is given by d af dT [fhv(T),T)] = aT +'Vf·X 1 f 2 =-K--+R+IXI 2T 3/2

T

= _!T- 3 / 2 K 2

(8.21)

by (8.20). The following lemma can be viewed as an infinitesimal Bishop-Gromov volume comparison result for the Ricci flow geometry. The striking part is that no curvature assumption is needed. 8.16 (Pointwise monotonicity along .c-geodesics). Suppose that (Mn,9 (T)), T E [0, T], is a complete solution to the backward Ricci flow with bounded curvature. (i) For any V E TpM and 0 < T < TV, LEMMA

(8.22)

d~

[(47rT)-n/2 e-£(')'V(T),T).cJV (T)]

~ 0,

where equality holds if equality in (7.139) holds.

8. APPLICATIONS OF THE REDUCED DISTANCE

392

(ii) For any V

E TpM and 0

< 7 < T,

(47l'7)-n/2 e-l('YV(1').1') I:- Jv (7) ::; 7l'-n/2e-IVI~(o,p).

(8.23)

Hence, even for a complete solution on a noncompact manifold, the reduced volume is well defined. PROOF.

(i) Recall from (7.139) that

(d~ logl:-Jv )

(7) ::;

2: - ~7-~K.

From this and (8.21), we compute

d~

[(47l'7)-n/2 e-l(-rv(1'),1') I:-Jv (7)]

= (47l'7)-n/2 e-l(-rv(1'),1') I:- JV (7)

(-!!...27 -

df d7

+ ~ log I:- J V ) d7

::; O.

(ii) It follows from (8.22) that for any 0
o+ V (7) = 1. (ii) The reduced volume is nonmcreasmg:

V (7I) ~ V (72)

(8.25)

for any 0 < 71 < 72 < T, and V(7)::; 1 for any 7 E (O,T). (iii) EquaZzty m (8.25) holds if and only zf (M,g (7)) is isometric to Euclidean space (l~n, 9lE), regarded as the Gaussian soliton. PROOF.

(i) From equation (7.131) it follows that lim

1'-->0

n(pO)

'

(7) = TpMn.

2. REDUCED VOLUME FOR RlCCI FLOW

393

Since 0Cp,O) (Tl) = 0 (Tl) ::J 0 (T2) for Tl < T2, we have limT-+o+ XnCT) = 1, where Xn denotes the characteristic function of the set O. We compute lim T-+O+

V (T) =

lim ( (47rT)-n/2 e-e(rVCT),T) C Jv (T) dx (V) JnCT)

T-+O+

= (

lim [xn(T) (47rT)-n/2 e-e C'YVCT),T) C Jv (T)] dx (V)

JTpM T-+O+

= (

1 . 7r-n/2e-1V12 dx (V)

= 1,

JTpM

where we used (8.24).

(ii) From (8.22), we have for any 0 < Tl < T2 and V

E

0(T2),

T;n/2e-eC'YVCTl),TI)CJv (Td 2: T;n/2e-eC'YVCT2),T2)CJv (T2) , so

V (T2) = (

(47rT2)-n/2e-lC'YV(T2),T2) C Jv (T2) dx (V)

JnCT2) :::; ( (47rTl)-n/2e-lC'YVCT1),Tl) C Jv (Tl) dx (V) Jnb)

:::; (

(47rTl)-n/2e-l('YV(TI),Tl) CJv (Tl) dx (V)

In(Tl)

= V (Tl) , where we used 0 (Tl) ::J 0 (T2)' Note that from (8.25) and (i).

V (T) :::;

1 for any T > 0 follows

(iii) We prove this statement in two steps by first showing that g(T) is a shrinking gradient Ricci soliton and then showing that (M, 9 (T)) is Euclidean space. If V (Td = V (T2) for a pair of times 0 < Tl < T2, then for T E (Tl' T2) and V E O(T), we have that equality in (8.22) holds:

d~ [(47rT)-n/2 e-£C'YVCT),T) CJv (T)]

= 0,

which, by the proof of Lemma 8.16, implies that we have equality in (7.139). Hence, by (7.140), we get (8.26)

Rc (-Tv (T), T)

+ (Hesse) (TV(T), T) = 9 (-Tv (T), T) 2T

for all V E O(T) and T E [T1,T2J, and where e is Coo at (TV(T),T) for all such (V, T). Since V (Tl) = V (T2) , we have 0 (Tl) = 0 (T2)' Suppose there exists VI E TpM such that Tvl :::; Tl. Since 0 (Tl) -# 0, there exists V2 E TpM such that Tv2 > Tl. Since the function V I-t Tv is a continuous function, there exists V3 E TpM such that TV3 E (Tl' T2) . Thus V3 E 0 (Tl) - 0 (T2) , which is a contradiction. Therefore, for every V E TpM, we have Tv> Tl, so that 0 (Tl) = TpM (and hence 0 (T) = TpM

394

8.

APPLICATIONS OF THE REDUCED DISTANCE

for all 7 E [TI,72])' This implies f. is Coo on M x [TI,72] and (8.26) holds on M x [71,72], Thus 9 (7) is a shrinking gradient Ricci soliton. Given 70 E [TI,72] , by Proposition 1.7, the shrinking gradient Ricci soli9 (70) , \l f. (70) , - ~) may be put into a canonical timeton structure dependent form (1.11) defined for all t < 70,

(M,

g(t) = f (t) r.p(t)*g (70) , 70 where g(t) is a solution of the Ricci flow, by (1.10), i.e., f (t) = 70 - t (c = - ';0)' and r.p(t) is a I-parameter family of diffeomorphisms with r.p (0) = idM . By the uniqueness of complete solutions of the Ricci flow with bounded curvature (see Chen and Zhu [82]) and since g(O) = 9 (70) , we have 9 (7) =

9 (70 - 7),

so that (8.27) Since IRm [g (7)]1 :::; Co < (8.27) we have

00

for 7 E [0, T] (we just use this for 7 small), by

sup IRm [g (70)]1 = sup IRm [r.p(70 - 7)*g (70)]1 M M 7

7

:::; - sup IRm [g (7)]1:::; Co70 M 70 for all 7 E (0,70]' Hence IRm [g (70)]1 == 0. Since 0(70) = TpM, M is diffeomorphic to jRn. Part (iii) follows since a flat shrinking gradient Ricci soliton on jRn must be the Gaussian soliton. 0 REMARK 8.18. (i) The Riemannian analogue of Corollary 8.17(i) is lim Vol B (p, r) = 1. wnrn

r-+O

(ii) Note that for the shrinking gradient Ricci soliton 9 (7) in subsection 7.3 of Chapter 7, the metric 9 (0) is not well-defined. The monotonicity of the reduced volume can be easily generalized to the following. For any fixed measurable subset A c TpM, we can define D(A, 7) to be the set of vectors V E A such that TV > 7, i.e.,

D(A, 7)

~

{V

E

A: TV > 7} = An 0(7).

It is clear that D(A,7) satisfies D(A,72)

c D(A,71) if 72 > 71.

COROLLARY 8.19 (.c-relative volume comparison). Suppose (Mn, g(7)), 7 E [0, T], is a complete smooth solution to the backward Ricci flow with the curvature bound IRm (x, 7)1 :::; Co < 00 for (X,7) E M x [0, T]. Define for any 7 E (0, T) and any measurable subset A c TpM, VA

(7)

~

{ } £:,. exp(D(A,r))

(41l'7)-n/2 exp [-f. (q, 7)] dJ.Lg(r) (q).

394

8.

APPLICATIONS OF THE REDUCED DISTANCE

for all l' E h,T2])' This implies f is Coo on M x h, T2J and (8.26) holds on M x [1'1, T2J . Thus 9 (1') is a shrinking gradient Ricci soliton. Given TO E h, T2J , by Proposition 1.7, the shrinking gradient Ricci soliton structure 9 (TO) , V f (TO) , - r~) may be put into a canonical timedependent form (1.11) defined for all t < TO,

(M,

g(t) = f (t) t.p(t)*g (TO), TO where g(t) is a solution of the Ricci flow, by (1.10), i.e., f (t) = TO - t (c = - ';0)' and t.p(t) is a I-parameter family of diffeomorphisms with t.p (0) = idM . By the uniqueness of complete solutions of the Ricci flow with bounded curvature (see Chen and Zhu [82]) and since g(O) = 9 (TO), we have

9 (TO - 1'),

9 (1') =

so that

t.p(TO - T)*g (TO) = TO 9 (1')

(8.27)

for

l'

E (0, TOJ.

l'

Since IRm [g (T)JI :::; Co < (8.27) we have

00

for

sup IRm [g (To)JI M

l'

E [0, TJ (we just use this for

l'

small), by

= sup IRm [t.p(TO - T)*g (To)JI M

l'

l'

:::; - sup IRm [g (1')]1 :::; Co-

TO

TO O. Since 0(1'0) = TpM, M is

M

for all l' E (0, TOJ. Hence IRm[g(To)JI = diffeomorphic to ]Rn. Part (iii) follows since a flat shrinking gradient Ricci soliton on ]Rn must be the Gaussian soliton. 0 REMARK 8.18. (i) The Riemannian analogue of Corollary 8.17(i) is lim VolB (p, r)

= 1.

wnrn

r-.O

(ii) Note that for the shrinking gradient Ricci soliton 9 (1') in subsection 7.3 of Chapter 7, the metric 9 (0) is not well-defined. The monotonicity of the reduced volume can be easily generalized to the following. For any fixed measurable subset A c TpM, we can define D(A, 1') to be the set of vectors V E A such that TV > 1', i.e.,

D(A, 1')

~

{V

E

A: TV > T} = An 0(1').

It is clear that D(A,T) satisfies D(A,T2)

c D(A, 1'1) if 1'2 > 1'1.

COROLLARY 8.19 (C-relative volume comparison). Suppose (Mn,g(1')), l' E [0, TJ, is a complete smooth solution to the backward Ricci flow with the curvature bound IRm (x, 1')1 :::; Co < 00 for (x,1') E M x [0, TJ. Define for any l' E (0, T) and any measurable subset A c TpM, VA

(1')

~

(471'T)-n/2 exp [-f (q, T)J dJ.tg(r) (q).

{

J.c

r

exp(D(A,r))

2.

Then for any

71

REDUCED VOLUME FOR RICCI FLOW

395

< 72,

PROOF. By the definition of the C-Jacobian we know that for any L1 function f on M

r

JL,. exp(D(A,r))

f(y) dJ.tg(r)(Y)

=

r

JD(A,r)

f(C r exp(V))C JV(7) dx(V).

(We have used this change of variables formula for A = TpM in previous sections.) We have

VA

(72) =

S

r

JD(A,"T2)

r r

JD(A,"T2)

S

(47r72)-n/2e-l(-YV("T2),"T2)CJv(72)dx(V) (47r7I)-n/2 e -l(w(rt},TJ.) CJv (71)

dx (V)

(47r71)-n/2e-l(w(71),rl) C Jv (7I)

dx (V)

JD(A,71)

= VA (71).

o The above can be thought of as a relative volume comparison theorem for the Ricci flow. This is along the lines of the generalization by Shunhui Zhu in [384] (see also Theorem 1.135 in [111] for example). 2.4. Monotonicity of reduced volume revisited. Now we give another proof of the monotonicity of the reduced volume without using the C-Jacobian. Recall that under the evolution equations

a = 2I4j,

a7 gij with

7

a

2 n a7f - 6f + IVfl - R+ 27

= 0,

> 0, we have

d~ 1M 7- n/ 2e- f dJ.tg = 0. In comparison, by (7.146), the reduced distance .e is a subsolution to the above equation for f. We use this fact to give another proof of the monotonicity of reduced volume. THEOREM 8.20 (Monotonicity of the reduced volume: second proof). Let (Mn, g (7)) , 7 E [0, T], be a complete solution to the backward Ricci flow satisfying the curvature bound IRm (x, 7)1 Co < 00 for (x, 7) EM x [0, T]. Then for any 7 E (0, T), the reduced volume V (7) is differentiable and nonincreasing:

s

(8.28)

396

8. APPLICATIONS OF THE REDUCED DISTANCE

PROOF. We justify the differentiation under the integral sign in equality (8.17). Consider the difference quotient for the reduced volume integrand:

Note that

if (T) = (47r)-n/2

(8.29)

dd

T

lim ( 0. There exists a constant Cl > 0, depending only on rO, n, T, and sUPMx [0,T/2] RCg(t), and there exists qo E M such that

£ (q, To) :::; Cl

for every q E Bg(o) (qO, ro) .

(ii) (if lower bound) Suppose there exist rl > 0 and VI > Volg(o) Bg(o) (w, rl) ;::::

°such that

VI

for all w EM. Then there exists a constant C2 > 0, depending only on rl, VI, n, T, and supM x [O,T /2] Rc g( t), such that

if (To) ;:::: C 2 · PROOF. (i) By Lemma 7.50 there exists qo EM such that 3

£ (qO, To - T/2) = min f (q, To - T/2) :::; ~2. qEM

For any q E Bg(o) (qO, ro) , let f3 : [To - T /2, Tol ~ M be a constant speed minimal geodesic from qo to q with respect to 9 (0) . Defining Co

3This corresponds to time t

~

sup RCg(t), Mx[0,T/2]

= T /2.

3.

NO LOCAL COLLAPSING VIA REDUCED VOLUME MONOTONICITY

399

3. A weakened no local collapsing theorem via the monotonicity of the reduced volume In this section, (Mn,g (t)) , t E [0, T), shall denote a complete solution to the Ricci flow with T < 00 and SUPXEM, tE[O,td IRmg (x, t)1 < 00 for any tl < T (Le., the curvatures are bounded, but possibly not uniformly as t --+ T, as in the case of a singular solution). We fix a time To E (t, T) and a basepoint Po EM. Let g(1')~g(To-1').

Then (Mn, 9 (1')), l' E [0, To], is a solution to the backward Ricci flow with initial metric 9 (0) = 9 (To) and bounded sectional curvature. Let C b) denote the C-Iength of a curve ,,(, let L : M x (0, To] --+ 1R denote the L-distance, let f. : M x (0, To] --+ 1R denote the reduced distance, and let V : (0, To] --+ (0,00) denote the reduced volume, all with respect to 9 (1') and the basepoint (po, 0) .

3.1. A bound of the reduced distance. The following lower bound of the reduced volume will be used in the proof of the Weakened No Local Collapsing Theorem 8.26. LEMMA 8.22 (Lower bound for

V at initial time).

(i) (f. upper bound) Fix an arbitrary ro > O. There exists a constant Cl > 0, depending only on ro, n, T, and SUPM x [0,T/2] RCg(t), and there exists qo E M such that

f. (q, To) ~ C1

for every q E Bg(o) (qO, ro) .

(ii) (V lower bound) Suppose there exist rl > 0 and VI > Volg(o) Bg(o) (w,rl) 2:

°such that

VI

for all w EM. Then there exists a constant C2 > 0, depending only on rI, VI, n, T, and SUPM x [0,T/2] RCg(t), such that

V (To) 2: C2· PROOF. (i) By Lemma 7.50 there exists qo EM such that 3

f.(qO, To - T /2) = min f. (q, To - T /2) ~ ~. qEM 2 For any q E Bg(o) (qO, ro), let f3 : [To - T /2, To] --+ M be a constant speed minimal geodesic from qo to q with respect to 9 (0). Defining

Co

~

sup Mx[0,T/2]

3This corresponds to time t

= T /2.

RCg(t),

400

8. APPLICATIONS OF THE REDUCED DISTANCE

we have 1·1;(7') ::; eCoT I'I;(TO) = eCOT 1'1~(0) acting on vector fields for [To - T /2, To] . We can estimate £, (13) as follows: £,

E

(;3)

::; {TO .jT (Rg(7') (13 (7)) lTo-T/2

d

+ 1 12 ) 7

g(7')

d7

::; -2 (T~/2 - (To - T /2)3/2) sup Rg(t) 3 Mx[0,T/2] =

7

~ (T~/2 _ (To _ T /2)3/2)

(

+ eCoT iTO

sup Rg(t) Mx[0,T/2]

To-T/2

.jT Id1312 d 7

g(O)

d7

+ 4eCoT (dg(o) ~;' qo) )2)

(G

COT ::;3~T3/2 no+ 4e T2 r 5) . Let a : [0, To - T /2] - t M be a minimal £'-geodesic with a (0) a (To - T /2) = qo. Then

= Po

and

= 2JTo - T/2· f (qO, To - T/2) ::; nJT/2.

£, (a)

Consider the concatenated path: 'Y (7)

== (a '-' 13)(7) = { a (7) if t

E [0, To - T/2]' 13(7) iftE[To-T/2,To].

.

This path is well defined and piecewise smooth. We have 1 f (q, To) ::; 2JTo£' (T)

::;

1

= 2JTO [£, (a) + £, (;3)]

~ [nJT/2 + ~ T 3/ 2 ( nCo + 4e~~ r5) ]

~ Cl (ro, n, T, Mx[0,T/2] sup Rc get») . (ii) Choosing

ro = rl in (i), we have

V (To) = 1M (41l'To)-n/2 e-f.(q,TO)d/Lg(o) (q) 2: {

1Bg(o) (qo,rI)

2: {

1B9

(0)

(41l'To)-n/2 e-f.(q,TO)d/Lg(o) (q)

(41l'To) -n/2 e -Cl d/Lg(O) (q) (qO,rl)

2: VI (41l'T)-n/2 e- Cl

~ C2 (Vl,r 1,n,T, Mx[0,T/2] sup RCg(t»). o

3.

NO LOCAL COLLAPSING VIA REDUCED VOLUME MONOTONICITY

401

3.2. The weakened no local collapsing theorem. In Lemma 8.9 we saw how the reduced volume of a static metric bounds the volume ratios of balls from below. Similarly, the reduced volume monotonicity for solutions of the Ricci flow enables one to prove a weakened form of the no local collapsing theorem, which we first encountered in Chapter 6 using entropy monotonicity. DEFINITION 8.23 (Strongly K-collapsed). Let K > 0 be a constant. We say that a solution (Mn,g(t)), t E [O,T), to the Ricci flow is strongly K-collapsed at (qO, to) EM x (0, T) at scale r > 0 if (1) (curvature bound in a parabolic cylinder) IRmg (x, t)1 ::; for all x E Bg(to) (qO, r) and t E [max {to - r2, O} ,to] and (2) (volume oj ball is K-collapsed)

fx

Volg(to)

Bg(to) (qO,

r)
- -r2-

and

404

8. APPLICATIONS OF THE REDUCED DISTANCE

along

IV,

and hence we can use (7.48) to get 4

Therefore by Holder's inequality,

(t

ltv (T)lg.(T) dT) 2

~ loTI T- 1/2dT loTI v'Tliv (T)I;.(T) dT

~ 2v'cr loTI T- 1/ 2W (T)I;.(T) dT ~ (2v'cr)2 (e6(n-l)Cc- 1 / 2 +

We get the last inequality by choosing

C~C

(e6(n-l)c -

12(n-1)2

CI ~ ~

1))

such that

4(e6(n-l)ft l/2 + 12 (n~t2- 1)2 (e6(n-l)f _ 1)) 0, Lemma 8.35, after parabolic rescaling 9 (T) by Ti, yields the curvature bound 5- 1 = 5(n,E,A)-1 for gri(O) on B gTi (l) (qri ,VE- 1) x [A-1,A]. Applying Lemma 8.35 with E = 1 and A = 2, we obtain

For any E

IRmgTi (q, 1)1 ::; 5- 1 = 5 (n, 1,2)-1

for q E B gTi (l) (qri' 1).

