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IV2uI2 > 0
i.e., IVUI2 is subharmonic and hence constant by the maximum principle. Moreover,
IV2uI = 0, that is, u is a totally geodesic mapping. By using the strict negativity of Kx, we know that the rank dim(u.(TTT2)) < 1
dzET2.
This implies Im(u) is contained in some closed geodesic. Therefore a and b are homotopic to multiples of some closed geodesic.
4.2 Nonpositively curved metric spaces In order to study more complicated group actions by the theory of harmonic maps, we want to allow the image space X to be singular. In fact, we will consider a singular metric space X with nonpositive curvature. Suppose X C Rk is a closed set, and fl is a bounded, smooth domain in M. We want to find energy minimizing maps from ft to X with specified boundary data cp : Oft -- X.
Let HI(ft,X) = {v E HI(ft,Rk)
:
v(x) E X a.e. x E ft} where HI(fl,Rk)
is the Hilbert space of Rk-vector-valued L2 functions on fl with first derivatives in L2. Recall that the energy of u E HI (ft, X) is given by
E(u) =
In
IVu12 dp
where d IA is the volume element of M. The relation u = cp on 80 is to be understood
in the HI (ft, X) trace sense.
Lemma. Suppose V : Oft -' X occurs as the trace of some map in HI(ft,X). Then there exists u E HI (ft, X) such that u = V on On, and E(u) < E(v) for all v E H1(Sl, X) with v = ,p on On.
LECTURE 4. HARMONIC MAPPING INTO SINGULAR SPACES
153
Proof. (direct method) Let {ui } be a minimizing sequence of maps in H' (S2, X) with ui = V on BSI. Since a bounded subset of H' (ft, X) is weakly compact, there is a subsequence again denoted {ui } which converges weakly to a map u E H' (fl, X). On the other hand, E is sequentially weakly lower-semicontinuous. Therefore
E(u) = inf{E(v)
:
v E H' (fl, X),
v = ip on BSI}.
Note that if every pair of points in X can be joined by at least one Lipschitz curve, then Lemma 4.3 implies that any pair of points x, y in X can be joined by an energy minimizing curve y parametrized by arclength. Using the infemum of the length of such curves -y one obtains an intrinsic metric on the space X. This should not be confused with the topologically equivalent,but smaller extrinsic distance Ix - y1. Next, we explain the generalized notion of nonpositive curvature which we use. We assume the X is metrized as above so that any two points x, y in X may be joined by a unique curve in X whose length in Rk is dist(x, y). Consider a geodesic triangle in X, wxy and a corresponding triangle in lit2, Oxy, with the same side lengths. Let y(s) denote the minimizing curve parametrized by arclength between
x and y. Then the squared distance D(s) = dist2(y(s),w) corresponds to D(s) = 189 + (1- 8)212 = s2 + as + Q which is the unique quadratic polynomial determined by boundary conditions D(0) = 1x12 = Id(x,w)12 and 13(t) = 1912 = d(y,w)12. Note that U "(s) = 2. The nonpositive curvature condition of X is that the corresponding inequality D"(s) > 2 holds, in a weak sense. Since D(s) is only Lipschitz, x = y(0),
and y = y(t), the condition D"(s) >_ 2 means that f t S"(s)D(8)d8 >_ 2 f t S(s)ds
V S E C,,00 (0, e), s> 0.
Definition. X has nonpositive curvature if, for any three points w, x, y E X, the inequality D"(s) > 2 holds. There are two elementary examples of non-positively curved singular spaces: (1) 1lcees. A tree is a connected and simply connected graph. One determines an intrinsic metric on a tree by assigning a length to each edge. (2) Surfaces with cone metric: On the disk, the metric is smooth and flat away from the origin. The sign of the curvature at 0 is determined by the cone angle a at 0; namely,
1. a < 27r b positive curvature at 0. 2. a = 27r zero curvature at 0. 3. a > 21r negative curvature at 0. One can often understand these two examples by approximating them by a sequence of smooth surfaces {Xi}, and hope that some properties of harmonic mappings into X, can be inherited. In fact, here is a question due to M. Wolf [W]. X Degeneration question: Consider a sequence of smooth manifolds X. where convergence is in the sense of the distance functions. If of : Cl - Xj are harmonic maps with bounded energy, E(u,) < c, what do the limits of the uj's look like?
154
R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS
Suppose each Xi has nonpositive curvature. As before, by virtue of the Bochner formula,
2o(IVuuI2) > Rico(Vui, Vui) > -clVuil2. I Vu, I2 is thus a subsolution for the operator i + 2C. We can apply the DeGiorgiNash-Moser mean-value inequality ([GT]) to conclude that
-
Sup IVUj 12 0). e.g., If 4' = zdz2, then the map which projects, in a suitable way, the regions between the maximal negative curves onto the maximal positive curves defines a harmonic map from the disk to the tree. See Figure 1.
Figure 1
0>0 Suppose X is a surface with a negatively curved cone metric. e.g., We may obtain such an X in R3 by choosing any curve (embedded) in S2 of length 0 > 27r
and taking a cone over it with vertex at the origin. This gives a cone angle 0. Let g be a cone metric on X, u*g is a real symmetric (2, 0) tensor on D2 and 0 + F + Fdzdz. Here 4' = ip(z)dz2 is the Hopf differential of u, and F is the energy density of u. We can view Fdzdz as a conformal metric; its sketch factor is constant in all directions. On the other hand, 4'(v v) < 0 along the direction v of minimal sketch, and 4'(w w) > 0 along the direction w of maximal sketch. Moreover, vlw. The direction fields v, w, define the positive and negative foliations of 4' away from the discrete set I0+-' {0}. Recent work of E. Kuwert[K] analyzes the possible collapse of u along the negative foliation of 4'.
Theorem ([K]). Assume that the boundary map
8d(p(s), 0(s))
for a< t.
By virtue of the convexity of distance function, we have the following proposition concerning energy convexity.
Proposition 5.1. Suppose X is a Riemannian simplicial complex of nonpositive curvature, and uo and u1 are Lipschitz maps from Il to X. Then if ut is the geodesic homotopy from uo to ul we have
2E(ut) > 2 fn IVd('uo,u1)t2dµ weakly on [0,1], i.e.,
fo
1
E(ut)("(t)dt > 2fa IVd(uo, u1)12 f 1 C(t)dt `d S E C ([0,1], R) C > 0. 0
Proof. (See [GS], also). First consider the one-dimensional case in which we have Lipschitz curves 7o, -ti : (-6,6) --+ X and a geodesic homotopy yt for 0 < t < 1. Assume that 6 = 0 is a point of differentiability for -yo, -y1 and 7e for a.e. t E (0,1). We fix s = 0 and calculate Iaso We can replace -yo, 71 by the corresponding constant speed geodesics. Let e(s) be the length of the curve t -, -yt(s) and observe that t(s) = d(ryo(s), y1(s)) is a Lipschitz function of s, and reparametrize the homotopy by setting ryr(s) = ry(rs/t(s) for r E [0,e(s)]. Thus r H 7y, (s) is now a unit speed geodesic. For any h the function r ,- d2(ryr(h),7y.(0)) is convex because X has nonpositive curvature. At any r for which d -y,(0) exists we have
lim h-2d2 (7r(h),7r(0)) =I
h-.0
12. dyr (0) ds
Since d (0) exists at r = 0, and r = 1(0), it follows that there is a sequence h. tending to zero such that r ,- hi 2d2 (ry . (h,), ryr (0)) converges uniformly on [0,1(0)] In to a convex function which agrees a.e. with the function r H I2 (0).
particular, we can assume that I t I2 (0) is convex in r. Now by the chain rule we have
(o) =
-re(o)-2Z(0)r&(0)(0)+
Wary.,/1(0)(0)
or in terms of t d ds
7t (0) =
dit (0) + ds
te(o)-1
de(o) d1ft (0). ds dt
LECTURE 5. ENERGY CONVEXITY OF MAPS TO AN NPC METRIC SPACE
159
For any r1, r2 E (0,1(0)) with r1 2I
d(°)
I2
'(0)12
= 2 (4-d(io(s)7i(s))
2 I
s=
since IV(t)12 is convex.
Now to prove the result in higher dimensions, observe that the map (x, t) ut(x) is Lipschitz. Thus, for almost every line parallel to the t-axis, it is differentiable at a.e. point of the line. At such points of differentiability the previous results tell us 2
dt2IVu`I2 > 21Vd(uo,u1)I2
in the weak sense. Thus, if (E Co (0,1) and C > 0 we have, for a.e. x E M
f
I
IVutl2(x)C""(t)dt > 2J IVd(uo(x),ul(x))I2C(t)dt.
o 0
o
Integrating w, t, and x we get f' IVutI2S"(t)dµdt > 2 r1I
n
I.
IVd(uo,ul)I2d, I
(t)dt.
o
Corollary S.Z. Assume X C It" is the same as the above proposition. Suppose uo and ul are Lipschitz and e > 0 is small enough so that E(u1) < Eo + e for i = 0, 1 where Eo = inf {E(u) : u E H' (fl, X), u = rp on &Z}. Then fn d2(uo, ul) < ce with a constant c depending only on Poincare constant fl.
R. SCHOEN. BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS
160
Proof. Apply the proposition and the Poincare inequality (since d(uo, u1) = 0 on ice) to get 2E(ut) 2! cind2(uo,u1)dµ weakly on [0,1]. By the convexity of the energy, `da E 10, 1]
Eo < E(ua) < aE(uo) + (1 - a)E(u1) < Eo + e. Now, use the fundamental theorem of calculus and an appropriate test function to get the required result.
Corollary 5.3. Suppose X is simply connected NPC. There is a unique energy minimizing map u : 0 -, X wsth given Lipschitz boundary data.
Proof. Suppose uo and u1 are two minimizers with the same boundary data W. Applying Corollary 5.2 with e = 0, we can conclude that fey ld(uo, u1)12dp = 0. It follows that uo = u1. This completes the proof.
5.4 Monotonicity We may choose a smooth variation of the domain of the form u, = uoFT, where
.,(x) = (1 + r£(x))x, where r is small and l; is a smooth, compactly supported approximation to the characteristic function of a ball B,(0). Since u is a minimizer, we can deduce from a first variation argument, as in L. Simon [Si], that
(5.1)
(2 - n)
IVuI2dµ + a f Be (o)
z
IVu12dE = B (0)
2o. f
Be
I
car
for a.e. a. One can integrate the identity w.r.t a to get the usual monotonicity formula for the normalized energy. Th derive the monotonicity of another useful quantity, we will combine (5.1) with an inequality having to do with the convexity of the distance function in the target. For example, if X is a smooth NPC space and u is a smooth harmonic map, then we have, by the chain rule, for Q E X, fixed Ad2 (u(x), Q) = tr Vdd2(u(x), Q) (Vu, Vu) R
_
Hessd2(u,Q)(Ve,,u,
21VuI2.
a=1
Since Hessp d2(p. Q)(v, v) > 2]v12 for all v E TpN i.e., d2(u(x), 0) is a strongly subharmonic function. For NPC metric space X, we need to make this proof variational since this is the only tool available.
LECTURE 5. ENERGY CONVEXITY OF MAPS TO AN NPC METRIC SPACE
161
Fix Q E X. Given p E X. There exist a unique geodesic y(t) parametrized with constant speed on [0,11 such that y(0) = Q, and y(1) = p. Define R,\,Q : X -+ X by RA,Q(p) = ry(A). Then Ra,Q is a Lipschitz and
contracting map. In fact d(Ra,Q(1h),Ra,Q(p2)) 5 Ad(p1,p2)
R1,Q = Identity RO,Q(p) = 0 V P E X. So R,\,Q is a retraction map of X onto Q with finite Lipschitz constant.
Proposition 5.5. Suppose X is a NPC metric space. If u E H' (St, X) is minimizing and u(fl) is a compact subset of X, then Od2(u(x),Q) > 21Vu12 weakly.
Proof. Fort E Co (St) with t > 0, consider the deformation u.(x) = R1_Tf(S),Q (u(x)) where r > 0. Then u, = u near Oft, and E(u,) has a minimum at r = 0. Therefore
d dt
E(uT) > 0 .
IT.o
On the other hand,
8 = D8 R1-=((x),Q(u(x)) - Ti 8
11
12
tai
I2 -I D &u (x)R1-*(,Q(u) a R1-rt.Q(u(x))+T2 FX
Lt
- 2TOx DOxt (x)R1-*(,Q(u(x)) 2
2
ax i) 18 R1-*((x),Q(u(x)) I
Note that RA,Q(p) = d(Ra,Q(p),Q)y1(Ra,Q(p)) where y is the unit speed geodesic from Q to p. We have that D. RA,Q (P) . 811 R,\,Q (P) = Dvd2 (gA,Q (P), Q). 2
Using the contracting property of RA,Q, we also have I D 8x; (x)R1-T(Q(u) I< (1-
Therefore
E(u1)
0. Remark. Since the function involved is only Lipschitz, our derivative is taken almost everywhere.
LECTURE 6 The Order Function We define the order function for u E H'(11, X) as follows: Let X E f Z and 0 < o < dist(x,BfZ). If u $ Q on 8B, (x), then o fB.(x) IDuI2d,L
ord(x, o, Q) =
Q)dE.
fea, (x)d2(u,
Proposition 6.1. Suppose X is a NPC space and u E Hl (Il, X) is a locally minimizing map. Then, for any x E ft, either u = Q near x, or ord(x, o, Q) is monotonically increasing.
Proof. For x E ft given, we assume that ord(x, o, Q) is defined for o small; otherwise, from Proposition 5.4 it follows that d2(u(y), Q) is subharmonic, and u Q near x. Define E(o) = fB,,(x) JVuI2dp, 1(c) = foB,(.) d2(u, 8)dE. From the monotonicity formula (5.1), we compute the logarithmic derivative of E,
E(s)
o
E(a) JoBe(x)
or
Recall from Proposition 5.4 we have that Od2(u, Q) > 21Vu12. Integrating this inequality over B,(x) using a smooth approximation of the characteristic function of B,(x), we get that d2 21Bo(=) IVu12d z < JOB,,.(z) (u(),Q)dE 8r
2E(a)
sf denote polar coordinates in Br.(x0). We will say that a
Lipschitz map l : B.ro(xo) - X is essentially homogeneous of degree 1 if there is a nonnegative function A : S"-1 - R and an assignment 'y to each E Sii-1 of unit speed geodesic in X with 'yy(0) = 8(0), y (A(l;)'y) = £('y) for x = ryt; E B.ro(xo). In other words, a map is essentially homogeneous of degree 1 if the restriction of u to each ray is a constant speed geodesic in X. For xo E l and o > 0 such that Bo(xo) CC II, we consider the error with which u can be approximated by degree 1 essentially homogeneous maps. Define
R(xo, o) = inf
sup
d(u(x), e(x)) : l is essentially homogeneous of degree 1
zEBe(xo)
Note that R(xo, o) < supXEB,(xo) d(u(x), u(xo)) < Lip(u)v.
LECTURE 7. SMOOTHNESS RESULTS FOR HARMONIC MAPS
171
Definition. A minimizing map u : fl -' X is intrinsically differentiable on a compact subset K C St provided there exist ro > 0, c > 0 and 0 E (0, 1] such that R(x, o) C cot+13R(x, ro) for all x E K and o E (0, ro). The constants c, 0, ro depend only on K, A, X and the total energy of u.
Definition. A subset S C X is essentially regular if for any minimizing map u : A - X with u(fl) C S, the restriction of u to any compact subset of it is intrinsically differentiable.
Suppose that X0 is a totally geodesic subcomplex of X and e : R" -, Xo is an essentially homogeneous degree 1 map, we have the following concept.
Definition. I is said to be effectively contained in Xo if t-1(XI) is of codimension at least one in R" where X1 is the subcomplex of Xo consisting of simplices which
are faces of a simplex in X but not in X0. Notice that X1 C X0 is of at least codimension one. We are ready to state the following theorem:
Theorem 7.5. Let u : A -> X be a minimizing map. Let xo E fl and ro > 0 be such that B,.o(xo) CC A. Let Xo C X be a totally geodesic subcomplex, and let e : B,.o(xo) - Xo be an essentially homogeneous degree I map. Assume that Xo is essentially regular near p = £(xo). There exists bo > 0 depending only on Q, 12, X, Xo such that if l is effectively contained in Xo and SupB., (xo) d(u(x), e(x)) < bo, then u is intrinsically differentiable near xo, and there exist o < bo < ro such that u(Bs, (xo)) S Xo. The proof is rather technical. We omit it, and refer to [GS].
LECTURE 8 Order 1 Points and Partial Regularity The next result convinces us that a point x E ft where the order of u at x, ord°(x) = 1 and where its approximating map u. has rank k = dimX is a regular point, provided that a little regularity condition holds on X. Roughly, this regularity condition means that there exist an isometric totally geodesic embedding i : B, (x) C ft - X corresponding to that of u.. Notice that if X is an F-connected complex, then this condition holds.
Theorem 8.1. If X is an F-connected complex, and u : ft X is a minimizing map, then any point x E ft where ord" = 1 and where u, has rank k = dim X is a regular point. We also omit the proof and refer to [GS].
Suppose that X is F-connected, denote Xp as the tangent cone of X at po which is also F-connected. Let J : R'" -+ Xpo be an isometric totally geodesic embedding for 1 < m < k. It can be easily seen that J(Rm) is contained in at least one k-flat. We need the following geometric result.
Theorem 8.2. Let X k be F-connected, and let X0 be the union of all k-dimensional flats in Xp which contains J(Rm). Then the subcomplex Xo is totally geodesic and is isometric to R'" x X1 where X 1 is F-connected of dimension k - n. Moreover J is effectively contained in Xo. Combining these theorems we can then prove the simplification result.
Theorem 8.3. Let X be F-connected. Then the following properties hold:
1. For any positive integer u and a compact set Ko C X there exist eo = eo(Ko, u) > 0 such that for any minimizing map u : fZ" -' X with u(f2) K0, we have either ord(x) = 1 or ord(x) > 1 + co for all x E ft.
2. Let u : 11 -, X be a minimizing map, and let x0 E with ord(xo) = 1. There exists a totally geodesic subcomplex X0 of X, (xo) which is isometric to R'" X Xj for some 1 < m < min{n, k} and some F-connected X1 of
dimension k - m such that u(B,(xo)) C X0 for some o > 0. Moreover, if we write u = (ul,u2) = B,,,(xo) - R""' x X1, then ui is harmonic with rank m at xo, and ord"2(xo) > 1 3. X is essentially regular. 173
174
R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS
Proof. (Sketch) (i) Denote k = dim X. If k = 1, i.e., X is a tree, we will show explicitly that co only depends on n. In fact, if ord"(xo) > 1 and u(xo) is not a vertex of X, then u is smooth near xo and thus ord"(xo) > 2 since ord"(xo) is integer. If u(xo) is a vertex with p edges emanating from po = u(xo), then p > 2 by the maximum principle. If p = 2, then, again, ord" (xo) > 2. We assume that p > 3, and consider every homogeneous approximating map u, : R" -+ XP. If we choose an edge e emanating from po and introduce an arc length parameter 8 along e, then on Oe = u.-1{e} the function he = s(u.) is a homogeneous harmonic function of degree a = ord"(xo). Of course, Oe is the cone over a region De C S"-1. It follows that heIDe is a first eigenfunction of De and the corresponding first eigenvalue AI (De) = a(a + n - 2). We thus decompose S"-1 into p disjoint regions all with the same eigenvalues. We can choose one such region D. that vol(De) < " S"-1 P Standard results about eigenvalues then imply that there exist bn > 0 such that ,\I (De) = a(a + n - 2) > n - 1 + bn. In particular, a > 1 + co with co = co(n) > 0. Property 2 is an easy consequence of Theorem 8.1 and Theorem 8.2. To establish 3 we work by induction on k. For k = 1 the result follows easily. Assume that k > 2 and all F-connected complexes of dimension less than k are essentially regular. By a compactness argument, it suffices to prove the following result for any xo E 13. There exists ro > 0 such that for or E (0, ro] R(xo,or) < ar1+BR(xo,ro)
for constants c, ro, A depending on x0, E(u), Il, Xo. There are two cases to consider.
