Mean Curvature Flows and Homotopy of Maps Between Spheres

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1 arxiv:math/0304v1 [math.dg 19 Feb 003 Mean Curvature Flows and Homotopy of Maps Between Spheres Mao-Pei Tsui & Mu-Tao Wang February 19, Abstract Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A corollary is that any area-decreasing map between unit spheres (of possibly different dimensions) is homotopic to a constant map. 1 Introduction Let Σ 1 and Σ be two compact Riemannian manifolds and M = Σ 1 Σ be the product manifold. We consider a smooth map f : Σ 1 Σ and denote the graph of f by Σ; Σ is a submanifold of M by the embedding id f. In [17, [18, and [19, the second author studies the deformation of f by the mean curvature flow (see also the work of Chen-Li-Tian [). The idea is to deform Σ along the direction of its mean curvature vector in M with the hope that Σ will remain a graph. This is the gradient flow of the volume functional and a stationary point is a minimal map introduced by Schoen in [1. In [19, the second author proves various long-time existence and convergence results of graphical mean curvature flows in arbitrary codimensions under assumptions on the Jacobian of the projection from Σ to Σ 1. This quantity is denoted by Ω in [19 and Ω > 0 if and only if Σ is a graph over Σ 1 by the implicit function theorem. A crucial observation in [19 is that Ω is a monotone quantity under the mean curvature flow when Ω > 1. 1

2 In this paper, we discover new positive geometric quantities preserved by the graphical mean curvature flow. To describe these results, we recall the differential of f, df, at each point of Σ 1 is a linear map between the tangent spaces. The Riemannian structures enables us to define the adjoint of df. Let {λ i } denote the eigenvalues of (df) T df, or the singular values of df, where (df) T is the adjoint of df. We say f is an area decreasing map if λ i λ j < 1 for any i j at each point. Under this condition, the second author proves the Bernstein type theorem [1 and interior gradient estimates [ for solutions of the minimal surface system. It is also proved in [3 that the set of graphs of area-decreasing linear transformations forms a convex subset of the Grassmannian. We prove that this condition is preserved and the following global existence and convergence theorem. Theorem A. Let Σ 1 and Σ be compact Riemannian manifolds of constant curvature k 1 and k respectively. Suppose k 1 k and k 1 + k > 0. If f is a smooth area decreasing map from Σ 1 to Σ, the mean curvature flow of the graph of f remains the graph of an area decreasing map, exists for all time, and converges smoothly to the graph of a constant map. The following is an application to determine when a map between spheres is homotopically trivial. Corollary A Any area-decreasing map from S n to S m with m is homotopically trivial. We remark that the result when m = is proved by the second author in [0 using a somewhat different method. The higher homotopy groups π n (S m ) has been computed in many cases and it is known that homotopically nontrivial maps do exist when n m. Since an area-decreasing map may still be surjective when n > m, we do not know any topological method that would imply such a conclusion. We would like to thank Professor R. Hamilton, Professor D. H. Phong and Professor S.-T. Yau for their constant advice, encouragement and support. Preliminaries In this section, we recall notations and formulae for mean curvature flows. Let f : Σ 1 Σ be a smooth map between Riemannian manifolds. The graph