Since g(O) is K-noncollapsed on all scales, we have gri(O) is K-noncollapsed on B gTi (l) (qri' 1) and the injectivity radius estimate inj gTi(l) (qr,) ~ 51 (n, K). Now we can apply Hamilton's Cheeger-Gromov-type Compactness Theorem 3.10 to the sequence of solutions gri (0) to the backward Ricci flow to get

(Mn,gri(O),qrJ ~ (M~,goo(O),qoo)

for 0 E [A-1,A].

The limit is a complete solution to the backward Ricci flow. Since each gri (0) satisfies the trace Harnack estimate, the limit goo (0) satisfies the trace Harnack estimate. Note that goo(O) is K-noncollapsed on all scales, has nonnegative curvature operator, and satisfies inj goo(l) (qoo) ~ 51 (n, K) . To finish the proof of Theorem 8.32, we need to show that for each 0, goo (0) is a nonflat shrinking gradient Ricci soliton.

4.3. The limit of reduced length fdq,O). Let fdq, 0) ~ fgTi (q, 0), [A -1, A] , denote the reduced distance of the solution gr; (0) with respect to the basepoint Po. LEMMA 8.36 (Limit of the reduced distance). (i) The limit

oE

exists in the Cheeger-Gromov sense on Moo x [A-I, A].5 (ii) The limitfoo(q,e) is a locally Lipschitz junction on Moo x [A-1,A] and V goo((J)foo(q, 0) and a~'f (q, e) exist a.e. on Moo x [A-I, A]. (i) Suppose i oexPgoo (l,q) (x)

,0)

by Theorem D6.2.7 in [202]. Hence

converges to V'goo(1,q)foo (exP9oo (l,q) (x)

,0)

Bgoo (l,q) (0, l6 injgoo(l) (q)) and V' gTi«(J)fi (cJ>i eXPgoc (l,q) (x) ,0) converges to V' goo «(J)foo (exP 9oo (l,q) (x) ,0) a.e. on B goo (l,q) (0, l6 injgoo(l) (q)) .

a.e. on

0

Because q is an arbitrarily point on Moo, we have proved that

(8.49)

lV'gT/il;Ti«(J) (cJ>i (-) ,0)

-t

lV'gocfool!oo«(J) (-,0)

a.e. on Moo for each 0 E (A-I, A) . From 2~+1V' gTi«(J)fiI2_RgTi«(J)+j = 0 and (8.49) we know ~ (cJ>i (-),0) converges a.e. on Moo for each 0 E (A-I, A) . Since fi (cJ>i (-) ,0) converges to foo (·,0) uniformly, we conclude

Bfi ( () ) BO cJ>i· ,0

-t

Bfoo ( ) BO ·,0

a.e. on Moo for each 0 E (A-I, A) .

°

(ii) For any smooth compactly supported function cp(q, 0) ~ on Moo x (A -1, A), for i large enough we can extend cp (cJ>; 1 (ql) , 0) by 0 to a smooth function, still denoted by cp (cJ>;1 (ql) ,0) , which has compact support on M x (A-I, A). Using the Lipschitz test function e- fi (qllJ)CP (cJ>;1 (ql) ,0) in (7.146), we get

and

i~l 1M (~~ +

V'gT/i· V'gTiCP - RgTi

+ ~)

x e-fi(~i(q),(J)cp (q, 0) dJ.L~:gTi«(J) (q) Taking the limit i

- t 00,

we obtain (8.48). The lemma is proved.

dO ~

O. 0

4. BACKWARD LIMIT OF ANCIENT I\:-SOLUTION IS A SHRINKER

413

4.4. The limit of the reduced volume. Note that the limit goo(O) is defined for 0 E [A-I, A] . Instead of considering the reduced volume of this limit, we define the function

which will play the role of the reduced volume; formally this is the reduced volume using the limit function foo. The fact that Voo(O) is finite follows from the following lemma. LEMMA

8.38.

(i) We have

(ii) Voo(O) is a constant contained in (0,1) . (iii) For any 1/J(O) which has compact support in (A-I, A), fA f (47rO)-n/2 e-£oo (q,8) 1/J' (O)dJ1,goo (8) (q)dO JA-l JMoo

(8.50)

= fA Voo (O)1/J'(O)dO = 0, JA-l

where 1/J' (0) = d1/J / dO. (i) Let Ti be a subsequence such that both gTi and fi(q, 0) converge. By (8.23), we have PROOF.

(47rTio)-n/2e-l'(')'V(Ti8),Ti8) .c Jv (TiO) ::; (47r) -n/2 e -lvl~(o).

Then by Lebesgue's dominated convergence theorem and (8.40), we have .lim V(TiO) t-+oo

= f lim ((47rTio)-n/2 e- l (')'V(Ti 8),T;8) .cJv (TiO) dJ1,g(o) (V))

iJRn t-+oo

= f JMoo

(47rO)-n/2.lim (Ti -n/2 e - l i ("fy"fiv(8),8) dJ1,g(Ti 8)) t-+oo

= f (47rO)-n/2 e- loo (q,8) .lim dJ1,gT(8)

JJRn

= f

t-+oo

(47rO)-n/2

e-£oo (q,8)dJ1,goo (8)

•

(q)

= Voo(O).

JMoo

Since

V(T)

is a monotone decreasing function, we have

(8.51) In particular, Voo(O) is independent of O.

414

8. APPLICATIONS OF THE REDUCED DISTANCE

(ii) Note that V(oo) < 1 follows from Corollary 8. 17(iii). To see V(oo) 0, we compute using (8.46) and 0 = 1 that V(Ti)

>

= f (47r)-n/2 e- li (q,1)dJ.Lgd1)(q) 1M

'

~ f

IfB gT,.

(47r)-n/2 e- li (q,1)dJ.L (1)

T

(1)(q)

g,

(qTi ,e:- 1/2 )

~ (47rA)-n/2 e-o- 1 VoIgT, (1) B gTi (l) (qTi,c- 1/ 2)

= (471' A) -n/2 e _0-1 . Ti-n/2

U

I g(Td B g(Ti) ( qTi' Ti1/2 C-1/2)

vO

•

By Lemma 8.35, we have R (q, Ti) ~ 0~1 on the ball Bgh) (qTi' Til/2c-1/2). It follows from g(T) being ~-noncollapsed on all scales (choosing the scale r = min { Ti1/ 2C 1/2, Ti1/ 201/ 2} ) that I B g(Ti) ( qTi' Ti1/2 C-1/2) Ti-n/2'\T vO g(Td

Hence

~ ~.

(mIll . { c -1/2 , ud/2})n .

V(Ti) ~ (47rA)-n/2e-O-l~. (min {c- 1/ 2,01/2})n

and V(oo)

> O.

(iii) For any 1jJ(O) which has compact support in (A-I, A), we compute

(47rO)-n/2 e- loo (q,(J) 1jJ' (O)dJ.Lg oo ((J) (q)dO j A f A-l1Moo

= jA Voo(O)1jJ'(O)dO = Voo(O) fA 1jJ'(O)dO = O. A-I

lA-I

o 4.5. The limit is a shrinking gradient soliton. Let 1jJ(O) ~ 0 be a smooth function only of 0 with compact support in (A-I, A). Applying Stokes's theorem for Lipschitz functions, we get

L~1 iMoo (47rO)-~ (a~; - Rgoo + ;) e- loo (q,(J)1jJ (0) dJ.Lgoo((J) (q) dO = jA f A-11M

=0,

(47rO)-~ e- loo (q,(J)1jJ' (0) dJ.Lgoo((J) (q) dO

4. BACKWARD LIMIT OF ANCIENT It-SOLUTION IS A SHRINKER

415

where 'l/J(O) 2: 0 is an arbitrarily smooth function with compact support in (A-I, A). For any smooth compactly supported 6t + 1HI (M; Z2) I, where H l (M;Z2) is the first homology group of M with Z2 coefficients, then any n disjoint essential 2-spheres contain a parallel pair. Here is the basic argument. Suppose {SI, ... ,Sn} is a collection of n essential disjoint 2-spheres in M. By isotopy and topological surgeries, one may find a new collection of n essential disjoint 2-spheres, still denoted by {SI, ... , Sn}, that are in 'nice position' with respect to the triangulation. Here, 'nice position' means that the intersection of each 2-sphere with each tetrahedron consists of a disjoint union of geometric triangles and quadrilaterals. These are called normal surfaces. There are only seven normally isotopic triangles and quadrilaterals in a tetrahedron. This shows that inside a tetrahedron (73, all but at most six components of (73 \ (SI U ... U Sn) are parallel regions. It follows from this fact by a simple computation that there are two parallel 2-spheres if n > 6t + 1HI (M; Z2) I· 1.3. Irreducible 3-manifolds. Irreducible 3-manifolds joined in connected sums may be regarded as building blocks for 3-manifolds. Most familiar 3-manifolds are irreducible. For instance, the complement of a knot in S3 is irreducible by Alexander's theorem. Also, if a covering space N of a 3-manifold M is irreducible, then M is irreducible. This is due to the fact that 2-spheres are simply connected. Thus any 2-sphere in M can be lifted to a 2-sphere in N. Now using the irreducibility of N, one produces a 3-ball in N bounding the lifted 2-sphere. Using the Brouwer fixed point theorem, one then shows that the 3-ball is mapped injectively to M by the covering map. This proves M is irreducible. In particular, if the universal cover of a 3-manifold is R3, then the manifold is irreducible. This shows, for instance, that all flat (e.g., SI x SI X SI) and all hyperbolic 3-manifolds are irreducible. A deep result of Meeks and Yau [263] shows that the converse is also true. Namely, if a manifold is irreducible, then all covering spaces of it are irreducible. In [264] Milnor proved that the connected sum decomposition of a compact 3-manifold is unique up to self-homeomorphism of the 3-manifold. But the decomposition in Kneser's theorem is in general not unique up to isotopy of the 3-manifold. This is due to the action of the diffeomorphism group of the 3-manifold on the decomposition. For instance, the manifold (S2 x SI )#(S2 X SI) has many nonisotopic essential separating 2-spheres, due to the large diffeomorphism group of the manifold. 2. Incompressible surfaces and the geometrization conjecture 2.1. Incompressible surfaces and Haken manifolds. The success of the study of 2-spheres in 3-manifolds prompted people to look for more general surfaces. Evidently, surfaces that can be contained inside a coordinate chart are not going to be interesting. Haken introduced the following

436

9. BASIC TOPOLOGY OF 3-MANIFOLDS

important concept. A compact, connected, properly embedded surface F in a 3-manifold M is said to be incompressible if one of the following conditions holds:

(1) F i= 52 or B2 and the inclusion map induces an injective homomorphism in the fundamental groupj or (2) F = 52 is an essential 2-spherej or (3) F = B2 and 8F is not null homotopic in 8M. An orientable Haken 3-manifold is a compact and irreducible 3-manifold M that admits a two-sided incompressible surface. This definition is the same as saying that M is a compact, orient able , and irreducible manifold which contains an incompressible surface other than lRJID2 • The reason is that if a compact, irreducible, orientable 3-manifold M contains JRJP>2 as an incompressible surface, then M = JRJP>3. Also, if a compact surface is incompressible in M and is one-sided, then the boundary of a regular neighborhood of the surface is a two-sided incompressible surface. Furthermore, since we assume the 3-manifold is orientable, two-sided surfaces are the same as orient able surfaces. Haken manifolds constitute a huge portion of all of 3-manifolds. For instance, if a compact orientable 3-manifold M is irreducible and has nonempty boundary, then M is Haken. Also, if a closed, irreducible 3-manifold has positive first Betti number, it is Haken. The homeomorphism classification of Haken manifolds is considered to be solved. It is due to the deep work of F. Waldhausen [363] in 1968. Among the many results he proved, the following stands out as one of the most striking. THEOREM 9.2 (Waldhausen). Two homotopically equivalent closed Haken manifolds are homeomorphic. 2.2. Torus decompositions and the geometrization conjecture. A Seifert 3-manifold (also called a Seifert space) is a compact 3-manifold admitting a foliation whose leaves are 51. Although this was not the original definition by Seifert in 1931, subsequent work of Epstein [136] shows that this simpler definition is equivalent to Seifert's original formulation. If a compact 3-manifold admits an 51 action without global fixed points (i.e., no point is fixed by all elements in 51), then the manifold is a Seifert space. Seifert 3-manifolds have been classified. In particular, there exist Seifert manifolds which are irreducible, non-Haken 3-manifolds and have infinite fundamental group. In 1976, Thurston constructed closed hyperbolic 3manifolds that are not Haken. Suppose M is a closed, irreducible, and orient able 3-manifold. A natural step after the connected sum decomposition is to look for incompressible tori. This is called the torus decomposition. A compact, irreducible 3-manifold is called geometrically atoroidal if every incompressible torus is isotopic to a boundary component. (If the manifold is closed, this simply means that there are no incompressible tori.) The torus decomposition theorem of Jaco

2.

INCOMPRESSIBLE SURFACES AND GEOMETRIZATION CONJECTURE

437

and Shalen [224] and Johannson [225] says the following. (For simplicity, we state the result for closed manifolds only.) THEOREM 9.3 (Jaco and Shalen, Johannson). Suppose M is a closed, orientable, and irreducible 3-manifold. Then there exzsts a possibly empty disjoint union of incompressible tori in M that decomposes M into pieces which are either Seifert 3-manifolds or geometrically atoroidal 3-manifolds. Furthermore, the minimal such collection of tori is unique up to isotopies. Thurston's work on the geometrization of 3-manifolds addresses the geometries underlying the Seifert pieces and the geometrically atoroidal pieces. Thurston proved that if N is a non-Seifert, geometrically atoroidal manifold appearing in the decomposition above (so that it has nonempty boundary), then the interior of the manifold N admits a complete hyperbolic metric of finite volume. Also, it is proved that the interior of any compact Seifert 3-manifold admits a complete, locally homogeneous Riemannian metric of finite volume. Thus, the remaining issue is the geometry of a closed, irreducible, geometrically atoroidal 3-manifold. The geometrization conjecture of Thurston for a closed, irreducible, orientable 3-manifold M states that there is an embedding of a (possibly empty) disjoint union of incompressible tori in M such that every component of the complement is either a Seifert space or else admits a complete Riemannian metric of constant curvature and finite volume. Using Thurston's theorem and the torus decomposition theorem of Jaco and Shalen and of Johannson, one can reduce the geometrization conjecture to the following form. Suppose M is a closed, irreducible, orientable 3manifold without any incompressible tori. Conjecture I: If the fundamental group of M is infinite and does not contain any subgroup isomorphic to Z EEl Z, then M admits a hyperbolic metric. Conjecture II: If the fundamental group of M contains a subgroup isomorphic to Z EEl Z, then M is a Seifert space. Conjecture III: If the fundamental group of M is finite, then M admits a spherical (constant positive curvature) metric. Topologists have made great progress toward resolving these conjectures. First of all, if the manifold is Haken, Conjecture I was shown to be valid by Thurston. (See also McMullen [262] and Otal [294].) Conjecture II for Haken manifolds was shown to be valid by the work of Gordon and Heil [158], Johannson [225]' Jaco and Shalen [224], Scott [318] and Waldhausen [364]. Conjecture II for non-Haken manifolds was solved affirmatively by Casson and Jungreis [61] and Gabai [149] in 1992. Furthermore, Gabai, Myerhoff, and Thurston [150] proved that if a closed, irreducible 3-manifold M is homotopic to a hyperbolic 3-manifold N, then M is homeomorphic to N. This gives evidence that Conjecture I holds. Note that Conjecture III implies the Poincare conjecture. Indeed, if M is a simply-connected 3-manifold, then by Kneser's theorem, we may

438

9 BASIC TOPOLOGY OF 3-MANIFOLDS

assume that M is irreducible. Since M has a trivial fundamental group, it is geometrically atoroidal. (By definition, if a manifold contains an incompressible torus, its fundamental group must contain Z EB Z, the fundamental group of the torus.) Thus, if Conjecture III holds, M admits a metric of constant positive sectional curvature. Thus M must be S3. 2.3. Examples of 3-manifolds and their geometries. There are eight locally homogeneous Riemannian geometries in dimension 3. Besides the well-known constant curvature metrics S3, JR3, and '}-£3, the rest of the five geometries are given by the standard metrics on S2 xJR, '}-£2 xJR, SL(2, JR), nil, and sol. (The Ricci flow of homogeneous metrics on S3 is discussed in Section 5 of Chapter 1 in Volume One.) H~e is a description of the three non product geometries. The geometry SL(2, JR) may be regarded as the universal cover of the Lie group SL(2, JR) with a metric invariant under left multiplication. The nilpotent 3dimensional Lie group nil is the Heisenberg group of strictly upper-triangular 3 x 3 matrices. (The Ricci flow on nil is discussed in Section 7 of Chapter 1 in Volume One.) The geometry sol is that of the 3-dimensional solvable Lie group defined as the semi-direct product of JR2 with JR, where the action of

t E JR on JR2 is given by the matrix

(~ e~t).

(The Ricci flow on sol is

discussed in Section 7 of Chapter 1 in Volume One.) A closed, orient able manifold with S2 x JR geometry must be either S2 x Sl or lRJP>3#JRF. It is interesting to note that JRF#JRIP3 is the only connected-sum 3-manifold admitting a locally homogeneous metric. The product of a closed surface of genus at least two with the circle admits an '}-£2 x Sl metric, i.e., the product of their standard metrics given by the uniformization theorem. The unit tangen!..bundle over a surface of negative Euler characteristic is a 3-manifold with SL(2, JR) geometry. More generally, any circle bundle over a surface of negative Euler characteristic admits the SL(2, JR) geometry if the Chern class of the bundle is nonzero. A circle bundle over the torus with nonzero Chern class is a 3-manifold with nil geometry. A torus bundle over the circle whose monodromy is a linear map with distinct real eigenvalues has sol geometry. It can be shown that any closed 3-manifold with one of these five geometries is finitely covered by one of the examples just mentioned. For more information, see the excellent reference [319]. A closed 3-manifold with sol geometry is not a Seifert 3-manifold. It admits a nontrivial torus decomposition coming from the torus fiber. On the other hand, any manifold admitting one of the six geometries S3, JR3, S2 X JR, '}-£2 X JR, SL(2, JR), or nil is a Seifert manifold. We will say a compact 3-manifold M is a topological graph manifold if it admits a torus decomposition such that each complementary piece is a Seifert manifold. (This is a standard definition in low-dimensional topology.)

3. DECOMPOSITION THEOREMS AND THE RICCI FLOW

439

In particular, this implies that the boundary of M is a possibly empty collection of tori. Graph manifolds appear in the work of Cheeger and Gromov as those 3-manifolds admitting an oF-structure. Note that the definition of graph manifold that appears in Cheeger and Gromov's work [74] is slightly different from the one above. To be more precise, a graph manifold in the sense of Cheeger-Gromov is a closed 3-manifold admitting a decomposition by (not necessarily incompressible) tori such that each complementary piece is a Seifert space. It can be shown that if M is a graph manifold in the sense of Cheeger-Gromov, then it is either a topological graph manifold or else a connected sum of topological graph manifolds with 8 2 x 8 1 factors and lens spaces. At any rate, there are no fake 3-spheres or fake 3-balls embedded inside a Cheeger-Gromov graph manifold. 3. Decomposition theorems and the Ricci flow Recall that a solution (M3, g( t)) , t E [0, T), to the Ricci flow is said to develop a singularity at time T E (0,00) if the norm of the Riemann curvature tensor becomes infinite at some point or points of the manifold as t / T. (See Corollary 7.2 of Volume One.) A typical situation in which a finite time singularity develops is the neckpinch. It is important to note that the formation of neckpinch singularities may be triggered more by the (local) nonlinearity of the Ricci flow PDE than by the (global) topology of the underlying manifold. In any case, here is a heuristic description. (For precise statements, see Section 5 of Chapter 2 in Volume One, as well as the recent papers of Angenent and one of the authors [7, 8].) Suppose a 3-manifold M contains a separating 2-sphere. Then under the Ricci flow, a region homeomorphic to 8 2 x lR may develop in M such that, as t / T, the sectional curvatures become infinite precisely along the hypersurface identified with 8 2 x {O}. In this evolution, the geometry of the region identified topologically with 8 2 x lR asymptotically approaches the cylinder 8 2 x lR with its standard product metric. Hamilton developed a program of applying Ricci flow techniques to general 3-manifolds and analyzed the singularities which may arise (see especially [186], [189], and [190]).1 Some of Hamilton's ideas are as follows; we first consider [186]. Via point-picking arguments and assuming an injectivity radius estimate, one dilates about singularities and takes limits using the Cheeger-Gromov-type compactness theorem for solutions of the Ricci flow to obtain so-called singularity models, which are nonflat ancient solutions of the Ricci flow. In dimension 3 these ancient solutions have nonnegative lSome other papers in which Hamilton developed his program to approach Thurston's geometrization conjecture by Ricci flow are as follows: characterizing spherical space forms [178], weak and strong maximum principles for systems [179]' ancient 2-dimensional solutions and surface entropy monotonicity [180] (as used in [186]), matrix Harnack estimate [181] and its applications to eternal solutions [182], and the compactness theorem [187]. (These are only partial descriptions that reflect aspects of the papers' relevance to Hamilton's 3-manifold program.)