First suppose ord"(xo) > 1. Then from 1 we know that ord"(xo) > 1 + Co. This implies, by lecture 6, that sup d(u(x),u(xo)) 5 cal+eo sup d(u(x),u(xo)) XEB., (;o) Bro (xo)
for some constant c and ro > 0. Therefore, the desired decay on R(xo, o) follows. In the remaining case ord(xo) = 1, the result follows immediately from 2, and the inductive assumption.
8.4 Proof of Singular Set Estimate Now, we start to prove the main Theorem 7.4. The estimate on the Hausdorff dimension of S(u) is an application of the basic argument of Federer dimensional reduction [F2]. For any subset E C fZ and any real number a E (0, n), we recall the definition of Hausdorff (outer) measure p{" (E) = inf
{r: : U B,&o) E a
o0
{-1
i=1
and of the Hausdorff dimension
dime E = inf{s : 7{"(E) = 0}.
LECTURE 8. ORDER 1 POINTS AND PARTIAL REGULARITY
175
We observe that S(u) = So U . . . U Sk where ko = min{n, k -1} and S. consists
of those singular points having rank j, where the rank at xo is the rank of the approximating map it. if ord(xo) = 1, and the rank is zero if ord(xo) > 1. We will first show dime So < n - 2. In fact, if we define So = {x E 0, ord"(x) > 1}. Then we actually show that dim,{ So < n - 2. Notice that $o C So. In order to do this, we need the following lemma whose proof is very standard.
Lemma 8.5. If {ui} is a sequence of minimizing maps from BI to X with E(uj) and Image (ui) uniformly bounded, then a subsequence of {ui} converges uniformly
on compact subsets of B1(0) to a minimizing map u : BI - X, and N8(So(u) n B*(0)) > lim f'(So(ui) n B,.(0))
for all r E (0, 1). In particular, dimw(So(u)) > lim dimw(So(u)). We now show that dim So (u) < n - 2. Suppose s E [0, n] with 7 i° (So(u)) > 0. Then by [F1] we may find x0 E fl such that
l a- w*(So(u) n B.,(x)) > 2-'. 0-0
Let u.: R" -
be a homogeneous approximating map for u at xo. Let a = ord"(xo) so that it, is of degree a. Since xo E So(u), we have a > 1 + co. We may apply Lemma 8.5 to suitable rescalings {ui} of it near xo to conclude that ?1 (So(u.)) > 0. Since So(u.) is a cone, it follows that there is x E S"-1 fl So(u.) such that
11'm o-w(So(u.) fl B,(xl)) > 2 °. 0-0 Let ul be a homogeneous approximating map for it, at x1. Then ord"1(x1) > 1+eo. It is easy to we that derivative of u1 is zero along ray t "- tx1. If we choose
coordinates in which x1 = (0,..., 0, 1), then Rk = 0. Therefore the restriction of u1 i denoted u1, to
R"-1 is a
homogeneous map of degree a > 1 + co. We then have So(ul) = So(ul) x R
and thus ?V-I (So('01)) > 0. Ifs > n - 2, we may repeat this argument inductively and produce finally an go > 0 and a minimizing map v : R2 homogeneous of degree a > 1 + Eo such that xa-("-2)(So(v)) > 0. Thus repeating the argument again will produce a geodesic w with degree > 1 and contradict with ord'(xo) = 1 for all xo.
We now show by induction on k = dim X that dim S(u) < n - 2. For k = 1 we have S = So, and we have established this case. Assume that k > 2 that the conclusion is true for F-connected complexes of dimension less than k. Let
176
R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS
xo E Sm -So for a minimizing map u : f2 --+ X1. We then have ord"(xo) = 1, and by 2 in Theorem 8.3 there is a bo > 0 such that u(B6. (xo)) C Xo = Rm x X1 with X 1 an F-connected of dimension k - 7n. Thus we have u = (u1, u2) where ul : BQO (xo)
R"', U2 : Boo(xo) --+ Xl are both minimizing. S,,,(u) n Bo(xo) S S(u2) n B,,(xo) n B,0(xo)) < n - 2 and thus By inductive assumption we then have
dime S(u) < n - 2. To prove 2 we use induction on k = dim X again. Fbr k = 1, we have S = So and dim% S < n - 2. Let e > 0 and d > n - 2. Let 122 be a fixed domain of D with 121 CC 122 cc f2, and choose a finite covering {Br, (xj)}j_1 of So n f21 satisfying xj E So and Erg < e, and B4r, (xj) C 122. Let on 0 \ B2r, (xj) 1 a:5 Sj < 1 on B2r,(xj) \ Br,(xj) . 0
Brj (xj )
Then IVS, I < 2r; 1. Define +p = min{cp, : j = 1... P} and observe that W = 0 near so n S21 and Cpl = 1 on f2 \ Ujt=1 B2r, (xj). Now let Wo = e and observe
f IVVuIIV boldp =
2f
pIVVuIIVVldp 1/2
52(f
Ra.,(x,)
jvlVVuIIVul-102dj4
(f IVuIIV'l2dp ` Uj_, R2.' (-j)
1/2
by Schwartz inequality. On the other hand, a result for harmonic maps (see [ES]) implies that on regular set we have 20IVu12 > IVVU12 - cIVu12.
1
For j = 1, ... , e let p3
on B2r, (xj)
a < pj < 1 on Bar, (x.) \ B2r, (xj) with 12\B4r,(xj) 0
I V Pj 1 5
. Define
p = max{pj, j
= 1, ... , t}
and observe that p = 1 on Uj=I B2,., (x,), and p = 0 outside Uj=1 Bar, (xj). We therefore have
f
IVVuI2loul-'92dp < f lVV I2IVul-1,p2p2dp. B2,,(x,}
0
LECTURE 8. ORDER I POINTS AND PARTIAL REGULARITY
177
An equality in [SY( implies that on regular set we have (1- e)IVVuI2 > IVIIVUI2 for some e,, > 0. Therefore we have AIVuI >
cIVUI
on fl \ So. Using integration by parts, we have En
f
n
.5-2
f
inn
n
This implies fn IVVUI2IVuI-1p2,p2di
r;2 j_I
fa..j(;j)
t IVuldµ
1+eo and therefore SUPB3r (ii) IVul < crjeo. Thus we have
I
ff1
IVVuIIVa'oldµ 5
cF,rJ -2+e0 < e j=1
provided n - 2 < d < n - 2 + co. Now we can assume that 2 holds for maps into Fconnected complexes of dimension less than k. We cover (S - t=I Br,(x;)) n S21 with balls {B,., (yy) : p = 1,... } such that in B,, (yy) the map can be written u = (u1, u2) as in (ii) of Theorem 8.3. By the inductive assumption, there exists a function Op vanishing near V n B,., (yy) and identically one outside a slightly larger neighborhood with
fn IVVuIIV+GpIdu5 2-Pe. We finally set 0 = m1n{,0o, v/1 i ... , . 0.) and conclude
in IVVuIIV,PIdp 5 E fn IVVuIIV &Idp < 2E. P=Q
LECTURE 9 Rigidity Results via Harmonic Maps In this chapter we prove some rigidity theorems for discrete groups with the help of the theory developed in the previous lectures. In particular, we prove our p-adic superrigidity results for lattices in some groups of rank one.
First, let's review briefly the history of rigidity. We assume that (Mg) is complete (compact or non-compact with finite volume) locally symmetric space of
non-compact type, i.e., in a neighborhood of any point, M is isometric to k, a simply-connected globally symmetric space with nonpositive sectional curvature.
Note that the isometry group of k is a semi-simple Lie group of non-compact type. A typical example of non-compact type locally symmetric space is hyperbolic space X (i.e., KX =_ -1, and consequently X is locally isometric to the hyperbolic
disc (H",gH)). For M a given smooth manifold, we consider Mo = {g : (M,g) is locally symmetric} and define an equivalence relation in Mo by saying that, for g', 92 E Mo, 9i ^' 92 if and only if there exist a diffeomorphism F : M -, M such that gl = F'g2. We thus consider the moduli space Mo/ Diff(M). For example, if dim M = 2, then it is well-known that ,Mo/ Diff (M) is a (6g - 6) dimensional space where g = genus of M. The situation is different for locally symmetric manifolds of dimension larger than two whose universal cover is irreducible. In 1960 Calabi-Vesentini ([CV]) proved the local rigidity result (in the Kahler case) which says that there is no nontrivial curve in Mo. In 1970, Mostow ([M]) proved the rigidity theorem (for the compact case) which says any two locally symmetric structures in Mo are equivalent. In particular, the moduli space Mo/ Diff(M) is a point. In the 1970's, G. Marqulis ([Ma]) proved his celebrated "superrigidity" for lattices in groups of real rank at least two. One analytic approach to prove Mostow rigidity theorem is to use harmonic map theory. In fact, in 1979, Y.T. Siu ([Siu]) used harmonic maps to prove rigidity in the Hermitian locally symmetric case; in the 1980's, Sampson ([Sa]) proved a vanishing theorem of harmonic maps from Kahler manifold into an arbitrary manifold with a certain negative curvature assummption. Now, we formulate both Archimedian superrigidity and p-adic superrigidity in our setting. One can refer to [Ma] or [Zim] for details on superrigidity.
179
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R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS
(i) Archimedian Case: Let M be locally symmetric, N be symmetric space of non-compact type. Denote the universal cover of M as M so that M = M/I' with r = ir1(M). Let p : r - isom(N) be a homomorphism. Consider a p-equivariant map W : M -' N i.e., .p o ry = p(y) o p Vy E F. Then Archimedian superrigidity concerns whether there exists a totally geodesic p-equivariant map u : M - N. (ii) p-adic Case. Let M be as above. We replace N by an F-connected complex X such that isom(X) = p-adic Lie group. (Notice that there exists a procedure due to Bruhat-Tits (see [Br}) for starting with a p-adic Lie group G and constructing a building X such that GC- isom(X)). Then p-adic superrigidity concerns whether there exists a constant p-equivariant map, i.e., whether p(r) lies in the isotropy subgroup of a point.
Theorem 9.1 (Margulis). If rank (M) > 2, then both superrigidity properties stated in (i) and (ii) are true. Note that rank (M) is the dimension of a maximal flat in M, i.e., a totally geodesic submanifold isometric to Rk. Therefore, H2 has rank 1; H2 x H2 has rank 2, SL(n, R)/SO(n, R) has rank n - 1. Margulis also showed that (i) and (ii) imply the arithmeticity of r (see [Ma] or [Zim]). Margulis's result left questions unanswered for rank (M) = 1. In fact, it is known that superrigidity fails for lattices in the isometry groups of real and complex hyperbolic space(see Introduction in [GS]). A result of K. Corlette ([C]) showed that Archimedian superrigidity (i) holds for quaternionic hyperbolic HQ and Cayley hyperbolic spaces HCB. Here, we show that p-adic superrigidity (ii) holds for H$ and He s
Theorem 9.2 (Gromov-Schoen). In the case of M = HQ or HC"., Corlett's vanishing theorem can be derived to prove p-adic superrigidity (ii). Consequently the corresponding r are arithmetic. We will sketch the proof of this theorem in the remainder of this lecture. First, we need to prove the existence of a finite energy equivariant harmonic map. Let X be a Euclidean building associated to a p-adic Lie group H(=- isom(X)). Then X has a compactification X = X UOX such that any h E H acting isometrically on X extends as a homeomorphism to X. Moreover, if {Ps} is a sequence from X with lim,-. P; = P E OX, and if {Q;} is another sequence from X with d(P;,Q;) < c independent of i, then lim; Q; = P. Finally, the isotropy group of P E OX is a proper algebraic subgroup of H (see [Br]). Let M be a complete Riemannian manifold, M be its universal covering manifold such that m = M/r. Suppose we have a homomorphism p : r - H. Then we have the following existence result.
Theorem 9.3. Suppose p(l') is Zariski dense in H (i.e., p(T) is not contained in a proper algebraic subgroup of H), and suppose there exists a Lipschitz p-equivariant map v : M - X with finite energy. Then there is a Lipschitz equivariant map u of least energy and the restriction of u to a small ball about any point is minimizing. Proof. See Theorem 7.1 in [GS].
LECTURE 9. RIGIDITY RESULTS VIA HARMONIC MAPS
181
Using the result of our main theorem, we now prove the following extension of Corlette's vanishing theorem [Cl.
Theorem 9.4. Let w be a parallel p -form on M, and assume that u is a finite energy equivariant harmonic map into an F-connected complex X. In a neighborhood of any regular point of u, the form w A du satisfies d* (w A du) = 0.
Proof. Suppose xo E M is regular point for u. Then there exists 60 > 0 such that u(B6a (xo)) C k-flat F. The calculation of [C] then implies dd' (w A du) = 0 in B6o (xo). Note that sets R(u) and S(u) are 17-invariant, and we define Ro =
R(u)/t, so = S(u)/t. We then have from Theorem 7.4 that dimSo < n - 2, and for any compact subdomain III C M there exists a sequence of nonnegative Lipschitz functions {t/ii} which vanish in a neighborhood of So n NI and tend to 1 on M \ (So n S21) such that
ilim f 00 m
0.
Let p be a nonnegative Lipschitz function which is one on BR(xo) and zero outside B2R(xo) with Iopl < 2R-1. We then apply Stoke's theorem on M using the identity r/iip2 (w A du, dd-(w A du)) = 0. Thus we obtain
1' p2lld(A du)II2dp = ± fM (*d(ip2) A *(w A du), d`(w A du)) dµ. IM
This implies
fM
A du)II2du < CfM (,Oip(Vpl + IV ',Ip2) IVullld'(w A du)II
Using Young's inequality, we then have
f t,b p2lld'(w Adu)II2du < C M
fM"IVpI2IVuI2dii+C f p2I0llvuldµlvVul. M
Therefore, r, IId*(w A du)II2d/A < cR-2E(u) + e(R)
IJV0ijjVVujdjA.
M
By first choosing R large and then taking i to infinity, we finally have that d' (w A du) __ 0 on R(u). We now derive a consequence of this result.
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Theorem 9.5. A finite energy equivariant harmonic map from either I or KCn. into an F-connected complex is constant. See Theorem 7.4 in [GS] for the proofs. Let M be 1I (or HC,), so that isom(M) is (n, 1) (or F4 2D). We then have Lemma 9.6. There exists a finite energy Lipschitz equivariant map. See Lemma 8.1 in [GS] for the proofs. Combining all these results will give Theorem 9.2.
References A. Ancona, Negatively curved manifolds, elliptic operators, and Martin boundary, Ann of Math 125 (1987), 495-536. M. Anderson, The Dirichlet problem at infinity for manifolds of negative [A] curvature, J. Diff Geom 18 (1983), 701-721. [AS] M. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann of Math 121 (1985), 429-461. W. Ballman, On the Dirichlet problem at infinity for manifolds of nonpositive [B] curvature, Forum Math 1 (1989), 201-213. [Br] K. Brown, Buildings, Springer-Verlag, Heidelberg and New York, 1988. [Ch] J. Chen, On energy minimizing mappings between and into singular spaces, preprint (1992). [Cl] S. Y. Cheng, Liouville theorem for harmonic maps, Proc Symp Pure Math 36 (1980), 147-151. [An]
, The Dirichlet problem at infinity for nonpositively curved manifolds, Comm Anal Geom. 1 (1993), 101-112. [CY] S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm Pure Appl Math 28 (1975), 333-354.
[C2]
[Cho] H. I. Choi, On the Liouville theorem for harmonic maps, Proc AMS 85 (1982), 91-94. K. Corlette, Archimedean superrigidity and hyperbolic geometry, Ann of Math 135 (1992), 165-182. [CV) E. Calabi and A. Vesentini, On compact, locally symmetric Kdhler manifolds, Ann of Math 71 (1960), 472-507. [C]
[EL] [ES]
J. Fells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. J. Fells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 88 (1964), 109-160.
[Fl]
H. Federer, Geometric Measure Theory, Springer-Verlag, Heidelberg and New York, 1969.
[F2]
, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing fiat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970), 767-771. M. Gromov and R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, I. E. H. S. Publications.
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(GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Heidelberg and New York, 1977. [HL] R. Hardt and F. H. Lin, Harmonic maps into round cones and singularities of nematic liquid crystals, Mat. Zeit. 213 (1993), 575-593. [J] J. Jost, Equilibrium maps between metric spaces, Calc of Var and PDE 2 (1994), 173-204. (KS]
N. Korevaar and R. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm Anal Geom 1 (1993), 561-659. [K] E. Kuwert, Harmonic maps between flat surfaces with conical singularities, To appear, Mat. Zeit. (1995). [Li] P. Li, A lower bound for the first eigenvalue of the Laplacian on a compact Riemannian manifold, Indiana Math J 28 (1979), 1013-1019. [LY1] P. Li and S. T. Yau, Estimates of eigenvalues of a compact Riemannian manifold, Proc Symp Pure Math 36 (1980), 205-239. [LY2] , On the parabolic kernel of the Schrodinger operator, Acta Math 156 (1986), 153-201.
F. H. Lin, On nematic liiquid crystals with variable degree of orientations., Comm. Pure Appl. Math. 44 (1991), 453-468. [LS] T. Lyons and D. Sullivan, Function theory, random paths and covering spaces, J Diff Geom 19 (1984), 299-323. [Ma] G. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Heidelberg and New York, 1989. [M] Mostow, Strong Rigidity of Locally Symmetric Spaces, Ann. of Math. Studies 78, Princeton University Press, Princeton, 1973. [P] A. Preissman, Quelques proprigtes globales des espaces de Riemann, Comment Math Helv 15 (1942-43), 175-216. [Sa] J. H. Sampson, Applications of harmonic maps to Kiihler geometry, Contemp Math 49 (1986), 125-133. [S] R. Schoen, Analytic aspects of the harmonic map problem, Seminar in Nonlinear Partial Differential Equations M. S. R. I. vol.2 (S. S. Chern, eds.), Springer, Heidelberg and New York, 1985. [SU) R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, 3 Diff Geom 17 (1982), 307-335. [SY] R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, Boston, 1994. [Se] T. Serbinowski, Boundary regularity of harmonic maps to nonpositively curved metric spaces, Comm Anal Geom 2 (1994), 139-154. [Sim] L. Simon, Lectures on Geometric Measure Theory, Centre for Mathematical Analysis, Australian National University, 1984. [Siu] Y. T. Siu, the complex analyticity of hasrmonic maps and the strong rigidity of compact Kahler manifolds, Ann. of Math. 112 (1980), 73-112. [Su] D. Sullivan, The Dirichlet problem at infinity for a negatively curved manifold, J Diff Geom 18 (1983), 723-732. [W] M. Wolf, Harmonic maps from a surface to R-trees, Math. Z. 218 (1994), [L]
577-593. [Y]
S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm Pure Appl Math 28 (1975), 201-228.
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W. Ziemer, Weakly Differentiable Rnctions, Springer-Verlag, New York, 1989.
[Zim] R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhauser, Boston/ Basel/Stuttgart, 1984.
Singularities of Geometric Variational Problems Leon Simon
IAS/Park City Mathematics Series Volume 2, 1996
Singularities of Geometric Varational Problems Leon Simon LECTURE 1 Basic Introductory Material Introductory Remarks These lectures are intended as a brief introduction, at graduate level, to the techniques (principally analytic and measure-theoretic) needed in the study of regularity and singularity of minimal surfaces and energy minimizing maps (sometimes loosely referred to as harmonic maps-see the discussion of terminology in 1.1 and 1.3 below).
