The extension and convergence of mean curvature flow in higher codimension
aa r X i v : . [ m a t h . DG ] A p r THE EXTENSION AND CONVERGENCE OF MEANCURVATURE FLOW IN HIGHER CODIMENSION
KEFENG LIU, HONGWEI XU, FEI YE, AND ENTAO ZHAO
Abstract.
In this paper, we first investigate the integral curvature conditionto extend the mean curvature flow of submanifolds in a Riemannian manifoldwith codimension d ≥
1, which generalizes the extension theorem for the meancurvature flow of hypersurfaces due to Le-ˇSeˇsum [12] and the authors [25,26]. Using the extension theorem, we prove two convergence theorems for themean curvature flow of closed submanifolds in R n + d under suitable integralcurvature conditions. Introduction
Let F : M n → N n + d be a smooth immersion from an n -dimensional Riemann-ian manifold without boundary to an ( n + d )-dimensional Riemannian manifold.Consider a one-parameter family of smooth immersions F : M × [0 , T ) → N satis-fying ( (cid:0) ∂∂t F ( x, t ) (cid:1) ⊥ = H ( x, t ) ,F ( x,
0) = F ( x ) , where (cid:0) ∂∂t F ( x, t ) (cid:1) ⊥ is the normal component of ∂∂t F ( x, t ), H ( x, t ) is the meancurvature vector of F t ( M ) and F t ( x ) = F ( x, t ). We call F : M × [0 , T ) → N themean curvature flow with initial value F : M → N . This is the general form of themean curvature flow, which is a nonlinear weakly parabolic system and is invariantunder reparametrization of M . We can find a family of diffeomorphisms φ t : M → M for t ∈ [0 , T ) such that ¯ F t = F t ◦ φ t : M → N satisfies ∂∂t ¯ F ( x, t ) = ¯ H ( x, t ). Wewill study the (reparameterized) mean curvature flow (cid:26) ∂∂t F ( x, t ) = H ( x, t ) ,F ( x,
0) = F ( x ) . (1.1)In [2], Brakke introduced the motion of a submanifold by its mean curvature inarbitrary codimension and constructed a generalized varifold solution for all time.For the classical solution of the mean curvature flow, most works have been doneon hypersurfaces. Huisken [9, 10] showed that if the second fundamental form isuniformly bounded, then the mean curvature flow can be extended over the time.He then proved that if the initial hypersurface in a complete manifold with bounded Mathematics Subject Classification.
Key words and phrases.
Mean curvature flow, submanifold, maximal existence time, conver-gence theorem, integral curvature.Research supported by the National Natural Science Foundation of China, Grant No. 11071211;the Trans-Century Training Programme Foundation for Talents by the Ministry of Education ofChina, and the China Postdoctoral Science Foundation, Grant No. 20090461379. geometry is compact and uniformly convex, then the mean curvature flow convergesto a round point in finite time. Many other beautiful results have been obtained,and there are various approaches to study the mean curvature flow of hypersurfaces(see [4, 7], etc.). However, relatively little is known about the mean curvature flowsof submanifolds in higher codimensions, see [16, 17, 20, 21, 22] etc. for example.Recently, Andrews-Baker [1] proved a convergence theorem for the mean curvatureflow of closed submanifolds satisfying suitable pinching condition in the Euclideanspace.On the other hand, Le-ˇSeˇsum [12] and Xu-Ye-Zhao [25] obtained some integralconditions to extend the mean curvature flow of hypersurfaces in the Euclideanspace independently. Later, Xu-Ye-Zhao [26] generalized these extension theoremsto the case where the ambient space is a Riemannian manifold with bounded ge-ometry.For an n -dimensional submanifold M in a Riemannian manifold, we denote by g the induced metric on M . Let A and H be the second fundamental form andthe mean curvature vector of M , respectively. In this paper, we first generalize theextension theorems in [12, 25, 26] to the mean curvature flow of submanifolds in aRiemannian manifold with bounded geometry. Theorem 1.1.
Let F t : M n → N n + d ( n ≥ be the mean curvature flow solution ofclosed submanifolds in a finite time interval [0 , T ) , where N has bounded geometry.If(i) there exist positive constants a and b such that | A | ≤ a | H | + b for t ∈ [0 , T ) ,(ii) || H || α,M × [0 ,T ) = (cid:16)R T R M t | H | α dµ t dt (cid:17) α < ∞ for some α ≥ n + 2 ,then this flow can be extended over time T . Let ˚ A be the tracefree second fundamental form, which is defined by ˚ A = A − n g ⊗ H . Denote by || · || p the L p -norm of a function or a tensor field. We obtain thefollowing convergence theorems for the mean curvature flow of closed submanifoldsin the Euclidean space. Theorem 1.2.
Let F : M n → R n + d ( n ≥ be a smooth closed submanifold. Thenfor any fixed p > , there is a positive constant C depending on n, p, V ol ( M ) and || A || n +2 , such that if || ˚ A || p < C , then the mean curvature flow with F as initial value has a unique solution F : M × [0 , T ) → R n + d in a finite maximal time interval, and F t converges uniformlyto a point x ∈ R n + d as t → T . The rescaled maps e F t = F t − x √ n ( T − t ) converge in C ∞ to a limiting embedding e F T such that e F T ( M ) is the unit n -sphere in some ( n + 1) -dimensional subspace of R n + d . Theorem 1.3.