440

9. BASIC TOPOLOGY OF 3-MANIFOLDS

sectional curvature. By the strong maximum principle, the universal covers of the ancient solutions either split as the product of a surface solution with ~ or have positive sectional curvature in which case they are diffeomorphic to either 8 3 or ~3.2 In the case of splitting, Hamilton proved that the ancient surface solution is either a round shrinking 8 2 (and the universal cover of the singularity model is hence geometrically a shrinking round product cylinder; by definition we say that in this case a neck singularity forms) or it has a backward limit which is the cigar soliton. Note that Perelman's no local collapsing theorem rules out the last case of a cigar. In the case when the universal cover of the singularity model has positive sectional curvature and is diffeomorphic to ~3, the covering is trivial. This ancient solution is either Type I or has backward limit which is a steady Ricci soliton on a topological ~3. In the latter case, the asymptotic scalar curvature ratio is infinite, and by dimension reduction, there exists a sequence of points tending to spatial infinity for which the corresponding dilations of the solution limit to an ancient product solution, which again must be a shrinking round cylinder.3 In this case the singularity model is expected to be the positively curved and rotationally symmetric Bryant soliton and the forming singularity is expected to be a degenerate neckpinch. In summary, we should have that at the largest curvature scale the dilations yield the Bryant soliton, whereas at lower scales dilations yield round product cylinders. This agrees with the fact that the dimension reduction of the Bryant soliton, and more generally a 3-dimensional gradient steady soliton with positive curvature which is /'i,noncollapsed at all scales, is a round product cylinder. On the other hand, the former case of a Type I ancient solution with positive sectional curvature, if it exists, also dimension reduces to a round cylinder. Thus a consequence of Hamilton's 3-dimensional singularity formation theory and Perelman's no local collapsing theorem is that if a finite time singularity forms on a closed 3-manifold, then either M is diffeomorphic to a spherical space form or a neckpinch forms. In [186] Hamilton studies 3-dimensional singularity formation by considering regions in the solution where the scalar curvature is comparable to its spatial maximum. He studies the regions where the scalar curvature is not comparable to its spatial maximum by the technique of dimension reduction. For example, when a 3-dimensional steady Ricci soliton singularity model forms, Hamilton proved an injectivity radius estimate to obtain a second limit which is either a shrinking round product cylinder or the product of a cigar with R (Again no local collapsing rules out the latter case.) The 2When the universal cover of the singularity model has positive sectional curvature and is diffeomorphic to 8 3 , M admits a metric with positive sectional curvature (e.g., 9 (t) for t large enough) and hence is topologically diffeomorphic to a spherical space form. In this case the singularity model must be geometrically a shrinking spherical space form with its underlying manifold diffeomorphic to M. 3With the help of no local collapsing.

3. DECOMPOSITION THEOREMS AND THE RICCI FLOW

441

regions with curvature comparable to their spatial maximum are geometrically close to a 3-dimensional steady Ricci soliton, whereas some regions with curvature not comparable to their spatial maximum are geometrically close to a shrinking round product cylinder. In [189] Hamilton developed surgery theory and formulated a version of Ricci flow with surgery. Although this theory was developed for solutions on closed 4-manifolds with positive isotropic curvature, its higher aim was clearly a surgery theory for 3-manifolds. Indeed, the class of 4-manifolds with positive isotropic curvature is flexible enough to allow for connected sums. Many of the techniques developed in [189] applied to the setting of the Ricci flow on closed 3-manifolds. Limiting arguments and the study of ancient solutions were developed by Hamilton with the aim of enabling surgery. A contradiction argument using limiting techniques was proposed to show that for suitable surgery parameters, the set of surgery times is discrete, and in particular, do not accumulate in finite time. Unfortunately, as was known to some mathematicians working in the field of Ricci flow and as pointed out in [298], there was an error in this part of Hamilton's argument. In the recent work of Perelman [297], [298], building on Hamilton's theory, Ricci flow behavior (especially singularity formation) on 3-manifolds is carefully examined and classified. The overall picture is subtle and technical, with some of the foundations being discussed in this volume. In the following two simplified examples (the first of which continues our discussion above), we try to convey some of its topological flavor, omitting most of the details. The formation of neckpinch singularities (as described above) in a certain sense reflects the topological connected-sum decomposition of the underlying manifold. Indeed, suppose a neckpinch with two ends occurs on a region identified with 8 2 x lR. Hamilton proposes a surgery process as follows [189].4 One does surgery near the large ends of the long, thin tubes in that part of the manifold identified with 8 2 x JR, capping these off with round 3-balls. Note that Hamilton's theory predicts the existence of such tubes where the curvature is very large at the center and slowly decreases as one moves away from the center along the relatively very long length of the tube. In fact his theory predicts that as one approaches the singularity time, the tube becomes arbitrarily close to an exact cylinder and its size slowly increases as one moves away from the center to an arbitrarily much larger but still very small size. 5 Note that Perelman's surgery process in [298] is a modification of the surgery process proposed earlier by Hamilton. 6 4More precisely, he considers the 4-dimensional version of this. 5More precisely, the tube is conformally close to a round product cylinder, where the conformal factor changes very slowly as one moves away from the center. 6Huisken and Sinestrari have considered an analogue of Hamilton's surgery theory for the mean curvature flow.

442

9. BASIC TOPOLOGY OF 3-MANIFOLDS

After the surgery, one continues the Ricci flow on the resulting (possibly disconnected) manifold, taking the glued (smoothed) metric as initial data. Heuristically, this surgery procedure corresponds to Kneser's sphere decomposition theorem. It is known that a 2-sphere removed from the neck may bound a 3-ba11. Thus for an arbitrary initial 3-manifold M, there is no guarantee that this surgery process will stop in finitely many steps. The finiteness theorem of Kneser is in some sense related to a finiteness conjecture of Hamilton for Ricci flow. The latter states that if one runs the unnormalized Ricci flow on a closed 3-manifold and performs geometrictopological surgeries whenever the Ricci flow develops a finite time singularity, then after finitely many such surgeries, the Ricci flow will have a nonsingular solution for all time. (Since one discards S3 and S2 x SI factors, this solution may well be empty.) Using Kneser's finiteness theorem, one sees that if Hamilton's conjecture does not hold, then all but finitely many geometric-topological surgeries at finite time singularities of the Ricci flow must split off 3-spheres. (In this regard, see the following recent papers: Perelman [299] and Colding and Minicozzi [116].) In this context, Hamilton's finiteness conjecture may be regarded as a geometric refinement of Kneser's theorem. Another type of Ricci flow behavior on 3-manifolds reflects the torus decomposition. This was first noticed in the work of Hamilton [190]. For simplicity, we recall Hamilton's formulation. He assumes that a solution to the normalized Ricci flow on a closed 3-manifold M exists for all time t E [0, 00) with uniformly bounded curvature. In this scenario, as time approaches infinity, it may happen that a manifold M can be decomposed into two parts M = Mthin U Mthick. In the components of Mthin, the metrics are collapsing with bounded curvature. (Recall that a manifold is said to be collapsible if it admits a sequence of Riemannian metrics of uniformly bounded curvature and volumes tending to zero.) In the components of Mthick, the metrics converge to complete hyperbolic metrics of finite volume. Furthermore, using minimal surface techniques, Hamilton proves that the fundamental group of Mthin n Mthick injects into the fundamental group of M. By the work of Cheeger and Gromov on collapsing manifolds, one concludes that Mthin is a Cheeger-Gromov graph manifold and hence a connected sum of Seifert spaces and sol-geometry manifolds. As a consequence, Hamilton was able to establish the geometrization conjecture under the (restrictive) hypotheses of long-time existence and uniformly bounded curvature. Perelman's recent work [298]' in conjunction with Shioya and Yamaguchi [332], claims to establish a similar picture without the assumption of bounded curvature. 4. Notes and commentary

We would like to thank Ian Agol for carefully reading this chapter and for pointing out mistakes in a draft version. We also note that Milnor's

4. NOTES AND COMMENTARY

443

[267] and Morgan's [272] are two excellent survey papers introducing Ricci flow on 3-manifolds and giving an overall picture of the current state of the research in the field.

APPENDIX A

Basic Ricci Flow Theory Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics. - Joseph Fourier

In this appendix we recall some basic Ricci flow notation, formulas, and results, mostly from Volume One. Unless otherwise indicated, all page numbers, theorem references, chapter and section numbers, etc., refer to Volume One. Some of the results below are slight modifications of those stated therein. If an unnumbered formula appears on p. of Volume One, we refer to it as (Vl-p. v.); if the equation is numbered 0 .•, then we refer to it as (Vl-O .•). The reader who has read or is familiar with Volume One may essentially skip this chapter, referring to it only when necessary.

v.

1. Riemannian geometry 1.1. Notation. Let (M, g) be a Riemannian manifold. Throughout this appendix we shall often sum over repeated indices and not bother to raise (or lower) indices. For example, aijbij ~ gikgjiaijbki.

• If a is a l-form, then a~ denotes the dual vector field. Conversely, if X is a vector field, then X b denotes the dual l-form. • T M, T* M, A2T* M, and S2T* M denote the tangent, cotangent, 2-form, and symmetric (2, D)-tensor bundles, respectively. • r, V', and .6. denote the Christoffel symbols, covariant derivative, and Laplacian, respectively. • R, Rc, and Rm denote the scalar, Ricci, and Riemann curvature tensors, respectively. • r often denotes the average scalar curvature (assuming M is compact). • The upper index on the Riemann (3, l)-tensor is lowered into the 4-th position: ~jki = R7]k gmi . • ..\ ~ f.l ~ 1/ denote the eigenvalues of the Riemann curvature operator of a 3-manifold, in decreasing order. • d = dist, diam, and inj denote the Riemannian distance, diameter, and injectivity radius, respectively. • L, A = Area, and V denote length, area, and volume, respectively. 445

A. BASIC RICCI FLOW THEORY

446

• trg denotes the trace with respect to 9 (e.g., of a symmetric (2,0)-

tensor).

• sn usually denotes the unit n-sphere.

• For tensors A and B, A * B denotes a linear combination of contractions of the tensor product of A and B. • ~ denotes an equality which holds on gradient Ricci solitons. • ~ denotes an equality which holds on expanding gradient Ricci solitons.

1.2. Basic Riemannian geometry formulas in local coordinates. In Ricci flow, where the metric is time-dependent, it is convenient to compute in a local coordinate system. Let (Mn,g) be an n-dimensional Riemannian manifold. Almost everywhere we shall assume the metric 9 is complete. Let {xi} be a local coordinate system and let 8i ~ a~.. The components of the metric are gij ~ 9 (8i , 8 j ) . The Christoffel symbols for the Levi-Civita connection, defined by "ai8j ~ rfj8k, are k

r ij

(Vl-p. 24)

="21 g kl (8i g j l + 8 j g i l

where (gij) is the inverse matrix of curvature (3, I)-tensor, defined by

(gij) .

- 8lgij ) ,

The components of the Riemann

8 8) 8 . l 8 R ( 8x i '8x j 8x k =;= R ijk 8x l

(Vl-p. 286)

'

are (Vl-p. 68) The Ricci tensor is given by (Vl-p. 92)

D

. -

.L"'IJ -

P RPpij -_!l Up r ij

-

!l·rPpj

u~

+ r ijq r ppq -

q rP r pj iq'

The scalar curvature is R = gij ~j. If M is oriented and the local coordinates { xi} ~1 have positive orientation, then the volume form is (Vl-p. 70) The Bianchi identities. (1) First Bianchi identity: (Vl-3.l7)

0 = ~jkl + ~klj

+ ~ljk'

(2) Second Bianchi identity: (Vl-3.l8) where we take "Rm, cyclically permute the first three indices of "q~jkl (components), and sum to get zero.

447

1. RIEMANNIAN GEOMETRY

(3) Contracted second Bianchi identity: . 1 (VI-3.13) ,p~J = 2'ViR, which is obtained from (VI-3.18) by taking two traces (e.g., multiplying by gqlgjk and summing over q, e,j, k). 1.3. Cart an structure equations. For metrics with symmetry, such as rotationally symmetric metrics, it is convenient to calculate with respect to a local orthonormal frame, also called a moving frame. Let {ed ?=1 be a local orthonormal frame field in an open set U c Mn. Denote the dual orthonormal basis ofT* M by {wi} ~=1 so that 9 = E?=1 wi 0 wi. The connection I-forms w{ E 0 1 (U) are defined by n

'VXei ~ LwI (X) ej, j=1

(Vl-p. 106a)

for all i = 1, ... , n and X E Coo (T Mlu). They are antisymmetric: wI The first and second Cartan structure equations are

-w;.

(Vl-p. 106b)

dw i

(Vl-p. 106c)

Rm ~~

=

= wj !\ w;,

= O~ ~

j = dw1~ - w~~ !\ wk·

The following formula is useful for computing the connection I-forms:

(A.l)

k

wi (ej)

= dw i (ej, ek) + dwJ. (ei' ek) - dw k (ej, ei).

1.4. Curvature under conformal change of the metric. Let 9 9 be two Riemannian metrics on a manifold Mn conformally related by 9 = e2u g, where u : M - t R If {ed?=1 is an orthonormal frame field for g, then {ei}~1 , where ei = e-uei, is an orthonormal frame field for g. The Ricci tensors of 9 and 9 are related by and

(A) .2

if (- -)

c el, ei = e

-2u ( Rc (ee, ei) + (2 - n) 'Vel'VeiU - 8li flu ) + l'Vul 2 (2 _ n) 8il~ - (2 - n) eR.{u) edu) .

Tracing this, we see that the scalar curvatures of 9 and (A.3)

R = e- 2u (R - 2 (n -

9 are related by

1) flu - (n - 2) (n - 1) l'VuI 2 )

.

For derivations of the formulas above, which are standard, see subsection 7.2 of Chapter 1 in [111] for example. 1.5. Variations and evolution equations of geometric quantities. The Ricci flow is an evolution equation where the variation of the metric (i.e., time-derivative of the metric) is minus twice the Ricci tensor. More generally, we may consider arbitrary variations of the metric. Given a variation of the metric, we recall the corresponding variations of the Levi-Civita connection and curvatures. (In Volume One, see Section 1 of Chapter 3 for the derivations, or see Lemma 6.5 on p. 174 for a summary.)

A. BASIC RICCI FLOW THEORY

448

A.I (Metric variation formulas). Suppose that 9 (s) is a smooth I-parameter family of metrics on a manifold Mn such that :sg = v. (1) The Lem-Civita connection r of 9 evolves by LEMMA

(VI-3.3)

= ~ £p ( 'V'/ilkVjp + 'V'j'V'pVik - 'V'i'V'pVjk - 'V'j'V'kVip )

(VI-p. 69a)

2g

q -RqiJkVqp - R ijpVkq

.

(3) The Ricci tensor Rc of 9 evolves by

8 8s Rjk

(VI-3.5) (VI-p. 69b)

1 = '2 gpq ('V' q'V'jVkp + 'V' q'V' kVJp - 'V' q'V' pVjk - 'V' j 'V' kVqp) =

-~ [~LVjk + 'V'j'V'k (trgv) + 'V'J (8v)k + 'V'k (8v)j]

,

where ~L denotes the Lichnerowicz Laplacian of a (2, a)-tensor, which is defined by (~LV)jk ~ ~Vjk

(VI-3.6)

+ 2gqpR~jkVrp -

gqPRjpvqk - gqPRkpVjq.

(4) The scalar curvature R of 9 evolves by (VI-p. 69c) (VI-p. 69d)

where V ~ gi j vij is the trace of v. (5) The volume element df.L evolves by

8 V 8s df.L = '2 df.L.

(VI-p. 70)

(6) Let "Is be a smooth family of curves with fixed endpoints in M n and let Ls denote the length with respect to 9 (s). Then :s Ls ("(s)

(VI-3.8)

=

~

1

v (T, T) dO'

1.

-1

('V'TT, U) dO',

18

-'-- a1s -'-- as' a th T -;h were O' zs. arc leng, au' an d U -;1.6. Commuting covariant derivatives. In deriving how geometric quantities evolve when the metric evolves by Ricci flow, commutators of covariant derivatives often enter the calculations. (For the following, see p. 286 in Section 6 of Appendix A in Volume One.) If X is a vector field, then

(VI-p. 286a)

1. RIEMANNIAN GEOMETRY

449

If 0 is a I-form, then

(VI-p. 286b) More generally, if A is any (p, q)-tensor field, one has the commutator (VI-p. 286) [V',~, V'.J J

Al1···lq ....:... kl···kp --;-

V'.V' ·Al1···lq ~

kl···kp

J

V' ·V'·Al1···lq

-

~

J

kl···kp

q

=

p

~ R~r:

L..J

Al1· .. lr-l mlr+l· .. l q _ ~ R"!': Al1 .. ·lq ~Jm kl· .. k p L..J ~Jks kl···ks-l m ks+l' ·kp ·

r=1

8=1

1. 7. Lie derivative. Because of the diffeomorphism invariance of the Ricci flow, the effect of infinitesimal diffeomorphisms on tensors, e.g., the Lie derivative, enters the Ricci flow. (See p. 282 in Section 2 of Appendix A in Volume One.) The Lie derivative of the metric satisfies

(£Xg) (Y, Z) = 9 (V'yX, Z)

(VI-p. 282a)

+ 9 (Y, V' zX)

for all vector fields X, Y, Z. In local coordinates

(£Xg)ij = (£Xg)

(VI-p. 282b) In particular, if X

(a~i' a~j )

= V'iXj + V'jXi.

= V'I is a gradient vector field, then (£Vjg)ij = 2V'iV'jf.

1.8. Bochner formulas. (See p. 284 in Section 4 of Appendix A in Volume One.) The rough Laplacian denotes the operators ~:

where T%M ~ (VI-p. 284a)

Coo (T%Mn)

®P T* M 0 ®q T M,

~

Coo (T%Mn) ,

defined by

n

(~A)(Y1"'" Yp; 01, ... , Oq) =

L

(V'2 A) (ei' ei, Y1, ... , Yp; 01, ... , Oq)

i=1

for all (p, q)-tensors A, all vector fields Y1, ... , Yp, and all covector fields 01 , . .. , Oq, where {ed~=1 is a (local) orthonormal frame field. The Hodgede Rham Laplacian -~d : [!P (M) ~ [!P (M) is defined by -~d ~

(VI-p. 284b)

d8 + 8d.

In particular, if 0 is a I-form, then ~dO

(VI-p. 284c) For any function

1:M

=

Rc (0) .

