Homotopy of area decreasing maps by mean curvature flow
aa r X i v : . [ m a t h . DG ] F e b HOMOTOPY OF AREA DECREASING MAPS BYMEAN CURVATURE FLOW
ANDREAS SAVAS-HALILAJ
AND
KNUT SMOCZYKAbstract.
Let f : M → N be a smooth area decreasing mapbetween two Riemannian manifolds ( M, g M ) and ( N, g N ). Underweak and natural assumptions on the curvatures of ( M, g M ) and( N, g N ), we prove that the mean curvature flow provides a smoothhomotopy of f to a constant map. Introduction
Given a continuous map f : M → N between two smooth manifolds M and N , it is an interesting problem to find canonical representatives inthe homotopy class of f . One possible approach is the harmonic mapheat flow that was defined by Eells and Sampson in [ES64]. Providedthat M and N both carry appropriate Riemannian metrics, they provedlong-time existence and convergence of the heat flow, showing thatunder these assumptions one finds harmonic representatives in a givenhomotopy class. This approach is applicable usually when the targetspace is negatively curved. However, in general one can neither expectlong-time existence nor convergence of the flow, in particular for mapsbetween spheres, since the flow usually develops singularities.Another way to deform a smooth map f : M → N between Riemannianmanifolds ( M, g M ) and ( N, g N ) is by deforming its corresponding graphΓ( f ) := { ( x, f ( x )) ∈ M × N : x ∈ M } , in the product space M × N via the mean curvature flow . A graphicalsolution of the mean curvature flow can be described completely interms of a smooth family of maps f t : M → N , t ∈ [0 , T ), f = f ,where 0 < T ≤ ∞ is the maximal time for which the smooth graphicalsolution exists. Mathematics Subject Classification.
Primary 53C44, 53C42, 57R52, 35K55.
Key words and phrases.
Mean curvature flow, homotopy, area decreasing,graphs, maximum principle.The first author is supported by the grant of EΣΠA : PE1-417.
In case of long-time existence of a graphical solution and convergencewe would thus obtain a smooth homotopy from f to a minimal map f ∞ : M → N as time t tends to infinity. Recall that a map is calledminimal, if and only if its graph is a minimal submanifold of M × N .The first result in this direction is due to Ecker and Huisken [EH89].They proved long-time existence of entire graphical hypersurfaces in R n +1 . Moreover, they proved convergence to a flat subspace, if thegrowth rate at infinity of the initial graph is linear. The crucial obser-vation in their paper was that the scalar product of the unit normalwith a height vector satisfies a nice evolution equation that can be usedto bound the second fundamental form appropriately.The complexity of the normal bundle in higher codimensions makes thesituation much more complicated. Results analogous to that of Eckerand Huisken are not available any more without further assumptions.However, the ideas developed in the paper of Ecker and Huisken openeda new era for the study of the mean curvature flow of submanifolds inRiemannian manifolds of arbitrary dimension and codimension (seefor example [Wan01], [Wan01b], [Wan02], [CLT02], [SW02], [Smo02],[TW04], [Smo04], [MW11], [LL11], [CCH12], [CCY13] and the refer-ences therein).A map f : M → N is called weakly length decreasing if f ∗ g N ≤ g M and strictly length decreasing , if f ∗ g N < g M . Hence a length decreasing maphas the property that its differential shortens the lengths of tangentvectors. A smooth map f : M → N is called weakly area decreasing ifits differential d f decreases the area of two dimensional tangent planes,that is if k d f ( v ) ∧ d f ( w ) k g N ≤ k v ∧ w k g M , for all v, w ∈ T M . If the differential d f is strictly decreasing thearea of two dimensional tangent planes, then f is called strictly areadecreasing . Analogously, we may introduce the notion of weakly and strictly k-volume decreasing maps .In [Wan02, TW04], Wang and Tsui studied deformations of smoothmaps f : M → N between Riemannian manifolds under the meancurvature flow. Under the assumption that the initial map is strictlyarea decreasing, M and N are compact space forms with dim M ≥ M and sec N satisfysec M ≥ | sec N | , sec M + sec N > , EAN CURVATURE FLOW 3 they proved long-time existence of the mean curvature flow of the graphand convergence of f to a constant map. Recently, Lee and Lee [LL11]generalized the result of Wang and Tsui by showing that the sameresult holds true provided that M and N are compact Riemannianmanifolds whose sectional curvatures are bounded bysec M ≥ σ ≥ sec N for some strictly positive number σ > Theorem A.