3 of f is an embedded submanifold Σ in M = Σ 1 Σ. At any point of Σ, the tangent space of M, TM splits into the direct sum of the tangent space of Σ, T Σ and the normal space NΣ, the orthogonal complement of the tangent space TΣ in TM. There are isomorphisms TΣ 1 TΣ by X X + df(x) and TΣ NΣ by Y Y (df) T (Y ) where (df) T : TΣ TΣ 1 is the adjoint of df. We assume the mean curvature flow of Σ can be written as a graph of f t for t [0, ǫ) and derive the equation satisfied by f t. The mean curvature flow is given by a smooth family of immersions F t into M which satisfies ( F t ) = H where H is the mean curvature vector in M and ( ) denotes the projection onto the normal space NΣ. Notice that we do not require F is in the t normal direction since the difference is only a tangential diffeomorphism (see for example White [4 for the issue of parametrization). By the definition of the mean curvature vector, this equation is equivalent to ( F t ) = (Λ ij M F x i F x j ) where Λ ij is the inverse to the induced metric Λ ij = F, F on Σ. x i x j In terms of coordinates {y A } A=1 n+m on M, we have Λ ij F F x j x = i Λij ( F A x i x + F B F C j x i x j ΓA BC) y A where Γ A BC is the Chistoffel symbol of M. By assumption, the embedding is given by the graph of f t. We fix a coordinate system {x i } on Σ 1 and consider F : Σ 1 [0, T) M given by F(x 1,, x n, t) = (x 1,, x n, f n+1,, f n+m ). We shall use i, j, k, l = 1 n and α, β, γ = n+1 n+m for the indexes. Of course f α = f α (x 1,, x n, t) is time-dependent. Therefore F = fα and t t y α Λ ij ( F A x i x j + F B x i F C x j ΓA BC) y A = Λij ( f α x i x j 3 f γ y α+γl ij y l+ fβ x i x j Γα βγ y α).

4 of Thus the mean curvature flow equation is equivalent to the normal part [ fα t Λij ( f α x i x + fβ f γ j x i x j Γα βγ) y α Λij Γ l ij y l is zero. Now given any vector a i + b α, the equation that the normal part is y i y α zero is equivalent to b α a i fα x i = 0 (.1) for each α. Therefore we obtain the evolution equation for f f α t Λij ( f α x i x + fβ f γ j x i x j Γα βγ + f α Γl ij ) = 0. (.) xl where Λ ij f is the inverse to g ij + h α f β αβ x i x j, y α and g ij = y i, y j and h αβ = are the Riemannian metrics on Σ y β 1 and Σ, respectively. This is a nonlinear parabolic system and the usual derivative estimates do not apply to this equations. However, the second author in [19 identifies a geometric quantity in terms of the derivatives of f α that satisfies the maximum principle; this quantity and its evolution equation are recalled in the next section. 3 Two evolution equations In this section, we recall two evolution equations along the mean curvature flow. The basic set-up is a mean curvature flow F : Σ [0, T) M of an n dimensional submanifold Σ inside an n + m dimensional Riemannian manifold M. Given any parallel tensor on M, we may consider the pullback tensor by F t and consider the evolution equation with respect to the time-dependent induced metric on F t (Σ) = Σ t. For the purpose of applying maximum principle, it suffices to derive the equation at a space-time point. We write all geometric quantities in terms of orthonormal frames keeping in mind all quantities are defined independent of choices of frames. At any point p Σ t, we choose any orthonormal frames {e i } i=1 n for T p Σ t and {e α } α=n+1 n+m for N p Σ t. The second fundamental form h αij is denoted by 4

5 h αij = M e i e j, e α and the mean curvature vector is denoted by H α = i h αii. For any j, k, we pretend h n+i,jk = 0 if i > m. When M = Σ 1 Σ is the product of Σ 1 and Σ, we denote the projections by π 1 : M Σ 1 and π : M Σ. By abusing notations, we also denote the differentials by π 1 : T p M T π1 (p)σ 1 and π 1 : T p M T π (p)σ at any point p M. The volume form Ω of Σ 1 can be extended to a parallel n-form on M. For an oriented orthonormal basis e 1 e n of T p Σ, Ω(e 1,, e n ) = Ω(π 1 (e 1 ),, π 1 (e n )) is the Jacobian of the projection from T p Σ to T π1 (p)σ 1. This can also be considered as the pairing between the n-form Ω and the n-vector e 1 e n representing T p Σ. We use Ω to denote this function as p varies along Σ. By the implicit function theorem, Ω > 0 at p if and only if Σ is locally a graph over Σ 1 at p. The evolution equation of Ω is calculated in Proposition 3. of [19. When Σ is the graph of f : Σ 1 Σ, the equation at each point can be written in terms of singular values of df and special bases adapted to df. Denote the singular values of df, or eigenvalues of (df) T df, by {λ i } i=1 n. Let r denote the rank of df. We can rearrange them so that λ i = 0 when i is greater than r. By singular value decomposition, there exist orthonormal bases {a i } i=1 n for T π1 (p)σ 1 and {a α } α=n+1 n+m for T π (p)σ such that df(a i ) = λ i a n+i for i less than or equal to r and df(a i ) = 0 for i greater than r. Moreover, e i = { 1 1+λ i (a i + λ i a n+i ) becomes an orthonormal basis for T p Σ and { 1+λ 1 (a n+p λ p a p ) p e n+p = a i a n+p if 1 i r if r + 1 i n if 1 p r if r + 1 p m (3.1) (3.) becomes an orthonormal basis for N p Σ. In terms of the singular values λ i, Ω = 1 n i=1 (1 + λ i ) (3.3) 5