~ ~

~ V' 1 =

(A.4)

~O -

V' ~I + Rc (V' f)

and (A.5)

~ 1\1112

= 21V'V' 112 + 2 Rc (V'I, V' f) + 2V'1 . V' (~f) ,

A. BASIC RICCI FLOW THEORY

450

where the dot denotes the metric inner product, i.e., X . Y gijXiyj.

=

If -9tgij (

= (X, Y) =

-2~j on M x (a,w) and f: M x (a,w) -+~, then

~-

:t) IV fl2

= 21VV fl2 + 2V f . V ( ( ~ -

!)

f) .

1.9. The cylinder-to-ball rule. The following is an obvious modification of Lemma 2.10 on p. 29 of Volume One.

°

LEMMA A.2 (Cylinder-to-ball rule). Let < L ::; 00 and let 9 be a warped-product metric on the topological cylinder (0, L) x sn of the form

9 = dr 2 + w (r)2 gcan, where w : (0, L) -+ ~+ and gcan is the canonical round metnc of radius 1 on sn. Then 9 extends to a smooth metric on B

(6, L)

(as r -+ 0+) if and

only if

(V1-2.16)

lim w(r)

r ......O+

= 0,

lim w' (r) = 1,

(V1-2.17)

r ...... O+

and

d2k w

(V1-2.18)

lim d 2k (r) = r ...... O+ r

°

for all kEN.

1.10. Volume comparison. We recall the Bishop-Gromov volume comparison (BGVC) theorem. THEOREM A.3 (Bishop-Gromov volume comparison). Let (Mn,g) be a complete Riemannian manifold with Rc ~ (n - 1) K, where K E ~. Then for any p E M, VoIB(p,r) VolKB(PK,r) zs a nomncreasing function of r, where PK is a point in the n-dimensional simply-connected space form of constant curvature K and YOlK denotes the volume in the space form. In particular

(A.6) for all r > 0. Given p and r is isometric to B(PK, r).

If Rc

~

> 0,

equality holds in (A.6) if and only if B(p, r)

0, we then have the following.

COROLLARY A.4 (BGVC for Rc ~ 0). If (Mn,g) is a complete Riemannian manifold with Rc ~ 0, then for any p E M, the volume ratio Vol~~p,r) is a nonincreasing function of r. We have Vol~~p,r) ::; Wn for all r > 0, where Wn is the volume of the Euclidean unit n-ball. Equality holds if and only if (Mn, g) is isometric to Euclidean space.

1.

RIEMANNIAN GEOMETRY

451

As a consequence, we have the following characterization of Euclidean space. COROLLARY A.5 (Volume characterization ofll~n). If (Mn, g) is a complete noncompact Riemannian manifold with Rc 2:: 0 and if for some p EM, lim VolB (p, r) = r-+oo

Wn ,

rn

then (M, g) is isometric to Euclidean space. The following result about the volume growth of complete manifolds with nonnegative Ricci curvature is due to Yau (compare with the proof of Theorem 2.92). COROLLARY A.6 (Rc 2:: 0 has at least linear volume growth). There exists a constant c (n) > 0 depending only on n such that zf (Mn, g) is a complete Riemannian manifold with nonnegative Ricci curvature and p E Mn, then Vol B (p, r) 2:: c (n) VolB (p, 1) . r for any r E [1,2 diam (M)).1 The asymptotic volume ratio of a complete Riemannian manifold (Mn,g) with Rc 2:: 0 is defined by

AVR(g)~ lim VoIB(p,r),

(A.7)

r-+oo

wnrn

where Wn is the volume of the unit ball in ]Rn. By the Bishop-Gromov volume comparison theorem, AVR (g) ~ 1. Again assuming Rc (g) 2:: 0, we have for 8 2:: r, (1) 8n - 1

A (8) ~ A (r) r n where A (8)

~

1'

Vol8B (p, 8),

(2) (A.8)

VoIB(p,r) > Wnrn

A(8) nwn 8 n -

> AVR(g). 1 -

We have the following relation between volume ratios and the injectivity radius in the presence of a curvature bound (see for example Theorem 5.42 of [111]). THEOREM A.7 (Cheeger, Gromov, and Taylor). Given c > 0, ro > 0, and n EN, there exists /'0 > 0 such that if (Mn , g) is a complete Riemannian manifold with Isect I ~ 1 and if p E M is such that VolB (p, ro) n

rO

2:: c,

lWe allow the noncompact case where diam (M) =

00.

A. BASIC RICCI FLOW THEORY

452

then inj (p) 2:

LO·

1.11. Laplacian and Hessian comparison theorems. Given K E lR and r > 0, let (n - 1) ..JK cot ( ..JKr )

HK (r)

~

{

(n _ 1)

°

~~th ( VJKlr)

if K

> 0,

if K

= 0,

if K < 0,

2:!K.

where if K > we assume r < The function HK (r) is equal to the mean curvature of the (n - 1)-sphere of radius r in the complete simplyconnected Riemannian manifold of constant sectional curvature K. THEOREM A.8 (Laplacian comparison). Let (Mn,g) be a complete Riemannian manifold with Rc 2: (n - 1) K, where K E lR. For any p E M n and x E Mn at which dp (x) is smooth, we have

Ildp (x) ::; HK (dp (x)) .

(A.9)

On the whole manifold, the Laplacian comparison theorem (A.9) holds in the sense of distributions. That is, for any nonnegative Coo function tl' If u = H is a fundamental solution centered at a point X E M, then taking tl ---t 0 implies the Cheeger-Yau estimate:

I (y, t) ~ d (~'ty)2

(A.14)

In terms of u, the positive solution to the heat equation, on a complete Riemannian manifolds with nonnegative Ricci curvature, we have

8

8t logu -1V'logul and

2

n

= ~logu ~ - 2t

{d

U (X2' t2) > (t2) ---:----'----'- -n/2 exp - (Xl, X2)2} . U(XI,tl) - tl 4(t2-tl) For a fundamental solution u H (y, t)

= H,

~ (41rt)-n/2 exp { _ d (~ty)2} .

1.13. Calabi's trick. In this subsection we give an example of Calabi's trick which is useful in the study of heat-type equations and analytic aspects of the Ricci flow. In particular, a slight modification of the discussion below applies to the proof of the local first derivative of curvature estimate for the Ricci flow (see Theorem A.30). First, let us recall some facts about the distance function. Let (Mn, g) be a Riemannian manifold. Given p EM, the distance function r (x) ~ d (x, p) is Lipschitz on M with Lipschitz constant 1. Let Cut (p) denote the cut locus of p and let Cp ~ {V E TpM : d (p,expp (V)) =

IVI},

A. BASIC RICCI FLOW THEORY

454

so that Cut (p) zero. We have

= expp (aCp ). The cut locus is a closed set with measure expplintCp : Cp \aCp

is a diffeomorphism. Let field on TpM -

aIar =

--t

M\ Cut (p)

I!I L:~1 xi a~i denote the unit radial vector

{5} . If x rt. Cut (p) U {p}, then r is smooth at x, 'V'r (x) =

(exppL alar, and l'V'r (x)1 = 1. Suppose that we have a function F : M x [0, T) differential inequality

(~ at -

(A.15)

--t

lR which satisfies the

2 < - C - F2

6.) F

for some constant C. For an example of such a function, see the proof of the local first derivative of curvature estimate in Part II of this volume. If M is a closed manifold, then we can apply the maximum principle to F to obtain the estimate

F (x, t) :::; C coth (Ct) , where the RHS is the solution to the ODE 1ft = C2 - P with limt'\.o f (t) = +00. On the other hand, if M is noncompact, then one way of obtaining an estimate for F is to localize the equation by introducing a cut-off function. In particular, suppose (M n , g) satisfies Rc ~ - (n - 1) K for some K > O. Let 'f/ : [0, 00) --t lR be a smooth nonincreasing function satisfying 'f/( s) = 1 for 0 :::; s :::; ~ and 'f/(s) = 0 for s ~ 1. We may assume 0 ~ 'f/' ~ -6Jri

(A.16)

- CoJri :::; 'f/" :::; Co,

and

where Co is a universal constant. 2 Given p E M and A 'f/ ( d(~t»)· Recall that in M\ ({p} U Cut (p))

'V'd (', p) 1 = 1,

1

> 0, define ¢(x)

6.d (" p) :::; (n - 1) v'K coth ( v'Kd ( " p)) ,

where the Laplacian estimate follows from (A. g) and the assumption Rc - (n - 1) K. Hence at points x rt. Cut (p) , ¢ is smooth and (A.17)

~

1'V'¢1 2 :::;C¢,

6.¢=

~

~'f/,.6.d+ 12'f/"I'V'dI2~-CV¢,

where C depends on A (and where we used (A.16), 'f/' :::; 0, and the fact that the support of 'f/' is contained in B (p, A) \B (p,

(d(:»)

4) ).

We calculate that for x ¢

(:t -

6.) (¢F) = ¢2

rt. Cut (p) ,

(:t -

6.) F - ¢ (6.¢) F - 2¢'V'¢· 'V' F

:::; -2'V'¢. 'V' (¢F)

+ ¢2C2 -

2Let ( be a cut-off function with 0 ~ (' ~ -3 and constant, and define 'f/ = (2.

¢2 F2 - ¢F6.¢ + 2FI'V'¢12

ICI

S; C, where C is a universal

455

1. RIEMANNIAN GEOMETRY

(the above calculation holds for any C 2 function 0). If (V 3 ,go) zs a closed Riemannian 3-orbifold of positive Ricci curvature, then a unique solution g (t) of the normalized Ricci flow wzth g (0) = go exists for all t > 0, and as t ---+ 00, the g(t) converge to a metric goo of constant positive sectional curvature. In particular, V 3 is dzJJeomorphic to the quotient of 53 by a finite group of isometries. One of the main ideas in the proof of Theorem A.26 is to apply Theorem A.25 to obtain pointwise curvature estimates which lead to the curvature tending to constant as the solution evolves. From now on we shall assume that A (t) 2:: J.L (t) 2:: v (t) are solutions of the ODE system (VI-6.32). The evolutions of various quantities and their applications to the Ricci flow on closed 3-manifolds via the maximum principle for systems are given as follows.

(1) d dt (v

(A.20)

+ J.L) = v 2 + J.L2 + (v + J.L) A 2:: 0,

with the inequality holding whenever J.L + v 2:: preserved in dimension 3 under the Ricci flow.

o.

So Rc > 0 is

(2) (A.21) If go has positive Ricci curvature, then so does g (t) and there exists a constant C 1 < 00 such that (A.22)

A (Rm)

~

C 1 [v (Rm)

+ J.L (Rm)].

(3) If v + J.L > 0, then

! Cv +~ ~:)l-' ) log

= 8 (v + A _ J.L) _ (1 _ 8) (v + J.L) J.L + (J.L - v) A + J.L2 V+J.L+A J.L2 ~ 8 (v + A - J.L) - (1 - 8) -v---'+-J.L-+-A

+ A-

J.L ~ A ~ 2C1 J.L and ~ 2:: vt;..f1: 2:: enough so that 1~6 ~ 12b2, we have Since v

6b1

1

d ( -log dt (v

A-V

+ J.L + A)1-6

)

~

o.

,

choosing 8

> 0 small

463

2 BASIC RICCI FLOW

So if go has positive Ricci curvature, then there exist constants C fJ > 0 such that A (Rm) - 1/ (Rm) C (A.23) Rl-li ::;.

< 00 and

We shall call (A.23) the 'pinching improves' estimate. Next we consider estimates for the derivatives of Rm.

2.5. Global derivative estimates. For solutions to the Ricci flow on a closed 3-manifold with positive Ricci curvature, we have the following estimate for the gradient of the scalar curvature. (See Theorem 6.35 on p. 194 of Volume One.) THEOREM A.28 (3-dimensional gradient of scalar curvature estimate). Let (M 3 ,g (t)) be a solution of the Ricci flow on a closed 3-manifold with g (0) = go. If Rc (go) > 0, then there exist {J,"8 > 0 depending only on go such that for any {3 E [0, {J], there exists C depending only on {3 and go such that

IV'RI2

R3

< {3R-8/2 + C R- 3 . -

After a short time, the higher derivatives of the curvature are bounded in terms of the space-time bound for the curvature. (See Theorem 7.1 on pp. 223-224 of Volume One.) THEOREM A.29 (Bernstein-Bando-Shi estimate). Let (Mn, g (t)) be a solution of the Ricci flow for whzch the maximum principle applies to all the quantztzes that we consider. (This is true in particular if M is compact.) Then for each 0: > 0 and every mEN , there exists a constant C (m, n, 0:) dependzng only on m, and n, and max {o:, I} such that if IRm (x, t)lg(t) ::; K

0:

for all x E M and t E [0, K],

then for all x E M and t E (0, ~],

(Vl-p.

224 )

()I l"mR v m x, t g(t)::;

C(m,n,o:)K tm / 2

With all of the above estimates and some more work, one obtains Theorem A.26. Finally we mention that an important local version of Theorem A.29 is the following. THEOREM A.30 (Shi-Iocal first derivative estimate ). For any 0: > 0 there exists a constant C (n, K, r, 0:) depending only on K, r, 0: and n such that if Mn zs a manifold, p E M, and g (t), t E [0,7], 0 < 7 ::; 0:/ K, zs a solutwn to the Ricci flow on an open neighborhood U of p containing Bg(o) (p, r) as a compact subset and if IRm (x, t)1 ::; K for all x E U and t E [0,7],

A BASIC RICCI FLOW THEORY

W-I

then

Inv

(A.24)

R

( )1 C (n, K, r, a) = C(n, vXr, a)K m y, t:s; jt jt

for all (y,t) E Bg(D) (p,r/2) x (0,7]. Gwen m addition {3 > 0 and"( > 0, if also "(/vX:S; r:S; {3/vX, then there exists C(n,a,{3,"() such that under the above assumptwns K IV'Rml < C (n, a, {3, "() ;;

-

m

Bg(D)

vt

(p,r/2) x (0,7].

For a proof and applications, see W.-X. Shi [329], [330], Hamilton [186], [111], or Part II of this volume.

2.6. The Hamilton-Ivey estimate. The following result reveals the precise sense in which all sectional curvatures of a complete 3-manifold evolving by the Ricci flow are dominated by the positive sectional curvatures. (See [186] or Theorem 9.4 on p. 258 of Volume One.) THEOREM A.31 (3d Hamilton-Ivey curvature estimate). Let (M3,g(t)) be any solution of the Rzccz flow on a closed 3-mamfold for 0 :s; t < T. Let v (x, t) denote the smallest eigenvalue of the curvature operator. If infxEM v (x, 0) ~ -1. then at any pomt (x, t) EM x [0, T) where v (x, t) < 0, the scalar curvature is estimated by (A.25)

R

~ Ivl (log Ivl

+ log (1 + t) -

3) .

2.7. Ricci solitons. If 9 is a Ricci soliton, then

for some p E (VI-5.16)

Rc-!!"'g = LXqg n lR and I-form X. Under this equation we have

~ (R -

p)

+ (V' (R -

p), X)

+ 21Rc _~gI2 + ~ (R -

p)

= O.

Using this formula, in Proposition 5.20 on p. 117 of Volume One, the following classification result for Ricci solitons was proved. (See Chapter 1 of this volume for the relevant definitions.) PROPOSITION A.32 (Expanders or steadies on closed manifolds are Einstein). Any expanding or steady Rzcci soliton on a closed n-dzmenswnal manifold is Emstem. A shrinking Riccz solzton on a closed n-dzmenswnal manifold has positive scalar curvature. In dimension 2, all solitons have constant curvature. tion 5.21 on p. 118 of Volume One.)

(See Proposi-

PROPOSITION A.33 (Ricci solitons on closed surfaces are trivial). If (M2, 9 (t)) is a self-similar solution of the normalized Rzcci flow on a Riemanman surface, then g(t) == g(O) is a metric of constant curvature.

3. BASIC SINGULARITY THEORY FOR RICCI FLOW

465

3. Basic singularity theory for Ricci flow The knowledge of which geometry aims is the knowledge of the eternal - Plato Geometry is knowledge of the eternally existent. - Pythagoras And perhaps, posterity will thank me for having shown it that the ancients did not know everything - Pierre Fermat

In this section we review some basic singularity theory as developed by Hamilton and discussed in Volume One. 3.1. Long-existing solutions and singularity types. For the following, see pp. 234-236 in Section 1 of Chapter 8 in Volume One. DEFINITION A.34. • An ancient solution is a solution that exists on a past time interval (-00, w). • An immortal solution is a solution that exists on a future time interval (a, 00 ) . • An eternal solution is a solution that exists for all time (-00, (0). DEFINITION A.35 (Singularity types). Let (Mn, 9 (t)) be a solution of the Ricci flow that exists up to a maximal time T ~ 00.

• One says (M, 9 (t)) forms a Type I singularity if T < sup

(T - t) IRm (', t)1
O. In all dimensions, we have the following. (See Proposition 9.20 on p. 274 of Volume One.) PROPOSITION A.43 (Trace Harnack estimate). If (Mn, 9 (t)) is a solution of the Ricci flow on a complete manifold with bounded positive curvature operator, then for any vector field X on M and all times t > 0 such that the solution exists, one has (Vl-p.274)

aR

R

at + t + 2 (VR,X) + 2Rc(X,X)

;::: O.

The proof of Proposition A.43 will be given in Part II. When n = 2, by choosing the minimizing vector field X = - R- 1 V R, it can be seen that (Vl-p.274) is equivalent to (Vl-p.169b). One also has Corollary 9.21 on p. 274 of Volume One, namely COROLLARY A.44 (Trace Harnack consequence, tR monotonicity). If (Mn,g(t)) is a solution of the Ricci flow on a complete manifold with bounded curvature operator, then the function tR is pointwzse nondecreasmg for all t ;::: 0 for which the solution exists. If (M,g (t)) zs also ancient, then R ztself is pointwise nondecreasing.

A BASIC RICCI FLOW THEORY

468

3.4. Surface entropy formulas. The surface entropy N is defined for a metric of strictly positive curvature on a closed surface M2 by (Vl-p. 133) Let

f

N(g)

~

r

t:.f

=R

Rlog Rdp,. 1M2 be the potential function, defined up to an additive constant by

(VI-5.8)

- r.

(See Lemma 5.38 on p. 133 and Proposition 5.39 on p. 134 of Volume One.) PROPOSITION A.45 (Surface entropy formula). If (M2, 9 (t)) 2S a solutwn of the normalized Ricci flow on a compact surface wzth R (., 0) > 0, then (VI-5.25) (Vl-p. 134)

dN dt

=_

r

1M2

=_

r

r

IV'RI2 dA+ (R-r)2dA R 1M2

V'R + R V' fl2 dA 1M2 R

- 2

1

fM21V'V' f - ~t:.f . gl2 dA

::; o. 3.5. Ancient 2-dimensional solutions. 3.5.1. Examples. (See pp. 24-28 in Section 2 of Chapter 2 in Volume One.) Hamilton's cigar soliton is the complete Riemannian surface (lR?, 9E), where . dx @ dx + dy @ dy (VI-2.4) gE =;= 1 + x2 + y2 This manifold is also known in the physics literature as Witten's black hole. In polar coordinates (VI-2.5)

1 + r2

If we define

(Vl-p. 25a)

s

~ arcsinh r = log (r + J 1 + r2)

,

then we may rewrite gE as (VI-2.7)

gE

= ds 2 + tanh2 sd(P.