Let M and N be two compact Riemannian manifolds.Assume that m = dim M ≥ and that there exists a positive constant σ such that the sectional curvatures sec M of M and sec N of N and theRicci curvature Ric M of M satisfy sec M > − σ, Ric M ≥ ( m − σ ≥ ( m −
1) sec N . If f : M → N is a strictly area decreasing smooth map, then themean curvature flow of the graph of f remains the graph of a strictlyarea decreasing map and exists for all time. Moreover, under the meancurvature flow the area decreasing map converges to a constant map. Remark 1.1.
The above theorem generalizes the results in [TW04]and [LL11] since the curvature assumption is more general.
Remark 1.2.
In [SHS12] we studied minimal graphs generated bylength and area decreasing maps between two Riemannian manifolds.From the examples presented in [SHS12, Subsection 3.6], it follows thatthe imposed curvature assumptions in Theorem A are optimal.
Remark 1.3.
According to Theorem A, any strictly area decreasingmap between two compact Riemannian manifolds ( M, g M ) and ( N, g N )satisfying these curvature assumptions must be null-homotopic. Sucha result fails to hold for 3-volume decreasing maps since Guth [Gut07]showed that there are infinitely many non null-homotopic 3-volumedecreasing maps between unit euclidean spheres.At this point let us say some words about the proof of Theorem A.Since the manifold M is assumed to be compact, short time existenceof the mean curvature flow is guaranteed. Moreover, it follows bycontinuity that there is an interval where the solution remains a graph.The first step in our proof is to show that the assumption of being area ANDREAS SAVAS-HALILAJ AND KNUT SMOCZYK decreasing is preserved by the mean curvature flow. As a consequenceof this fact, it follows that the norm of the differential of the initialmap remains bounded in time. This fact implies that the deformationof the graph via the mean curvature flow remains a graph as long as thesolution exists. The second step is to prove that the flow exists for longtime. In general, this can be achieved by showing that the norm of thesecond fundamental form remains bounded in time. However, such abound is not available. Following ideas developed by Wang and by Tsuiand Wang in [Wan02], [TW04] we introduce an angle-type function on M . We show then that under our curvature assumptions this functionsatisfies a nice differential inequality involving also the squared norm ofthe second fundamental form. The idea now is to compare the norm ofthe second fundamental form with this angle-type function. Followingthe same strategy as in [Wan02] one can verify that there are no finitetime singularities of the flow. At this point a deep regularity theoremof White [Whi05] is needed. Hence in this way it is shown that theflow exists for all time. Going back to the evolution equation of thespecial angle-type function we conclude that under our assumptionsthe solution converges smoothly to a constant map at infinity.The organization of the paper is as follows. In Section 2 we recall somebasic facts from the geometry of graphs. In Section 3 we provide theevolution equations and the basic estimates which are used in the proofof our result. In Section 3 we give the proof of Theorem A.2. Geometry of graphs
The purpose of this section is to set up the notation and to give somebasic definitions. We shall follow closely the notations in [SHS12].2.1.
Basic facts.
Let ( M, g M ) and ( N, g N ) be Riemannian manifoldsof dimension m and n , respectively. The induced metric on the ambientspace M × N will be denoted by g M × N or by h· , ·i , that isg M × N = h· , ·i := g M × g N . The graph of a map f : M → N is defined to be the submanifoldΓ( f ) := { ( x, f ( x )) ∈ M × N : x ∈ M } of M × N . The graph Γ( f ) can be parametrized via the embedding F : M → M × N , F := I M × f , where I M is the identity map of M .The Riemannian metric induced by F on M will be denoted byg := F ∗ g M × N . EAN CURVATURE FLOW 5
The two natural projections π M : M × N → M and π N : M × N → N are submersions, that is they are smooth and have maximal rank. Thetangent bundle of the product manifold M × N , splits as a direct sum T ( M × N ) = T M ⊕ T N.