Since it is technically simpler, we concentrate almost exclusively on energy minimizing maps, but the reader should keep in mind that essentially all the results discussed in these lectures have very close analogues for minimal surfaces. The first 3 lectures are meant to be essentially self-contained, assuming no prior knowledge about harmonic maps; the main analytic tool used in the first 3 lectures is the Schoen-Uhienbeck regularity theorem. We defer the proof of this until the second series of lectures (given in week 3 of the RGI). The last lecture in this present series touches on more recent work. This is too lengthy to be covered in any detail in the available time, but we do state the main results in a self-contained way, and prove a few things which give at least some hint of the kinds of techniques which are involved.
1.1 Definition of Energy Minimizing Map Assume that fl is an open subset of R", n > 2, and that N is a smooth compact Riemannian manifold of dimension p > 2 which is isometrically embedded in some
Euclidean space RP. We look at maps u of fl into N; such a map will always be thought of as a map u = (ul, ... , uP) : fl - RP with the additional property that u(fl) C N. Consider such a map u = (u1, ... , uP). We do not assume that u is 'Mathematics Department, Stanford University, Stanford, CA 94305
E-mail address: lasCgauss. stanford. edn ® 1996 American Mathematical Society 187
188
L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS
smooth-in fact we make only the minimal assumption necessary to ensure that the energy of u is well-defined. Thus we assume only that Du E Li (SZ), and then the energy EBo(y) (u) of u in a ball BP(Y) - {X : IX - YI < p} with BP(Y) C ft is defined by
EB,(Y)(u) = B,(Y)
IDuI2.
Notice that here Du means the n x p matrix with entries D;u'(- 80/8x'), and I2 = E: .l Ej=1(D;u!)2. We study maps which minimize energy in St in the I
sense that, for each ball BP(Y) C ft, 8B,(Y)(U) < eB,(Y)(W),
for every w : BP(Y) -+ RP with Dw E L2(BP(Y)), with w(B,,(Y)) C N, and with w =- u in a neighbourhood of 8BP(Y). Such u will be called an energy minimizing map into N.
1.2 Definition of Regular and Singular Set Given an energy minimizing map u in the sense of 1.1 above, the regular set regu of u is defined simply as the set of points Y E ft such that u is smooth in some neighbouhood of Y; thus reg u is an open subset of fl by definition. The singular set sing u of u is then defined to be the complement of reg u in fZ. Thus
sing u=f\regu, and sing u is a closed subset of R.
1.3 The Variational Equations Suppose u is energy minimizing as in 1.1, suppose BP(Y) C Il, and suppose that for some 6 > 0 we have a 1-parameter family {u,},E(-6,6) of maps of BP(Y) into N such that Du, E L2(fl) and u, _- u in a neighbourhood of 8BP(Y) for each a E (-6, 6), and uo = u. Then by definition of minimizing we have £B,,(y)(u,) takes a minimum at a = 0, and hence
whenever the derivative on the left exists. The derivative on the left is called the first variation of £B,(y) relative to the given family; the family {u,} itself is called an (admissible) variation of u. There are two important kinds of variations of u: Class 1: Variations of the form (i)
U. =llo(u+SC),
LECTURE 1. BASIC INTRODUCTORY MATERIAL
189
where ( = ((1,... ,(P) with each (' E Cr(B,(Y)) where II is the nearest point projection onto N. (Here and subsequently denotes the COO functions with compact support in BP(Y)).) Notice that this nearest point projection onto N is well-defined and smooth in some tubular neighbourhood {x E RP :
dist(X, N) < ao} for some ae > 0, and hence ua defined in (i) is an admissible variation for 1al < vo. We recall the general facts that the induced linear map dIIy gives orthogonal projection of RP onto the tangent space of N at Y E N, and the Hessian Hess fly has the properties that v1 Hess IIy (v2i v3) is a symmetric function of vi, v2, v3 E RP and is related to the second fundamental form of N via the identity v1 Hess IIy (v2, v3) = -1 E vo, Ay (vT,, v1 where the sum is over all permutations al, a2, 03 of the integers 1,2,3 and where vT means orthogonal projection onto the tangent space of N at Y. On the other hand by using a Taylor series expansion for II it is straightforward to check that Diu, = Diu+s((Di()T +( Hess fu(Diu,.))+O(82), where ( )T means orthogonal projection into the tangent space at the image point u(X), and hence for such a variation * implies the integral identity
1.3(i)
J0 i=1
(Diu DDS - ( Au(D;u, D,u) = 0
for any ( as above. Notice that if u is C2 we can integrate by parts here and use the fact that ( is an arbitrary C°° function in order to deduce the equation
6U+
n
E Au(Diu, Diu) = 0, i=1
where Au means simply (tut, ... , tuP). The identity 1.3(i) is called the weak form of the equation 1.3(i)'; of course if u is not C2 the equation 1.3(i)' makes no sense classically, and must be interpreted in the weak sense 1.3(i). It is worth noting (although we make no specific use of it here), that, in case u E C2, 1.3(i) says simply
(Du)T = 0
at a given point X E B,(Y), where (Au)' means orthogonal projection of Lu(X) onto the tangent space Tu(X)N of N at the image point u(X). Class 2: Variations of the form
u,(X) = u(X + s((X)), where ( = ((1,... ,(n) with each (i E C°°(B,,(Y)). Then D,u,(X) = F,'=, Diu(X + s() + sD;('DJu(X + a(), and hence after making the change of variable ( = X + s( (which gives a C°° diffeomorphism of
190
L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS
BB(Y) onto itself in case Ia1 is small enough) in this case * implies
1.3(ii)
JB(Y) i,7=1
(IDuI26i, -
2Diu DJu)D,C' = 0.
The identities 1.3(i), (ii) are of great importance in the study of energy minimizing maps. Notice that if u E C2 we can integrate by parts in 1.3(ii) in order to deduce that 1.3(i) implies 1.3(ii); it is however false that 1.3(i) implies 1.3(ii) in case Du
is merely in L2 (and there are simple examples to illustrate this). One calls a map u into N which satisfies 1.3(i) a "weakly harmonic map", while a map which satisfies both 1.3(i) and 1.3(ii) is usually referred to as a "stationary harmonic map". Thus the above discussion thus proves that energy minimizing implies stationary harmonic. We shall not here discuss weakly harmonic maps, but we do mention that such maps admit far worse singularities (see e.g. [RT1, 2]) than the energy minimizing maps . (Except in the case n = 2 when there are no singularities at all-we show this below in the case of minimizing maps, and refer to recent work of F. H61ein [HF] for the general case of weakly harmonic maps.)
1.4 The Monotonicity Formula An important consequence of the variational identity 1.3(ii) is the "monotonicity identity" 1.4(i)
f IDuI2 =2f p2'" IB P(Y) IDuI2 - 02-"B.,(Y) ,(Y)\B.(Y)
R2_"I ft I2
VR
'
valid for any 0 < o, < p < po, provided B ft (Y) c 0, where R = IX - Y J and 8/8R means directional derivative in the radial direction IX - Y I -1(X - Y). Since it is a key tool in the study of energy minimizing maps, we give the proof of this identity.
Proof. First recall a general fact from analysis-Viz. if a. are L' functions on BA(Y) and if fBn°(Y) Eni a'D,,S = 0 for each C which is C°° with compact support in BPo (Y), then, for almost all p E (01 po), fs,(Y) E, -j a,D,C = fes,(Y) n' a( for any C E C°°(-ffP(Y)), where a = (al,... , a") and r)(=- p-1(X - Y)) is the outward pointing unit normal of 8B,,(Y). (This fact is easily checked by approximating the characteristic function of the ball Bp(Y) by CO° functions with compact support.) Using this in the identity 1.3(ii), we obtain (for almost all p E (O,po))
that
Io(Y) is=1
(IDuI26i, - 2Diu Dju)Di(' _ (IDuI26i,, - 2Diu Dju)p 1(X `
IOB(Y) i)=1
- Y')(-'.
LECTURE 1. BASIC INTRODUCTORY MATERIAL
191
In this identity we choose (3(X) = Xj - Yj, so DCj = b;j and we obtain
(n - 2)
JB9(Y) IDuI2 = P-I j Bv(Y)
(IDu2 - 210u/8R12).
Now by multiplying through by the factor pl_n and noting that LB,, f = for almost all p, we obtain the differential identity d
p2_n
JBp(y) IDuI2
dP C
I
=2d
r
T p-
R2-n I R
fB, f
12
for almost all p E (0,po). Since JB0 f is an absolutely continuous function of p (for any LI-function f), we can now integrate to give the required monotonicity identity. Notice that since the right side of 1.4(i) is non-negative, we have in particular
that 1.4(ii)
p2-n f
IDuI2 is an increasing function of p for p E (0, po), P(Y)
and hence that the limit as p - 0 of p2_n ',,(y) IDuI2 exists.
1.5 The Density Function We define the density function eu of u on 11 by
eu(Y) = limp2-n P1o
IDuI2.
Bo(Y)
(As we mentioned above, this limit always exists at each point of f1 for a minimizing map u.) We shall give a geometric interpretation of this below. For the moment, notice that the density eu is upper semi-continous on Cl; that is
1.5(ii)
Yj - Y E 0
eu(Y) > limsupeu(Yj). j-.oo
Proof. Let e > 0, p > 0 with p + e < dist(Y, 851). By the monotonicity 1.4(ii) we have $, (1'j) 0 be given. There are constants co = eo(n, N, A) > 0 and C = C(n, N, A) such that the following holds for any p > 0, e E (0, 1]: If B(1+E)p(Y) C fl, u,v : B(l+E)P(Y)\Bp(Y) - BY, if u, v have L2 gradients Du, Dv with p2-" fB{1+.)o(Y)\Bo(Y)(IDui2 + IDvJ2) < A, u(X),v(X) E N for X E B(1+.)p(Y))\Bp(Y), and if a-2np-" fB(1+.)v(Y)\B,(Y) Iu - VI2 < e0, then there is w on B(1+,)p(Y)\BB(Y) such that w = u in a neighbourhood of 8Bp(Y), w = v in a neighbourhood of 8B(1+i)p(Y), w(B(1+1)p(Y)\Bp(Y)) c N, and IDw12
IB(I+.)(Y)\fi(Y)
0. -(0)
Thus any tangent map of u at Y has scaled energy constant and equal to the density of u at Y; this is also a nice interpretation of the density of u at Y. Furthermore if we apply the monotonicity formula 1.4(i) to V then we get the identity
0 = v2-"
JB.,(0)
T2-" r I
JB.(O)
R2-n I O1112,
IDwp 12 =
JB,(O)\B(O)
OR
so that Oco/OR = 0 a.e., and since W has L2 gradient it is correct to conclude from this, by integration along rays, that
W(AX) - W(X) VA > 0, X E R".
2.2(ii)
This is a key property of tangent maps, and enables us to use the further properties of homogeneous degree zero minimizers. (See 2.3 below.) We conclude this section with another nice characterization of the regular set of u: 2.2(iii)
Y E reg u b 3 a constant tangent map V of u at Y
(and in this case of course the tangent map is unique). To prove 2.2(iii), note that by Corollary 2 of 1.7 we have Y E regu . . 9 (Y) = 0, but e (Y) = 0 e- 'P const. by 2.2(i).
2.3 Properties of Homogeneous Degree Zero Minimizers Suppose o : R" - N is a homogeneous degree zero minimizer (e.g. a tangent map of u at some point Y); thus W(AX) - V(X) for all A > 0, X E R". We first observe that the density ep(Y) is maximum at Y = 0; in fact by the monotonicity formula 1.4(i), for each p > 0 and each Y E R",
2
f
P(y)
R2-n I _ RV 12
+
p2-"
JBP(Y)
LECTURE 2. TANGENT MAPS
199
where Ry(X) - IX - YI and 8/8Ry = IX - YI-1(X - Y) D. Now BB(Y) C Bp+l yl (0), so that P2_n f
B(y)
P2_"
IDWI2 0, so fl {X eu(X) = a} is a discrete set. Remark. Here "dim" means Hausdorff dimension; thus dim S. < j means simply that Nf+,, (Sj) = 0 for each e > 0. Before we give the proof of this lemma, we note the following corollary.
Corollary. dim sing u < n - 3. Proof. By 2.4(ii), sing u = Sn-3, hence the lemma with j = n - 3 gives precisely dim sing u < n - 3 as claimed.
Proof of Lemma 1. We first prove that So fl {X : eu(X) = a} is a discrete set for each a > 0. Suppose this fails for some a > 0. Then there are Y, Yf E So rl {X :
eu(X) = a} such that Yj 0 Y for each j, and Yj --+ Y. Let pf = IY1 - Yj and consider the scaled maps uy, p, . By the discussion of 2.1 there is a subsequence pj,
such that uy p,, -# tp, where V is (by definition) a tangent map of u at Y; also,
by 2.2 we have e,p(0)=eu(Y)=a. Let i = JY, -Y1-1(Yt - Y)(E Sn-1). We can suppose that the subsequence f is such that E,, converges to some t E S"-1. Also (since the transformation
LECTURE 2. TANGENT MAPS
201
X *-4 Y + p, X takes Yj to E,) O, (Y;) = Ou, of (f j) = a for each j, hence by the upper semi-continuity of the density (as in 1.8) we have Op(t) > a. Thus since O,,(X) has maximum value at 0 (by 2.3(i)), we have Op(1) = 010(0) = a, and hence t E S(ip), contradicting the fact that S(W) = {0} by virtue of the assumption
that Y E& Before we give the proof of the the fact that dim S; < j, we need a preliminary lemma, which is of some independent interest. In this lemma and subsequently we use ly P to be the map of R" which translates Y to the origin and homotheties by
the factor p-'; thus
IIY,P(X)=p 1(X-Y). Lemma 2. For each Y E S and each 6 > 0 there is an e > 0 (depending on u, Y, b) such that for each p E (0, eJ
GYP{X E BP(Y) : 9u(X) > 9u(Y) - e} C the 6-neighbourhood of Lyp for some j-dimensional subspace Ly,P of R".
Proof. If this is false, then there exists b > 0 and Y E S, and sequences pk 10, ek 10 such that (1)
{X E B1(0) : 9u,.,pk (X) > Ou(Y) - ek} ¢ the b-neighbourhood of L
for every j-dimensional subspace L of R". But uy,P,, -, gyp, a tangent map of u at Y, and 0. (Y) = O,P(0). Since Y E S,, we have dimS(V) < j, so (since S(W) is the set of points where 9,o takes its maximum value 9,,(0)), there is a j-dimensional subspace Lo D S(W) (Lo = S(ip) in case j) and an a > 0 such that (2)
O,,(X) < O,p(0) - a
for all X E B1(0) with dist{X, Lo} > 6.
Then we must have, for all sufficiently large k', that (3)
OuY,k, (X) < 0,x(0) - a b'X E B1(0) with dist{X, Lo} > 6.
Because otherwise we would have a subsequence {k} C {k'} with Ou,,,,. (Xk) > 010(0) - a for some sequence Xk E B1 (0) with dist{Xk, Lo} > b. Taking another subsequence if necessary and using the upper semi-continuity result of 1.8, we get Xk --# X with 9,,(X) > 8,(0) - a, contradicting (2). Thus (3) is established. But (3) says precisely that, for all sufficiently large k',
{X E B1(0) : 9u, ok (X) > O,,(0) - a) C the b-neighbourhood of Lo, thus contradicting (1).
Completion of the proof of Lemma 1: We decompose S. into subsets S.,t, i E {1,2,. .. }, defined to be the set of points Y in S1 such that the conclusion of
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Lemma 2 above holds with e = i-1. Then, by Lemma 2, Sj = Ui>1Sj,i. Next, for each integer q > 1 we let 5,,,,q = {X E Sj,, : 9,,(X) E
and note that 8j = Ui,gSj,i,q. For any Y E Sj,, we have trivially that
Sj,i,g C {X : 9u(X) > 6,(Y) - 1/i}, and hence, by Lemma 2 (with e = i-1), for each p< i-1 r)Y.p(Sj,i,q n B4(Y)) c the 6-neighbourhood of Ly,
for some j-dimensional subspace Ly,,, of R. Thus each of the sets A = Sj,i,q has the "6-approximation property" that there is po (= i-1 in the present case) such that, for each Y E A and for each p E (0, p0], *
i?y,, (AnBP(Y)) C the 6-neighbourhood of Ly,p
for some j-dimensional subspace Ly,, of R". In view of the arbitrariness of 6 the proof is now completed by virtue of the following lemma:
Lemma 3. There is a function /3 : (0, oo) - (0, oo) with limt jo /3(t) = 0 such that if 6 > 0 and if A is an arbitrary subset of R" having the property * above, then gj+P(6)(A) =0. This lemma is quite easy to prove, using the fact that there is a fixed constant C" such that for each or E (0,1) we can cover the closed unit ball B1 (0) of Rj with a finite collection of balls {B,(Yk)}k=1,...,q in Rj with radii o and centers Yk E B1(0) such that Qoj < C,,. In view of the arbitrariness of o, it then follows that for each f3 > 0 we can find o = 0(/3) E (0,1) such that there is a cover of RIM by balls {B,,(Yk)}k=1,...,Q such that Qoj+A < 2. More generally, if L is any j-dimensional subspace of R" and 6 E (0,1/8) there is /3(6) (depending only on n, 6) with /3(6) 1.0 as 6 j 0 such that the 26-neighbourhood of L n B1(0) can be covered
by balls B,(Yk), k = 1,... ,Q with centers in L n B1(0) and with Qoj+00) < 12' By scaling this means that for each R > 0 a 26R-neighbourhood of L n BR(0) can be covered by balls B,R(Yk) with centers Yk E L n BR(0), k = 1, ... , Q such that Q(oR).+0(6) < !Ri+0(6). The above lemma follows easily from this general fact by using successively finer covers of A by balls. The details are as follows: Supposing without loss of generality that A is bounded, we first take an intial cover of A by balls BPo/2(Yk) with A n BPo/2(Yk) # 0, k = 1,... Q, and let For each k pick Zk E An B,/2(Yk). Then by * with p = po To = Q(po/2)?+p(6).
there is a j-dimensional affine space Lk such that A n BPo (Zk) is contained in the 6-neighbourhood of Lk. Notice that LknB,./2(Yk) is a j-disk of radius < po/2, and so by the above discussion we can cover its bpo-neighbourhood by balls B,poi2(Zj,t), P = 1,... , P, such that P(opo/2)j+0(6) < 1(po/2)j+0(6) Thus A can be covered
LECTURE 2. TANGENT MAPS
203
by balls B,,.12(Wt), k = 1, ... , M, such that M(opo/2)j+0(6) < !To. Proceeding iteratively we can thus for each q find a cover by balls B,1,12(Wk), k = 1, ... , Rq, such that Rp(ogpo/2)J+#(6) < 2-qTo.
LECTURE 3 The Top-Dimensional Part of Sing u Here u continues to denote an energy minimizing map from 11 C R" into N C RP; the discussion is mainly only relevant when there are actually genuine "(n - 3)dimensional parts" of singular set in the sense that there are points Y E sing u at
which there are tangent maps V with dim S(,p) = n - 3. But the reader should keep in mind that all the discussion here carries over with an integer m < n - 4 in place of n - 3 if the target manifold N happens to be such that all tangent maps V of energy minimizing maps into N have dim S(w) = m < n - 4 (one such case is in fact mentioned later in this lecture, when dimN = 2 and N has genus > 1). Since the discussion is essentially identical in this case, there is no conceptual loss of generality in adopting the definition of top dimensional part in the following section.
3.1 Definition of Top-dimensional Part of the Singular Set We define the top dimensional part sing. u to be the set of points Y E sing u such that some tangent map V of u at Y has dim S(V) = n - 3. Notice that then by definition we have sing u\sing.u C and hence by Lemma I of the last lecture we have 3.1(i)
dim(sing u\sing. u) < n - 4.
To study sing. u further, we first examine the properties of homogeneous degree zero minimizers W : R" -' N with dim S(W) = n - 3.