The scalar curvature of gE is

4

(Vl-p. 25b)

RE

= 1 + r2 =

4 cosh2 s

16

(e s

+ e- s )2·

(See pp. 31-34 in subsection 3.3 of Chapter 2 in Volume One.) Let h be the flat metric on the manifold M2 = lRxst, where st is the circle of radius 1. Give M2 coordinates x E lR and () E st = lR/27rZ. The Rosenau

3

BASIC SINGULARITY THEORY FOR RICCI FLOW

469

solution or sausage model (see [311] or [141]) of the Ricci flow is the metric g = u . h defined for t < 0 by (Vl-2.22)

u(x,t) = .x-I sinh (-.xt) , cosh x + cosh .xt

where .x > O. LEMMA A.46 (Rosenau solution and its backward limit). The metric defined by (Vl-2.22) for t < 0 extends to an anczent solution with positive curvature of the Rzcci flow on S2. The Rosenau solution is a Type II ancient solution which gives rzse to an eternal solution zf we take a lzmzt lookmg infimtely far back in time. In particular, if one takes a lzmit of the Rosenau solution at either pole x = ±oo as t ~ -00, one gets a copy of the cigar soliton. Note that the sausage model is an ancient Type II solution which encounters a Type I singularity. 3.5.2. Classification results. The following provides a characterization of the cigar soliton. (See Lemma 5.96 on p. 168 of Volume One.) LEMMA A.47 (Eternal solutions are steady solitons, 2d case). The only ancient solution of the Ricci flow on a surface of strictly positive curvature that attains its maximum curvature in space and time is the cigar (lR2,g~ (t)). The following classifies 2-dimensional complete ancient Type I solutions. (See Proposition 9.23 on p. 275 of Volume One.) PROPOSITION A.48 (Nonflat Type I ancient surface solution is round S2). A complete ancient Type I solution (N2, h (t)) of the Riccz flow on a surface zs a quotzent of ezther a shrmking round S2 or a flat ]R2. We have the following result for 2-dimensional Type II solutions. (See Proposition 9.24 on p. 277 of Volume One.) PROPOSITION A.49 (Type II ancient solution backward limit is a steady, 2d case). Let (M2, g (t)) be a complete Type II anczent solutwn of the Rzccz flow defined on an interval (-00, w). where w > O. Assume there exists a function K (t) such that IR I :S K (t). Then either g (t) is flat or else there exists a backwards limzt that is the cigar solzton. Combining the above results, we obtain the following. lary 9.25 on p. 277 of Volume One.)

(See Corol-

COROLLARY A.50 (Ancient surface solutions). Let (M 2 ,g (t)) be a complete ancient solution defined on (-00, w), where w > o. Assume that its curvature is bounded by some functwn of time alone. Then either the solution is flat or it is a round shrinkmg sphere or there exists a backwards limit that is the cigar.

470

A BASIC RICCI FLOW THEORY

3.6. Necklike points in Type I solutions. (See Section 4 in Chapter 9 of Volume One.) We say that (x, t) is a Type I c-essential point if c (Vl-p. 262a) IRm(x,t)1 ~ T-t > O. We say that (x, t) is a O. Then either (M, 9 (t)) is isometric to a spherical space form or else there exists a constant c > 0 such that for all 7 E (-00,0] and 0, there are x E M and t E (-00,7) such that (x, t) is an ancient Type I c-essential point and a such that for each t E (0, J), the set

°

Image (Rm [g (t)])

c

A2T* M

%s a smooth subbundle which is invariant under parallel translation and constant in time. Moreover, Image (Rm [g (x, t)]) is a L%e subalgebra of A2T;M ~ so (n) for all x E M and t E (0, J).

As an application of the strong maximum principle we have the following classification result due to W.-X. Shi. THEOREM A.54 (Complete noncompact 3-manifolds with Rc ~ 0). If (M3, g(t)) , t E [0, T), is a complete solution to the Ricci flow on a 3manifold with nonnegative sectional (Ricci) curvature, then for t E (0, T) the universal covering solution (;\it 3 ,g(t)) is either

(1) ]R3 with the standard flat metric, (2) the product (.N2, h (t)) x ]R, where h (t) is a solution to the Ricci flow with positive curvature and.N2 is diffeomorphic to either S2 or]R2 or (3) g(t) and g(t) have positive sectional (Ricci) curvature and hence ;\it3 is diffeomorphic to S3 or]R3 (in the former case M3 is diffeomorphic to a spherzcal space form). 4.2. Hamilton's matrix Harnack estimate. Motivated by the consideration of expanding gradient Ricci solitons, Hamilton proved the following (see [181]). THEOREM A.55 (Matrix Harnack estimate for RF). If (M n , g( t )), t E [0, T), %s a complete solution to the Ricci flow with bounded nonnegative curvature operator, then for any I-form WE COO (AIM) and 2-form U E Coo (A2 M) , we have (A.26) where and P pij

=i= V'p~j

-

V' i!lpj.

REMARK A.56. See the discussion in Section 2 of Chapter 1 for a motivation for defining Mij and Ppij. Chposing an orthonormal basis of cotangent vectors {w a } ~=I at any point (x, t), letting W = wa and U = wa A X for any fixed I-form X, and summing over a, yield the trace Harnack estimate for the Ricci flow (Proposition A.43).

A. BASIC RICCI FLOW THEORY

472

The following is a generalization of the trace Harnack estimate (see [105] and [290]). THEOREM A.57 (Linear trace Harnack estimate). Let (Mn,g(t)) and h (t) , t E [0, T), be a solution to the linearized Ricci flow system:

8

8t gij

=

-2~),

8

-hi)' at = (b..Lh) t).. such that (M, 9 (t)) is complete with bounded and nonnegative curvature operator. h (0) 2: 0, and [h (t)[g(t) ~ C for some constant C < 00. Then h (t) 2: 0 for t E [0, T) and for any vector X we have (A.27) where H = gij hi) . Indeed, (A.27) generalizes Hamilton's trace Harnack estimate since we may take hi) = Rij (under the Ricci flow we have Bt~j = (~L RC)i)·

4.3. Geometry of gradient Ricci solitons. The asymptotic scalar curvature ratio of a complete noncompact Riemannian manifold (Mn, g) is defined by ASCR (g) = lim sup R (x) d (x, 0)2, d(x.O)--->oo

where 0 E M is a choice of origin. This definition is independent of the choice of O. Theorem 9.44 on p. 354 of [111]: THEOREM A.58 (Asymptotic scalar curvature ratio is infinite on steady solitons, n 2: 3). If (Mn, g, f) , n 2: 3, is a complete steady gradient Riccz soliton with sect (g) 2: 0, Rc (g) > 0, and if R (g) attains its maximum at some pomt, then ASCR (g) = 00. Theorem 8.46 on p. 318 of [111]: THEOREM A.59 (Dimension reduction). Let (Mn,g(t)), t E (-oo,w), 0, be a complete noncompact ancient solution of the Riccz flow with bounded nonnegative curvature operator. Suppose there exzst sequences Xi E M , 1r'. ---> 00 ,and A- ---> 00 such that dO(p,Xi) > A-1. and 1. Ti -

w

>

(A.28)

R(y, 0) ~ r;2

for all y E BO(Xi, Ari).

Assume further that there exists an injectivity radius lower bound at (Xt' 0); namely, injg(O) (Xi) 2: &ri for some & > o. Then a subsequence of solutions (Mn, r;2g(r;t), Xi) converges to a complete limit solution (M~, goo (t), xoo) which is the product of an (n - l)-dzmenswnal solutwn (wzth bounded nonnegative curvature operator) with a line.

4.

MORE RICCI FLOW THEORY AND ANCIENT SOLUTIONS

4;3

Theorem A.58 says the following. (See Chapter 6 for a definition of K-noncollapsed. ) COROLLARY A.50 (Dimension reduction of steady solitons). If (Mn,g,f) , n ~ 3, zs a complete steady gradient Riccz sohton whzch is K-noncolla]JsPc! on all scales for some K > and 1.f sect (g) ~ 0, Rc (g) > 0, and zf R VI) attams its maxzmum at some point, then a dilatzon about a sequence of points tendmg to spatial infinity at time t = converges to a complete solutzon (M~, goo (t), x oo ) which zs the product of an (n - l)-dzmenszonal solutwn5 with JR.

°

°

Proposition 9.45 on p. 355 of [111]:

°

PROPOSITION A.51 (Rc > expanders have AVR > 0). If (Mil, 9 (t)) , t > 0, is a complete noncompact expanding gradient Ricci soliton with Rc > 0, then AVR (g (t)) > 0. Theorem 9.56 on p. 362 of [111]: THEOREM A.52 (Steady or expander with pinched Ricci has R exponential decay). If (Afn , g) is a gradient Riccz soliton on a noncompart mamfold with pmched Ricci curvature in the sense that Rij ~ =:RgtJ for some E > 0. where R ~ 0, then the scalar curvature R has exponential decay. Theorem 9.79 on pp. 375-376 of [111]: THEOREM A.63 (Classification of 3-dimensional gradient shrinking solitons with Rm ~ 0). In dimension 3, any nonflat complete shrmkmg gradzent Ricci soliton with bounded nonnegative sectional curvature is ezther a quotzent of the 3-sphere or a quotient of 8 2 x R

4.4. Ancient solutions. Theorem 10.48 on p. 417 of [111]:

°

THEOREM A.54 (Ancient solution with Rm ~ and attaining sup R is steady gradient soliton). If (Mn , 9 (t)) , t E (-00, w) , is a complete solutzon to the Ricci flow with nonnegative curvature operator. posztive Rzccz curvature, and such that SUPMx (-oo,w) R ·lS attained at some pomt in space and tzme, then (M n, 9 (t)) zs a steady gradient Rzcci soliton. Analogous to the above result is the following:

°

THEOREM A.65 (Immortal solution with Rm ~ and attaining sup tR is gradient expander). If (Mn,g (t)), t E (0,00), is a complete solution to the Ricci flow with nonnegative curvature operator, positive Rzcci curvature, and such that sUPMx(O,oo) tR is attained at some point in space and time, then (Mn,g(t)) zs an expanding gradient Ricci soliton. Proposition 9.29 on p. 344 of [111]: 5With bounded nonnegative sectional curvature.

A. BASIC RICCI FLOW THEORY

474

PROPOSITION A.66 (n 2: 2 backward limit of Type II ancient solution with Rm 2: and sect> 0). Let (Mn,g(t)) , t E (-oo,w) , w E (0,00], be a complete Type II ancient solution of the Rzcci flow with bounded nonnegative curvature operator and positive sectional curvature. Assume either

°

(1) M is noncompact, (2) n is even and M is orientable, or (3) 9 (t) is K-noncollapsed on all scales. Then there exists a sequence of pmnts and times (Xi, ti) with ti -> -00 such that (M,gi (t) ,Xi), where gi(t) ~ ~g (ti + Hilt), limits in the Coo pointed Cheeger-Gromov sense to a complete nonflat steady gradient Ricci soliton (M~, goo (t) ,x oo ) with bounded nonnegative curvature operator.

Theorem 9.30 on p. 344 of [111]: THEOREM A.67 (Ancient has AVR = 0). Let (Mn,g(t)), t E (-00,0], be a complete noncompact nonflat anczent solution of the Rzccz flow. Suppose g(t) has nonnegative curvature operator and sup (x,t)EM x (-00,01

IRm g(t) (x) Ig(t) < 00.

Then the asymptotic volume ratio AVR(g(t))

=

°

for all t.

Theorem 9.32 on p. 345 of [111]: THEOREM A.68 (Type I ancient has ASCR = 00). If(Mn,g(t)), -00 < t < w, is a complete noncompact Type I ancient solution of the Ricci flow with bounded positive curvature operator, then the asymptotic scalar curvature ratio ASCR(g(t)) = 00 for all t. 5. Classical singularity theory In this section we continue the discussion of Hamilton's singularity theory and recall some further results concerning the classifications of singularities, especially in dimension 3. An exposition of some of these results, which were originally proved by Hamilton in [186] and [190]' is given in [111]. One of the differences between Hamilton's and Perelman's singularity theories is that in Hamilton's theory, singularities are divided into types, e.g., for finite time singularities, Type I and Type IIa. In Perelman's theory, a more natural space-time approach is taken where singularity analysis is approached via the reduced distance function. Throughout most of this section we shall consider the case of dimension 3. We first consider the case of Type I singularities, which was essentially treated in Volume One (see Theorems A.51 and A.52 above). Applying the compactness theorem and the classification of Type I ancient surface solutions to Theorem A.51 yields the following.

5.

CLASSICAL SINGULARITY THEORY

475

THEOREM A.69 (3d Type I - existence of necks). If (M3,g(t)) is a Type I singular solution of the Rzcci flow on a closed 3-manifold on a maximal time interval ~ t < T < 00, then there exists a sequence of points and times (Xi, ti) with ti - t T such that the corresponding sequence of dilated solutions (M3, gi (t), Xi) converges to the geometric quotient of a round shrinking product cylinder S2 x ~.

°

For the rest of this section we consider Type IIa singular solutions. In this case we invoke Perelman's no local collapsing theorem (see Chapter 6). This has the following two effects on Hamilton's theory. It enables one to apply the compactness theorem to the dilation of Type IIa singular solutions. It rules out the formation of the cigar soliton as a product factor in a singularity model. Classical point picking plus the no local collapsing theorem yield the following result (see Proposition 8.17 of [111]). PROPOSITION A.70 (Type IIa singularity models are eternal). Choose any sequence Ti / T. For a Type IIa singular solution on a closed manifold satisfying JRmJ ~ CR+C

(A.29)

and for any sequence {(Xi, td}, where ti

(A.30)

(Ti - ti) R (Xi, ti)

=

-t

T, satisfying6

max (Ti - t) R (X, t) ,

Mx[O,TiJ

the sequence (M, gi (t), Xi) , where

(A.31)

gi (t) ~ ~g (ti

+ Rilt)

with ~ ~ R (Xi, td,

preconverges to a complete eternal solution

(M~, goo (t), xoo), t

E

(-00,00),

with bounded curvature. The singularity model (Moo, goo (t)) zs nonflat, noncollapsed on all scales for some I'\, > 0, and satisfies R(goo (t)) = 1 = R(goo)

sup

1'\,-

(Xoo, 0) .

Moo x ( -00,00) If n = 3, then the singularzty model has nonnegative sectional curvature. In particular, we have that if (M3, g (t)) is a Type IIa singular solution of the Ricci flow on a closed 3-manifold, then by Theorem A.64, the

smgularity model

(M~, goo (t)) obtained in Proposition A.70 zs a steady

gradient soliton. Since scales for some I'\,

(Moo, goo (t))

is nonflat and I'\,-noncollapsed on all

> 0, the sectional curvatures of goo (t) are positive. 7 Now

by Theorem A.58, the asymptotic scalar curvature ratio of

(Moo, goo (t))

is

6This is a special case of the point picking method described in subsection 4.2 of Chapter 8 in Volume One. 7In the splitting case, we obtain a cigar, contradicting the no local collapsing theorem.

4;6

A BASIC RICCI FLOW THEORY

equal to infinity. Thus we can apply dimension reduction, Theorem A.59, to get a second limit which splits as the product of a surface solution and a line. This second limit is a shrinking round product cylinder 52 x R8 Hence, as a consequence of Hamilton's singularity theory and Perelman's no local collapsing theorem, we have the following result, which complements Theorem A.59. THEOREM A.71 (3d Type IIa - existence of necks). If (M 3 ,g(t)) is a Type IIa singular solutwn of the Ricc~ flow on a closed 3-mamfold, then there exists a sequence of points and times (Xi, ti) such that the corresponding sl'quence (/\/1. Yi (t) ..1"/) COrlveryes to a round shrinking product cylinder 52 x

JR. A precursor to the above result is Theorem 9.9 in Volume One, which basically says that even for a Type IIa singular solution on M3 x [0, T), there exists a sequence of points (Xi, ti) with ti ~ T whose curvatures satisfy (T - ti) [Rm (Xi, ti)[ ~ C for some C > 0 independent of i and at the points (Xi, ti) the curvature operators approach that of 52 xlR after rescaling. However Theorem 9.9 in Volume One does not directly imply the existence of a cylinder limit because, for the sequence (M, gl (t) , Xi) , it not a priori clear that the curvatures are bounded in space at finite distances from X t independent of i, even at time O. The reason for this is that globally, the curvature of gi (0) , whose norm is 1 at X t , may be unbounded since the solution is Type IIa whereas the point (Xi, tt) may, for example, have curvature (T - ti) [Rm (Xi. td[ :S C for some C < 00 independent of i. Since finite time singularities are either Type I or Type IIa, we obtain the existence of necks for all finite time singular solutions on closed 3-manifolds.

80t herwise we again get a cigar limit.

APPENDIX B

Other Aspects of Ricci Flow and Related Flows 1. Convergence to Ricci solitons Given that convergence to a soliton plays a role in proving the convergence of the Ricci flow on compact surfaces (see Chapter 5 ill Volume One), it is reasonable to ask if noncompact steady solitons playa role as limiting geometries for the Ricci flow on complete surfaces (or higher-dimensional manifolds) . Since steady soliton solutions are, by definition, evolving by diffeomorphism, even if we start near a soliton metric, we cannot expect pointwise convergence of the Ricci flow unless we take the diffeomorphisms into account. We use the following notion of convergence [373]: DEFINITION B.l. Let g(t) satisfy the Ricci flow on a noncompact manifold Mn for 0 :S t < 00 and let 9 be a metric on M. We say that g(t) has modified subsequence convergence to 9 if there exist a sequence of times ti ---? 00 and a sequence of diffeomorphisms ¢t of M such that ¢ig(fi) converges uniformly to 9 on any compact set.

Since the Ricci flow is conformal on surfaces and preserves completeness, it makes sense to begin with metrics in the conformal class of the Euclidean metric, i.e., metrics of the form (B.1)

We can describe the overall 'shape' of such metrics in terms of the aperture and circumference at infinity. First, the aperture is defined by

A( 9 ) -;- l'1m L(aB (0, r)) , 1'~00 21Tr ...!...

°

where the B (0, r) are geodesic balls about some chosen point E IR? (The choice does not affect the value of the limit.) For example, the flat metric has aperture one, the cigar metric has aperture zero, and surfaces in 1R3 that are asymptotic to cones have aperture 0:/(21T), where 0' is the cone angle. The circumference at infinity is defined by

Coo(g)

~

sup inf {L(aD) I K is compact, D is open and KeD}. K

D

For flat and conical metrics, Coo is infinite, while Coo is finite for the cigar. Let R_ = max{ -R, a}. 477

478

B

OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS

THEOREM B.2 (Wu [373]). Let go be a complete metric on ]R2, of the form (B.I), with Rand lV'ul (measured using go) bounded, and I R_ dJ-l finite. Then a solution to the Ricci flow exists for all time, the aperture and circumference at infinity are preserved, and the metric has bounded subsequence convergence as t ---t 00. If R(go) > 0 and A(go) > 0, then the limit is flat. If R(go) > 0 and Coo(go) < 00, then the limit is the cigar. Note that Coo being finite implies that A = O. However, there are plenty of complete metrics with A = 0 and Coo = 00, for which the limit of the Ricci flow is not classified. An example of a surface of positive curvature with A = 0 and Coo = 00 is the paraboloid (M2,g) ~ {(x,y,z) E]R3: z = x 2 +y2}. OUTLINE OF PROOF. Short-time existence follows from the BernsteinBando-Shi estimates. As with the Ricci flow on compact surfaces (see Section 3 of Chapter 5 in Volume One), long-time existence is proved by using a potential f such that D.f = R (where D. denotes the Laplacian with respect to g) and examining the evolution of the quantity h ~ R + lV'fI2.

In fact, for metrics of the form (B.I) we can use

8u

-

at

f = -u. Because

=D.u=-R

and

%t lV'ul 2 = D.1V'uI 2 -

21V' 2uI 2,

we have 8h = D.h _ 21MI2

8t

'

where M is the symmetric tensor with components Mij

= V'iV'jU + ~Rgij.