The four metrics g M , g N , g M × N and g are related byg M × N = π ∗ M g M + π ∗ N g N , (2.1)g = F ∗ g M × N = g M + f ∗ g N . (2.2)Additionally, we define the symmetric 2-tensorss M × N := π ∗ M g M − π ∗ N g N , (2.3)s := F ∗ s M × N = g M − f ∗ g N . (2.4)The Levi-Civita connection ∇ g M × N associated to the Riemannian met-ric g M × N on M × N is related to the Levi-Civita connections ∇ g M on( M, g M ) and ∇ g N on ( N, g N ) by ∇ g M × N = π ∗ M ∇ g M ⊕ π ∗ N ∇ g N . The corresponding curvature tensor R M × N on M × N with respect tothe metric g M × N is related to the curvature tensors R M on ( M, g M )and R N on ( N, g N ) byR M × N = π ∗ M R M ⊕ π ∗ N R N . The Levi-Civita connection on M with respect to the induced metricg = F ∗ g M × N is denoted by ∇ , the curvature tensor by R and the Riccicurvature by Ric.2.2. The second fundamental form.
Let F ∗ T N denote the tangentbundle of N along M . Note that by definition the fibers of F ∗ T N at x ∈ M coincide with T F ( x ) N . The differential d F of F is then a sectionin the bundle F ∗ T N ⊗ T ∗ M . In the sequel we will denote all fullconnections on bundles over M that are induced by the Levi-Civitaconnection on N via the immersion F : M → N by the same letter ∇ .The covariant derivative of d F is called the second fundamental form of the immersion F and it will be denoted by A . That is A ( v, w ) := ( ∇ d F )( v, w ) = ∇ g M × N d F ( v ) d F ( w ) − d F ( ∇ v w ) , for any vector fields v, w ∈ T M . The second fundamental form A isa symmetric tensor and takes values in the normal bundle N M of thegraph Γ( f ). Since N M is a subbundle of F ∗ T N , the full connection ∇ can be used on N M . By projecting to the normal bundle again, weobtain the connection on the normal bundle N M of the graph, which ANDREAS SAVAS-HALILAJ AND KNUT SMOCZYK will be denoted by the symbol ∇ ⊥ . If ξ is a normal vector of the graph,then the symmetric tensor A ξ given by A ξ ( v, w ) := h A ( v, w ) , ξ i is called the second fundamental form with respect to the direction ξ .The trace of A with respect to the metric g is called the mean curvaturevector field of Γ( f ) and it will be denoted by H := trace g A. Note that the mean curvature vector field H is a section of the normalbundle of Γ( f ). In the case where H vanishes identically, the graphΓ( f ) is called minimal .By Gauß’ equation the curvature tensors R and R M × N are related bythe formulaR( v , w , v , w ) = ( F ∗ R M × N )( v , w , v , w )+g M × N (cid:0) A ( v , v ) , A ( w , w ) (cid:1) − g M × N (cid:0) A ( v , w ) , A ( w , v ) (cid:1) , (2.5)for any v , v , w , w ∈ T M . Moreover, the second fundamental formsatisfies the
Codazzi equation ( ∇ u A )( v, w ) − ( ∇ v A )( u, w ) = R M × N (cid:0) d F ( u ) , d F ( v ) (cid:1) d F ( w ) − d F (cid:0) R( u, v ) w (cid:1) , (2.6)for any u, v, w on T M .2.3.
Singular decomposition.