3.2 Homogeneous Degree Zero ap with dim S(V) = n - 3 Let V : R" -+ N be any homogeneous degree zero minimizer with dim S(q) _ n-3. Then, modulo a rotation of the X-variables which takes S(W) to {0} x R"--3, we have 3.2(i)
Ox, y) = -POW, 205
L. SIMON. SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS
206
where we use the notation X = (x, y), x E R3, V E Rs-3, and where Wo is a homogeneous degree zero map from R3 into N. We in fact claim that 3.2(u)
sing,po = {0}
and hence (PoIS2 E C°°,
so that e,oIS2 is a smooth harmonic map of S2 into N. To see this, first note that sing Wo D {0}, otherwise go, and hence V, would be constant, thus contradicting the hypothesis dim S(V) = n - 3. On the other hand if a 0 0 with t E sing coo, then by homogeneity of Wo we would have {A : A > 0) C sing o, and hence {(AC Y)
: A > 0, y E R"-3} C singp.
But the left side here is a half-space of dimension (n - 2), and hence this would give fl' 2 (sing cp) = oo, thus contradicting the fact that 7f'2(sing) = 0 by Corollary 3 of Lecture 1. Thus 3.2(ii) is established.
We also note that if W (j) is any sequence of homogeneous degree zero minimizers
with coU)(x,y) - po')(x) for each j, and if limsup1.O° fB,(0) IDp(')I2 < oo, then lim sup sup lDtWo) I< 00 j-.oo S2
for each e > 0. Indeed this follows easily from the compactness theorem (Lemma 1 of 1.7) and from the fact that all singular sets of minimizers have dimension < n - 3; the details are left as an exercise. It then follows that for a particular co with
fs2IDvoI2 3 and N real analytic we can also establish the discreteness as follows. Let M(O) by the harmonic map operator on S2. Thus M(O) = AO + E3 A*(V.*, V:'+G), where Atp = (4tp ...... A 9) and V,ti I = (VjPI,... ,V,iiD), 2IN12
LECTURE 3. THE TOP-DIMENSIONAL PART OF SING u
207
With Q &119011g thl 1aplacian on $2, and with V1I such that (V1(1 Vif1 V3() = the S2-gradient off for any C' scalar function f on S2; thus lp is a smooth solution
of MM = 0 on S2 if and only if the homogeneous degree zero extension of 1p to R3\{0} is a smooth solution of the variational equation 1.3(i) of Lecture 1, and in fact M(+y) = 0 is exactly the variational equation (analogous to 1.3(i)) corresponding to the the energy functional es2(0) = fs, IVV)12, Where [V0[2 =
E IIV I2.
Now let 00 be any smooth solution of M(tk) = 0 on S2 and v E T, where T
denotes set of smooth RP-valued function on S2 with the property v(x) E T,yo(y)N for each x E S2. (Thus T is a the set of smooth sections of the pull-back by too of the tangent bundle of N.) The linearized operator Gvov of M(+') at tb = +Go is defined by
zvo(v) = ZM('0s))1'=0, where 0, is any 1-parameter family of smooth maps of 52 into N with ili, (x) varying smoothly in the joint variables (x, s) E S2 x (-e, e) for some e > 0, and with v = j t/isl,=o. (Of course such a family always exists for any given v and the derivative on the right is independent of which particular family is used, so long as ids ti, 1=o = v.) The linearized operator G,yo always has non-trivial kernel;
in fact if 0, is any 1-parameter family of smooth harmonic maps of S2 into N (that is, M(P,) = 0 Vs) with ip,(x) varying smoothly in the joint variables (x, s) E then v is automatically such a S2 x (-e, e) for some e > 0, and if v = solution, by definition of to.. In particular by using 0, = e'Atf10, where A is any skew-symmetric transformation of R3, we get a linear space of solutions spanned by the special solutions 3.2(iv)
v(x) _- x`DJ,po(x) - x'Djt,bo(x),
i, j = 1, 2, 3
(In computing D,00 we assume 1(,a is extended as homogeneous degree zero to R3\{0}.) Similarly by considering the homotheties of S2 (which are conformal and hence preserve harmonicity), we get a family generated by the special solutions 3.2(v)
Dit/io(x),
i = 1, 2, 3.
Now if K denotes the L2 projection of T onto the kernel of C,r,o, then the operator N(v) = M(II(Oo + v)) + K(v), for v E T with [vI < b where b > 0 is small enough to ensure that the nearest point projection n onto N is smooth in the b-neighbourhood of N, then the linearization of .A( at 0 is just Gee + K, which has trivial kernel. Then using the implicit function theorem (applied on the appropriate Holder spaces-we refer to the the discussion of [SL2, pp.537--5401 for the details)
together with the fact that, by smoothness of N, there is 61 E (0, b) such that any smooth map t' : S2 -, N with 10 - 001 < bI can be uniquely represented in the form +P = II(Oo + v) for v E T, we can find a real analytic embedding W of a ball B,,(0) C ker G,y into T such that all solutions of M (tk) = 0 are contained in
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L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS
P(B,(0)), and the set of such solutions is precisely W applied to the set of points f E B. (0) such that V f (f) = 0, where f is the real analytic function on Bs (0) defined by f(O) - Cs. (*(0)) for 0 E B,(0). (In fact' is just the restriction to B, (0) of the local inverse A(-', which exists by the inverse function theorem.) Then since real analytic functions f have the property that f is constant on the connected components of the set where V f = 0, and since there is d < a such that at most one connected component of the set of points where V f = 0 intersects the ball B. (0), we deduce that the set of energies of smooth harmonic maps : S2 -' N is discrete. We actually note here that the above construction of the embedding W and the map f makes sense in the smooth case, and shows that the solutions near 00 are contained in the manifold W(B0(0)) which has dimension equal to the dimension of the kernel ker 4,,0. We shall want to refer to the following integrability condition:
f = const.
3.2(vi)
Notice that by the above discussion this is equivalent to saying that all of the manifold 'Y(B,(0)) corresponds to solutions of .M(II(tyo ++1')) = 0 and the energy
E is constant on'I(B,(0)). That is, it is equivalent to the requirement that there is a > 0 such that the set of solutions of M(v) = 0 with (v - OoIca < a forms a manifold of dimension equal to the dimension of ker£,yo. (This explains why 3.2(vi) is called an integrability condition.) Notice that by the Schauder theory for elliptic equations the condition Iv-001c3 < or is equivalent to the condition Iv - tbo J', < a modulo a fixed multiplicative constant. Using the definition of sing, u, the above discussion in particular implies 3.2(vii)
{eu(Y) : Y E sing, u} is discrete
whenever either dim N = 2 or N is real-analytic or when the integrability condition 3.2(vi) holds.
3.3 The Geometric Picture Near Points of sing. u Let K be a compact subset of fl and Y E sing, u f1 K and let W be a tangent map of u at Y with dim S(W) = n - 3. As in 3.2, we can assume without loss of generality (after making an orthogonal transformation of the X variables which takes S(W) to {0} x Rn-3), that 3.3(i)
cV(x, y) ° Vo(x),
X E R3, U E Rn-3.
By definition of sing, u, there is a sequence pj 10 such that 3.3(ii)
lim pn
3-00
J
Iu -W12 = 0, P (Y)
LECTURE 3. THE TOP-DIMENSIONAL PART OF SING u
209
so for p = p3 with j sufficiently large we can make the scaled L2-norm p-" fB0(y) Iu- WI2 as small as we wish. On the other hand we claim that for any homogeneous degree zero minimizing maps 'p : R" -. N as in 3.3(i) and any ball B,(Y) with BPo (Y) C fl we have the estimate 3.3(iii) sing u fl BP12(Y) C {X : dist(X, (Y + {0} x R"3)) < b(p)p}
`dp < po,
b(p)=C(p"f p Iu-'pl2)I11", where C depends only on n, N, A with A any upper bound for
p02_" fB,, (y)
IDuI2. In
view of 3.3(ii), this perhaps suggests that the possibility that the top dimensional
part of the singular set is contained in a C' manifold (or at least a Lipschitz manifold) of dimension n-3. But there is a problem in that 3.3(11) only guarantees
that b(p) is small when p is proportionally close to one of the pj, and, without further input, we cannot conclude very much about the structure of singe u from this-see the discussion in 3.4 below. We conclude this section with the simple proof of 3.3(iii). We assume Y = 0.
Proof of 3.3(111). Let p < po and Z = ({, n) E sing. u r Bp/2(0). Take o = Qolti, with 60 < 2 to be chosen. By the Schoen-Uhlenbeck regularity theorem there is co = eo(n, N, A) > 0 such that
f0:5 O-" f.,(Z) lu - w(Z)I2 B
(1)
0 and each Y E sing, u there is W as above, an orthogonal transformation Q of R" and a py,6 > 0 such that 3.3(iii) holds for all p:5 py,6i with Q independent of p. We claim that such a property implies that sing, u is contained in a countable union of (n - 3)-dimensional Lipschitz graphs: To be precise, we could apply the case j = n - 3 of the following lemma:
Lemma. Let j E (1,... , n - 1). Suppose 6 E (0, ] and A is a subset of R" such that at each point Y E A there is a j-dimensional2 subspace Ly of R" and py > 0 such that *
AnBB(Y)c{X : dist(AnB,(Y),Y+Ly))<Sp}
dp 0 such that sing. u n BB(Y) is contained in an (n - 3)-dimensional Lipschitz graph for each Y E sing. u n K. We see in the next lecture that there are stronger conditions on the L2-norm which guarantee much stronger results in certain cases.
LECTURE 4 Recent Results Concerning sing u Recall that in the previous lecture we identified a "top dimensional part" sing. u of the singular set sing u. Here we further refine this set by defining
sing.u={YEsing. U : Notice that by 3.2(vii) of the previous lecture we know that if either dim N = 2 or if N is real-analytic then for each compact K C fI there is is a finite set F = {al, ... , aq} such that
K n sing. u= 0, a g .F. Thus
singu n K = (uQ 1sing., u) u ro,
with dim ro < n - 4; our aim is to give conditions which ensure that each sing, u has closure which is (n - 3)-rectifiable in a neighbourhood of each of its points.
4.1 Statement of Main Known Results If dim N = 2, we can show that each of the sets closure sing. u is locally (n - 3)rectifiable (i.e. countably (n - 3)-rectifiable with locally finite (n - 3)-dimensional Hausdorff measure (so that for each Y E closure sing. u there is o > 0 such that 7in-3(B,(Y)nclosuresing. u) < oo). As a matter of fact, a bit more can be proved:
Theorem 1. If dim N = 2, then sing u is countably (n - 3)-rectifiable, and, for each a > 0, S. is (n - 3)-rectifiable in a neighbourhood of each of its points, where
S. = {X E sing u
:
0.(X) > a} (D closure sing. u). (S. is closed by upper
semicontinuity 1.5(ii).) Furthermore, if N = S2 with its standard metric, or if N is S2 with a metric which is sufficiently close to the standard metric of S2 in the C3 sense, then sing u can be written as the disjoint union of a properly embedded (n - 3)-dimensional C'4'-
manifold and a closed set S with dim S < n - 4. If n = 4 then S is discrete and 213
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L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS
the CI.µ curves making up the rest of the singular set have locally finite length in compact subsets of ft.
Remark. In case n = 4 and N = S2, Hardt & Lin [HL] have proved, by different methods than those to be described in these lectures, that the singular set is a union of arcs with endpoints forming a discrete set. In case dim N > 3 we unfortunately can only get information about the part of the set sing. u consisting of points Y E sing. u such that all tangent maps ap of u at Y with dim S(W) = n - 3 have the following "integrability property", in which we assume that we have made an orthogonal transformation of the X variables to ensure, as in 3.2(i), that cp(x, y) = apo(x). Then we require:
t
the condition 3.2(vi) holds with apo in place of 'o.
Unfortunately this integrability condition is not always satisfied in case dim N > 3, so the following theorem in general fails to establish rectifiability of the entire singular set, even if N is real analytic.2
Theorem 2. Suppose dim N > 3 and suppose that the integrability property t holds for all tangent maps ap of u with dim S(ap) = n - 3. Then sing u is countably
(n - 3)-rectifiable, and the set S. = {X : Au(X) > a} has finite measure in a neighbourhood of each of its points for each a > 0. We want to give some brief indications of the kinds of techniques which are needed to prove such results. Without explaining the terminology (for which we refer to [SL4]), we want to mention here again that there are analogous results (with analogous techniques of proof) for minimal surfaces. For example, the following theorem about mod 2 minimizing surfaces is proved in [SL4]:
Theorem. If M is an n-dimensional mod 2 minimizing current in an open subset f2 of some (n + k)-dimensional smooth Riemannian manifold, and if fI has zero mod 2 boundary in 0, then the singular set sing M of M is (n - 2)-rectifiable, and can be decomposed sing M = u, oS,, where W 2(So) = 0 and where each Sj, j ? 1 has locally finite 9{"-2-measure.
4.2 Preliminary Remarks on the Method of Proof "Blowing Up" We initially suppose 0 E singu and work in balls B9(0). To begin, we note that the argument used in 3.3 to prove that singu n Bp/2(0)) is contained in a (6p)-neighbourhood of {0} x R"-3, where 6 = C(p-" fBPio? [u - apl2)1/", assuming
that S(W) = {0} x R"-3, actually gives more information. Namely, we can use that argument together with the regularity theorem 1.6 and the inequality 3.2(iii) in order to deduce that for any eo small enough (depending only on n, N, A) we have for each t > 0 4.2(i)
ptIDt(u - ap)I < Cr eo in B,/2 (0)\{X : dist(X, {0} x R"-3) < bp}
21n the meantime, this integrability condition $ has been shown to be unnecessary in case the target N is real analytic; see (SL6]
LECTURE 4. RECENT RESULTS CONCERNING SING u
215
W > 0, where b = C(eo'p " fBp(o) lu - w12)I/n, with C depending only on n, N, A, provided 6 is small enough depending on n, N, A. Thus W smoothly approximates u away (at least distance bp) from the singular axis {0} x Rn-3
Now analogous to the discussion of the linearized operator L. on S2 (in the previous lecture), we can also discuss linearizing the harmonic map operator M(u) _ Au + E" I Au(Diu, Diu) at gyp: 4.2(ii)
whenever 41/, is a family of smooth maps of BR, (0)\{X : dist(X, {0} x Rn-3) < 6,), varying smoothly in both X, s, where R, j oo and 6, 10 as IsI 10, with tfio = cp, and where v = To p. 1,=0. Notice that then v(X) E Tp(X)N for each X E R"\{0} x Rn. Then in view of the definition 4.2(ii) and the inequalities 4.2(i) (with B = 0, 1, 2)
it is clear that the difference u - W satisfies, in the "good" region B.,2(0)\{X dist(X, {0} x R"-3) < by}, an equation of the form C,p(u - w)T = E,
4.2(iii)
where vT(X) means orthogonal projection of v onto T,(X)N, where IEI < C(ID(u(p)I2+p 2Iu-WI2) < Ceo on {X E BP/2(0) : dist(X, {0} x Rn-3) 2!6p), and where C depends only on n, N, A. In view of the fact that IEI is small relative to u - gyp, this suggests that we should try to approximate u -,p by solutions I(i of the linear equation t, = 0; this is essentially the idea of "blowing up", going back to De Giorgi (in his work on oriented boundaries of least area).
In fact our initial aim is to show that such an approximation can be made provided we have the following "no 6-gaps at radius p" hypothesis for the part S+ = {X E singu : 9u(X) > 9,p(0)} of the singular set: 4.2(iv)
({0} x Rn-3) n Bp(0) C UZES+B6p(Z)
notice in particular that if this hypothesis fails (for a given 6 > 0), then there is a point Yo = (0, yo) E ({0} x R"-3) n Bp(0) such that S+ n B6p(Yo) = 0. Thus, if IYoI < p/2 and if p -n f8P(o) Iu - w12 < e2 with a small enough (depending only on 6, n, N, A), then by 3.3(iii) we have 4.2(v)
S+ n p I(B6P(Yo)) n BP/2(0) = 0,
where p is the orthogonal projection of R" onto {0} x R"-3. This explains the terminology "no 6-gaps at radius p". (We explain how to handle the alternative when there are 6-gaps at radius p as in 4.2(v) later.) Subject to the above no 6-gaps hypothesis 4.2(iv), it is in fact possible to prove the approximation 4.2(vi)
(Bp)-"
I
B, (o)
Iu -'P -
I2
4p n fBP(o) Ju -'PI2,
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L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS
where 0 is fixed E (0, 2), and where 10 is a solution of L = 0 of very explicit (and controlled) type-we show that such an inequality holds with i/' of the special form n-3 2
V,(x, y) = E E aijgp'Diwwo(x) +'d'o(t), j=1 i=1
where ail are constants, w = IxI-1x E S2, where i
is a smooth solution on 52 of
the equation G,,sk = 0 as in 3.2, and where Ia,sl+supS2 I1GoI ea(o), then p2-n
f
BOP(Z)
12
IDvuI2 + J e v(Z)
R22 n 18 < Cp n 1
Bp(z)
lu - wz I2,
where WZ(X) =W(X+Z), X E Rn, Rz = IX-ZI, a/ORz = and C depends only on n, N, A. We give the proof of this lemma at the end of this section, but for the moment we record some corollaries.
Corollary 1. Under the same hypotheses, for any a E (0,1), Iu - pzI2 4.,.(Z)
W°
< cpn+° /
JBo(Z)
I.u - PzI2,
LECTURE 4. RECENT RESULTS CONCERNING SING u
217
where C depends only on n, N, A, a.
This corollary is a direct consequence Lemma 1 together with the calculus inequality P
J
R°-1 f2(R) dR< 0°
R°-1f2(R) dR+C° rP RI+°(f'(R))2 dR, I/2
0
valid for any bounded CI function on (0, p). We apply this to f (R) = u(Z+Rw) (pz(w), where w = IX - ZI-'(X - Z) E Sn-1.
Corollary 2. Under the same hypotheses, if Z = ({,7l) E B0 , and if Bp(0) C (Zo we have
p-2If,I2+p n r B, (Z)
Iu-wZl2 $,,(0) and such that any given point X ties in no more than C,, of the balls B,(Z,,), and such that (1)
{(x, y) E BP,2(0) : Ixl < a/2} C u,Q IB,(Z.,).
By virtue of the no 6-gaps hypothesis at radius p, this can evidently be done with (2)
Q5 C(P/a)n-3,
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L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS
where C depends only on n. Now by virtue of Corollary 1 we have for each Z = Z.,
1Bev(z)
Iu - cpZI2 < R"-a
Cp-n
it, - SPZI2,
JBy/2(Z)
and hence by Corollary 2
(3) o-n+a JB.
Iu-VI2.
IU_-pZI2 < Cp n+a Js,2(Z) IU_,PZI2 < Cp n+a Bp(0)
(Z)
Notice also that
Iu-wPzI2S21u-wit+2IV-cPzl2,
(4)
and, by the same computation (based on the inequality 3.2(iii)) that we used in the proof of 3.3(iii), we know that
XEB,(Z),
ISP(X)-,Pz(X)I S CIxi-1o,
where C depends only on N, A, and hence (4) together with another application of Corollary 2 gives
lu - wz I2 < 2Iu _ WI2 + CIxI-2pn 1
Iu _ o m) into a part which is contained in an embedded CI," manifold and a part which can be covered by a countable collection of balls B,,, (Yk) with Ek °k 3 < (10),;n-3, and such that a similar decomposition can be made starting with any of the balls B,,, (A;) in place of Bp(Y). Here fl is a fixed constant E (0,1), and we need to take e small enough depending only on n, N, A. Thus after j iterations we get we get that S+ is contained in the union of j embedded CI.v manifolds together with a set which is covered by a family of balls B,,,(Yk) with Eko'k-3 < (1 - Q).pn-3. Thus S is contained in the countable union of C14 manifolds together with a set of (n - 3)-measure zero. This iterative procedure has the additional property that it controls the sum of the measures of the embedded manifolds, thus giving the local finiteness result stated in the theorems of 4.1. Finally to prove the additional conclusions of Theorem 1 in case N is S2 or metrically sufficiently close to S2, we need to use a result of Brezis, Coron, & Lieb [BCL] which asserts that a homogeneous minimizer ' from R3 into the standard S2 is such that 0192 is an orthogonal transformation. This result is easily seen (for topological reasons) to imply that there we can encounter no 6-gaps (at any radius) in the iterative argument described above, and hence we obtain an inequality like 4.5(iii) uniformly for non-isolated points Y of S+, assuming that Y E Bpo/2(Y°), where po" fB'.'(Y0) lu -,p(°)I2 is sufficiently small, where ,p(°) with dim S((°)) _ n - 3 as above. This evidently implies the stated Cl," property of S+.