Long-time bounds for lV'ul and R (and higher derivatives) follow. The bounds on IRI imply that the metric remains complete; in particular, the length of a given curve at time t > 0 is bounded above and below by multiples of its length at time zero. By a theorem of Huber [210], the hypothesis I R_ dJ-l < 00 implies that I RdJ-l ::; 47l'X (M) on a complete surface M. In particular, I IRI dJ-l is finite, and this is preserved by the Ricci flow. Finite total curvature, together with bounds on IV'RI at any positive time, imply that R decays to zero at infinity. One then shows that Coo (g) and A (g) are preserved under the flow. In the special case when R(go) > 0, limt-+oo eu(x,y,t) exists pointwise and is either identically zero or positive everywhere. In the latter case, 8u/8t = - R implies that 00 Rdt is bounded, and hence the limiting metric is flat. In the general case, we may define diffeomorphisms (/>t(x, y) = e- u(0,0,t)/2(x, y), so that g(t) = c/>;g(t) is constant at the origin. Then uniform bounds on

10

1.

CONVERGENCE TO RICCI SOLITONS

479

the derivative of the conformal factor for 9 give subsequence convergence, on any compact set, to a metric g. If R(go) > 0 and Coo < 00, then using the Bernstein-Bando-Shi estimates, one can show that after a short time T, JIR2 IMI 2dJ.L is bounded uniformly in time (where dJ.L indicates measure with respect to the evolving metric). The evolution equation for IMI 2dJ.L then implies that

1 (l2 00

J

21V'M12

+ 3RIMI2dJ.L) dt < 00.

Thus, either M vanishes or R vanishes for the limiting metric g. If M = 0, then 9 is a gradient soliton; by Proposition 1.25, it is either flat or the cigar metric. However, Coo < 00 precludes a flat limit. If R(go) > 0 and A > 0, then one may use the Harnack inequality to show that tRmax is bounded. It follows that the limit 9 is flat in this 0 case. One may also study the Ricci flow for a solution of the form (B. 1) in terms of the nonlinear diffusion equation satisfied by the conformal factor v ~ eU , 8v (B.2) 8t = .6. log v, where .6. is the standard Laplacian on ]R2. (Note that v the cigar, where we write x = (x, y).)

= l/(k + Ix1 2) for

REMARK B.3. As pointed out by Angenent in an appendix to [373], the equation (B.2) is a limiting case of the porous medium equation

8v = .6.vm 8t as the positive exponent m tends to zero. For, substituting t = gives (B.3)

r:

~~ = .6. (vm and taking m

~

T /m

in (B.3)

1)

0 gives 8v/8T = .6. (log v).

In connection with Ricci flow on ]R2 we also have the following result, which says that metrics that start near the cigar converge to a cigar (in the same conformal class), but under weaker assumptions than in the above theorem. THEOREM B.4 (Hsu [206]). Suppose that vo E Lfoc(]R2) for p > 1, 0 ~ vo ~ 2/(,6lxI 2) for,6 > 0, and vo - 2/(,6(lxI 2 + ko)) E Ll(]R2) for ko > O. Then there exists a unique positive solution of (B.2), defined for 0 < t < 00, such that limt-+o v = Vo in LIon any compact set and such that

t~~ e2{3t v (e 2{3t m

Ll(]R2), for some kl > o.

x, t)

= ~(lxI2 + k1)-1

ISO

B.

OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS

In higher dimensions, there are not many results. However, consider complete warped product metrics on ]R3, of the form (B.4) where gean is the standard metric on S2. (Recall that the sectional curvatures VI. V2 of such metrics are given in terms of'l17 by (1.58).) For such metrics, we call prove convergence to a soliton if the curvature is positive and bounded and the manifuld "opens up" like a paraboloid. THEOREM B.5 (Ivey [220]). Let 9 be a complete metric on form (B.4). Suppose

]R3

of the

(B.5) and s'uppose

(B.6) for positzve constants C and Z, and (B.7)

liminf ('--'00

(aDr w

2)

> O.

Then the solution of the Rz('cz flow 'Wzth g(O) addztion. (B.~)

limsup (-->=

(aDr w 2 )

=9

exists for all time. If, zn

< 00,

then the flow converyes to a mtationally symmetric steady gradzent soliton, zn the sense of the C=-Cheeger-Gromov topology (see Theorem 3.10).

Intuitively, the condition (B. 7) means that the area of a sphere centered at the origin grows at least as fast as for a paraboloid, while (B.8) means that the sphere area grows no faster than a paraboloid. In other words, the metric becomes fiat as r - t 00, but not too fiat. (By contrast, the result of Shi [330], which gives convergence of the Ricci fiow to a fiat metric, assumes that sectional curvatures fall off like r-(2+c).) The condition V2 :::; ZVl means that as the sectional curvature Vl along the planes tangent to the spheres becomes fiat as r - t +00, the sectional curvature V2 of the perpendicular planes also becomes fiat. Of course, for the Bryant soliton of subsection 3.2 of Chapter 1, V2 falls off much faster than VI. However, it is not difficult to construct other metrics that satisfy the conditions in the theorem; see [220] for details. For metrics on ]R3 of the form (B.4), the warping function satisfies a quasilinear heat equation

aw

at = w

, , 1 - (w l )2 -

w

=

-(Vl

+ V2)W.

1. CONVERGENCE TO RICCI SOLITONS

481

However, the time-derivative a/at under the Ricci flow and the radialderivative a/or do not commute; in fact,

a a oJ [ at' or = 2V2 or . We now outline the proof of Theorem B.5; again, details may be found in [220]. First, long-time existence is proved by obtaining an estimate, depending only C and Z, for Iww"'l. Then, convergence to a soliton is proved by examining the evolution of the quantity

Q ~ R+ ((VI +V2)W/w,)2. Comparing with (1.58) shows that Q coincides with R+ 1\7112 when 9 is the Bryant soliton. Thus, Q is constant for the soliton. In general, it evolves by the equation

(B.9)

(8t8) - ~ Q

- 2

where

(8Q )2

(ww')2

= - (1

+ (w')2 +WV2)2 8r

( WV2 w'

WW'

)

oQ

+ 1 + (w')2 + WV2 (VI + V2) or'

~ = (~)2 8r

_

2W ' W

~ 8r

is the Laplacian with respect to 9 for functions depending on rand t only. Given (B.5) and (B.6), the paraboloid condition (B.8) is equivalent to Q having a positive lower bound. In fact, (B.6) implies that (B.10)

(ww')2 ~ (1

+ Z)2/Q.

As part of the proof of long-time existence, one shows that conditions (B.5) and (B.6) persist in time. These, together with (B.10), imply that

_~) ~ _ A (oQ) 2 _ (~ at Q or

B 1 8Q I

Or

for some positive constants A, B. Applying the maximum principle to the corresponding inequality for rP = l/Q shows that the lower bound on Q persists in time. Finally, existence of a limiting metric is proven using the compactness theorem, and showing that it is a nontrivial soliton comes by appealing to Theorem A.64 and the positive lower bound for Q. REMARK B.6. The generalization of Theorem B.5 to rotationally symmetric Ricci flow in higher dimensions should be straightforward. It may even be possible to generalize at least the long-time existence to the nonrotationally-symmetric case by finding a generalization for the quantity Q (open problem). It would also be interesting to find conditions under which the flow converges to the product of the cigar metric with the real line.

482

B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS

2. The mean curvature flow In this section we give a brief introduction to the mean curvature flow and some monotonicity formulas. It is interesting to compare these monotonicity formulas for the mean curvature flow to those for the Ricci flow. We also refer the reader to the books by Ecker [132] and X.-P. Zhu [386]. 2.1. Mean curvature flow of hypersurfaces in Riemannian manifolds. Let ('pn+l, 9P) be an orientable Riemannian manifold and let M n be an orientable differentiable manifold. The first fundamental form of an embedded hypersurface X : M ---+ P is defined by 9 (V, W) ~ 9p (V, W)

for V, W E TxX (M). More generally, for an immersed hypersurface, we define 9 (V, W) ~ (X*V,X*W) for V, W E TpM. Let 1/ denote the choice of a smooth unit normal vector field to M. The second fundamental form is defined by h (V, W) ~ (Dvl/, W)

=-

(DvW, 1/)

for V, WE TxX (M), where D denotes the Riemannian covariant derivative of (P, 9P) and ( , ) ~ 9p ( , ). To get the second equality in the line above, we extend W to a tangent vector field in a neighborhood of x and use (1/, W) == O. In particular, 0 = V (1/, W) = (Dvl/, W) + (DvW, 1/). The mean curvature is the trace of the second fundamental form: n

H~ Lh(ei,ei), i=l

where {ed is an orthonormal frame on X (M) . A time-dependent immersion X t = X (', t) : M ---+ P, t E [0, T), is a solution of the mean curvature flow (MCF) of a hypersurface in a Riemannian manifold if

(B.ll)

ax (p, t) = H. . (x (p, t)) ~ -H (p, t) . 1/ (p, t), at

p E M, t

E

[0, T),

where ii is called the mean curvature vector. When the X t are embeddings, we define M t ~ X t (M). From now on we shall assume that the X t are embeddings, although for the most part, the following discussion holds for immersed hypersurfaces. Let {xi} ~=1 denote local coordinates on M so that {¥xt} are local coordinates on M t . We have

9ij

~ 9 (~~, ~~) = (~~, ~~),

hij

~ h (aaX , aax) xt xJ

=

/aax,DMI/) = xt ax)

\

- /

\

DM aax, ,1/) . ax' x J

2.

THE MEAN CURVATURE FLOW

483

Note that H = gij h ij and

(B.12)

D Mil = hjkg

ke

axJ

ax ax e'

We have the following basic formulas for solutions of the mean curvature flow. LEMMA B. 7 (Huisken). The evolution of the first fundamental form (induced metric), normal, and second fundamental form are given by the following:

a

(B.13)

at gij = -2Hhij ,

(B.14)

DltY at

(B.15)

ata hij =

= \lH, ke \l/VjH - Hhjkg hei

= t:"hij -

+ H (Rmp )lIijll 2Hhikhkj + Ihl 2 hij

+ hij (Rc p) hik (Rc P ) jk - hkj (Rc P )ik + 2 (Rm P ) kije hke + hki (Rm P ) II kj II + hkj (Rm P ) IIkill - Di (Rcp)jll - D j (RcP)ill + DII (RCP)iJ'

(B.16)

111/ -

a

(B.17)

at H = t:"H + Ihl

(B.18)

!!.-dH

2

H

+H

(Rc p)1II/ ,

at ,.. = -H 2dH,..,

where \l is the covariant derivative with respect to the induced metric on the hypersurface and dp, is the volume form of the evolving hypersurface.

REMARK B.8. Technically, we should consider these tensors (or sections of bundles) as existing on the domain manifold M. However, we shall often view them as tensors on the evolving hypersurface M t = X t (M). The time-derivative of the unit normal is expressed slightly differently since it is actually the covariant derivative of II in the direction ~~ along the path t I--t X t (p) for p E M fixed. On the other hand, if we view 9 and h as on the fixed manifold M, we have the ordinary time-derivatives, whereas if we view 9 and h as on the evolving M t , then the time-derivatives are actually Da. at

PROOF. While carrying out the computations below, keep in mind that the inner product of a tangential vector with a normal vector is zero. The

-18!

B OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS

evolution of the metric is given by

%t gil = ( D%;. (aa~) , ~~ ) + ( ~~ , D:~ (aa~)) = -H (DQLV' aT' =

-2Hh i.!,

aa~) x - H(aa~' x Dax ax) v) J

g;) (g;, v) = O. This is (B.13).

where we used (v, = Since (~~, v) = 0 and

X 0= aat (v, aax~X ) = (Daxv, (-Hv)) at aax~ ) + \/v,DcJx a,' the normal v evolves by

··aHaX Dax v = gtJ_. at axt -. ax) = V' H,

(B.19)

where V' H is the gradient 011 the hypersurface of H. This is (B.14). The evolution of the second fundamental form is (we use [~"'i, g~] = 0)

~ hiJ = -aat (Dax v) ax' aa~, Xl

ot

(Daxat (Dax aX) axJ,Daxv) J ,v) - (Dax axi ax axi ax at = _ (Dax (DQJ£ (ax)) ,v) _ (Rmp (ax, ax) ax,v) ax' ax) at at axt ax.! ax ,Daxv) - (DQL ax' axl at = -

(Dax ax' (aa~v xl + HDax ax) v) ,v) + H(Rmp (v, aa~) xt aax, xJ v) - (DQL V'H) ax' aa~, xJ kC ax k aH 2H = aa·a . + Hhjkg ( Daxa ",v ) +H(Rmp)ViJV-fiJ·a k xt xJ ax' x< x = V'iV'jH - Hhjklfhfi + H(Rmp)vijv' =

where we used (B.ll), (B.19), and (B.12), and where

ax ' ax) fiJk = 9kf ( D QL -a ax"" ax' x J are the Christoffel symbols and V' is the covariant derivative with respect to the induced metric on the hypersurface. This is (B.15).

2 THE MEAN CURVATURE FLOW

-185

'fracing the above formula, we see that the mean curvature evolves by

This is (B.17). Equation (B.18) follows from (B.13) and

Finally we rewrite the evolution equation for hij to see (B.16). Recall that the Gauss equations say that for any tangent vectors X, Y, Z, W on M, (RmM) (X,Y,Z,W)

=

(Rmp) (X,Y,Z,W)

+ h (X, W) h (y, Z) -

h (X, Z) h (Y, W) .

'fracing implies (in index notation)

(RCM)il = (RcP)if - (Rmp)vifv

+ Hhif -

h;e·

The Codazzi equations say for any X, Y, Z E T M, (\lxh) (Y, Z) - (\lyh) (X, Z)

=-

(Rmp (X. Y) Z, v) .

In index notation, this is \l ihkj

= \l khij

- (Rm P )ikjv .

'fracing over the Y and Z components in the hypersurface directions, we have \l H - div (h)

=-

Rcp (v).

To obtain (B.16) from (B.15) we note that, by using the Codazzi equations and Gauss equations, we obtain

(B.20) where Rcp (v)

~

(Rcp))vdx) is a I-form on M. Note that

(rp

)r = -~v (:~, ~;) = j

-hi)

and similarly k

(rp )iv

1 kl / ax ax) axi ' axl

="29 v \

k£

=9

hit·

486

B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS

Thus, considering Rc P as a 2-tensor on 'P and changing its covariant derivative to the one with respect to gp in the formula (B.20), we have

'h'VjH

= \h'Vkhkj - Di (Rcp)jv - (fp)rj (Rcp)w - (fp)7v (RCP)jk = \1 /iJ khkj - Di (Rc p) jv

+ hij (Rcp)w -

hf (RcP)jk'

Commuting derivatives and applying the Codazzi and Gauss equations, we compute

\1i\1jH = \1k\1ihkj - (RCM)uh£j - (RmM)ikj£hk£ - Di (RcP)jv

+ hij (Rcp)w - hf (RcP)jk = \1 k\1 khij - \1 k (Rm P )ikjv - (Hhu - h7£) h£j - (hi£hkj - hijhk£) h k£

+ (Rm P )vuv h£j - (Rm P )ikj£ hk£ - Di (Rcp)jv + hij (Rcp)w - hf (RcP)jk = b..hij - Hh7j + Ihl 2 hij - Dk (Rm P )ikjv + hki (Rmp)vkjv + hkk (RmP)ivjv - h1 (Rmp)ikj£ - (Rc p)u h£j + (Rm P )vuv h£j - (Rm P )ikj£ hk£ - Di (Rcp)jv + hij (Rcp)w - hf (RCP)jk' - (Rc p)u h£j

where h~ ~ hikhkj and in the third line (Rmp )ikjv dx i 0 dx k 0 dx j is considered as a 3-tensor. This is known as Simons' identity. Hence, under the mean curvature flow

ata hij = \1i\1jH -

k£ Hhjkg h£i + H (Rmp )vijv

= b..hij - 2Hh;j

+ Ihl 2 hij

+ Dv (RcP)ij - Di (Rcp)jv (Rc P )i£ h£j + (Rm P ) vi£v h£j - (Rm P )ikj£ hkl

- Dj (RCP)iv -

+ hij (Rcp)w - hf (RcP))k + hki (Rmp)vkjv - h1 (Rmp)ikj£' where we used the second Bianchi identity:

o

2

THE MEAN CURVATURE FLOW

487

EXERCISE B.9. Compute gt (lhl2 - ~H2) . See §5 of Huisken [212] for a study of under what conditions on (P, 9P) the pinching estimate

Ihl 2- .!. H2 ~ C . H 2- t5 n

holds for some 8 > 0 and C
. E ~ such that at some time

(B.44)

More generally, a solution to the XCF is a cross curvature breather if there exist times t1 < t2, a diffeomorphism 'P : M -+ M, and a > 0 such

500

B OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS

that (B.45) We have the following nonexistence result. B.27 (XCF breathers are trivial). If (M3, 9 (t)) zs a cross curvature breather with negative sectwnal curvature, then 9 (t) has constant sectional curvature. LEMMA

PROOF.

We have

d r1M ·gZJciJdJ-l > 0,

dt Vol (g (t)) =

so that the breather equation (B.45), i.e., 9 (t2) = w.p*g (tJ) for tl < t2, implies a > 1. On the other hand, J (g (t2)) = a 1/ 2J (g (tJ)) 2': 0, which contradicts the monotonicity formula (B.39) unless J (g (t2)) = J (g (td) = 0, in which case 9 (t) has constant sectional curvature. 0 4. Notes and commentary

Lemma B.7 is Lemma 3.3, Theorem 3.4 and Corollary 3.5(i) in [212]. See l\la and Chen [259] for a study of the cross curvature flow for certain classes of metrics on sphere and torus bundles over the circle. For work on the stability of the cross curvature flow at a hyperbolic metric, see Young and one of the authors [235].

APPENDIX C

Glossary adjoint heat equation. When associated to the Ricci flow, the equation is

au

at + Dou -

Ru

= O.