As in [SHS12], for any fixed point x ∈ M , let λ ( x ) ≤ · · · ≤ λ m ( x )be the eigenvalues of f ∗ g N with respect to g M . The correspondingvalues λ i ≥ i ∈ { , . . . , m } , are usually called singular values of thedifferential d f of f and give rise to continuous functions on M . Let r = r ( x ) = rank d f ( x ) . Obviously, r ≤ min { m, n } and λ ( x ) = · · · = λ m − r ( x ) = 0 . It is wellknown that the singular values can be used to define the so called singular decomposition of d f . At the point x consider an orthonormalbasis { α , . . . , α m − r ; α m − r +1 , . . . , α m } EAN CURVATURE FLOW 7 with respect to g M which diagonalizes f ∗ g N . Moreover, at the point f ( x ) consider an orthonormal basis { β , . . . , β n − r ; β n − r +1 , . . . , β n } with respect to g N such thatd f ( α i ) = λ i ( x ) β n − m + i , for any i ∈ { m − r + 1 , . . . , m } .Then one may define a special basis for the tangent and the normalspace of the graph in terms of the singular values. The vectors e i := α i , ≤ i ≤ m − r, √ λ i ( x ) ( α i ⊕ λ i ( x ) β n − m + i ) , m − r + 1 ≤ i ≤ m, (2.7)form an orthonormal basis with respect to the metric g M × N of thetangent space d F ( T x M ) of the graph Γ( f ) at x . Moreover, the vectors ξ i := β i , ≤ i ≤ n − r, √ λ i + m − n ( x ) ( − λ i + m − n ( x ) α i + m − n ⊕ β i ) , n − r + 1 ≤ i ≤ n, (2.8)give an orthonormal basis with respect to g M × N of the normal space N x M of the graph Γ( f ) at the point F ( x ). From the formulas above,we deduce thats M × N ( e i , e j ) = 1 − λ i λ i δ ij , ≤ i, j ≤ m. (2.9)Consequently, the eigenvalues µ , µ , . . . , µ m of the symmetric 2-tensors with respect to g, are µ := 1 − λ m λ m ≤ · · · ≤ µ m := 1 − λ λ . As it was observed in [TW04, LL11], for any pair of indices i, j we have µ i + µ j = 2(1 − λ i λ j )(1 + λ i )(1 + λ j ) . Hence, the graph is strictly area decreasing, if and only if the tensor sis strictly 2-positive.
ANDREAS SAVAS-HALILAJ AND KNUT SMOCZYK
Moreover we get,s M × N ( ξ i , ξ j ) = − δ ij , ≤ i ≤ n − r, − − λ i + m − n λ i + m − n δ ij , n − r + 1 ≤ i ≤ n. (2.10)and s M × N ( e m − r + i , ξ n − r + j ) = − λ m − r + i λ m − r + i δ ij , ≤ i, j ≤ r. (2.11)2.4. Area decreasing maps.
For any smooth map f : M → N itsdifferential d f induces the natural map Λ d f : Λ T M → Λ T M ,Λ d f ( v, w ) := d f ( v ) ∧ d f ( w ) , for any v, w ∈ T M.
The map Λ d f is called the 2- Jacobian of f . The supremum norm of the 2-Jacobian is defined as the supremum of q f ∗ g N ( v i , v i ) f ∗ g N ( v j , v j ) − f ∗ g N ( v i , v j ) , where { v , . . . , v m } runs over all orthonormal bases of T M . A smoothmap f : M → N is called weakly area decreasing if k Λ d f k ≤ strictly area decreasing if k Λ d f k < . The above notions are expressedin terms of the singular values by the inequalities λ i λ j ≤ λ i λ j < , for any 1 ≤ i < j ≤ m , respectively. On the other hand, as alreadynoted in the previous section, the sum of two eigenvalues of the tensors with respect to g equals1 − λ i λ i + 1 − λ j λ j = 2(1 − λ i λ j )(1 + λ i )(1 + λ j ) . Hence, the strictly area-decreasing property of f is equivalent to the2- positivity of s.The 2-positivity of a tensor T ∈ Sym( T ∗ M ⊗ T ∗ M ) can be expressed asthe positivity of another tensor T [2] ∈ Sym(Λ T ∗ M ⊗ Λ T ∗ M ). Indeed,let P and Q be two symmetric 2-tensors. Then, the Kulkarni-Nomizuproduct P ⊙ Q given by( P ⊙ Q)( v ∧ w , v ∧ w ) = P( v , v ) Q( w , w ) + P( w , w ) Q( v , v ) − P( w , v ) Q( v , w ) − P ( v , w ) Q( w , v ) EAN CURVATURE FLOW 9 is an element of Sym(Λ T ∗ M ⊗ Λ T ∗ M ). Now, to every elementT ∈ Sym( T ∗ M ⊗ T ∗ M ) let us assign an element T [2] of the bundleSym(Λ T ∗ M ⊗ Λ T ∗ M ), by settingT [2] := T ⊙ g . We point out that the Riemannian metric G of Λ T M is given byG = g ⊙ g = g [2] . The relation between the eigenvalues of T and the eigenvalues of T [2] is explained in the following lemma:
Lemma 2.1.
Suppose that T is a symmetric -tensor with eigenvalues µ ≤ · · · ≤ µ m and corresponding eigenvectors { v , . . . , v m } with respectto g . Then the eigenvalues of the symmetric -tensor T [2] with respectto G are µ i + µ j , ≤ i < j ≤ m, with corresponding eigenvectors v i ∧ v j , ≤ i < j ≤ m. Evolution of Graphs Under The Mean Curvature Flow
In the present section we shall derive the evolution equations of someimportant quantities. We mainly follow the setup and presentationused in [Smo12, SHS12].3.1.