The fact that the exceptional set sing u\sing. u is discrete in the case n = 4 involves a simple scaling and compactness argument together with several applica-
tions of the result in the previous paragraph. The details are given in (SL5] (see [HL]).
Acknowledgments. Partially supported by NSF grant DMS-9207704 at Stanford University. It is a pleasure to thank Tatiana Toro for her invaluable assistance in the preparation and correction of these notes.
References (AW] (A]
[BCL]
W. Allard, On the first variation of a varifold, Annals of Math. 95 (1972), 417-491. F. Almgren, Q-valued functions minimizing Dirichlet's integral and the regularity of of area minimizing rectifiable currents up to codimension two, Preprint. H. Brezis, J.-M. Coron, & E. Lieb, Harmonic maps with defects, Comm. Math. Physics 107 (1986), 82-100.
LECTURE 4. RECENT RESULTS CONCERNING SING u
223
[DeG]
E. De Giorgi, Frontiers orientate di misura minima, Sem. Mat. Scuola
[GT]
Norm. Sup. Pisa (1961), 1-56. D. Gilbarg & N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, 1983.
[G]
E. Giusti, Minimal surfaces and functions of bounded variation, Birk-
[HL]
[HF]
[JJJ
hauser, Basel, Boston, 1984. R. Hardt & F: H. Lin, The singular set of an energy minimizing harmonic map from B4 to S2, Mansucripta Math. 69 (1990), 275-287. F. H61ein, Reularite des applications faiblement harmoniques entre une surface et une varietee Riemannienne, C. R. Acad. Sci. Paris Ser. I Math. 312 (1991), 591-596.
J. Jost, Harmonic maps between Riemannian manifolds, Proceedings of the Centre for Mathematical Analysis, Australian National University 3 (1984).
[Luck1] S. Luckhaus, Partial Holder continuity for minima of certain energies among maps into a Riemannian manifold, Indiana Univ. Math. J. 37 (1988), 349-367. [Luck2] S. Luckhaus, Convergence of Minimizers for the p-Dirichlet Integral, To appear, 1991. [MCB] C. B. Morrey, Multiple integrals in the calculus of variations, Springer Verlag, 1966. [RJ
[RT1] [RT2] [SS]
[SU)
[SL1]
R. E. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta. Math. 104 (1960), 1-92. T. Riviere, Everywhere disconinuous harmonic maps from B3 to S2, C. R. Acad. Sci. Paris Ser. I Math. 314 (1992), 719-723. T. Riviere, Axially symmetric harmonic maps from B3 to S2 having a line of singularities, C. R. Acad. Sci. Paris Ser. I Math. 313 (1991), 583-587. R. Schoen & L. Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), 741-797. R. Schoen & K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982), 307-336. L. Simon, Proof of the basic regularity theorem for harmonic maps, this volume.
[SL2] [SL3] [SL4] [SL5] [SL6]
[WB]
L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Annals of Math. 118 (1983), 525-572. L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Australian National University 3 (1983). L. Simon, Cylindrical tangent cones and the singular set of minimal submanifolds, Journal of Differential Geom. 38 (1993), 585-652. L. Simon, On the singularities of harmonic maps, in preparation. L. Simon, Rectifiability of the singular set of energy minimizing maps, Calculus of Variations and PDE 3 (1995), 1-65. B. White, Preprint.
Proof of the Basic Regularity Theorem for Harmonic Maps Leon Simon
IAS/Park City Mathematics Series Volume 2, 1996
Proof of the Basic Regularity Theorem for Harmonic Maps Leon Simon LECTURE I Analytic Preliminaries This second series of lectures is meant as an elementary introduction to the basic regularity theorem for harmonic maps, which we used frequently in the first lecture series [SL1]. We deal here with energy-minimizing maps rather than minimal surfaces, because the basic regularity theory for these (while being very analogous to the theory for minimal surfaces) is technically simpler than the corresponding theorems for minimal surfaces. Interestingly enough though, the regularity theory for harmonic maps into compact Riemannian manifolds was not established until much later than the basic regularity theory for minimal surfaces. (The work of Giaquinta-Giusti and Schoen-Uhlenbeck was done in the early 1980's, whereas the regularity theorem for area minimizing hypersurfaces was proved by De Giorgi [DG] in the early 1960's.) The explanation for this (apart from the difficulty in proving the energy inequality discussed in Lecture 3 below) is perhaps that the close parallel between regularity theory for area-minimizing surfaces and regularity theory for energy minimizing maps (e.g. monotonicity and harmonic approximation as a key ingredients in the proof of the main regularity lemma) was not widely appreciated until the work of Schoen-Uhlenbeck [SU] and Giaquinta-Giusti [GG]. The plan of this series of 4 lectures is as follows: Lecture 1: Analytic Preliminaries. Lecture 2: Proof of the Schoen-Uhlenbeck theorem modulo a "reverse Poincare" inequality.
Lecture 3: Proof of the reverse Poincar6 inequality used in Lecture 2. Lecture 4: Higher regularity and other mopping-up operations. I Mathematics Department, Stanford University, Stanford, CA 94305
E-mail address: 1msigauss . stanf ord. edu © 1996 American hint hemaUral Sor,rty 227
228
L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS
1.1 Holder Continuous Functions Recall that ifflC R" is open and if aE (0, 11, we saythatu :
fl -Ris
uniformly Holder continuous with exponent a on Nl (written u E C°,°(St ), if there
is a constant C such that Iu(X) - u(Y)l < CIX - YI° for every X,Y E fl. There are various reasons why Holder continuity turns out to be so important in geometric analysis and PDE. We mention two reasons here: (1) (Scaling.) Notice that if Iu(X) -u(Y)I < 1IX -YI° for every X,Y E fl and
if for given R > 0 we define the scaled function u(X) = R-°u(RX) for X E Sl a {R-'Y : Y E fl}, then II (X) -u(Y)I 0 is a constant, and p2-n /
B1(Y)
IDuI2 < #2(p/R)2.,
dY E BR/2(Xo), P E (0, R/2).
Then u E Co"(BR12(Xo)), and in fact VX,Y E BRI2(Xo)
Iu(X) - u(Y)I s C13(IX - YI/R)°,
Proof. Let Ay,,, = (wp")-1 fB,(y) u. The Poincare inequality gives
p -n f (Y)
Iu
_,\Y",12
Cp2(p/R)2o
0. Remark: Notice that, once we have established this, we will have completed the proof that u E C°'°(BR/4(Xo))-subject to the same hypotheses as in the above lemma-by virtue of the Remark (2) following the regularity lemma of 2.1.
3.2 A Lemma of Luckhaus, and Some Corollaries The following lemma is due to Luckhaus [Luckl] (see also [Luck2]), and extends the Lemma 4.3 of [Sin.
Lemma 2. Suppose N is an arbitrary compact subset of RA, n > 2 and u, v S"-1 -, RP with Vu, Vv E L2 and u(S"-1), v(S"'1) C N. Then for each e E (0,1) there is a w : S"-1 x [0, el RP such that Vw E L2, wIS"'1 x {0} _ U, wIS"-1 x {e} = v, IowJ2
-
< Ce
f
Sn-1
(IVuI2 + IVvI2) + Ce-1 J 239
n-I
lu - vl2
240
L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS
and
dist2(w(x, s), N) < CE'-II(J
lu - v12)1/2
IVul2 + IVvl2)'/2(J
+ Cc"
f
S^-'
lu - v12
for a e. (x,8) E Sn-1 x [0, E1. Here V is the gradient on S"`1 and V is the gradient on the product space S"-' x [0, e1. We give the proof of this in the next lecture, but for the moment we want to establish two useful corollaries:
Corollary 1. Suppose N is a smooth compact manifold embedded in RP and A > 0. There is Co = eo(n, N, A) > 0 such that the following holds:
If E E (0, 1], u : S"-' -' R' with Vu E L2, fs^_, lVul2 < A, u(S"-1) c N, and if there is A E RP such that F--2n fs^_, lu - Ale < 4, then there is a w Sn' x [0, E1-+ RP such that Vw E L2, w(Sn-' x [0, e]) C N, WIS"-' X {0} = u,
wIS"-1
J ^-' x fo.e) IVW12 < CE J3^-' IVul2 +Ce_1
x {E} _- const.,
Sn-'
lu - Alt,
C = C(n, N)
Proof. First note that, since u(S"-1) C N, dist2(A, N) < lu(x) _A 12 for each x E Sn-'. Thus, assuming for the moment that co E (0,11 is arbitrary, by integrating over S"-1 we obtain IS,,-11
dist2(A, N)
0 at a point of M, then u = cont. If the sectional curvature KN of N is negative, then u = const or u(M) is covered by a closed geodesic.
Proof. Integrate (1.5) over M to obtain IVduI2 = 0, Ric(M)(du, du) = 0, (R(N) (du, du)du, du) = 0 on M under the above assumptions.
0
For most of our purposes it suffices to note a weaker Bochner-type estimate.
Tb state this we return to the setting of maps u: M - N C RN. Differentiating (1.2) in direction xµ and taking the scalar product with ux.. with respect to y, on account of (1.1) we obtain (1.6)
-LMe(u) + IV2uI2 < IRicMle(u) +C(e(u))2,
where the second term on the right arises from estimating .1PV (Ac(u)(Vu,Vu)M(Vk ° u))xµ . u.-
= Ak(u)(Vu,Vu)M70'(dvk(u)ux- u..) = (A(u)(Vu, Vu)M)2 < C(e(u))2.
PART 1. THE EVOLUTION OF HARMONIC MAPS
265
Weakly harmonic maps Let
H1'2(M; N) = {u E H1.2(M; R" ); u(x) E N for almost everyx E M. where H1,2(M;R") is the standard Sobolev space of L2-mappings U: M - R" with distributional derivative Du E L2. That is, HI.2(M; N) is the space of maps u: M - N with finite energy E(u). It was observed by Schoen-Uhlenbeck (124] that in general HI.2(M; N) as defined above is larger than the weak closure of C' (M; N) in the H1"2-norm IIUIIHI.2 = j (IHH2 + IVu12) dvoiM, r
which in turn is larger than the strong closure of C°°(M; N) in H1,2(M; N). However, if m = dim M = 2 these spaces all coincide. By a result of Bethuel [7] the same is true if ir2(N) = 0. The respective relations between these spaces for general M and N were analyzed by Bethuel-Zheng (9] and Bethuel [7].
Definition 1.2. A map u E HI "2(M; N) is weakly harmonic if u satisfies (1.2) in the distribution sense.
Example 1.2. The map u : B1(0) C R' - S"`-1, given by u(x) = Izl belongs to H1'2(Bl(0);S"`-I) form > 3 and weakly solves (1.2), that is
-Du = IVuI2u.
Existence of harmonic maps As in Hodge theory, where one seeks to realize a de Rham cohomology class by a harmonic differential form, a basic existence problem for harmonic maps is the following:
Homotopy problem: Given a map uo : M -, N is there a harmonic map u homotopic to uo? This question, as we shall see below, has an affirmative answer if the sectional curvature KN of N is non-positive (Eells-Sampson [38]), or if m = 2 and v2(N) = 0 (Lemaire (102], Sacks-Uhlenbeck [121]). However, for N = S2 and m = 2, we have the following counterexamples:
Example 1.3. (Lemaire [102], Wente [151]): If u: B1(0) C R2 -' S2 is harmonic It., then u coast. and U 10H, (o) =
Example 1.4. (Eells-Wood [39]): If u: T2 - 52 is harmonic, then degu # ±1. In higher dimensions (m > 3), hardly any result is known for the homotopy
problem unless KN < 0. However, there are various existence results for the Dirichlet problem.
Dirichlet problem:Suppose OM 96 0 and let uo: M -+ N be given. Is there a harmonic map u : M - N such that u = uo on 8M? The Dirichlet problem can be attacked using variational methods. By minimizing E among the class H,I,a (M; N) = {u E HI.2(M; N); u = uo on OM}
266
M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS
one obtains a weakly harmonic map u satisfying the desired boundary condition. Similarly, one could attempt to solve the homotopy problem by minimizing E in a given homotopy class. However, Lemaire's example shows that in general homotopy classes are not weakly closed in H1"2(M; N). This is made explicit by the following example, whose construction relies on the fact that the conformal group on S4 acts non-compactly on HI2(S2, S2).
Example 1.5. The mappings ua = v, I ° D,,
irp : S2 . S2,
where ap : S2 \ {p} - R2 denotes stereographic projection and Dax = Ax is dilation by a factor A, are homotopic to the identity id = ul : S2 -, S2. But
ua-'u00(x)P (A-'oo), weakly in H1.2(S2; S2). (Incidentally, by conformal invariance of E in dimension m = 2, all ux are harmonic!)
Regularity The direct methods in general only yield weakly harmonic maps. In fact, the (singular) map x --' of Example 1.2 is minimizing from B1(0) C Rm into Sm-I for its boundary values, if m > 3 (Brezis-Coron-Lieb [13], Lin [105]). A partial regularity theory for energy minimizing maps was developed by Schoen-Uhlenbeck
[123], [1241 showing that in general an energy minimizing map u is smooth on an open set whose complement, the singular set Sing(u), has Hausdorff-dimension < m - 3 and is discrete if m = 3, which is best possible. Even if we regard the map u(x) = >zf as a map u: B1(0) C Rn -* Sm-I C N = Sm C R"1+1, for example, whence there is no topological reason for a singularity, u is still minimizing if m ? 7 (Jager-Kaul [82], Baldes [61). Moreover, Hardt-Lin-Poon [71] have constructed
examples of minimizing harmonic maps u: B1(0) C R3 --# S2 with cylindrical symmetry whose boundary data u l aB, (e) : 8B1(0) 95 S2 -' S2 have degree 0 and such that u possesses an arbitrarily large number of singular points on the axis of symmetry, and Rivii re [119] has exhibited weakly harmonic maps R3 -+ S2 with line singularities. By contrast, if m = 2, energy-minimizing harmonic maps are smooth (Morrey [110], [111], Giaquinta-Giusti [51]). Moreover, Grater [66] proved smoothness of
conformal weakly harmonic maps. This result was extended by Schoen [122] to harmonic maps which are stationary with respect to variations of parameters in the domain, hence possessing a holomorphic Hopf differential IU"12 - 2iux . uY)dz2 (au)2dz2 = (IuxI2 in suitable conformal parameters z = x + iy on M. Finally Halein [72] recently has shown regularity of weakly harmonic maps in general.
Theorem 1.1 (Helein [72]). Let m = 2 and let u E H1.2(MN) be weakly harmonic. Then u E C' (M, N). Proof. For N = Sn-1, M = Bl(0) C R2 his proof exploits the equivalence of (1.2) and
n
-Du' _ E Vuf (u`Vu& - Ad Vu`), i = 1, ... , n, j=1
PART 1. THE EVOLUTION OF HARMONIC MAPS
267
because 0 = V1u12 = 2 E uj Vu'. He then observes that for any I< i, j < n there holds (1.7)
div(u'Vu7 - uuVu') = u'DuJ - u3 Du' = 0.
Hence there is a potential a'j E H1.2 such that
rot a'; = u' Dug - u' Vu' and (1.2) takes the form n
-Au' = E(u=ay - uyai ), j=1
where the right hand side is the sum of Jacobians of H 1.2-mappings. Continuity of u (and hence smoothness) then follows from results of Wente [150] and Brezis-Coron [12].
Realizing that (1.7) is a consequence of Noether's theorem and the symmetries of Sn-1, Helein then generalized this simple and beautiful idea to arbitrary target manifolds by an ingenious choice of rotated frame fields on u-'TN. Inspired by Htlein's result, Evans [41] for m > 3 has obtained partial regularity results for "stationary", weakly harmonic maps into spheres.
1.2. The Eells-Sampson result By Examples 1.3, 1.4 and 1.5 above we know that it may be difficult (if not altogether impossible) to solve the homotopy problem for harmonic maps by direct variational methods. To overcome these difficulties, Eells - Sampson [38) proposed to study the evolution problem
u1 - 0Mu = A(u)(Vu, Vu)M on M x [0, oo[ with initial and boundary data (1.9) u = uo at t = 0 and on 8M x [0, oo[ (1.8)
for maps u: M x [0, oo[-+ N C IR", the idea behind this strategy of course being that a continuous deformation u(., t) of uo will remain in the given homotopy class. Moreover, the "energy inequality" (see Lemma 1.1 below) shows that (1.8) is the t) for t - 00 (L2)-gradient flow for E, whence one may hope that the solution will come to rest at a critical point of E; that is, a harmonic map. For suitable targets, this program is successful.
Theorem 1.2 (Eells-Sampson [38]). Suppose M is compact, 8M = 0 and that the sectional curvature KN of N is non-positive. Then for any uo E COO (M; N) the Cauchy problem (1.8), (1.9) admits a unique, global, smooth solution u: M x oo suitably, converges smoothly to a harmonic map [0,oo[-+ N which, as t u0, E COO (M; N) homotopic to uo.
The proof uses three ingredients.
Lemma 1.1 (Energy inequality). For a smooth solution u of (1.8), (1.9) and any T > 0 there holds rT
E(u(T)) +
J0
[utI2 dvolM dt < E(uo).
268
M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS
Proof. Recall that A(u)(Vu, Vu) 1
Hence, upon multiplying (1.8) by ut
and integrating by parts, we obtain
I
Iutl2 dvolM + d E(u(t)) = 0
for any t > 0, and the desired estimate (in fact, with equality) follows upon integrating in t.
Lemma 1.2 (Bochner inequality). If K^' < 0, then for any smooth solution u of (1.8) with energy density e(u) there holds (1.10)
with a constant C depending only on the Ricci curvature of M.
Proof. To derive this estimate we use the equivalent intrinsic form
ut - trace(Vdu) = 0
(1.11)
of (1.8), where now u = (u',. .. , u') denotes the representation of u in suitable local coordinates on N. As in deriving (1.5) from (1.4) in the stationary case, we then conclude that (in normal coordinates around xo on M)
- Am) e(u) + IVdul2 + Rpuyo'uyfl = Rkjluxauyouyfluyd C
at (xo, t), where V, RQp, R,kil, respectively, denote the pull-back covariant deriv-
ative on T'M ® u-I TN, the Ricci curvature on M, and the Riemann curvature tensor on N. Prom this identity, the claim follows.