6 With all the notations understood, the following result is essentially derived in Proposition 3. of [19 by noting that (ln Ω) k = ( i λ ih n+i,ik ). Proposition 3.1 Suppose M = Σ 1 Σ and Σ 1 and Σ are compact Riemannian manifolds of constant curvature k 1 and k respectively. With respect to the particular bases given by the singular value decomposition of df, ln Ω satisfies the following equation. ( d dt ) ln Ω = h αik + λ i h n+i,ik + λ i λ j h n+j,ik h n+i,jk α,i,k k,i k,i<j + λ [ i (k 1 + λ 1 + k )( 1 (3.4) ) + k i i 1 + λ (1 n) j i j Next we recall the evolution equation of parallel two tensors from [15. The calculation indeed already appears in [17. The equation will be used later to obtain more refined information. Given a parallel two-tensor S on M, we consider the evolution of S restricted to Σ t. This is a family of timedependent symmetric two tensors on Σ t. Proposition 3. Let S be a parallel two-tensor on M. Then restriction of S to Σ t satisfies the following equation. ( d dt )S ij = h αil H α S lj h αjl H α S li + R kikα S αj + R kjkα S (3.5) αi + h αkl h αki S lj + h αkl h αkj S li h αki h βkj S αβ where is the rough Laplacian on two-tensors over Σ t and S αi = S(e α, e i ), S αβ = S(e α, e β ), and R kikα = R(e k, e i, e k, e α ) is the curvature operator of M. The evolution equations (3.5) of S can be written in terms of evolving orthonormal frames as in Hamilton [8. If the orthonormal frames are given in local coordinates by F = {F 1,, F a,, F n } (3.6) F a = Fa i. x i 6

7 To keep them orthonormal, i.e. g ij F i a F j b = δ ab, we evolve F by the formula t F i a = gij g αβ h αjl H β F l a. Let S ab = S ij F i af j b be the components of S in F. Then S ab satisfies the following equation ( d dt )S ab = R cacα S αb + R cbcα S αa + h αcd h αca S db + h αcd h αcb S da h αca h βcb S αβ. (3.7) 4 Preserving distance-decreasing condition In this section, we show the condition df < 1, or each singular value λ i < 1, is preserved by the mean curvature flow. The tangent space of M = Σ 1 Σ is identified with TΣ 1 TΣ. Let π 1 and π denote the projection onto the first and second summand in the splitting. We define the parallel symmetric two-tensor S by S(X, Y ) = π 1 (X), π 1 (Y ) π (X), π (Y ) (4.1) for any X, Y TM. Let Σ be the graph of f : Σ 1 Σ 1 Σ. S restricts to a symmetric two-tensor on Σ and we can represent S in terms of the orthonormal basis (3.1). Let r denote the rank of df. By (3.1), it is not hard to check π 1 (e i ) = a i 1 + λi, π (e i ) = λ ia n+i 1 + λi and π 1 (e i ) = a i, π (e i ) = 0 for r + 1 i n. for 1 i r, (4.) Similarly, by (3.) we have π 1 (e n+p ) = λ pa p 1 + λ p, π (e n+p ) = a n+p 1 + λ p and π 1 (e n+p ) = 0, π (e n+p ) = a n+p for r + 1 p m. for 1 p r, (4.3) 7