Mean curvature flow.
Let M and N be Riemannian manifolds, f : M → N a smooth map and F := ( I M , f ) : M → M × N . Then,by a classical result, there exists a maximal positive time T for whicha smooth solution F : M × [0 , T ) → M × N of the mean curvature flow dFdt ( x, t ) = H ( x, t )with initial condition F ( x,
0) := F ( x )exists. Here H ( x, t ) denotes the mean curvature vector field at thepoint x ∈ M of the immersion F t : M → M × N , given by F t ( x ) := F ( x, t ) . In this case we say that the graph Γ( f ) evolves by mean curvature flow in M × N . Let Ω M be the volume form on the Riemannian manifold ( M, g M ) andextend it to a parallel m -form on the product manifold M × N bypulling it back via the natural projection π M : M × N → M , that isconsider the m -form π ∗ M Ω M . Define now the time dependent smoothfunction u : M × [0 , T ) → R , given by u := ∗ Ω t , where here ∗ is the Hodge star operator with respect to the inducedRiemannian metric g andΩ t := F ∗ t ( π ∗ M Ω M ) = ( π M ◦ F t ) ∗ Ω M . Note that the function u is the Jacobian of the projection map from F t ( M ) to M . From the implicit map theorem it follows that u > φ t : M → M and a map f t : M → N such that F t ◦ φ t = ( I M , f t ) . In other words the function u is positive if and only if the solution ofthe mean curvature flow remains a graph. From the compactness of M and the continuity of u , it follows that F t will stay a graph at leastin an interval [0 , T g ) with T g ≤ T . In general T g can be strictly lessthan T . However, as we shall see in the sequel, under our curvatureassumptions T g = T .3.2. Evolution equations.
In this subsection we shall compute theevolution equations and estimate various geometric quantities that wewill need in the proof of our main result. In order to control the smallesteigenvalue of s, let us define the symmetric 2-tensorΦ := s − − c c g , where c is a time-dependent function.The evolution of the symmetric 2-tensor Φ under the mean curvatureflow is given in the following lemma. EAN CURVATURE FLOW 11
Lemma 3.1.
The evolution equation of the tensor Φ for t ∈ [0 , T g ) isgiven by the following formula: (cid:0) ∇ ∂ t Φ − ∆Φ (cid:1) ( v, w ) = − Φ(Ric v, w ) − Φ(Ric w, v ) − m X k =1 (s M × N − − c c g M × N )( A ( e k , v ) , A ( e k , w )) −
41 + c m X k =1 (cid:0) f ∗ t R N − c R M (cid:1) ( e k , v, e k , w )+ c ′ (1 + c ) g , where { e , . . . , e m } is any orthonormal frame with respect to g .Proof. The proof is straightforward and similar to [SHS12, Lemma 3.2].What we just need to take into account here, is that (cid:0) ∇ ∂ t g (cid:1) ( v, w ) = − M × N (cid:0) H, A ( v, w ) (cid:1) and (cid:0) ∇ ∂ t s (cid:1) ( v, w ) = s M × N (cid:0) ∇ v H, d F ( w ) (cid:1) + s M × N (cid:0) ∇ w H, d F ( v ) (cid:1) . This completes the proof. (cid:3)
Lemma 3.2.
Under the assumptions made in Theorem A, the strictlyarea decreasing property is preserved under the mean curvature flow forany time t ∈ [0 , T g ) . Proof.