Note that in the parametric setting of (1.8) and for general targets, upon differentiating (1.8) in direction x' , multiplying by uZ> and summing over 1 < a < m, by orthogonality A(u)(Vu, Vu) I TuN we obtain (1.12)
t& - AM e(u) + lV2ul2 < CMe(u) +CN(e(u))2,
and CM, CN, respectively, denote Constants where IV2ul2 = depending only on the Ricci curvature of M and the second fundamental form of N. The final ingredient is Moser's [112] sup-estimate for sub-solutions of parabolic equations. Let
Denote PR(zo) the cylinder
PR(zo)={z= (x, t); Ix - xol< R, to-R2 0 can be chosen uniformly for a set of initial data which is compact in {uo} + Ho'2(M; N). In particular, if uom E COO (M; N) converges to uo in H1'2(M;N), and if (um) is the corresponding sequence of local solutions (1.8) for initial and boundary data (uo,,), by the above a-priori estimate we have IIUrII12,TI(M) < C (E(uo) + 11411y..2) and the sequence (um) weakly accumulates at a function u E VT1(M). By Rellich's theorem, moreover, VUm --+ Vu in L2(M)
for almost every t, and it is easy to pass to the limit in equation (1.8); by the preceding remarks about regularity, in fact, u solves (1.8) classically in Mx]O,Ti]. Finally, by Lemma 1.1, u achieves its initial data continuously in H1 ,2(M; N). Uniqueness: The space of functions with bounded VT (M)-norm is a uniqueness
class. Indeed, if u, v E VT (M) weakly solve (1.8) with u(0) = uo = v(0), their difference w = u - v satisfies Iwt - AMWI < CIwI(IVu12 + IVvI2) + CIVWI(IVuI + IVvI).
Testing with w and integrating by parts, we obtain for almost every t > 0
2 JM
Iw(t)12 dvolM +
0G
r
J
fM
fM1w12(1Vu12
C ft
ff
ds
+ IVv12) dvolM ds
t
+C
0
(ft
IwIIvwl(IVul + IVvl) dvolM ds
M
1/2
(ft
lAS Iw14 dvolM d/
(ftrM
+C
1 4
1w14
dvolM
\ O//
ds/)
( pt
) f (1u1+ IVvI4) dvolM ds M
(o
t
)'/2
\Jo fu
IVwI2 dvolM ds 1/4
fM(IVu14 + LVvI4) dvolM ds)
)1/2
('
< C e(t) I (fo, fM
14 dvolM
+
L
< Ce(t)
[su p
f Iw(s)12 dvolM + f
t 0
fo,
fMIhh11201M ds I Vw12
fxf
dvolht ds0 0 small enough
on account of Lemma 1.4, where e(t)
and such that IIw(t)IIL2(M) = sup0 0 be the first singular time, and let Sing(l) = {xo E M;VR > 0: limsupE(u(t);BR(xo)) > e1}. t/i
Sing() is finite. Indeed, let x1, ... , xK E Sing(). Choose R > 0 such that B2R(xi)f1 B2R(x f) = 0 (i j), and fix r E [t - 2 °Eu , f[, where C is the constant in Lemma 1.5. Then by lemma 1.5 K
Ke1
E(u(t); B2R(xi)) 1=1
K,
< E(uo) -
lim limsupE(u(t);BR(xi))
R-.O t/l, < E(uo) - Kiel. i=1
PART 1. THE EVOLUTION OF HARMONIC MAPS
277
Similarly, let K2, K3, ... be the number of concentration points at consecutive
times t2 < t3 < ..., and let uj = limt,F, u(t) for j = 2,3,... Then by induction we obtain
E(uj) S E(u,-1)-Kje1 !5 ...
SE(uo)-(Kl+...+KK)ei,
and it follows that the total number K of concentration points, hence also the number of concentration times tj, is finite; in fact, K < E(uo)el 1 Smoothness: Let t = Ij for some j. To see that u is smooth up to time t away from Sing() we present an argument based on scaling, as proposed by Schoen [122] in the stationary case. By working in a local conformal chart we may assume that M is the unit disc B or half disc B+ = {(xI, x2) E B; x2 > 0};
moreover, by scaling we may assume t > 1. Finally, we shift time so that t = 0. The solution u then is defined on a domain containing M x [-1, 0]. For R > 0, Zo = (xo, to) denote
PR )(zo)_{z=(x,t);xEB(+),Ix-xol N be the smooth nearest-neighbor projection of a tubular neighborhood Ua of N in R" and let wk = 7r vk. Then wk E C°°(R2; N) and wk - u in H' 2 (R'; N). Invert wk along the circle of radius p to obtain 4Dk(x) = Wk
(zj)
.
Then by conformal invariance of Dirichlet's integral
lim upE(zuk; B,(0)) = limsupE(wk; R2 \ Bp(0)) = E(ii;R2 \ B,(0)) < 4
k-00 Moreover
k-00
wk = wk = Uk
on BB,,
and upon replacing uk by wk on Bp(0) for some sufficiently large k we obtain a new
map v:M -Nsuchthat
E(u(tk)) - E(v) = E(uk; Bp(0)) - E(wk; B,,(0)) ?
4
That is,
E(v) < E(u(tk)) - 4 < E(uo) - 4 < inf {E(u); u N u0}. On the other hand, since a2(N) = 0 by assumption, u(tk) and v are homotopic. The contradiction shows that the flow cannot develop singularities in finite or infinite time. Hence the proof is complete. As a second application we present a key step in the proof of a result of MicallefMoore on a generalization of the Berger-Klingenberg-Rauch-Topogonov sphere theorem.
Theorem 1.7. Let N be a compact, simply-connected, n-dimensional Riemannian manifold and let k > 0 be the first integer such that ak(N) 96 0. Then there exists a non-constant harmonic sphere u: S2 - N having Morse index MI(u) < k - 2.
Recall that the Morse index of a non-degenerate critical point x of a C2functional f on a Hilbert space X is the maximal dimension of a linear sub-space
V c X such that d2f(x)I vxv < 0. Proof. Represent Sk = {
n) E R3 x Rk-2;
ItJ2
+ In12 = 1 }
S2 x Bk-2, where Bk-2 = BI (0; Rk-2) and S2 x {n} is collapsed for n E 8Bk-2. Let ho : Sk N represent a non-trivial homotopy class [ho] E irk(N). With respect to the above decomposition of Sk, for every n E Bk-2 this induces a map
uo(rl) = ho(.,rl): S2 - N. Note that u0(rl) - coast. for i E 8Bk-2. Let u(.;,1) be the corresponding solutions to the Cauchy problem (1.8) (1.9). First suppose supE(uo(rl)) < eo. n
PART 1. THE EVOLUTION OF HARMONIC MAPS
281
Then by Theorem 1.5 the flows q) are globally smooth and converge smoothly to constant maps as t - oo. The convergence is uniform in q. Indeed, for any t > 0 let µ(t) = SUPE(u(t;q)) > 0-
Note that the map t F-+ p(t) is non-increasing by Lemma 1.1. For a sequence tt - o 'I
select in E BI-' such that E(u(tt; qt)) = sup E(u(tt; q)). n
Then a sub-sequence qt -a any t < oo we have
and, by locally smooth dependence of u(.; q) on q, for
E(u(t;rh)) E(u(t;fl)) = tlim 0.0 For large I there holds t < ti, and Lemma 1.1 gives
E(u(t; qt)) >_ E(u(ti; qt)) = p(tt) Hence
slim µ(t) < Jim E(u(t; f )) = 0. 00
Thus E(u(t; q)) -+ 0 as t -+ oo, uniformly in q E BI-'. By Proposition 1.3 this implies smooth convergence. But then for large t the map ht : Sk ^-- S2 x Bk-2 9
(t, q) - u(e, t; q) E N is homotopic to a map h: (t;, q) '--+ u(q) E N where u E C°°(Bk'2, N). Since Bk-2 is contractible, u and therefore also ho is homotopic to a constant map, contradicting our assumption about he. Hence
ek = inf sup E(h(., tj)) J co > 0, h ho
17
where we take the infimum over all h homotopic to he. Consider the set ? t = {u : S2 -+ N;
u is non-constant, smooth and harmonic,
E(u) < 2ek}.
?{ is compact modulo separation of harmonic spheres u E W. Suppose any u E ?t has Morse index > k - 2. Then for each such u there exists a maximal sub-space V E H2.2(S2; u-1TN) of dimension > k - 2 such that d2E(u)lvxv < 0.
Moreover, there exist numbers po > 0, e > 0 with the following property: If a: U --+ N denotes nearest-neighbor projection in a tubular neighborhood U of N onto N, and if we denote
Vp = {tr(u+v);v E Bp(0;V)} C H2'2 (S2;N) then
sup E(u) =: µp uEBV1
is strictly decreasing for 0 < p < po and iz,,
_ 2
-2
for the index of any non-constant harmonic 2-sphere in N. In particular, if u is a harmonic sphere as constructed in Theorem 1.7, we obtain
k>2+ that is, Hk(N) = lrk(N) = 0 for 0 < k:5 R. By Poincare duality
Hk(N)=0,
0 0 is the first integer such that 7rk(N) = 0, by the Hurewicz isomorphism theorem we also have 7rk(N) °-` Hk(N).
Hence 7rk(N) = 0 for all 0 < k < n, and N is a homotopy sphere. If n > 4, by the resolution of the generalized Poincare conjecture therefore N is homeomorphic to a sphere.
Extensions and generalizations Theorem 1.5 has been extended to target manifolds N with boundary by ChenMusina (201. The same technique can be used to study evolution problems related to other two-dimensional variational problems. For instance, in Struwe (1391, Rey [118] the evolution problem for surfaces of prescribed mean curvature is investigated; Ma Li (1031 has studied the evolution of harmonic maps with free boundaries.
PART 1. THE EVOLUTION OF HARMONIC MAPS
283
1.5. Existence of global, partially regular weak solutions for m > 3. Earlier we observed that singularities must be expected even for energy-minimizing weakly harmonic maps, if m > 3, and hence for the evolution problem (1.8). The following result was obtained by Chen-Struwe [21].
Theorem 1.8. Suppose M is a compact m-manifold, 8M = 0. For any uo E H1'2(M; N) there exists a distribution solution u: M x [0, oo[- N of (1.8),(1.9), satisfying the energy inequality and smooth away from a closed set E such that for each t the slice E(t) = E n (M x {t}) is of co-dimension > 2. As t -' oo suitably, u(t) converges weakly to a weakly harmonic limit u°° which is smooth away from a closed set E(oo) of co-dimension > 2.
Originally, the estimate on the co-dimension of E was obtained in space-time, the above improvement is due to X. Cheng [23]. For manifolds M with boundary OM 0 0 a similar existence and interior partial regularity result holds; see Chen [18]. Boundary regularity is open. The proof of Theorem 1.8 rests on two pillars: A penalty approximation scheme for (1.8), developed independently by Chen [17], Keller-Rubinstein-Sternberg [93] and Shatah [127], and a monotonicity estimate for (1.8), due to Struwe [140].
Penalty approximation Consider the case N = the Cauchy problem
S"-1
C R". Given u0 E H1"2(M;Sn-1), K E N consider
ut - OMu + Ku(IIu1I2 - 1) = 0,
(1.17)
(1.18)
ult=o = uo
for maps u: M x [0, oo[-+ R. That is, we "forget" the target constraint and regard all maps u: M - R" as admissible; however, we "penalize" violation of the constraint Iu12 = 1 more and more severely, as K -+ oo. (1.17) is the L2-gradient flow for the functional
EK(U) = E(u) + K r
JM
- 1)2 dvolM.
(Iu12 4
Indeed, we have
Lemma 1.6. If u E C°0(Mx]0,T];R") solves (1.17), (1.18), then there holds
EK(u(T)) + fT f I8ut2 dvolMdt = EK(uo) = E(uo); 0
M
in particular, u attains its initial data continuously in H'2(M; N)
Proof. Multiply (1.17) by ut and integrate to obtain the energy estimate. Since Btu E L2(M x [0,71), clearly u(t) -, uo in L2(M) and weakly in H'.2(M, N) as t -' 0. Since also limsoupE(u(t)) < limsoupEK(u(t)) < E(uo),
t-
t-
we, in fact, also have strong H1,2-convergence.
Moreover, we have an L°° a -priori bound.
O
M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS
284
Lemma 1.7. If u E C°°(Mx)0,7);R") solves (1.17),(1.18), then IIUIItoo 0 such that for any solution u E C°°(Rm x [-1,0[;N) of (1.8) above, if fi(R) < to for some R > 0, then
supIvul
0 and C = C(m, N, Eo).
Proof. Scaling with R, we may assume R = 1. For 6 > 0 fix p E]0,6[, zo = (xo, to) E Pp such that
(6 - p)2 sup e(u) = omax6{(6 - v)2 sup e(u)}, Pp P. (e(u))(zo) = supe(u) = eo. PP
PART 1. THE EVOLUTION OF HARMONIC MAPS
287
First assume eo 1 < (L-2)' and scale v(x, t) = u(xo + e- 1 /2x, to + eo 1 t). Note that v E Coo (P; N) and sup e(v) = eo 1 P
e(u) < eo-1 sup e(u)
sup P.o _112(zo)
P§4.
< 4eo' sup e(u) = 4, Pv
while
(e(v))(0) = 1. By the Bochner inequality (1.12) therefore we have d
in P
dt
and Lemma 1.3 gives
r
1 = (e(v))(0) < C fP e(v)dz = Ceo
12
f
e(u) dx dt.
Denote G1 (z) = G(z - x) the fundamental solution with singularity at z = (x,1) and choose z = zo + (0, eo 1). Then
1 -1, IVuk12Gz° dx, 2 "'x{t0-r3} e = k( 1 - Ito , else. E is relatively closed. Indeed, if 2 E E, let
zt E E, zt -. 2. By definition of E we have iminf lim inf i-oo
( 2R2 JRmx{t-R z} 1VukI2G=, dx > to
for any R > 0. Since G, -. Gr uniformly away from z and since E(uk(t)) < E0 < oo, the limits l - oo, k - oo may be interchanged for fixed R > 0, whence lim inf 4iz (R) > to,
k-oo for all R > 0; that is, z E E. Next observe that for zo f E there is a sequence (uk) and some R > 0 such that 0 u (R) < to. Proposition 1.4 implies that C
sup Mud 0, uniformly in k, and similar bounds for higher derivatives. Thus we may pass to the limit k -, oo in (1.8) and find that u is a smooth solution of (1.8) away from E. In order to be able to assert that u extends to a weak solution across E we need to estimate the "capacity" of E, respectively its m-dimensional Hausdorff measure with respect to the parabolic metric 6((x, t), (y, s)) = Ix - y1 + For a set S C Rm x R the latter is defined as
It --81-
inf`Fr;';SC UQr,(z;),z,ES,r; 0 and let Qr; (zi), ri < R, be a cover of S. Since S is compact, we may assume that the cover is finite. Moreover, a simple variant of Vitali's covering lemma shows that
there is a disjoint sub-family Qr;(zi), i E 9, such that S C UiEJQ5r;(zi). Let zi = yi + (0, r?), i E J. Since J is finite, there exists k E N such that t; -6'r; eo < 4 (6r2) < C f
t;-462r, fR-
:5
C(b)r, m
IVuk12G;, dxdt
I Vuk 12 dx dt + Cb-m exp
J
1) (- 3262
t62r; ftj -463r?
IVuk12G,; dxdt R'"
< C(b)r, m f
I Vuk I2 dx dt + Cb2-m exp
(__!_) Eo,
ri (ZI )
for all i E 9, where we used the fact that
G, < 6-m exp
(....i) Gx,
/
on Rm x [ti - 4b2ri , ti - b2ri ] \ Qr. (zi)
and Theorem 1.10 to derive the last inequality. If b > 0 is sufficiently small, we have
C62-m exp
- 32b2 1
Eo
e
0, X(s) - 2b for s > 36, for maps u : M -+ R", then the sequence of approximate solutions (UK) to (1.8) defined by the gradient flow of Ek again satisfies an analogue of Theorem 1.10 and Proposition 1.4. Similar to Proposition 1.5 we then establish that a sub-sequence (UK) converges weakly to a partially regular weak solution u of (1.8),(1.9). Moreover, inequality (1.24) holds. See Chen-Struwe [21] for details. Let us now turn to some further consequences of the monotonicity formula
Nonuniqueness Coron [27] observed that for certain weakly harmonic maps uo : B3 -+ S2 the stationary weak solution u(x,t) = uo(x) of (1.8) does not satisfy (1.24), hence must be different from the solution constructed in Theorem 1.8. Slightly modified, we repeat his construction. Suppose uo E Hja (R3; S2) is weakly harmonic, uo(x) = uo (), and consider u(x, t) = uo(x). Then u weakly solves (1.8) and -O2(p) =
f rIvuoi2exp (-fx 4p2 _ I2)
dx < co
2
for any z E R3 x R, any p /> 0. Suppose that u satisfies (1.24). This implies (1.25)
P JR3
IVuoI2exp I - Ix4p212 1 dz
1), moreover, we can achieve that the center of mass Regarding uox) = uo (
q = f PVuo(x)I2xdv01s2 0 0. (Hence the map uo is not minimizing for its boundary values on B1(0) C R3; see Brezis-Coron-Lieb [13], Remark 7.6.) Denote
0(P, x) =
1 P
212 )
IVuot2 exp I - Ix -
dx
\\\
JJJ
for brevity. Note that
0(P,0) = foo US2 is independent of p > 0. Moreover, compute
o (P, 0) = =
j j
s
=9J
IVuot2 2p27 3
exp
I exl' (-4P2)
exp (-Q) do p
P
(_2)
(JIvHoI2dvs2). oo
= ao
ll
2p3
P sds
q
p
Hence for z = tq, 0 < p < r, if t > 0 is sufficiently small we obtain 0(p, x) = ao + t II PI2 + 0(t2) > 0(r, x) = ao + t 1!12 + 0(t2), contradicting (1.25). On the other hand, as in Theorem 1.8 we can construct weak solutions ii to (1.8) for initial data uo satisfying (1.24), showing that u # u and hence showing nonuniqueness in the energy class of weak solutions to (1.8), (1.9). Note that we have spontaneous symmetry breaking, since u cannot be of the
form u(x,t) = v (11,t). The latter map v would solve (1.8), (1.9) on S2 x [0,00[. Since uo : S2 - S2 is smooth and harmonic, by local unique solvability of (1.8),(1.9) on 52 x [0, oo[ for smooth data this would imply v(t) - uo. It remains an open problem to exhibit a class of functions within which (1.8), (1.9) possesses a unique solution. Certainly, the class of functions satisfying the strong monotonicity formula
ot(p) 5 qt(r) for all x and all 0 < p < r < OF is a likely candidate.
M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS
292
Development of singularities The most surprising aspect of the monotonicity formula is that it may be used to prove that (1.8), (1.9) in general will develop singularities in arbitrarily short time. The existence of singularities was first established by Coron-Ghidaglia [28]; see also Grayson-Hamilton [61]. These results were based on comparison principles for the reduced harmonic map evolution problem (1.15) in the equivariant setting. A deeper reason for the formation of singularities was worked out by Chen-Ding [19]. This is related to a result by White [152].
Theorem 1.11. Let M, N be compact Riemannian manifolds and consider a smooth map uo : M -+ N. Then inf{E(u);u E C°°(M,N),
u is homotopic to uo} > 0
if and only if the restriction of uo to a 2-skeleton of M is not homotopic to a constant.
Remark 1.2. In particular, there are examples of non-trivial homotopy classes of maps uo : M -+ N such that inf{E(u); u is homotopic to uo} = 0.
Example 1.6. Let u1 = id: S3
S3. Let 7r: S3 \ {p} - R3 be stereographic
projection, and let Da : 1R3 - R3, D,, (x) = Ax be dilation with A > 0. Then define U,\ = 7-1 . Da . 7C: S3 - S3.
Clearly, ua - u1 = id for all A > 0 and E(ua) - 0 (A -+ oo). The construction of Chen-Ding can be vastly simplified by combining Remark
1.10 with Proposition 1.4, as in Struwe [143] or [145]. Let M, N be compact manifolds, dim M = m > 3.
Theorem 1.12. For any T > 0 there exists a constant e = e(M, N, T) > 0 such that for any map uo: M -+ N which is not homotopic to a constant and satisfies E(uo) < e the solution u to (1.8), (1.9) must blow up before time 2T.