8 From the definition of S, we have In particular, the eigenvalues of S are Notice that S is positive-definite if and only if S(e i, e j ) = 1 λ i 1 + λ i δ ij. (4.4) 1 λ i, i = 1 n. (4.5) 1 + λ i λ i < 1 for any singular value λ i of df. Now, at each point we express S in terms of the orthonormal basis {e i } i=1 n and {e α } α=n+1 n+m. Let I k k denote a k by k identity matrix. Then S can be written in the block form S = ( ) S(e k, e l ) = 1 k,l n+m B 0 D 0 0 I n r n r 0 0 D 0 B I m r m r (4.6) where B and D are r by r matrices with B ij = S(e i, e j ) = 1 λ i δ 1+λ ij and i D ij = S(e i, e n+j ) = λ i δ 1+λ ij for 1 i, j r. We show that the positivity of S i is preserved by the mean curvature flow. We remark that a similar positive definite tensor has been considered for the Lagrangian mean curvature flow in Smoczyk [14 and Smoczyk-Wang [15. Lemma 4.1 The condition T ij = S ij ǫg ij > 0 for some ǫ 0 (4.7) is preserved by the mean curvature flow if k 1 k. Proof. We compute the evolution equation for T ij. From Proposition (3.) and t g ij = h αij H α, 8

9 we have ( d dt )T ij = h αil H α T lj h αjl H α T li + R kikα S αj + R kjkα S αi + h αkl h αki T lj + h αkl h αkj T li + ǫh αki h αkj h αki h βkj S αβ. (4.8) To apply Hamilton s maximum principle, it suffices to prove that N ij V i V j 0 for any null eigenvector V of T ij, where N ij is the right hand side of (4.8). Since V is a null eigenvector of T ij, it satisfies j T ijv j = 0 for any i, and thus N ij V i V j is equal to ǫh αki h αkj V i V j + R kikα S αj V i V j h αki h βkj S αβ V i V j. (4.9) Obviously, the first term of (4.9) is nonnegative. Applying the relation in (4.6) to the last term of (4.9) we obtain h αki h βkj S αβ V i V j = h n+pki h n+qkj S pq V i V j + h n+pki h n+qkj V i V j. 1 p,q r r+1 p,q m Since T pq 0 implies that S pq ǫg pq, we obtain h αki h βkj S αβ V i V j 0. In the next lemma we show that R kikα S αj is nonnegative definite whenever S ij is under the curvature assumption k 1 k. Lemma 4. R kikα S αj = λ i (1 + λ i ) [ (k 1 k )(n 1) + (k 1 + k ) k i 1 λ k 1 + λ k Proof. We follow the calculation of the curvature terms in [19. R(e α, e k, e k, e i ) k = k = k δ ij. (4.10) R 1 (π 1 (e α ), π 1 (e k ), π 1 (e k ), π 1 (e i )) + R (π (e α ), π (e k ), π (e k ), π (e i )) k 1 [ π 1 (e α ), π 1 (e k ) π 1 (e k ), π 1 (e i ) π 1 (e α ), π 1 (e i ) π 1 (e k ), π 1 (e k ) + k [ π (e α ), π (e k ) π (e k ), π (e i ) π (e α ), π (e i ) π (e k ), π (e k ). 9