For a fixed metric, the adjoint heat equation is just the backward heat equation ~~ + Dou = O. ancient solution. A solution of the Ricci flow which exists on a time interval of the form (-00, w) , where w E (-00,00]. Limits of dilations about finite time singular solutions on closed manifolds are ancient solutions which are K-noncollapsed at all scales. For this reason a substantial part of the subject of Ricci flow is devoted to the study of ancient solutions. asymptotic scalar curvature ratio (ASCR). For a complete noncompact Riemannian manifold, ASCR(g) = limsup R(x)d(x,O)2,

°

d(x,O)--->oo

where E Mn is any choice of origin. This definition is independent of the choice of EM. For a complete ancient solution of the Ricci flow 9 (t) on a noncompact manifold with bounded nonnegative curvature operator, ASCR (g (t)) is independent of time. The ASCR is used to study the geometry at infinity of solutions of the Ricci flow and in particular to perform dimension reduction when ASCR = 00. asymptotic volume ratio (AVR). On a complete noncompact Riemannian with nonnegative Ricci curvature, AVR is the limit of the monotone quantity Vol ~n(p,r) as r ~ 00. This definition is independent of the choice of p EM. Ancient solutions with bounded nonnegative curvature operator have AVR = O. Expanding gradient Ricci solitons with positive Ricci curvatures have AVR > O. backward Ricci flow. The equation is

°

a aT g = 2Rc. Usually obtained by taking a solution 9 (t) to the Ricci flow and defining T (t) ~ to - t for some to. Bernstein-Bando-8hi estimates (also BB8 estimates). Shorttime estimates for the derivatives of the curvatures of solutions of the Ricci flow assuming global pointwise bounds on the curvatures. Roughly, given 501

502

C. GLOSSARY

a solution of the Ricci flow, if IRm (t)lg(t) :::; K on a time interval [0, T) of length on the order of K- 1 , then IV m Rm (t)lg(t) :::;

~;:/~

on that interval. The BBS estimates are used to obtain higher derivative of curvature estimates from pointwise bounds on the curvatures. In particular, they are used in the proof of the Coo compactness theorem for sequences of solutions assuming only uniform pointwise bounds on the curvatures and injectivity radius estimates. Bianchi identity. The first and second Bianchi identities are

+ Rjkil + Rkijl = 0, ViRjklm + VjRkilm + VkRijlm = 0, Rijkl

respectively. The Bianchi identities reflect the diffeomorphism invariance of the curvature. In the Ricci flow they are used to derive various evolution equations including the heat-like equation for the Riemann curvature tensor. Bishop---Gromov volume comparison theorem. An upper bound for the volume of balls given a lower bound for the Ricci curvature. This bound is sharp in the sense that equality holds for complete, simply-connected manifolds with constant sectional curvature. Bochner formula. A class of formulas where one computes the Laplacian of some quantity (such as a gradient quantity). Such formulas are often used to prove the nonexistence of nontrivial solutions to certain equations. For example, harmonic i-forms on closed manifolds with negative Ricci curvatures are trivial. In the Ricci flow, Bochner-type formulas (where the Laplacian is replaced by the heat operator) take the form of evolution equations which yield estimates after the application of the maximum principle. bounded geometry. A sequence or family of Riemannian manifolds has bounded geometry if the curvatures and their derivatives are uniformly bounded (depending on the number of derivatives). breather solution. A solution of the Ricci flow which, in the space of metrics modulo diffeomorphisms, is a periodic orbit. Bryant soliton. The complete, rotationally symmetric steady gradient Ricci soliton on Euclidean 3-space. The Bryant soliton has sectional curvatures decaying inverse linearly in the distance to the origin and hence has ASCR = 00 and AVR = 0. It is also expected to be the limit of the conjectured degenerate neckpinch. Buscher duality. A duality transformation of gradient Ricci solitons on warped products with circle or torus fibers. Calabi's trick. A typical way to localize maximum principle arguments is to multiply the quantity being estimated by a cut-off function depending on the distance to a point. Calabi's trick is a way to deal with the issue of the cut-off function being only Lipschitz continuous (since the distance function is only Lipschitz continuous).

503

C. GLOSSARY

Cartan structure equations. Given an orthonormal frame {ei} and dual coframe {w j }, they are the identities satisfied by the connection 1forms

{w;} and the curvature 2-forms {Rm 1}:

The Cartan structure equations are useful for computing curvatures, especially for metrics with some sort of symmetry. They may also be used for general calculations in geometric analysis. Cheeger-Gromov convergence. (See compactness theorem.) Christoffel symbols. The components of the Levi-Civita connection with respect to a local coordinate system: .

k

k

'iJ 8 / 8x i8/8xJ = rij 8/8x . The variation formula for the Christoffel symbols is the first step in computing the variation formula for the curvatures. The evolution equation for the Christoffel symbols is also used to derive evolution equations for quantities involving covariant derivatives. cigar soliton. The rotationally symmetric steady gradient Ricci soliton on the plane defined by

The scalar curvature of the cigar is Rr.

=

4 2

1+x +y

2

2

= 4 sech s,

where s is the distance to the origin. Note that gr. is asymptotic to a cylinder and Rr. decays exponentially fast. Perelman's no local collapsing theorem implies the cigar soliton and its product cannot occur as a limit of a finite time singularity on a closed manifold. classical entropy. An integral of the form M flog fdJ-l. collapsible manifold. A manifold admitting a sequence of metrics with uniformly bounded curvature and maximum injectivity radius tending to zero. compactness theorem (Cheeger-Gromov-type). If a pointed sequence of complete metrics or solutions of Ricci flow has uniformly bounded curvature and injectivity radius· at the origins uniformly bounded from below, then there exists a subsequence which converges to a complete metric or solution. The convergence is after the pull-back by diffeomorphisms, which we call Cheeger-Gromov convergence. conjugate heat equation. (See adjoint heat equation.)

J

c.

504

GLOSSARY

cosmological constant. A constant c introduced into the Ricci flow equation:

o at gij = -2 (Rij + cgi)) .

i

The case c = is useful in converting expanding Ricci solitons to steady Ricci solitons. cross curvature flow. A fully nonlinear flow of metrics on 3-manifolds with either negative sectional curvature everywhere or positive sectional curvature everywhere. curvature gap estimate. For long existing solutions, a time-dependent lower bound for the spatial supremums of the curvatures. See Lemmas 8.7. 8.9, and 8.11 in [111]. curvature operator. The self-adjoint fiberwise-lillear map Rm: A2MTI ~ A2M

n

defined by Rm (a)i) ~ RijkfCtfk. curve shortening flow (CSF). The evolution equation for a plane curve given by

-ox = at

-/'\,1)

'

where K, hi the curvature and 1/ is the unit outward normal. It is useful to compare the CSF with the Ricci flow (especially on surfaces). degenerate neckpinch. A conjectured Type IIa singularity on the nsphere where a neck pinches at the same time its cap shrinks leading to a cusp-like singularity. Such a singularity has been proven by Angenent and Velazquez to occur for the mean curvature flow. DeTurck's trick. (See also Ricci-DeTurck flow.) A method to prove short-time existence of the Ricci flow using the Ricci-DeThrck flow. differential Harnack estimate. Any of a class of gradient-like estimates for solutions of parabolic and heat-type equations. dilaton. (See Perelman's energy.) dimension reduction. For certain classes of complete, noncompact solutions of the Ricci flow, a method of picking points tending to spatial infinity and blowing down the corresponding pointed sequence of solutions to obtain a limit solution which splits off a line. Einstein-Hilbert functional. The functional of Riemannian metrics: E (g) = RdJ.L, where R is the scalar curvature. Einstein metric. A metric with constant Ricci curvature. Einstein summation convention. The convention in tensor calculus where repeated indices are summed. Strictly speaking the summed indices should be one lower and one upper, but in practice we do not always bother to lower and raise indices. energy. (See Perelman's energy.) entropy. (See classical entropy, Hamilton's entropy for surfaces, and Perelman's entropy.)

IMn

c.

GLOSSARY

505

eternal solution. A solution of the Ricci flow existing on the time interval (-00,00). Note that eternal solutions are ancient solutions which are immortal. expanding gradient Ricci soliton (a.k.a. expander). A gradient Ricci soliton which is evolving by the pull-back by diffeomorphisms and scalings greater than 1. Gaussian soliton. Euclidean space as either a shrinking or expanding soliton. geometrization conjecture. (See Thurston's geometrization conjecture.) gradient flow. The evolution of a geometric object in the direction of steepest ascent of a functional. gradient Ricci soliton. A Ricci soliton which is flowing along diffeomorphisms generated by a gradient vector field. Gromoll-Meyer theorem. Any complete noncompact Riemannian manifold with positive sectional curvature is diffeomorphic to Euclidean space. Hamilton's entropy for surfaces. The functional E (g) = fM2log R· RdJ.t defined for metrics on surfaces with positive curvature. Hamilton-Iveyestimate. A pointwise estimate for the curvatures of solutions of the Ricci flow on closed 3-manifolds (with normalized initial data) which implies that, at a point where there is a sufficiently large (in magnitude) negative sectional curvature, the largest sectional curvature at that point is both positive and much larger in magnitude. In dimension 3 the Hamilton-Ivey estimate implies that the singularity models of finite time singular solutions have nonnegative sectional curvature. harmonic map. A map between Riemannian manifolds f: (Mn,g)---+ (Nm, h) satisfying tlg,hf = 0, where tlg,h is the map Laplacian. (See map Laplacian. ) harmonic map heat flow. The equation is %f = tlg,hf. Harnack estimate (See differential Harnack estimate.) heat equation. For functions on a Riemannian manifold: ~~ = tlu. This equation is the basic analytic model for geometric evolution equations including the Ricci flow. heat operator. The operator ~ - tl appearing in the heat equation. Hodge Laplacian. Acting on differential forms: tld = - (d& + &d) . homogeneous space. A Riemannian manifold (M n , g) such that for every x, y E M there is an isometry t- : M ---+ M with d x) = y. Huisken's monotonicity formula. An integral monotonicity formula for hypersurfaces in Euclidean space evolving by the mean curvature flow using the fundamental solution to the adjoint heat equation. immortal solution. A solution of the Ricci flow which exists on a time interval of the form (a, 00), where a E [-00,00). isoperimetric estimate. A monotonicity formula for the isoperimetric ratios of solutions of the Ricci flow. Examples are Hamilton's estimates

c.

506

GLOSSARY

for solutions on closed surfaces and Type I singular solutions on closed 3manifolds. Jacobi field. A variation vector field of a I-parameter family of geodesics. Kahler-Ricci flow. The Ricci flow of Kahler metrics. Note that on a closed manifold, an initial metric which is Kahler remains Kahler under the Ricci flow. ~-noncollapsed at all scales. A metric (or solution) which is ~­ noncollapsed below scale p for all p < 00. ~-noncollapsed below the scale p. A Riemannian manifold satisfying Vol~~x,r) ~ ~ for any metric ball B(x, r) with IRm I ~ r- 2 in B (x, r) and r < p. ~-solution (or ancient ~-solution). A complete ancient solution which is ~-noncollapsed on all scales, has bounded nonnegative curvature operator, and is not flat. In dimension 3, a large part of singularity analysis in Ricci flow is to classify ancient ~-solutions. L-distance. A space-time distance-like function for solutions of the backward Ricci flow obtained by taking the infimum of the £-length. The L-distance between two points may not always be nonnegative. i-distance function. (See reduced distance.) £-exponential map. The Ricci flow analogue of the Riemannian exponential map. £-geodesic. A time-parametrized path in a solution of the backward Ricci flow which is a critical point of the £-length functional. £-Jacobi field. A variation vector field of a I-parameter family of £-geodesics. £-length. A length-like functional for time-parametrized paths in solutions of the backward Ricci flow. The £-length of a path may not be positive. Laplacian (or rough Laplacian). On Euclidean space the operator ~ = E~l 8(~~)2. On a Riemannian manifold, the second-order linear differential operator ~ = gij·~h'Vj acting on tensors. Levi-Civita connection. The unique linear torsion-free connection on the tangent bundle compatible with the metric. (Also called the Riemannian connection. ) Lichnerowicz Laplacian. The second-order differential operator ~L acting on symmetric 2-tensors defined by (VI-3.6), i.e., ~LVij ~ ~Vjk

+ 2gqp R~jk vrp -

gqp R Jp Vqk - gqp R kp V Jq .

Lichnerowicz Laplacian heat equation. The heat-like equation = (~LV)ij for symmetric 2-tensors. linear trace Harnack estimate. A differential Harnack estimate for nonnegative solutions of the Lichnerowicz Laplacian heat equation coupled to a solution of the Ricci flow with nonnegative curvature operator, which generalizes the trace Harnack estimate. (See trace Harnack estimate.) &tVij

C. GLOSSARY

507

little loop conjecture. Hamilton's conjecture which is essentially equivalent to Perelman's no local collapsing theorem (e.g., the conjecture is now a theorem). Li-Yau-Hamilton (LYH) inequality. (See differential Harnack estimate.) locally homogeneous space. (See also homogeneous space.) A Riemannian manifold (Mn, g) such that for every x, y E M there exist open neighborhoods U of x and V of y and an isometry 1-: U - V with t-{x) = y. A complete simply-connected locally homogeneous space is a homogeneous space. locally Lipschitz function. A function which locally has a finite Lipschitz constant. logarithmic Sobolev inequality. A Sobolev-type inequality which essentially bounds the classical entropy of a function by the L2- norm of thf:l first derivative of the function. long existing solutions. Solutions which exist on a time interval of infinite duration. long-time existence. The existence of a solution of the Ricci flow on a closed manifold as long as the Riemann curvature tensor remains bounded. By a result of Sesum, in the above statement the Riemann curvature tensor may be replaced by the Ricci tensor. map Laplacian. Given a map f : (Mn, g) - (Nm, h) between Riemannian manifolds, Ag,hf is the trace with respect to 9 of the second covariant derivative of f. (See (3.39) in Volume One.) matrix Harnack estimate. In Ricci flow a certain tensor inequality of Hamilton for solutions with bounded nonnegative curvature operator. A consequence is the trace Harnack estimate. One application of the matrix Harnack estimate is in the proof that an ancient solution with nonnegative curvature operator and which attains the space-time maximum of the scalar curvature is a steady gradient Ricci soliton. maximum principle. (Also called the weak maximum principle.) The first and second derivative tests applied to heat-type equations to obtain bounds for their solutions. The basic idea is to use the inequalities Au ~ 0 and \7u = 0 at a spatial minimum of a function u and the inequality !JJt ~ 0 at a minimum in space-time up to that time. This principle applies to scalars, tensors, and systems. maximum volume growth. A noncom pact manifold has maximum volume growth if for some point p there exists c > 0 such that Vol B (p, r) ~ ern for all r > O. This is the same as AVR > 0 (when Rc ~ 0). mean curvature flow. The evolution of a submanifold in a Riemannian manifold in the direction of its mean curvature vector. modified Ricci flow. An equation of the form = -2 (Rc +VV f), where f is a function on space-time. Often coupled to an equation for f such as Perelman's equation Wt = -R - Af.

Itg

508

C. GLOSSARY

monotonicity formula. Any formula which implies the monotonicity of a pointwise or integral quantity under a geometric evolution equation. Examples are entropy and Harnack estimates. J.t-invariant. An invariant of a metric and a positive number (scale) obtained from Perelman's entropy functional W (g, I, T) by taking the infimum over all I satisfying the constraint fMn (47rT)-n/2 e- f dJ.t = 1. There is a monotonicity formula for this invariant under the Ricci flow. neckpinch. A finite time (Type I) singular solution of the Ricci flow where a region of the manifold asymptotically approaches a shrinking round cylinder sn-l X lR. Sufficient conditions for initial metrics on sn for a neckpinch have been obtained by Angenent and one of the authors. no local collapsing. (Also abbreviated NLC.) A fundamental theorem of Perelman which applies to all finite time solutions of the Ricci flow on closed manifolds. It says that given such a solution of the Ricci flow and a finite scale p > 0, there exists a constant K. > 0 such that for any ball of radius r < p with curvature bounded by r- 2 in the ball, the volume ratio of the ball is at least K.. We say that the solution is K.-noncollapsed below the scale p. v-invariant. A metric invariant obtained from the J.t-invariant by taking the infimum over all T > O. This invariant may be -00. null-eigenvector assumption. A condition, in the statement of the maximum principle for 2-tensors, on the form of a heat-type equation which ensures that the nonnegativity of the 2-tensor is preserved under this heattype equation. parabolic equation. In the context of Ricci flow, a heat-type equation (which is second-order). In general, parabolicity of a nonlinear partial differential equation is defined using the symbol of its linearization. Perelman's energy. The functional

This invariant appeared previously in mathematical physics (e.g., string theory) and I is known as the dilaton. Perelman's entropy. The following functional of the triple of a metric, a function, and a positive constant:

W(g,I,T) = fMn (T (R+

IVII2) + (f -

n)) (47rT)-n/2 e- f dJ.t.

Perelman's Harnack (LYH) estimate. A differential Harnack (e.g., gradient) estimate for fundamental solutions of the adjoint heat equation coupled to the Ricci flow. Perelman solution. The non-explicit 3-dimensional analogue of the Rosenau solution. The Perelman solution is rotationally symmetric and has positive sectional curvature. Its backward limit as t ~ -00 is the Bryant soliton.

C. GLOSSARY

509

Poincare conjecture. The conjecture that any simply-connected closed 3-manifold is diffeomorphic to the 3-sphere. (In dimension 3, the topological and differentiable categories are the same.) Hamilton's program and Perelman's work aim to complete a proof of the Poincare conjecture using Ricci flow. positive (nonnegative) curvature operator. The eigenvalues of the curvature operator are positive (nonnegative). potentially infinite dimensions. A device which Perelman combined with the space-time approach to the Ricci flow to embed solutions of the Ricci flow into a potentially Ricci flat manifold with potentially infinite dimension. preconvergent sequence. A sequence for which a subsequence converges. quasi-Einstein metric. The mathematical physics jargon for a nonEinstein gradient Ricci soliton. Rademacher's Theorem. The result that a locally Lipschitz function is differentiable almost everywhere. reaction-diffusion equation. A heat-type equation consisting of the heat equation plus a nonlinear term which is zeroth order in the solution. reduced distance. The distance-like function for solutions of the back(q, 7). (See L-distance.) Partly ward Ricci flow defined by f (q, 7) = motivated by consideration of the heat kernel and the Li-Yau distance function for positive solutions of the heat equation. reduced volume. For a solution to the backward Ricci flow, the timedependent invariant

2FrL

V (7) = {

iMn

(47rT)-n/2 e-£(Q,T)djLg(T) (q).

Ricci flow. The equation for metrics is %t 9ij = - 2~j. This equation was discovered and developed by Richard Hamilton and is the subject of this book. Ricci soliton. (See also gradient Ricci soliton.) A self-similar solution of the Ricci flow. That is, the solution evolves by scaling plus the Lie derivative of the metric with respect to some vector field. Ricci tensor. The trace of the Riemann curvature operator: n

Rc (X, Y)

= trace (X ~ Rm (X, Y) Z) =

L (Rm (ei' X) Y, ei) . i=l

Ricci-DeTurck flow. A modification of the Ricci flow which is a strictly parabolic system. This equation is essentially equivalent to the Ricci flow via the pull-back by diffeomorphisms and is used to prove short-time existence for solutions on closed manifolds. Riemann curvature operator. (See curvature operator.) Riemann curvature tensor. The curvature (3, i)-tensor obtained by anti-commuting in a tensorial way the covariant derivatives acting on vector

C. GLOSSARY

510

fields. Formally it is defined by Rm (X, Y) Z

'*' VxV'yZ - V'yVxZ - '\7[X,YjZ.

Rosenau solution. An explicit rotationally symmetric ancient solution on the 2-sphere with positive curvature. Its backward limit as t ~ -00 (without rescaling) is the cigar soliton. scalar curvature. The tra.ce of the Ricci tensor: R = L:~=l Rc (ei' ei) = gij~j.

sectional curvature. The number K (P) = (Rm (el' e2) e2, el) associated to a 2-plane in a tangent spaee P c TxM, where {el' e2} is an orthonormal basis of P. self-similar solution. (For the Ricci flow, see Ricci soliton.) Shi's local derivative estimate. A local estimate for the covariant derivatives of the Riemann curvature tensor. short-time existence. The existence, when it holds, of a solution to the initial-value problem for the rued flow on some nontrivial time interval. For example, for a smooth initial metric on a closed manifold. shrinking gradient Ricci soliton (a.k.a. shrinker). A gradient Ricci soliton which is evolving by the pull-back by diffeomorphisms and scalings legs than 1. singular solution. A solution on a maximal time interval. If the maximal titne interval [0, T) is flnite, then sup

IRml
o. Invent. Math. 89 (1987), no. 2, 225-246. [345] Tian, Gang. Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math. 101 (1990), no. 1, 101-172. [346] Tian, Gang. Kahler-Emstem metncs with positive scalar curvature. Invent. Math. 130 (1997), no. 1, 1-37. [347] Tian, Gang. Canomcal metrics m Kahler geometry. Notes taken by Meike Akveld. Lectures in Mathematics ETR Zurich. Birkhauser Verlag, Basel, 2000. [348] Tian, Gang; Yau, Shing-Tung. Kahler-Einstein metrics on complex surfaces wdh C 1 > o. Comm. Math. Phys. 112 (1987), no. 1, 175-203. [349] Tian, Gang; Yau, Shing-Tung. Complete Kahler mamfolds with zero Ricci curvature. I. J. Amer. Math. Soc. 3 (1990), no. 3, 579-609. [350] Tian, Gang; Yau, Shing-Tung. Complete Kahler manifolds with zero Rzccz curvature. II. Invent. Math. 106 (1991), no. 1, 27-60. [351] Tian, Gang; Zhang, Zhou. Preprint. [352] Tian, Gang; Zhu, Xiaohua. Umqueness of Kahler-Ricci sohtons on compact Kahler manifolds. C. R. Acad. Sci. Paris Ser. I Math. 329 (1999), no. 11, 991-995. [353] Tian, Gang; Zhu, Xiaohua. Umqueness of Kahler-Riccz solitons. Acta Math. 184 (2000), no. 2, 271-305. [354] Tian, Gang; Zhu, Xiaohua. A new holomorphic mvanant and umqueness of KahlerRzccz solztons. Comment. Math. Relv. 77 (2002), no. 2, 297-325. [355] Tian, Gang; Zhu, Xiaohua. Preprint. [356] Topping, Peter. Lectures on the Riccz flow. London Mathematical Society Lecture Note Series (No. 325). Cambridge University Press, 2006. [357] Topping, Peter. Diameter control under Ricci flow. Comm. Anal. Geom. 13 (2005) 1039-1055. [358] Topping, Peter. Rzcci flow compactness via pseudolocalzty, and flows with mcomplete imhal metncs. Preprint. [359] Tso, Kaising. On an Aleksandrov-Bakelman type maximum principle for secondorder parabolic equatzons. Comm. Partial Diff. Equations 10 (1985), no. 5, 543-553.