The proof follows steps in our previous paper [SHS12, Subsec-tion 3.5, Proof of Theorem D]. For the sake of completeness let usbriefly describe the idea of the proof. Since the initial map is strictlyarea decreasing, there exists a positive number ρ such thats [2] − ρ G ≥ . Here ρ is just the minimum of the smallest eigenvalue of s [2] at time 0.We claim now that the above inequality is preserved under the meancurvature flow. In order to show this, let us introduce the symmetric2-tensor M ε := s [2] − ρ G + ε t G , where ε is a positive number. Consider any T < T g . It suffices toshow that M ε > , T ] for all ε < ρ /T . Assumein contrary that this is not true. Then, there will be a first time t ∈ (0 , T ) such that M ε is non-negatively definite in [0 , t ] and thereis a null-eigenvector v for M ε at some point ( x , t ). Then, according to the second derivative criterion (see [Ham82, Theorem 9.1]), at thispoint it holds M ε ( v, v ) = 0 , (cid:0) ∇ M ε (cid:1) ( v, v ) = 0 and (cid:0) ∇ ∂ t M ε − ∆ M ε (cid:1) ( v, v ) ≤ . From the first condition we get that at ( x , t ) it holds λ m λ m − < λ i < , for any i ∈ { , . . . , m − } . Carrying out the same estimates as in[SHS12, Subsection 3.5, Proof of Theorem D], we get that at ( x , t ) itholds (cid:0) ∇ ∂ t M ε − ∆ M ε (cid:1) ( v, v ) ≥ ε > (cid:3) Proposition 3.3.
Under the assumptions of Theorem A, the solutionof the mean curvature flow remains a graph as long as the flow exists.Proof.
As we mentioned before, there exists a time T g such that F t is graphical for any t in the interval [0 , T g ). We claim that T g = T .Arguing indirectly, let us assume that T g < T and that F T g is notgraphical. Since s stays positive in time, there exists a positive universalconstant ε such that 1 − λ i λ j (1 + λ i )(1 + λ j ) ≥ ε > , for any 1 ≤ i < j ≤ m. In particular, we have ε (1 + λ i ) ≤ − λ i λ j λ j ≤ , for any 1 ≤ i < j ≤ m. From the above inequality we can see that1 + λ i ≤ ε , for every index 1 ≤ i ≤ m. Hence under the curvature conditionsof Theorem A, the singular values of d f t are bounded by a time-independent universal constant for every t ∈ [0 , T g ). For any fixedarbitrary x ∈ M , the continuity of u implies that u ( x, T g ) = lim t ր T g u ( x, t )= lim t ր T g p (1 + λ ( x, t )) · · · (1 + λ m ( x, t )) > . EAN CURVATURE FLOW 13
Thus, u ( x, T g ) > x ∈ M . This implies that the map F T g must be graphical,contradicting our initial assumption on T g . Therefore T g = T . Thiscompletes the proof. (cid:3) In the next lemma we give the evolution equation of the function u .The proof can be found in [Wan02] and for this reason will be omitted. Lemma 3.4.
The function log u evolves under the mean curvature flowfor t ∈ [0 , T ) according to ∇ ∂ t log u − ∆ log u = k A k + m X k =1 r X i =1 λ m − r + i A ξ n − r + i ( e i , e k )+ 2 X ≤ k ≤ m X ≤ i Under the assumptions made in Theorem A, for any t ∈ [0 , T ) there exists a positive number δ such that A : = k A k + m X k =1 r X i =1 λ m − r + i A ξ n − r + i ( e i , e k )+2 X ≤ k ≤ m X ≤ i Because the strictly area decreasing property is preserved underthe mean curvature flow, there exists a positive number δ such that λ i λ j ≤ − δ, for any t ∈ [0 , T ) and 1 ≤ i < j ≤ r . Thus, for any 1 ≤ k ≤ m , weobtain that X ≤ i A ≥ δ k A k + (1 − δ ) k A k − − δ ) X ≤ k ≤ m X ≤ i The next estimate will be crucial in the proof of Theorem A. This es-timate exploits a decomposition formula for the curvature componentsthat we obtained in [SHS12]. It makes it possible to relax the curvatureassumptions used in the paper by Lee and Lee [LL11] to those statedin our main