10 Notice that π (X), π (Y ) = X, Y π 1 (X), π 1 (Y ) since TΣ 1 TΣ. Therefore R(e α, e k, e k, e i ) k = (k 1 + k ) [ π 1 (e α ), π 1 (e k ) π 1 (e k ), π 1 (e i ) π 1 (e α ), π 1 (e i ) π 1 (e k ) k + k (n 1) π 1 (e α ), π 1 (e i ) Now use π 1 (e α ) = λ p π 1 (e p )δ α,n+p and S(e j, e n+p ) = λ jδ jp 1+λ j in (4.6), we have R kikα S αj = R n+p,kki S n+p,j α,k p,k {λ p (k 1 + k ) [ π 1 (e p ), π 1 (e k ) π 1 (e k ), π 1 (e i ) π 1 (e p ), π 1 (e i ) π 1 (e k ) = p,k } + λ p k (n 1) π 1 (e p ), π 1 (e i ) S n+p,j { [ = λ i δ ij (k 1 + k ) 1 + λ i (1 + λ i ) δ ij π 1 + λ 1 (e k ) i k δ } ij. + k (n 1) 1 + λ i Recall that π 1 (e k ) = 1 1+λ k R kikα S αj = λ i δ ij (1 + λ i ) and we obtain [ (k 1 + k )( k i This can be further simplified by noting (k 1 + k )( k i λ k ) + k (1 n) = (k 1 k )(n 1) where we use the following identity for each i ( k i λ k ) n 1 1 ) + k 1 + λ (1 n). k + (k 1 + k ) k i 1 λ k (1 + λ k ) (4.11) = 1 k i( λ k ) = 1 λ k (1 + λ k i k ). 10

11 5 Preserving area-decreasing condition In this section, we show that the area decreasing condition is preserved along the mean curvature flow. By (4.5), the sum of any two eigenvalues of S is 1 λ i 1 + λ i + 1 λ j 1 + λ j = (1 λ iλ j) (1 + λ i )(1 + λ j ). (5.1) Therefore the area decreasing condition λ i λ j < 1 for i j is equivalent to the two-positivity of S, i.e. the sum of any two eigenvalues is positive. We remark that curvature operator being two-positive is preserved by the Ricci flow, see Chen [1 or Hamilton [8 for detail. The two-positivity of a symmetric two tensor P can be related to the convexity of another tensor P [ associated with P. The following notation is adopted from Caffarelli-Nirenberg-Spruck [3. Let P be a self-adjoint operator on an n-dimensional inner product space. From P we can construct a new self-adjoint operator P [k = k 1 P 1 i i=1 acting on the exterior powers Λ k by P [k (ω 1 ω k ) = k ω 1 P(ω i ) ω k. i=1 With the definition of P [k, we have the following lemma. Lemma 5.1 Let µ 1 µ µ n be the eigenvalues of P with corresponding eigenvectors v 1 v n. Then P [k has eigenvalues µ i1 + +µ ik and eigenvectors v i1 v ik, i 1 < i < i k. Recall that the Riemannian metric g and S are both in TΣ TΣ, the space of symmetric two tensor on Σ. We can identify S with a self-adjoint operator on the tangent bundle through the metric g. Therefore S [ and g [ are both sections of (Λ (TΣ)) Λ (TΣ) associated to S and g respectively. We shall use orthonormal frames in the following calculation; this has the advantage that g is the identity matrix and we will not distinguish lower index and upper index. With the above interpretation and (5.1), we have the following lemma. 11

12 Lemma 5. The area decreasing condition is equivalent to the convexity of S [. To show that the area decreasing condition is preserved, it suffices to prove that the convexity of S [ is preserved. In fact, we prove the stronger result that the convexity of S [ ǫg [ for ǫ > 0 is preserved. We compute the evolution equation of S [ ǫg [ in terms of the evolving orthonormal frames {F a } a=1 n introduced earlier in (3.6). We will use indexes a, b, to denote components in the evolving frames. Denote S ab = S(F a, F b ) and g ab = g(f a, F b ) = δ ab. Since {F a F b } a<b form a basis for Λ TΣ, we have S [ (F a F b ) = S(F a ) F b + F a S(F b ) = S ac F c F b + F a S ac F c = c<d(s ac δ bd + S bd δ ac S ad δ bc S bc δ ad )F c F d and g [ (F a F b ) = c<d(δ ac δ bd δ ad δ bc )F c F d. (5.) We denote S [ (ab)(cd) = (S acδ bd +S bd δ ac S ad δ bc S bc δ ad ) and g [ (ab)(cd) = δ acδ bd δ ad δ bc. Thus the evolution equation of S [ ǫg [ in terms of the evolving orthonormal frames is ( d dt )(S acδ bd + S bd δ ac S ad δ bc S bc δ ad ǫδ ac δ bd + ǫδ ad δ bc ) = R eaeα S αc δ bd + R eceα S αa δ bd + R ebeα S αd δ ac + R edeα S αb δ ac R eaeα S αd δ bc R edeα S αa δ bc R ebeα S αc δ ad R eceα S αb δ ad + h αef h αea S fc δ bd + h αef h αec S fa δ bd + h αef h αeb S fd δ ac + h αef h αed S fb δ ac h αef h αea S fd δ bc h αef h αed S fa δ bc h αef h αeb S fc δ ad h αef h αec S fb δ ad h αea h βec S αβ δ bd h αeb h βed S αβ δ ac + h αea h βed S αβ δ bc + h αeb h βec S αβ δ ad. (5.3) Now, we are ready to prove that the area decreasing condition is preserved along the mean curvature flow. Lemma 5.3 Under the assumption of Theorem A, with S defined in (4.1)and S [ defined in (5.), suppose there exists an ǫ > 0 such that S [ ǫg [ 0 (5.4) holds on the initial graph. Then this is preserved along the mean curvature flow. 1