Proof. Suppose u E Coo (M x [0, 2T1; N) solves (1.8), (1.9). For z = (z, , T < t < 2T, R2 = T estimate $z (R) < CR2--E (u(t - R2)) 5 CR2-mE(uo) < co ife 0 is the constant in Proposition 1.4. Hence by Proposition 1.4 for z = (a, >, T < i:5 2T, we have the uniform a-priori bound
IVu(2)1< R =C(M,N,T). By the Bochner-type inequality (1.12) therefore we have d
AM) e(u) < Ce(u) on M x [T, 271
dt and Lemma 1.10 implies that
sup(e(u)) (x, 2T) < CE(uo) < Ce,
where C = C(M, N, T). Hence, if e = e(M, N, T) > 0 is sufficiently small, the image of u(2T) is contained in a convex, hence contractible, coordinate neighborhood on
293
PART 1. THE EVOLUTION OF HARMONIC MAPS
N and u(2T) is homotopic to a constant. But then also uo is homotopic to a constant map, a contradiction. Therefore u must blow up before time 2T.
0
Singularities of first and second kind Let u E C°°(Rm x [-1,0[; N) be a solution to (1.8) with an isolated singularity at the origin, and satisfying (1.24). If (1.26)
IVu(x,t)12 0,
0) = FO. Here we denote by ff(p, t) the mean curvature vector of Mt at a point x = F(p, t) E R". Equation (2.1) corresponds to the negative gradient flow for the volume of Mt. Indeed, if µt = t)'µ denotes the pull-back of the first fundamental form on Mt induced by the Euclidean metric µ , we have and
d
dtut =
-Haµt.
Recall that the first variation of volume Vol Mt =
JM=
r
dµ1
M
t
of the hypersurface Mt C R", when deformed in direction of a vector field t/i, is given by
f (9G, v) H d1im = Me
f4f,
('+G, L dpi"',
296
M. 8TRUWE, GEOMETRIC EVOLUTION PROBLEMS
where v denotes a unit normal vector field and H the corresponding mean curvature on Mt. Moreover, if A denotes the Laplacian in the pull-back metric At, we have
H=OF and (2.1) takes the form of a heat equation on M. The mean curvature flow was first investigated by Brakke [11], motivated by a study of grain boundaries in annealing metal, and later by Huisken [76]. While Brakke considered the problem in the general context of varifolds, Huisken approached the problem from the classical, parametric point of view (2.1). A weak form of (2.1) in terms of motion of level sets was later proposed by Osher-Sethian [116) and investigated in detail by Evans-Spruck [43], [44], [45], [46] and independently by Chen-Giga-Goto [22]. Finally, Ilmanen [79) has been able to relate the level-set flow and the Brakke motion of varifolds in the frame-work of geometric measure theory. We will trace a part of these developments. First we consider the parametric point of view. We will consider two model cases: the case when Mo is the boundary of a bounded region Uo C R" and the case when Mo is represented as an entire graph
Mo={(x',uo(x'));
x'ER"-II
Another special case is the case of plane curves or, more generally, curves on Riemannian surfaces. In the 80's Gage [49] and Gage-Hamilton [50] established that convex curves in the plane evolve smoothly to a nearly circular shape before they shrink to a 'round' point. Grayson [58], [59] extended these results to general closed embedded curves in the plane and on surfaces. In the latter case, another possible long-time behavior is the convergence towards a closed geodesic. AbreschLanger [1], Angenent [4) and Grayson [60] also studied immersed curves in the plane and obtained highly interesting examples of singular, self-similar behavior. However, in these notes we will study only the higher-dimensional case m > 2, where we will be able to observe singularities even if the original hypersurface is smoothly embedded, in contrast to the one-dimensional case.
2.2. Compact surfaces Let U0 be a compact set in R" with smooth boundary Mo and, say, outer unit normal v. If Uo is convex, then with our sign convention the mean curvature vector H is pointing inside Uo, and Mo, evolving by (2.1), is contracting.
Example 2.1. If Uo = BR(0), then Mt is a sphere of radius R(t), satisfying
=
-n-
R 1,
R(0) = Ro;
that is,
R(t) = R - 2(n - 1)t, and Mo shrinks to a point in finite time T = 2 "? I Example 2.2 (Angenent [5]). There exists a torus-like surface in R3 shrinking to a point by self-similar motion.
PART 2. EVOLUTION OF HYPERSURFACES BY MEAN CURVATURE
297
The evolution of spheres is in certain ways characteristic; moreover, their evolution gives nice "barriers" to control the evolution of more general hypersurfaces, due to the following
Theorem 2.1 (Comparison principle). If Mo and Mo are boundaries of smooth, relatively compact regions Uo C Uo CC R", respectively, and if Mo and Mo evolve
by mean curvature through families Mt = 8Ut, Mt = BUt, respectively, then for
t>0there holds UtCUt. IfUo0Uo,weeven have Ut000t. Heuristically, the comparison principle follows from the observation that if Mt were to "touch" Mt from "inside" at some point x, then the mean curvature of Mt at x would exceed the mean curvature of Mt at x, drawing Mt and Mt apart. A formal derivation, based on the maximum principle, will be given shortly. As a consequence of the comparison principle any compact hypersurface Mo = 8Uo evolving by mean curvature will become extinct in finite time T. Indeed, if
UoCBp,(xo)for some xo,Ro>0,we have T 0, T > 0 let Rp >R2 + 2(n - 1)T,
a= sup Iuol+Ro Blto (=o )
and choose balls Bo = BR. ((x'o, ±a)) "above" and "below" Mo as "barriers" to control u. Note that by time t the mean curvature flow will shrink the spheres
So = 8Bo to the concentric spheres St = 8Bt , where Bi = BR(t)((xo,±a)), with R(t)= Ro-2(n-1)t>RforO t, we have d
x(s) = -Hv =
x=
( IVVu
-
IVul
at the initial time s = t. On the other hand, the requirement u(x(s), s) = 'y for all s leads to 0=
d Z u(x(s),8)
Vu) +
au = -IVuI div (Dull + -u. 5i
Thus we say that the motion of level sets of u defines a generalized motion by mean curvature if u : R" x [0, oo[-. R satisfies (2.4)
69
u - (bij -
IVU12
/ U--,
= 0,
where i and j are now summed from 1 to n. Note that in the language of fluid mechanics, equation (2.4) corresponds to the Eulerian viewpoint while the parametric problem (2.1) corresponds to the Lagrangian one. In the present context, equation (2.4) seems to have first been proposed by Osher-Sethian [1161. For smooth functions u with Vu 34 0, equations (2.4) and (2.1) are in fact equivalent. However, if we want to model the motion of a closed hypersurface Mo bounding a compact region in R" by the evolving level sets rt = {u(x, t) = 0} of a solution of (2.4), it is clear that Vu must vanish at some point in the interior of Ft for each t and hence (2.4) cannot be interpreted classically on all of R" x (0, oo[. Moreover, by the geometric interpretation of (2.4), if u is a solution of (2.4), and if W is any continuous, one-to-one function, then %F(u) should be a solution of (2.4), as u and T(u) have the same level sets. To be able to include these cases a notion of weak solution is required. The appropriate notion of generalized solution is that of viscosity solution as in [29], [301, [40], [80], [83], and [106].
Definition 2.1. A function u E CO f1 LOO(R" x [0, oo[) is a weak sub- (super-) solution of (2.4) if the following holds: For each 0 E C°O(R"+a) and each (xo, to) E R" x]0, oo[ such that u - ¢ has a maximum (minimum) at (xo, to) we have (? 0) Ot - (bij - 7700-0-,-, < 0 for some 17 E R", InK < 1, and
n=
Vo(xo, to) IVOxo, to) I
if VO(xo, to)
0.
In particular, a weak sub-solution forces any smooth function whose graph "touches" the graph of u from "above" at (xo, to) to be a sub-solution at (x0, to), and similarly for super-solutions. In fact, for more general classes of equations one usually requires that ¢(xo, to) = u(xo, to) for comparison functions ¢. However, since translations u --+ u + e leaves (2.4) invariant, here this condition can be waived.
Often it may be convenient to represent admissible comparison functions as
0(x, t) = u(xo, to) + p(x - xo) + q(t - to) + 2 (z - zo)T R(z - zo) + o(Iz - zol2)
as z -+ zo, for p E R", q E R, R = (rij) a symmetric (n + 1) x (n + 1)-matrix, and z = (x, t).
M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS
302
Definition 2.2. A function u E co n L°°(R" x [0, ool) is a weak solution of (2.4) if u is simultaneously a weak sub- and super-solution. Let us now turn to some simple special cases.
Example 2.3. (The motion of spheres) Let uo(x) = Ro - 1x12. Then
u(x,t) = uo(x) - 2(n - 1)t satisfies (2.4) classically at all points where Vu 0 0, and the level surface
r t = {x E R' ; u(x, t) = 0}
is given by rt = BBR(i)(O)
with R(t) =
R-,0 - 2(n - 1)t as in the classical, parametric setting. To verify that u solves (2.4) also at xo = 0 it suffices to consider as comparison functions
0t (x, t) = u(x, t) f (z - zo)T R(z - zo), where z = (x, t), z o = (0, to), R = (ri,) >2 0. Then 0+ > u > r- near zo, V 0, and, letting rl denote an arbitrary unit vector in R', OP
(zo) _
- (bii - tli'Jj)WZx; = ut - (bii - 11070'u-,-, T- (bij -T(6,, - i7j%)rii
has the desired sign.
2.5. Uniqueness, comparison principles, global existence Thus, in a simple case we reobtain the classical motion by mean curvature. Note that in the example above we simultaneously recover the evolution of all spheres centered at the origin; moreover, the solution u remains smooth and is in fact globally defined even past the extinction time To = Ro/2(n - 1) of any individual sphere. However, it may not be obvious that this solution is unique; conceivably, after
passing through a "geometric" singularity a level surface might evolve in any one of several different ways, as is illustrated by examples of Brakke [11] for the mean curvature flow of varifolds. In fact, it is quite surprising that solutions to (2.4) are unique. Moreover, we shall see that the motion of a level surface 170 depends only on the surface, not on the function uo defining it. Finally, solutions persist for all time. Thus, (2.4) defines a unique global extension of the mean curvature flow of hypersurfaces to arbitrary level surfaces of continuous functions in R". We begin by the following observation. Theorem 2.4 (Naturality; [43], Theorem 2.8). If u E CO f1 LOO (R^ x [0, oo[) is a weak solution of (2.4) and if T : R --+ R is continuous, then v = W(u)
is a weak solution of (2.4).
Proof. First consider the case where T E COO with 'P' > 0 on R. C°°(R'+i) and suppose 0 = '+6(xo, to) - v(xo, to) R° for some constant RD > 0. Then there exists a weak solution u E CO n LO°(R" x [0, oo[) of (2.4) with initial data 0) = uo and
u(x,t) =yo Moreover, if uo is Lipschitz, so is
for lxl2 +2(n - 1)t> R t) and I[VUIIL-(R^x[o,oo[) : IIVu0IIL-(R^)
.
M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS
304
Proof. Assume for simplicity that uo is smooth. Extend uo to R"+1 by letting u0(x, xn+1) = u0(x) - Exn+1
Denote points in R"+1 by y = (x,xn+1). The level surface
to = {y E R"+1; uo(y) = 0} is a smooth graph
Hence, by Theorem 2.3 above, to `generates a smooth family of graphs
rt = { 1 X' !((x
t)) ; X E R" 1 ,
evolving by mean curvature. Any other level surface ro = {y;uo(y) = y}
is a parallel translate of to and evolves through parallel translates of I's. Hence uE (x, xn+1, t) = ue (x, t) - Exn+1
is a (classical) solution to (2.4) on R"+1 x [0, oo(. Computing explicitly, thus ue satisfies (2.5)
ut - 6ij -
uex, uex
(Due (2
+ E2)
ux,xj = 0
in R" x [0, oo[ with initial condition 0) = uo. Moreover, for fixed e > 0 by Theorem 2.3 we have local bounds for I Vue I on any slice R" x [0, 71. Thus, by the maximum principle for uniformly parabolic equations, we have uniform a-priori bounds IIuCIIL0(R,,x(O,ooi) = IIUOIIL.O(R°)
Finally, differentiating (2.5) with respect to xI, we have
It - b{j -
uex ue 'ZJ 2 IVueI +E2
ux,xixi
C(E) la'ds' I
IVux, I
and another application of the maximum principle gives the Lipschitz bound IIVudIIL°O(R°xj0,ooi) !5 IIVuoIIL,(R^),
uniformly in e > 0. Thus, passing to a suitable sequence e --+ 0 if necessary, ue - u boundedly and locally uniformly on R" x [0, oo[. A simple variation in the proof of the preceding compactness result shows that u weakly solves (2.4). The characterization of the support of (u - ryo)+ results from comparing the solutions ue above to the solutions yE (x, t) corresponding to the shrinking of spheres. 0
Theorem 2.7 (Comparison principle and uniqueness; (43], Theorem 3.2). If u, v E ConL°°(R" x [0, oo[) are weak solutions of (2.4) with initial data 0) = uo < vo = v(., 0) and if for some R > 0 both u and v are constant on Rn x [0, oo(n{IxI +
t > R}, then u < v. In fact, IIu - VIIL°°(R"x[0.oo() : IIu0 - VOIIL-(Rn)
PART 2. EVOLUTION OF HYPERSURFACES BY MEAN CURVATURE
305
Remark 2.3. The construction used above to prove global existence yields weak solutions u < v if the initial data are ordered correspondingly. This follows by applying the maximum principle to the approximate equations (2.5). However, the proof of the corresponding ordering relation among all weak solutions of (2.4) with initial data uo < vo is much more involved. An important concept in the proof is that of sup- and inf-convolution, defined as follows. For w E C° f l LO° (R" x [0, ooD,
e>0let
wE(z) = wc(z)
fER"x(O,oc(
(w()_!Iz_I2) e
fER^
(w()+!Iz_I2). e
sup
0.00(
Then inf w < wt < w < wt < sup w, and wt, wt - w locally uniformly on R" x 10,00f.
Moreover, sup- and inf-convolutions are regularizing in the sense that wt, wt are Lipschitz and almost everywhere twice differentiable. The latter assertion follows from the fact that, for instance, the function
z-+wt(z)+1x12=s{p{wt(t)+E (1x12-Iz-e12)}, being the supremum of a family of affine functions, is convex, hence almost everywhere twice differentiable by a theorem of Alexandroff. Finally, if w is a super-solution of (2.4), wt again is a super-solution. Similarly, if w is a sub-solution of (2.4), so is wt. Now the idea in the proof of the comparison
principle is as follows: Suppose by contradiction that u(z) > v(z) for some z E R"x]0,oo[. By hypothesis, u and v are constants for large 1x1 + t. Thus
max (u-v)=a>0
R" x (0,00(
is attained in R" x]0, oo[. By uniform local convergence uE -+ u, vt -, v then also
max (ut - vt)
R"x[0,o0(
a>0
-2
is attained in R"x]O,oo[ for sufficiently small e > 0. Finally, if we choose a > 0 small enough, also
max (ut - vt - at) > a4 > 0
R"x(o.oo(
will be attained at some interior point zo E R" x10, oo[. Suppose for simplicity that vt is twice differentiable near zo with Ovt (zo) 0. Then a suitable extension of (Vt +at) is admissible as a comparison function 4, for ut. Since ut - 0 by choice of zo has a local maximum at zo and ut is a weak sub-solution of (2.4), we find
vvl E
0 ? mt - 6,j
-
t
(v +a) - (aij - 10u12) v=exs, IV012
contradicting the fact that vt is a super-solution.
M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS
306
2.6. Monotonicity formula In order to obtain partial regularity results for (2.4) we may try to carry over the estimates obtained for harmonic maps by means of the monotonicity formula. In the following, we derive an analogous monotonicity estimate for the generalized mean curvature flow. In fact, in the parametric setting of problem (2.1) a monotonicity formula analogous to Theorem 1.10 was independently obtained by Huisken [771 in 1990.
For a given point zo = (xo, to) E R" x 10, oo[ let
G"' (x, t) =
I
1
exp
47r(to - t
t))an
(_ [x - xo[ a ` 4(to -
if t < to,
G,, (x, t) = 0 else. Observe that G,, is the fundamental solution to the backward heat equation in any n - 1- dimensional hyperplane through xo. (Any such hypersurface is a stationary solution of (2.1).) Consider a smooth function uo: R" - R which is constant for large [x[ and let u' be the unique global smooth solutions to the approximating problem (2.5) with initial data u° = uo constructed in the proof of Theorem 2.6 above.
Proposition 2.1. For any zo E R" x [0, oo[ the function
01(t) =
[VuE-12+,E2 G0 dx
JR^x{c}
is non-increasing and [2rui + t. Vuf 12
d
JR^x{c} 41rl2(iVudl2+E2)
dL
lVuE [a +E a G,
dx,
where r=t-to, f =x-xo. Proof. Translate zo to (0, 0) and for A > 0 scale u -, U,\ (X, t) = u(Ax, Alt) E-+Ea=AE.
Then ua solves (2.5) for Ea with u'(.,0) = uo.\ and 4 (-A2;u`) = 0af(-l;u1). Hence (2.6)
Wt--01 (t) _
I`(t;10) =
2
[t,
T- (0A'(- 1; ul))
Since u` is smooth, moreover d
tat(-1; ua) _
Vu, V( ua)+AE2Gdx [Vu I r+ - JR^x{-1} Vua V(a,,ua) Gdx
fR-X(-Il Vu' +A E where G = G(o.o) and we used the fact that A > 0. Integrating by parts and using the scaled equation (2.5), that is, div
Vu A
7u7\
uac E
)
VuT + A 7 '
PART 2. EVOLUTION OF HYPERSURFACES BY MEAN CURVATURE
307
and the relation VG(x, t) _ -&G(x, t), we thus obtain 4
)
(- 1; u`
JR"X(-1) -2t Mu
z Dua }
E
1 2 W \ ' + x qua 12
IVuj +,\2f2
JR^x{-l} 2AItl
_
12tui + x Vu` 12
IVuc 2 +E2
- JR^x{-A2} 2It13/2
d
(dAuG a)
dx
G dx
Gdx.
Thus, finally, we deduce from (2.6) that (2.7)
dv(t)0 0,s=0
-1,s 0 we have lim
to\T
^ x (T}
IDvIG, > C1 lim P1 -n r P-0 Bo(xo)
C2 > 0;
see [57], Lemma 3.5. On the other hand, the left hand side of this inequality is bounded by
4r°)
lim J IDvoiG0 0 to be determined, and let xo E 8rcnBp(t) be a point of density for the (n-1)-dimensional Hausdorff measure. Then
0 < C2 < lim
r
to\t JR^ x (t)
< lim
r
to\t JR^
0 to achieve that the right of the above inequality is smaller than C2. In view of the above results, it seems reasonable to extend the classsical motion of surfaces Mt by mean curvature beyond the first singular time T by letting Mt =
ert for t > T.