theorem. Lemma 3.6. Under the assumptions made in Theorem A we have B := m X l,k =1 (cid:0) λ l R M − f ∗ R N (cid:1) ( e l , e k , e l , e k ) ≥ . Proof. In terms of the singular values, we gets( e k , e k ) = g M ( e k , e k ) − f ∗ g N ( e k , e k ) = 1 − λ k λ k . Since 1 = g( e k , e k ) = g M ( e k , e k ) + f ∗ g N ( e k , e k )we derive g M ( e k , e k ) = 11 + λ k , f ∗ g N ( e k , e k ) = λ k λ k and 2g M ( e k , e k ) = 1 − λ k λ k + 1 , − f ∗ g N ( e k , e k ) = 1 − λ k λ k − . EAN CURVATURE FLOW 15 Note also that for any k = l we haveg M ( e k , e l ) = f ∗ g N ( e k , e l ) = g( e k , e l ) = 0 . We compute B ( l ) : = 2 m X k =1 (cid:16) λ l R M ( e k , e l , e k , e l ) − f ∗ R N ( e k , e l , e k , e l ) (cid:17) = 2 m X k =1 , k = l λ l sec M ( e k ∧ e l )g M ( e k , e k )g M ( e l , e l ) − m X k =1 , k = l σ N (d f ( e k ) ∧ d f ( e l )) f ∗ g N ( e k , e k ) f ∗ g N ( e l , e l ) . Here the terms σ N (d f ( e k ) ∧ d f ( e l )) are zero if d f ( e k ), d f ( e l ) are collinearand otherwise they denote the sectional curvatures on ( N, g N ) of theplanes d f ( e k ) ∧ d f ( e l ). Now the formula for g M ( e k , e k ) implies B ( l ) = m X k =1 , k = l (cid:18) − λ k λ k (cid:19) λ l sec M ( e k ∧ e l )g M ( e l , e l )+2 m X k =1 , k = l f ∗ g N ( e k , e k ) n(cid:0) λ l σ − σ N (d f ( e k ) ∧ d f ( e l )) (cid:1) f ∗ g N ( e l , e l )+ λ l σ (cid:0) g M ( e l , e l ) − f ∗ g N ( e l , e l ) (cid:1)o − λ l σ X k = l f ∗ g N ( e k , e k )g M ( e l , e l )= m X k =1 , k = l (cid:18) − λ k λ k (cid:19) λ l sec M ( e k ∧ e l )g M ( e l , e l )+2 m X k =1 , k = l λ l f ∗ g N ( e k , e k )g M ( e l , e l ) (cid:16) σ − σ N (d f ( e k ) ∧ d f ( e l )) (cid:17) + λ l σ m X k =1 , k = l (cid:18) − λ k λ k − (cid:19) g M ( e l , e l ) . We may then continue to get B ( l ) = 2 m X k =1 , k = l λ l f ∗ g N ( e k , e k )g M ( e l , e l ) (cid:16) σ − σ N (d f ( e k ) ∧ d f ( e l )) (cid:17) + λ l (cid:16) Ric M ( e l , e l ) − ( m − σ g M ( e l , e l ) (cid:17) + m X k =1 , k = l λ l λ l − λ k λ k (cid:16) σ M ( e k ∧ e l ) + σ (cid:17) . Now we can see that2 B = m X l =1 B ( l )= 2 m X l,k =1 , k = l λ l f ∗ g N ( e k , e k )g M ( e l , e l ) (cid:16) σ − σ N (d f ( e k ) ∧ d f ( e l )) (cid:17) + m X l =1 λ l (cid:16) Ric M ( e l , e l ) − ( m − σ g M ( e l , e l ) (cid:17) + X ≤ k Lemma 3.7. Under the assumptions of Theorem A, the function u satisfies the differential inequality ∇ ∂ t log u ≥ ∆ log u + δ k A k , for some positive real number δ. Proof. The proof follows by combining the estimates obtained in Lemma3.5 and 3.6 with the evolution equation of log u in Lemma 3.4. (cid:3) Proof of Theorem A Since the area decreasing property is preserved, the solution F staysin particular graphical for any t ∈ [0 , T ). 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Wang, Long-time existence and convergence of graphic mean cur-vature flow in arbitrary codimension , Invent. Math. (2002), no. 3,525–543.[Wan01a] , Mean curvature flow of surfaces in Einstein four-manifolds , J.Differential Geom. (2001), no. 2, 301–338.[Wan01b] , Deforming area preserving diffeomorphism of surfaces by meancurvature flow , Math. Res. Lett. (2001), no. 5-6, 651–661.[Whi05] B. White, A local regularity theorem for mean curvature flow , Ann. ofMath. (2) (2005), no. 3, 1487–1519. Andreas Savas-HalilajInstitut f¨ur DifferentialgeometrieLeibniz Universit¨at HannoverWelfengarten 130167 Hannover, Germany E-mail address: [email protected] Knut SmoczykInstitut f¨ur Differentialgeometrieand Riemann Center for Geometry and PhysicsLeibniz Universit¨at HannoverWelfengarten 130167 Hannover, Germany