13 Proof. Set M η = S [ ǫg [ + ηtg [. Suppose the mean curvature flow exists on [0, T). Consider any T 1 < T, it suffices to prove that M η > 0 on [0, T 1 for all η < ǫ T 1. If not, there will be a first time 0 < t 0 T 1 where M η = S [ ǫg [ + ηtg [ is nonnegative definite and has a null eigenvector V = V ab F a F b at some point x 0 Σ t0. We extend V ab to a parallel tensor in a neighborhood of x 0 along geodesic emanating out of x 0, and defined V ab on [0, T) independent of t. Define f = a<b,c<d V ab M η(ab)(cd) V cd, then by (5.), f equals a<b,c<d (S ac g bd + S bd g ac S ad g bc S bc g ad + (ηt ǫ)(g ac g bd g ad g bc ))V ab V cd. At (x 0, t 0 ), we have f = 0, f = 0 and ( d )f 0 where denotes the dt covariant derivative and denotes the Laplacian on Σ t0. We may assume that at (x 0, t 0 ) the orthonormal frames {F a } is given by {e i } in (3.1). In the following, we use the orthonormal basis {e i } to write down the condition f = 0 and f = 0 at (x 0, t 0 ). The basis {e i } diagonalizes S with eigenvalues {λ i } and we order {λ i } such that and S nn = 1 λ n 1 + λ n λ 1 λ λ n S = 1 λ 1 + λ S 11 = 1 λ λ 1. (5.5) It follows from Lemma (5.1) that {e i e j } i<j are the eigenvectors of M η. Thus we may assume that V = e 1 e. (5.6) At (x 0, t 0 ), the condition f = 0 is the same as This is equivalent to S 11 + S = ǫ ηt 0 > 0. (5.7) (1 λ 1 λ ) (1 + λ 1)(1 + λ ) = (ǫ ηt 0) > 0. 13

14 Thus λ 1 λ < 1 and λ i < 1 for i 3. (5.8) Next, we compute the covariant derivative of the restriction of S on Σ. So ( ei S)(e j, e k ) = e i (S(e j, e k )) S( ei e j, e k ) S(e j, ei e k ) = S( M e i e j ei e j, e k ) + S(e j, M e i e k ei e k ) = h αij S αk + h βik S βj. S jk,i = h αij S αk + h βik S βj. Recall that V ab is parallel at (x 0, t 0 ), V 1 = 1 and all other components of V ab is zero. At (x 0, t 0 ), f = 0 is equivalent to 0 = ep ((S ik δ jl + S jl δ ik S il δ jk S jk δ il + (ηt ǫ)(δ ik δ jl ǫδ il δ jk ))V ij V kl ) i<j,k<l = ep S 11 + ep S = h αp1 S α1 + h βp S β. Since S n+q,l = λqδ ql, we have 1+λ q λ 1 h 1 + λ n+1,p1 + λ h λ n+,p = 0 (5.9) for any p. By (5.3), at (x 0, t 0 ), we have ( d dt )f = η + R k1kαs α1 + R kkα S α + h αkj h αk1 S j1 + h αkj h αk S j h αk1 h βk1 S αβ h αk h βk S αβ. (5.10) The ambient curvature term can be calculated using Lemma 4. and we derive R k1kα S α1 + R kkα S α. k,α = (k 1 k )(n 1) i=1 λ i (1 + λ i ) + (k 1 + k ) 14 i=1 λ i (1 + λ i ) [ j i 1 λ j (1 + λ j ). (5.11)