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[360] Tsuji, Hajime. Existence and degeneratwn of Kiihler-Emstein metncs on mmimal algebrazc varietzes of general type. Math. Ann. 281 (1988), no. 1, 123-133. [361] Varadhan, S. R. S. On the behavwr of the fundamental solutzon of the heat equation with variable coefficients. Comm. Pure Appl. Math. 20 (1967),431-455. [362] Varopoulos, N. Hardy-Littlewood theory for semigroups. J. Funet. Anal. 63 (1985), 240-260. [363] Waldhausen, FriedheIm. On irreduczble 3-manifolds which are sufficiently large. Ann. of Math. (2) 87 (1968), 56-88. [364] Waldhausen, Friedholm. Cruppen mzt Zentrum und 3-dimenswnale Mannigfaltzgkezten. Topology 6 (1967), 505-517. [365] Wang, McKenzie Y.; Ziller, Wolfgang. Exzstence and nonexistence of homogeneous Emstem metrics. Invent. Math. 84 (1986), no. 1, 177-194. [366] Wang, Xu-Jia; Zhu, Xiaohua. Kiihler-Rzccz solitons on toric manzfolds with positzve first Chern class. Adv. Math. 188 (2004), no. 1,87-103. [367] Watson, N. A. A theory of temperatures in several vanables. Jour Proc. London Math. Soc. 26 (1973), 385-417. [368] Wei, Guofang. Curvature formulas m Sectwn 6.1 of Perelman's paper (math. DC/0211159). http://www .math. ucsb.edu/ -wei/Perelman.html [369] Weibel, Charles A. An mtroductwn to homologzcal algebra. Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994. [370] Weil, Andre. Introductwn Ii l'etude des variet€s kiihleriennes. (French) Publications de l'Institut de Mathematique de l'Universite de Nancago, VI. Actualites Sci. Ind., no. 1267. Hermann, Paris, 1958. [371] Wells, R. 0., Jr. Dzfferentzal analysis on complex manifolds. Second edition. Graduate Texts in Mathematics, 65. Springer-Verlag, New York-Berlin, 1980. [372] Wu, Lang-Fang. The Riccz flow on 2-orbzfolds wzth posztive curvature. J. Differential Geom. 33 (1991), no. 2, 575-596. [373] Wu, Lang-Fang. The Rzccz flow on complete ]R2. Comm. Anal. Geom. 1 (1993), no. 3-4, 439-472. [374] Yang, Deane. Convergence of Riemannian manifolds wzth integral bounds on curvature, I. Ann. Sci. Ecole Norm. Sup. (4) 25 (1992),77-105. [375] Yang, Deane. Convergence of Riemannian manifolds with mtegral bounds on curvature, II. Ann. Sci. Ecole Norm. Sup. (4) 25 (1992), 179-199. [376] Yau, Shing-Tung. On the curvature of compact Hermitian manifolds. Invent. Math. 25 (1974), 213-239. [377] Yau, Shing-Tung. Harmonic functwns on complete Riemannian manifolds. Comm. Pure Appl. Math. 28 (1975), 201-228. [378] Yau, Shing-Tung. Calabi's conjecture and some new results zn algebrazc geometry. Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798-1799. [379] Yau, Shing-Tung. On the Ricci curvature of a compact Kiihler manifold and the complex Monge-Ampere equatwn. I. Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411. [380] Yau, Shing-Tung. On the Harnack inequalitzes of partzal dzfferentzal equations, Comm. Anal. Geom. 2 (1994), no. 3, 431-450. [381] Yau, Shing-Tung. Harnack mequality for non-self-adJomt evolution equations, Math. Res. Lett. 2 (1995), no. 4, 387-399. [382] Ye, Rugang. On the i-function and the reduced volume of Perelman. http://www.math.ucsb.edu;-yer/reduced.pdf [383] Zheng, Fangyang. Complex differential geometry. AMS/IP Studies in Advanced Mathematics, 18. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2000. [384] Zhu, Shunhui. The comparison geometry of Ricci curvature. In Comparison Geometry, Grove and Petersen, eds. MSRI Publ. 30 (1997), 221-262.

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Index

adjoint heat equation, 201, 228 Alexander's Theorem, 433 almost complex manifold, 56 almost complex structure, 56 integrable, 56 ancient solution, 465 anti-holomorphic tangent bundle, 58 aperture, 477 approximate isometry, 150 arc length evolution, 457 variation formula, 448 Arzela-Ascoli Theorem, 137 associated 2-form, 61 asymptotic scalar curvature ratio, 472 is infinite on steady solitons, 472 asymptotic volume ratio, 253, 386, 451 of expanding soliton, 473 of Type I ancient solutions, 474 average nonlinear, 182 average scalar curvature, 6

breather solution, 41 Bryant soliton, 17, 440 Buscher duality transformation, 46 Calabi conjecture, 71 Calabi's trick, 453 Calabi-Yau metric, 72 canonical case, 73, 76 Cartan-Kahler Theorem, 10 center of mass, 182 Cheeger-Gromov convergence, 129 Cheeger-Gromov-Taylor, 159 Cheeger-Yau estimate, 453 Cheng's eigenvalue comparison, 269 Chevalley complex, 33 cigar soliton, 14, 102, 440, 468, 469 cigar-paraboloid behavior, 102 circumference at infinity, 477 Classic Dimension, xiv classical entropy, 124, 214 monotonicity, 215 Cohn-Vossen inequality, 17 collapsible manifold, 442 compactness of isometries, 155 compactness theorem for Kahler metrics, 139 for Kahler-Ricci flow, 140 for metrics, 130 for solutions, 131, 138 local version, 138 complex manifold, 56 complex projective space, 98 complex structure, 56 complex submanifold, 56 complexified cotangent bundle, 58 complexified tangent bundle, 58 concatenated path, 292 conjugate complex, 58

backward heat equation, 198 backward Ricci flow, 288 Bakry-Emery log Sobolev inequality, 218 Bianchi identity second,66 bisectional curvature, 65 nonnegative, 95, 96, 103, 118 Bishop volume comparison theorem, 450 Bishop-Gromov volume comparison, 450 boundary condition Dirichlet, 269 bounded geometry, 130 bounded variation locally, 367 531

Index

adjoint heat equation, 201, 228 Alexander's Theorem, 433 almost complex manifold, 56 almost complex structure, 56 integrable, 56 ancient solution, 465 anti-holomorphic tangent bundle, 58 aperture, 477 approximate isometry, 150 arc length evolution, 457 variation formula, 448 Arzela-Ascoli Theorem, 137 associated 2-form, 61 asymptotic scalar curvature ratio, 472 is infinite on steady solitons, 472 asymptotic volume ratio, 253, 386, 451 of expanding soliton, 473 of Type I ancient solutions, 474 average nonlinear, 182 average scalar curvature, 6

breather solution, 41 Bryant soliton, 17, 440 Buscher duality transformation, 46 Calabi conjecture, 71 Calabi's trick, 453 Calabi-Yau metric, 72 canonical case, 73, 76 Cart an-Kahler Theorem, 10 center of mass, 182 Cheeger-Gromov convergence, 129 Cheeger-Gromov-Taylor, 159 Cheeger-Yau estimate, 453 Cheng's eigenvalue comparison, 269 Chevalley complex, 33 cigar soliton, 14, 102, 440, 468, 469 cigar-paraboloid behavior, 102 circumference at infinity, 477 Classic Dimension, xiv classical entropy, 124, 214 monotonicity, 215 Cohn-Vossen inequality, 17 collapsible manifold, 442 compactness of isometries, 155 compactness theorem for Kahler metrics, 139 for Kahler-Ricci flow, 140 for metrics, 130 for solutions, 131, 138 local version, 138 complex manifold, 56 complex projective space, 98 complex structure, 56 complex submanifold, 56 complexified cotangent bundle, 58 complexified tangent bundle, 58 concatenated path, 292 conjugate complex, 58

backward heat equation, 198 backward Ricci flow, 288 Bakry-Emery log Sobolev inequality, 218 Bianchi identity second, 66 bisectional curvature, 65 nonnegative, 95, 96, 103, 118 Bishop volume comparison theorem, 450 Bishop-Gromov volume comparison, 450 boundary condition Dirichlet, 269 bounded geometry, 130 bounded variation locally, 367 531

532

conjugate heat equation, 201 connected sum decomposition, 433 convergence of maps C= on compact sets, 155 CP, 155 convergence on Il~?, 478 convex, 178 convex function, 367 coupled modified Ricci flow, 198 cross curvature breather, 499 cross curvature flow, 492 monotonicity formula, 496 short-time existence, 493 cross curvature soliton, 499 cross curvature tensor, 491

88- Lemma,

70 derivation of a Lie algebra, 33 derivative first, 366 second, 366 derivative of metric bounds, 132 diameter control, 268 differentiable function, 364 dilaton, 49, 191 dilaton shift, 49 dimension reduction, 440, 472 direct limit, 156 directed system, 156 Dirichlet boundary condition, 269 doubly-warped product, 26, 50 effective action, 49 Einstein metric, 5, 53 homogeneous, 32 standard type, 32 Einstein summation convention, 59 extended, 67 energy of a path, 286 energy functional, 44, 191 entropy, 124 entropy functional, 222 Euler-Lagrange equation for minimizer, 237 existence of minimizer, 238 entropy monotonicity for gradient flow, 226 for Ricci flow, 226 e-entropy, 229 essential 2-sphere, 433 eternal solution, 465

INDEX

Euclidean space characterization of, 451 exhaustion, 128 expanding breathers nonexistence, 213 expanding gradient soliton, 46 expanding soliton, 2 exploding soliton, 14 exponential map derivatives of, 175 exterior differential systems, 10 .1'-functional, 191 Fano manifold, 72 first Chern class, 64 first derivative, 366 first fundamental form, 482 first variation of .1',192 of W, 223 of We, 231 Frankel Conjecture, 73 fundamental solution of the heat equation, 453 funny way, 282 Futaki functional, 73, 98 Futaki invariant, 72 Gaussian soliton, 5 geometrically atoroidal, 436 geometrization conjecture, 437 197 gradient flow of gradient soliton structure, 4 graph space-time, 292 graph manifold Cheeger-Gromov, 439 topological, 438 Gromov-Hausdorff convergence, 38 Gross's logarithmic Sobolev inequality Beckner-Pearson's proof, 249

r,

Haken 3-manifold, 436 Hamilton's surface entropy formula, 217 gradient is matrix Harnack, 218 monotonicity, 217 harmonic map heat flow, 490 monotonicity formula, 490 Harnack inequality classical, 91 heat equation, 453 fundamental solution, 453 Hermitian metric, 57

INDEX

Hessian upper bound, 370 holomorphic coordinates, 55 holomorphic sectional curvature, 65 holomorphic tangent bundle, 58 homogeneous space, 32 Huisken's monotonicity formula, 487 Hamilton's generalization, 489 hyperbolic space, 269 immortal solution, 465 incompressible surface, 436 index form Riemannian, 345 Index Lemma Riemannian, 346 infinitesimal automorphism, 57, 97 injectivity radius estimate, 142, 159 local, 257 integral curves of \1£, 329 integration by parts for Lipschitz functions, 365 involutive system, 10 irreducible 3-manifold, 434 Jacobi field Riemannian, 346 Jacobian Riemannian, 359 Jensen's inequality, 235 Kahler class, 61 Kahler form, 61 Kahler identities, 61 Kahler manifold, 57 Kahler metric, 57 Kahler-Einstein metric, 72 Kahler-Ricci flow normalized, 76 K-collapsed at the scale r, 252 /'i- noncollapsed below the scale p, 252 Killing vector field, 11 Kneser Finiteness Theorem, 434 Koiso soliton, 44, 98 KRF,81 .c-cut locus, 356 L-distance, 293 and trace Harnack, 323 for Ricci flat solutions, 294 gradient of, 311 Hessian of, 318

533

is locally Lipschitz, 308 Laplacian Comparison Theorem, 321 time-derivative of, 312 triangle inequality, 328 .c-exponential map, 352 as T -+ 0,353 for Ricci flat solution, 353 .c-geodesic. 298 estimate for speed of , 303 existence for IVP, 305 on Einstein solution, 299 .c-geodesic equation, 298 .c-geodesics short ones are minimizing, 308 .c-Hopf-Rinow Theorem, 355 .c-index form. 359 .c-index lemma, 357 .c-Jacobi equation, 347 .c-Jacobi field, 346 time-derivative of, 351 .c-Jacobian, 360 of Ricci flat solution, 361 time-derivative of, 362 .c-Iength, 291 additivity, 292 first variation formula, 297 lower bound, 292 scaling property, 293 second variation of, 316 A-invariant, 204 lower bound, 206 monotonicity, 209 second variation of, 280 upper bound, 206 Laplacian for a Kahler metric, 67 Hodge-de Rham, 449 rough,449 Laplacian comparison Riemannian, 376 Laplacian Comparison Theorem, 452 for L-distance, 321 L-distance as T -+ 0,324 monotonicity of minimum, 324 supersolution to heat equation, 323 Levi-Civita connection, 446 evolution, 456 Kahler evolution, 77 of a Kahler metric, 62 variation formula, 448 Li-Yau inequality, 453 Lichnerowicz Laplacian, 448

534

for a Kahler metric, 70, 106, 118 Lie algebra cohomology, 33 Lie algebra square, 460 Lie derivative, 449 linear trace Harnack and We, 233 linearized Kahler-Ricci flow, 118 Lipschitz graph, 365 Little Loop Conjecture, 256, 257 locally bounded variation, 367 locally collapsing, 255 locally homogeneous geometries, 438 locally Lipschitz, 364 vector field, 365 logarithmic Sobolev inequality, 246 of L. Gross, 247 mapping torus, 37, 38 matrix Harnack formula for adjoint heat equation, 281 matrix Harnack quadratic, 9 for Kahler-Ricci flow, 109 maximal function, 265 maximum principle strong tensor, 107, 471 weak scalar, 458 weak tensor, 459 maximum volume growth, 96, 253, 507 mean curvature, 482 mean curvature flow, 482 convergence to self-similar, 489 evolution equations under, 483 measure, 196 minimal 'c-geodesic, 293 existence, 305 minimizer of :F Euler-Lagrange equation for, 206 existence of, 205 modified Ricci curvature, 194 modified Ricci flow, 197 modified Ricci tensor, 48 modified scalar curvature, 48, 194 evolution of, 203 first variation, 273 modified subsequence convergence, 477 mollifier, 367 Monge-Ampere equation complex, 75, 76 Monge-Ampere equation complex, 71 monotonicity of We, 233 monotonicity formula

INDEX

for static metric reduced volume, 382 for the gradient flow, 197 JL-invariant, 236 as T -+ 0,244 as T -+ 00, 243 finiteness, 237 monotonicity, 239 under Cheeger-Gromov convergence, 240 necklike point, 470 neckpinch, 439 Newlander-Nirenberg Theorem, 56 Nijenhuis tensor, 56 nilpotent Lie group, 34 NKRF,81 no local collapsing, 256, 440 improved version, 267 proof of, 258 normal holomorphic coordinates, 68 normal surfaces, 435 II-invariant, 236 monotonicity, 244 null eigenvector assumption, 459 open problem, 282 orbifold, 12, 141 bad, 12 parabola, 287 parallel 2-spheres, 434 Perelman's energy functional, 44 Perelman's entropy functional, 45, 222 Perelman's equations coupled to Ricci flow, 225 Perelman's Harnack quantity, 227 evolution, 227 pinching improves, 463 PL category, 433 Poincare conjecture, 437 pointed Riemannian manifold, 129 pointed solution to the Ricci flow, 129 pointwise convergence in CP , 128 uniformly on compact sets, 129 pointwise monotonicity along 'c-geodesics, 391 porous medium equation, 479 potential function, 76 potentially infinite dimensions and conformal geometry, 194 quasi-Einstein metric, 52

INDEX

Rademacher's Theorem, 364 real (p, p )-form, 59 real part of complex vector, 58 reduced distance, 326 for Einstein solution, 336 of a static manifold, 384 on shrinker, 343 partial differential inequalities, 327 regularity properties, 377 under Cheeger-Gromov convergence, 334 reduced volume for Ricci flow, 388 static, 381 reduced volume monotonicity heuristic proof, 389 proof of, 392, 395 Ricci breather, 203 Ricci curvature quasi-positive, 103 Ricci flow geometry, 391 Ricci flow on surfaces revisited, 145 Ricci form, 64 Ricci soliton, 2, 464 canonical form, 3 Ricci soliton structure, 4 Ricci tensor, 446 evolution, 457 of a Kahler metric, 64 variation formula, 448 Riemann curvature tensor variation formula, 448 Riemann tensor, 446 evolution, 456, 460 evolution for Kahler metric, 104 of a Kahler metric, 63 Riemannian distance function convexity of, 178 derivatives of, 175 Rosenau solution, 469 scalar curvature, 446 evolution, 457 Kahler evolution, 77 of a Kahler metric, 65 variation formula, 448 Schoenflies problem, 433 second derivative, 366 second fundamental form, 482 Seifert 3-manifold, 436 self-similar solution, 2

535

semisimple Lie group, 34 separating 2-sphere, 433 short-time existence compact manifolds, 457 noncompact manifolds, 458 shrinking breathers are gradient solitons, 242 shrinking gradient soliton, 45 shrinking soliton, 2 Simons' identity, 486 singularity model, 439 existence of, 143, 263 smooth category, 433 soliton Kahler-Ricci, 96, 97 space-time metric, 387 static reduced volume, 381 steady breathers nonexistence, 210 steady gradient soliton, 44 steady soliton, 2 Sterling's formula, 250 strongly K-collapsed, 401 supersolution, 374 support sense, 373 surface entropy, 468 surgery Ricci flow with surgery, 441 T-duality, 46 tensor of type (p, q), 59 3-manifolds with positive Ricci curvature revisited, 143 topological category, 433 torus decomposition, 436 trace Harnack quadratic, 467 for Kahler-Ricci flow, 109 Type I essential point, 470 Type I singularity, 489 Type III solution, 38 Dhlenbeck's trick, 459 Dniformization Theorem, 12,438 unitary frame, 65 vicosity sense, 374 volume doubling property, 266 volume form, 446 evolution, 457 Kahler evolution, 77 of a Kahler metric, 66 variation formula, 448 volume ratio, 382

536

W-functional, 222 warped product, 12, 46, 480 weak sense, 373 weakened no local collapsing, 401 Witten's black hole, 468 Yau's unifomization conjecture, 96

INDEX