2.8. Singularities A point zo = (xo, to) with xo E rc, is a singular point of a C2-solution u of (2.4) if Vu(zp) = 0. (Conversely, if so is regular, that is, Vu(zo) 0 0, then rt, is a C2-surface near xo.) Suppose, for instance, that A = V2u(zo) is positive definite. Then by (2.4) we
have a := ut = (bid -
0 at zo and hence for t < to close to to there
holds
rt
{x; (x - xo)T A(x - x0) = alto - t)1 . 2
Hence we expect the 0-level surfaces of the rescaled functions UR(X,
t) = u \XO + R'to + R2 /
at t = -1 to approach a smooth limit. More generally, consider the signum function ' above and for a weak solution u of (2.4) let v = 111(u). Moreover, for zo = (x0, to) E R" x [0, oo[, R > 0, let
x t VR(x, t) = v x0 + R, to +
R2
M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS
310
Suppose the level surfaces 1'R = {VR(-,t) = 0} are smooth. Then by the monotonicity estimate (2.8) for any T > 0 we have
f
0
T
2
JHR+ 2tvRl IDVRIGdt R°
IH
to
= fo_TR-2 JR. t
2
+
vI IDvIGzo dt 2r
0
(R - oo),
I
where v denotes the normal v = Tsf and H the mean curvature. Generalizing [77], we call a singularity of Type 1 if the surfaces TR smoothly converge to a smooth
surface Fr for every t < 0, as R - oo. Let H,,, v. denote the mean curvature and normal on Fr. Then by the above we have
2ItIHH = (x,v.). Represent rt ° = Ft (I'' 1). Then by (2.1) it follows that d (F, v_)
Ft=-H.v. _ -21t1
voo
whence up to tangential diffeomorphism 1'i° is contracting by self-similar motion. All smooth star-shaped hypersurfaces in R^ satisfying
H=(x,v)>0, and hence giving rise to self-similar solutions of (2.1) or (2.4), have been classified by Huisken [77], Theorem 5.1. In particular, we have:
Theorem 2.10 ([77], Theorem 4.1). If M"-1 C Ilt° is compact with non-negative mean curvature H = (x, v), then M is a sphere of radius n - 1. There seems to be a large variety of self-similar solutions of (2.1) that do not satisfy H > 0; in particular, the torus-shaped hypersurface of Example 2.2. Moreover, there are singularities that evolve at a different (faster) rate, which we call of Type 2. The study of singularities for the mean curvature flow is a very active field of current research.
PART 3 Harmonic maps of Minkowsky space
3.1. The Cauchy problem for harmonic maps Consider Mm+l = R x R', equipped with the Minkowsky metric 1
g = (gap) =
0 and let N be a compact Riemannian manifold, isometrically embedded in RI. By
analogy with harmonic maps of Riemannian manifolds, we define a map u: M'"+1 -+ N C R to be harmonic if it is stationary for the Lagrangian L(u; R) =
Jx l (u) dz,
where
1(u) =
29Q08Qu.80u`
= 2 (IVuI2 - lug12)
and where z = (t, x) = (x°, xl, ... , x"). Here, V = ( DiT , ... , 88 ) denotes the spatial derivatives and, for brevity, ea = g , a = 0, ... , m. Moreover, it will be convenient to denote D = (Oo...... m) _ (at, 0) and 8° = g°1980. For 0 E Co (M'"+l; u- TN) supported on a space-time domain R C M'+1 we have de
L ("N (u + CO); R) 11=0 = J O u O dz = 0, R
where _02
0
2-0
is the wave operator, acting component-wise on u; that is, u is harmonic if (3.1)
Ou 1 TuN. 311
M. 8TRUWE, GEOMETRIC EVOLUTION PROBLEMS
312
Suppose v1+1,. .. , v is a smooth (local) orthonormal frame field for the normal bundle T1N C TR". Then in the spirit of 1.1 we may write
u = E Akvk
u,
k
where Ak : M"'+I -. R is given by Ak = (E] u, vk a u) = -Sa (8°u, vk . U) + (8°u, So (vk ° u))
= A"(u)(BQu,8°u) = Ak(u)(Vu, Vu) - Ak(u)(ut,ut), with (3.2)
denoting the scalar product in R". Hence
Du = E(vk ° u)Ak(u)
(Sau,&,u)
= A(u)(Du,Du).
k
Introducing the "null form"
Q* 0) = VOW L' - ot0t, note that (3.2) also may be written in the form (3.3)
uk = a (u)Q(u`, uJ ),
k = 1, ... , n.
Finally, in local coordinates (u',. .. , uI) on N, (3.2) may be written (3.4)
uk = f'j (u)QW, uj ),
where I''if denote the Christoffel symbols on N. Note that Q(u, u) = 21(u), and Q is associated with the wave operator in the same way as the Dirichlet energy density is associated with the Laplacian. For hyperbolic problems it is natural to consider the Cauchy problem: Given
initial data uo,ul at t = 0 we seek u: M'+I -+ N C R" satisfying (3.2) and the initial condition (3.5)
Unt .o = uo,
ut 1t-o = ul
on R,".
The questions we consider are local and global existence of classical or weak solutions, and development of singularities. We will not consider the problem of determining the asymptotic behavior of solutions or scattering theory. Equation (3.2) shares the property of the homogeneous wave equation 0 = 0 that initial disturbances propagate with speed < 1. Thus, as far as the above topics are concerned, we may restrict our attention to initial data with "compact support"
in the sense that (3.6)
uo - const,
Ui = 0
on R' \ Sao
for some compact set Sao CC R"'. In what follows we discuss various mechanisms for proving existence for (3.2), (3.5).
PART 3. HARMONIC MAPS OF MINKOWSKY SPACE
313
3.2. Local existence Existence via the fundamental solution Consider the Cauchy problem for u: M"+1 -, R satisfying
u=f
in M'11
with initial data uo and u1. If m = 1, the solution is given by u(t, x) =21 (uo(x + t) + uo(x - t)) + 1 2
-t
x+a
:
l I f -a f(t - s, y) dy ds.
+2
Ifm=3,wehave u(t' x)
f'+t u1(v) dy
0
)
dt
(4ir , u0(x + t{) do
+f'
8
47r
S2
+
_L
J u1(x + tc) do
f (t - s, x + s£) do ds,
whereas, if m > 3, the representation formula involves also derivatives of f transverse to the backward light cone M(z) from z = (t, x), given by
M(z) _ {(s,y) ; t -s = Ix - vI} . Note that, even in dimension m = 1, there is no "gain" in derivatives. Thus, already
in dimension m = 1, for nonlinearities f = f (u, Du) as in (3.2) a simple-minded iteration procedure will fail to yield a local existence result due to loss of derivatives.
Energy method First consider a smooth solution u of the homogeneous wave equation
u=0
in M'"+I
having compact support on any slice it = cont.} Multiplying by ut we obtain the conservation law (3.7)
0 = attar - Auut =
d f utI1+IVu12\) dt
2
I
- div(Vuut).
Denote e(u) = IutI2 + IDuI2 = 1 IDuj2 2
2
the energy density of u and let
E(u(t)) = f
e(u) dx. t }xA^
Then, since u(t) has compact support, upon integrating the above conservation identity we obtain dtE(u(t)) = 0; in particular
E(u(t)) E(u(0)). Similarly, since derivatives v = D°u = 00 '0 ... &,W-u for any multi-index a = (ao,... , a,) again solve v = 0 with supp(v(t)) cc It'", we have E(D°u(t)) < E(D°u(0)).
M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS
314
Note that D°u(0) can be computed entirely in terms of uo and ul, in view of the equation utt = Au. 0. Observe that (3.7) remains true if instead of Du = 0 we only require On account of (3.1), the latter condition is satisfied for harmonic maps, and we obtain Lemma 3.1 (Energy inequality). Let u be a smooth solution of the Cauchy problem (3.2), (3.5) having compact support in the sense of (3.6). Then for all t we have
E(u(t)) < E(u(O)). (In fact, for smooth u equality holds.)
In order to obtain corresponding bounds for higher derivatives, however, we have to consider also inhomogeneous equations. It is useful to introduce the spacetime norms 00
1 1/4 I
/
OL+
,
1 < p, q < oo, and the corresponding spaces L9.P(Mm+1) = L9 (R; LP(Rm)). On a finite space-time cylinder SIT = [0, T] x Cl corresponding norms may be defined.
Consider now a smooth solution u: M+1 --, R of
Ou = f with initial data uo, ul, where u(t) has compact support for any t. Multiplying by ut we obtain
je(u) - div(Vuut) = fut 2 for all i. Hence we may assume that at least two distinct numbers r(') are finite. Since 2
EIryi'li < 2 + IoI < 2 + s
ryW < 1 + Iop < 1 + s,
i>O
then, if s > m/2, indeed we have
i< (21
i>O
-Y(i) - (s + 1) m
yi'l - 2(s + 1) ) o m2 2
3.3. Global existence Example 3.2. Geodesics (Sideris 1131]). If y: R -' N is a geodesic on N and if v: M"t+1 -, R satisfies
v=0,
then u = y
v solves
Ou = y'(v) Dv + y"(v)Q(v, v) 1 TuN, because y" 1 T.N. That is, u is harmonic. Note that u preserves the regularity of the initial data. Example 3.3. Let m = 1. In this case, global existence and regularity was established by Gu [68] and Ginibre-Velo [54]. A surprisingly simple proof was given by Shatah [127] based on the following observation: Multiply (3.1) by ut, respectively by ux, to obtain the system of conservation laws
0=Duut=Bte-8ym, 0=Duu,, =Ojm - Ore, where e = e(u) = z (IutI2 + Iu=12) is the energy density and m = m(u) = u=ut is the density of momentum; compare (3.7). Thus, e satisfies the linear wave equation
De=0.
PART 3. HARMONIC MAPS OF MINKOWSKY SPACE
319
and from the representation formula we obtain pointwise bounds on e, and hence on Du, in terms of the initial data. Higher regularity for smooth data then follows by using the energy inequality (3.8) to iteratively derive Gronwall-type estimates
d E(D°u(t)) < C + CE D- (u(t))) for any a.
Remark 3.4. For m = 3 and rather smooth initial data uo, ul whose norm is small in H', s > 10, Sideris [131] has constructed global solutions to (3.2), (3.5) in H' by combining the local existence result above with certain decay estimates that may be obtained by using the invariant norms of Klainerman [95]. The latter are defined by means of the generators of the Poincare group and scale changes. In addition to the standard differentials 8°, a = 0, ... , 3, denoted I',. .. , r4 in the following, these include the generators r5,. . . , r 10 of Lorentz and proper rotations
f3°6 =x°Y -xp8°,
0 0, w E S-'. Moreover, on M'"+I we consider the standard metric -dt2 + dr2 + r2d62,
written in terms of spherical coordinates (r, 0) on Rm. Consider equivariant maps
u: M+' - N of the form u(t, r, 0) = (h(t, r), w(6)) where w is a homogeneous harmonic polynomial of degree d > 0. Then
L(u) = 2 f {Ihri2 - Ihtl2 +
r
92(h)Ioew121
r'"-' dtdrd6,
and u is harmonic if and only if h : M'"+1 - R satisfies kf(h) = 0, Clh +
(3.10)
r2
where k = d(d + m - 2), and f (h) = g(h)g'(h). Observe that for N = S'" we have g(h) = sin h. More generally we require g(0) = 0, g'(0) = 1, g(h) > 0,
(3.11)
for h > 0, and g(-h) = -g(h).
Note that the ball of radius ho around 0 in N is convex iff g'(h) > 0 for
0 0. Moreover, h assumes the initial data (3.14) in the sense
that IIh(t, r) - aII6l. (R3) -+ 0
(t -+ 0),
b
(t - 0).
IIht(t, r) -
JIL2(R3) -> 0
Note that ho E Him, hi E L«. On the other hand, also the function
r>t
h(t,r)-f O(i),
r 0, showing that weak solutions are in general not unique. To verify that h solves (3.15), for any 0 we split 1
11100 {_hP + hrlr + r
_
sin 2h y r2 dr dt - Joo y(0, r)r2 1
b
i,(0,r) r2dr}+
...}r2drdt-j00
r
fo, f 0" e
111
f1f 0
{...}r2drdt=I+11.
0
Clearly, since Dh(t, r) = 0 for r < t, the second integral II = 0. Moreover, since h h for r > t, and since h satisfies (3.15) the first integral reduces to the boundary term
I=- 75 f (ht(t, t) + hr(t, t)) (t, t)t2 dt I
which also vanishes on account of ht+hr=_
t
ci/(t/+tq'\t/ (l (l (1
t) '( ) =0
for r = t.
Observe that h induces a solution u of (3.2) with E(u(t); BI(0)) < E(u(t); BI(0)) for any t E10, 11, where u is the solution corresponding to h. Hence there may be a chance of restoring uniqueness by some entropy principle.
PART 3. HARMONIC MAPS OF MINKOWSKY SPACE
323
Self-similar solutions, general theory More generally, self-similar solutions u(t, x) = v (A) to the harmonic map equation satisfy -vPP -
(3.16)
(m_p 1
+ (1 - pep) VP - p2(11
p2)
D,,,v J
where p, w denote spherical coordinates on Rm. This may either be verified by direct computations or by introducing similarity coordinates
T=
t2-r2,
r
p= !,
w=8
on {r < t} and writing the Minkowski metric on Mm+I as
ds2 = -dr2 + r2 {
dw2} -I,2)2 dp2 + 1 - p2 For u(t, r, 0) = v(-r, p, w) the Lagrangian then becomes (1
(3.17)
L(v) = 1
[ f_IVrl2 +
(1 - p2)2Iv I2 + (1 - p2)
Tmpm-I
dpdwdT. 2J T2 r2p2 l } (1 _ p2)(.+1)/2 In particular, if v = v(p,w) is stationary for L, we obtain (3.16). Note that (3.16) is an elliptic harmonic map problem on the m-dimensional hypersurface jr = 1} with the (hyperbolic) metric a
d802 = (1
vWI2
ip2)2 dp2 + 1
p2 dw2,
as was pointed out by Shatah-Tahvildar-Zadeh [130). Now observe that, if m = 3, for v to be regular (C2) at p = 1 we need
atp=1; that is, v(1, ) : 52 --+ N has to be harmonic. By the maximum principle (Jiiger-Kaul )81]) for harmonic maps into convex manifolds, therefore either v(1, ) - const. or
v(1, ) cannot be contained in a strictly convex part of N. Moreover, if v(1, ) cont., then for p < 1 sufficiently close to 1 the image of v(p, ) is contained in an arbitrarily small strictly convex part of N and again we may apply the Jiiger-Kaul maximum principle to conclude:
Theorem 3.5. If m = 3 and if u(t, x) = v (i) is a self-similar solution to the harmonic map equation (3.2), where v: R3 - N is smooth in a neighborhood of Bl (0) and such that the image v (BI (0)) is contained in a strictly convex part of
N, then v - cont. on B, (0). By Theorem 3.4 the above result is best possible in dimension m = 3. In case m = 2, due to the following result we can rule out self-similar solutions altogether.
Theorem 3.6. If m = 2 and if u(t, x) = v (1) solves (3.16), where v : D C R2 N is smooth in a neighborhood of BI (0), then v =- const. on B1 (0).
M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS
324
Proof. If m = 2 we may write (3.16) in the form (p
A"v
1 - p2vp)p + p
Multiplying by pvp E Wp
1 TN. P2
and integrating over w E S1, we obtain
(J' p(1-p2)Ivpl2dw- f IV"vI2d)
0.
Integrating in p, we find
JS,p(1-p2)Ivp12dw-f
v12dw=Co. JV
Inspection at p = 0 shows that Co = 0. Hence for p = 1 we obtain V ,,v = 0 that is, vIsa,(o) = cont. Finally, note that in dimension m = 2 Dirichlet's integral and Thence the harmonic map equation is conformally invariant. Theorem 3.6 thus is a consequence of Lemaire's result, Example 1.3
3.5. Global existence and regularity for equivariant harmonic maps for m = 2 The preceding examples of finite-time blow-up hardly leave any hope to achieve a satisfactory existence and regularity theory for the Cauchy problem for harmonic maps, except in dimension m = 2. In fact, we may state the following
Conjecture 3.1. If m = 2, then for any compactly supported initial data uo, ul with finite energy there exists a unique global weak solution u to the Cauchy problem (3.2) (3.5) satisfying the energy inequality. If E(u(0)) < co = CO(N) is sufficiently small, or if the range u(M2+1) lies in a strictly convex part of N or, more generally, does not contain the image of a harmonic sphere u: S2 - N, then u is globally smooth, provided uo and u1 are. At present the theory is still a long way from affording a proof of this conjecture
in general. Partial results, however, are known; in particular, Conjecture 3.1 has been rigorously established for equivariant harmonic maps into convex surfaces of revolution. In the following, we review these latter results, due to Shatah-Tahvildar-Zadeh [129]. A simplified proof was given by Shatah-Struwe [128]; moreover, Grillakis
[64] has recntly been able to relax the convexity assumption. Similar results for radially symmetric harmonic maps (m = 2) have been obtained by ChristodoulouTahvildar-Zadeh [25]. Thus we consider the Cauchy problem (3.10), that is, (3.18)
f
utt - Du +
r2 )
=0
in M2+1
for smooth, radially symmetric data (3.19)
ult--.o
= u0, ut 1t=o = u1
having compact support and such that uo(0) = 0 By uniqueness, also u will be radially symmetric u(t, x) = u(t, r), and u(t, 0) = 0 if u is smooth.
PART 3. HARMONIC MAPS OF MINKOWSKY SPACE
325
Regularity for small energy In a first step we shall show that this problem admits a unique smooth solution for all time provided the initial energy E(u(O))
= JR2
IuII2 +2IVuo 2
+ F(uo)1 dx r2
J
is sufficiently small. Here F(u) _ g2(u) = fo f (v) dv. Note that the energy inequality
E(u(t)) < E(u(0)) holds. Moreover, by radial symmetry
E(u(t))
= 2a ( Iut12 + I'url2 r+ F(u) r
2
0
} dr 111
_: 21rErad(u(t))Finally, it will be useful to denote
G(u) =
l
ix(Ia)
dx = 2,r f = F(u) dr = 27rGrad(u)
the potential energy of u.
Lemma 3.2. Given Co > 0, there are constants C1, el > 0 such that any solution u to (3.18), (3.19) with E(u(0)) < Co satisfies IIu(t)IIL.e 2. Then for any T > 0 we have the estimate IIVIIL-(Rmx(-T,T]) < C (IIfIIL9'(Rmx(-T,T]) + IIVOIIHI(R,) + IIt1IIH-I(Rm)) where
_ 1-
1
1
1
q
2
m+1
1
q
Hi = H'2 is the interpolation space between L2 and H1.2, H-4 its dual. Remark 3.6. Kapitanskii [90] was the first to note the importance of Strichartz' estimate for semi-linear wave equations with critical nonlinearity. Combining this result with the dilation estimates of Morawetz [109], it was possible to extend the regularity results of Struwe [142] and Grillakis [62] from m = 3 to m:5 5 (Grillakis [83]) and, more recently, even to m < 7 (Shatah-Struwe [128]).
Proof of Theorem 3.7. We apply Proposition 3.1 in dimension m = 4 with q = 3 , q' = to obtain bounds for to and its spatial gradient Vw; that is to,.. Interpolating between the two, we obtain bounds for "half a derivative" of to in Lq. More precisely, denote VTq = {u E L2(R'" x [-T,T]);u(t) E BQ'q(Rm) T
1/q
(LIIut)II, j dt
)
co.
But if Ixol # 0, given K E N, for T sufficiently close to to disjoint discs D(T; zk), zk = (to, --k), IxkI = 1X0 1, 1 < k < K, can be found, whence K
E(u(T)) > E E(u; D(T, zk)) > Keo k=1
for any K. This, however, contradicts our assumption that E(u(t)) < E(u(0)) < 00 for all t. Hence a first singularity must appear on the line x = 0. Suppose zo = (to, 0) is singular and shift time by to to achieve zo = 0. Denote K = K(0), KS = KS(0), etc. We need the following Morawetz-type dilation estimate:
Lemma 3.5. For a smooth solution u of (3.18) on a cone KS there holds z 1 uf(Z) F(a) F(u) dxdt + 4(s) j 1 1 - IXs IVUI2 + dx -' 0 Ixl2 S
ISI JK2IxI Ixll2 \
as
S-+0.
Proof. Multiply (3.18) by tut + xVu + u and use Remark 3.7 to control the z details. boundary terms; see Shatah-Struwe [128] for
Lemma 3.5 implies Theorem 3.8. Indgiven T < 0 consider the set AT
t E [T, O[ ; G(u; D(t)) =
f
F(u) dx > sup G(u, D(t)) - 62 T 0 by assumption, for large T < 0 Lemma 3.5 implies z
(t2 1- l
r
J
for any t E AT. In particular,
) IDu12 dx < 262
t(1-6)
u2(t)rdr < C6, and hence by the argument of Lemma 3.2 we obtain sup
Iu(t,x)I