15 This can be simplified as [ λ i (k 1 k )(n 1) (1 + λ + (k λ i i=1 i ) 1 + k ) (1 + λ i=1 i ) j>3 [ λ 1 1 λ + (k 1 + k ) (1 + λ 1) (1 + λ ) + λ 1 λ 1 (1 + λ ) (1 + λ 1) [ λ i = (k 1 k )(n 1) (1 + λ + (k λ i i=1 i ) 1 + k ) (1 + λ i=1 i ) j>3 [ (λ + (k 1 + k ) 1 + λ )(1 λ 1 λ ). (1 + λ 1) (1 + λ ) 1 λ j (1 + λ j ) 1 λ j (1 + λ j ) (5.1) This is nonnegative by equation (5.8). Using the relations in (4.6) again, the last four terms on the right hand side of (5.10) can be rewritten as h n+p,k1s 11 + h n+p,ks + h n+p,k1s pp + h n+p,ks pp p,k = (h n+1,k1s 11 + h n+,k1s 11 + h n+1,ks + h n+,ks k + h n+1,k1s 11 + h n+,k1s + h n+1,ks 11 + h n+,ks ) + h n+q,k1 S 11 + h n+q,k S + h n+q,k1 S qq + h n+q,k S qq. q 3,k (5.13) Since S ii S 11 for i, it is clear that (5.13) is nonnegative if S Otherwise, from (5.7), we may assume that S 11 < 0, S > 0 and S 11 + S > 0. (5.14) In particular, we have λ < λ 1 and λ 1λ < 1. From (5.9), we have h n+1,p1 = λ (1 + λ 1 ) λ 1 (1 + λ )h n+,p. Since λ < λ 1 and λ 1 λ < 1, we have λ (1+λ 1 ) λ 1 (1+λ ) < 1. Thus h n+1,p1 h n+,p for all p 1. (5.15) 15

16 Recall that S qq S for q 3. The right hand side of (5.13) can be regrouped as [ (4h n+1,k1 S h n+,k S ) + h n+,k1 (S 11 + S ) + h n+1,k (S 11 + S ) k + q 3,k [ h n+q,k1 (S 11 + S qq ) + h n+q,k (S + S qq ). This is nonnegative by (5.5),(5.14), and (5.15). Thus, we have ( d )f dt η > 0 at (x 0, t 0 ) and this is a contradiction. (1 λ i λ j ) Remark: The condition S [ ǫg [ 0 is equivalent to ǫ for all (1+λ i )(1+λ j ) i j. In particular, we have λ i 1 ǫ. This implies that the Lipschitz norm ǫ of f is preserved along the mean curvature flow. 6 Long time existence and Convergence In this section, we prove Theorem A. We show the quantity ln Ω satisfies the following differential inequality d dt ln Ω ln Ω + δ A c 3 ln Ω for some constant c 3 > 0 under the assumption λ i λ j 1 δ. Utilizing the blow-up analysis technique developed in [19, we can show the long time existence based on the fact that ln Ω is a dilation-invariant quantity that satisfies d dt ln Ω ln Ω + δ A. The convergence part follows from studying the differential inequality d dt ln Ω ln Ω c 3 ln Ω. Proof of Theorem A. Since λ i λ j < 1 for i j and Σ 1 is compact, we can find an ǫ > 0 such that (1 λ i λ j ) (1 λ i λ j ) (1+λ i )(1+λ j ) ǫ for all i j. By Lemma (5.3), the condition ǫ for all i j is preserved along the mean curvature (1+λ i )(1+λ j ) flow. In particular, we have λ i λ j 1 ǫ and λ i 1 ǫ. This implies Σ ǫ t 16

17 remains the graph of a map f t : Σ 1 Σ whenever the flow exists. Each f t has uniformly bounded df t. We look at the evolution equation (3.4) of ln Ω. The quadratic terms of the second fundamental form in equation (3.4) is λ ih n+i,ik + λ i λ j h n+j,ik h n+i,jk k,i<j α,i,k h αik + k,i = δ A + k,i λ ih n+i,ik + (1 δ) A + k,i<j λ i λ j h n+j,ik h n+i,jk. Let 1 δ = 1 ǫ. Using λ i λ j 1 δ, we conclude that this term is bounded below by δ A. By equation (4.11), the curvature term in (3.4) equals (k 1 k )(n 1) n i=1 λ i 1 + λ i + (k 1 + k ) n i=1 λ i 1 + λ i [ j i 1 λ j (1 + λ j ). (6.1) The second term on the right hand side of (6.1) can be simplified as [ n λ 1 λ n i j λ i 1 + λ i=1 i (1 + λ j i j ) = λ i λ j (1 + λ i=1 i j i )(1 + λ j ) = λ i + λ j (6.) λ i λ j (1 + λ i<j i )(1 + λ j ). This is non-negative because λ i λ j 1 δ. Thus ln Ω satisfies the following differential inequality: d dt ln Ω ln Ω + δ A. (6.3) According to the maximum principle for parabolic equations, min Σt ln Ω is nondecreasing in time. In particular, Ω min Σ0 Ω = Ω 0 is preserved and Ω has a positive lower bound. Let u = ln Ω ln Ω 0+c lnω 0 where c is a positive +c number such that ln Ω 0 +c > 0. Recall that 0 < Ω 1. This implies that 0 < u 1 and u satisfies the following differential inequality d dt u u + δ ln Ω 0 + c A. 17

18 Because u is also invariant under parabolic dilation, it follows from the blow-up analysis in the proof of Theorem A [19 that the mean curvature flow of the graph of f remains a graph and exists for all time. Using λ i 1 ǫ and λ ǫ i λ j 1 ǫ, it is not hard to show (k 1 + k ) i<j λ i + λ j λ i λ j (1 + λ i )(1 + λ j ) c 1 n λ i c 1 ln i=1 where c 1 is a constant that depends on ǫ, k 1 and k. Recall equation (3.3) and we obtain d dt ln Ω ln Ω c 3 ln Ω. n (1 + λ i ) (6.4) By the comparison theorem for parabolic equations, min Σt ln Ω is nondecreasing in t and min Σt ln Ω 0 as t. This implies that min Σt Ω 1 and max λ i 0 as t. We can apply Theorem B in [19 to conclude smooth convergence to a constant map at infinity. References [1 H. Chen, Pointwise quarter-pinched 4 manifolds. Ann. Global Anal. 9 (1991), [ J.-Y. Chen, J.-Y. Li and G. Tian, Two-dimensional graphs moving by mean curvature flow. Acta Math. Sin. (Engl. Ser.) 18 (00), no., [3 L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian., Acta Math. 155 (1985), no. 3-4, [4 K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105 (1991), no. 3, [5 J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964) i=1 18

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20 [19 M-T. Wang, Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension, Invent. Math. 148 (00) 3, [0 M-T. Wang, Subsets of grassmannians preserved by mean curvature flow. preprint, 00. [1 M-T. Wang, On graphic Berstein type results in higher codimensions. Trans. Amer. Math. Soc. 355 (003), no. 1, [ M-T. Wang, Interior gradient bounds for solutions to the minimal surface system. preprint, 00. [3 M-T. Wang, Gauss maps of the mean curvature flow. preprint, 00. [4 B. White, A local regularity theorem for classical mean curvature flow. preprint, 1999 (revised 00). Available at white/preprint.htm 0