Let F : M n → R n + d ( n ≥ be a smooth closed submanifold. Thenfor any fixed p > n , there is a positive constant C depending on n, p, V ol ( M ) and || H || n +2 , such that if || ˚ A || p < C , then the mean curvature flow with F as initial value has a unique solution F : M × [0 , T ) → R n + d in a finite maximal time interval, and F t converges uniformlyto a point x ∈ R n + d as t → T . The rescaled maps e F t = F t − x √ n ( T − t ) converge in XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 3 C ∞ to a limiting embedding e F T such that e F T ( M ) is the unit n -sphere in some ( n + 1) -dimensional subspace of R n + d . As immediate consequences of the convergence theorems, we obtain the followingdifferentiable sphere theorems. First let C be as in Theorem 1.2, we have Corollary 1.4.
Let F : M n → R n + d ( n ≥ be a smooth closed submanifold. If || ˚ A || p < C , for some p > , then M is diffeomorphic to the unit n -sphere. Similarly let C be as Theorem 1.3, we have Corollary 1.5.
Let F : M n → R n + d ( n ≥ be a smooth closed submanifold. If || ˚ A || p < C , for some p > n , then M is diffeomorphic to the unit n -sphere. We remark that in the above theorems and corollaries, we can replace the volume
V ol ( M ) by a positive lower bound of | H | in which case our method works withoutchange.The paper is organized as follows. In section 2, we introduce some basic equationsin submanifold theory, and recall the evolution equations of the second fundamentalform along the mean curvature flow. In section 3, by using the Moser iteration andblow-up method for parabolic equations, we prove Theorem 1.1. Theorems 1.2 and1.3 are proved in section 4. In section 5, we propose some unsolved problems onconvergence of the mean curvature flow in higher codimension.2. Preliminaries
Let F : M n → N n + d be a smooth immersion from an n -dimensional Riemannianmanifold M n without boundary to an ( n + d )-dimensional Riemannian manifold N n + d . We shall make use of the following convention on the range of indices.1 ≤ i, j, k, · · · ≤ n, ≤ A, B, C, · · · ≤ n + d, and n + 1 ≤ α, β, γ, · · · ≤ n + d. The Einstein sum convention is used to sum over the repeated indices.Suppose { x i } is a local coordinate system on M and { y A } is a local coordinatesystem on N . The metric g = P g ij dx i ⊗ dx j on M induced from the metric h , i on N by F is g ij = (cid:28) F ∗ (cid:16) ∂∂x i (cid:17) , F ∗ (cid:16) ∂∂x j (cid:17)(cid:29) . The volume form on M is dµ = p det( g ij ) dx .For any x ∈ M , denoted by N x M the normal space of M in N at point x , whichis the orthogonal complement of T x M in F ∗ T F ( x ) N . Denote by ¯ ∇ the Levi-Civitaconnection on N . The Riemannian curvature tensor ¯ R of N is defined by¯ R ( U, V ) W = − ¯ ∇ U ¯ ∇ V W + ¯ ∇ V ¯ ∇ U W + ¯ ∇ [ U,V ] W for vector fields U, V and W tangent to N . The induced connection ∇ on M isdefined by ∇ X Y = ( ¯ ∇ X Y ) ⊤ , for X, Y tangent to M , where ( ) ⊤ denotes tangential component. Let R be theRiemannian curvature tensor of M . KEFENG LIU, HONGWEI XU, FEI YE, AND ENTAO ZHAO
Given a normal vector field ξ along M , the induced connection ∇ ⊥ on the normalbundle is defined by ∇ ⊥ X ξ = ( ¯ ∇ X ξ ) ⊥ , where ( ) ⊥ denotes the normal component. Let R ⊥ denote the normal curvaturetensor.The second fundamental form is defined to be A ( X, Y ) = ( ¯ ∇ X Y ) ⊥ as a section of the tensor bundle T ∗ M ⊗ T ∗ M ⊗ N M , where T ∗ M and N M are thecotangential bundle and the normal bundle over M . The mean curvature vector H is the trace of the second fundamental form.The first covariant derivative of A is defined as( e ∇ X A )( Y, Z ) = ∇ ⊥ X A ( Y, Z ) − A ( ∇ X Y, Z ) − A ( Y, ∇ X Z ) , where e ∇ is the connection on T ∗ M ⊗ T ∗ M ⊗ N M . Similarly, we can define thesecond covariant derivative of A .Choosing orthonormal bases { e i } ni =1 for T x M and { e α } n + dα = n +1 for N x M , thecomponents of the second fundamental form and its first and second covariantderivatives are h αij = h A ( e i , e j ) , e α i ,h αijk = h ( e ∇ e k A )( e i , e j ) , e α i ,h αijkl = h ( e ∇ e l e ∇ e k A )( e i , e j ) , e α i . The Laplacian of A is defined by ∆ h αij = P k h αijkk .We define the tracefree second fundamental form ˚ A by ˚ A = A − n g ⊗ H , whosecomponents are ˚ A αij = h αij − n h αkk δ ij . Obviously, we have ˚ A αii = 0.Let R ijkl = g ( R ( e i , e j ) e k , e l ) , ¯ R ABCD = h ¯ R ( e A , e B ) e C , e D i ,R ⊥ ijαβ = h R ⊥ ( e i , e j ) e α , e β i . Then we have the following Gauss, Codazzi and Ricci equations. R ijkl = ¯ R ijkl + h αik h αjl − h αil h αjk ,h αijk − h αikj = − ¯ R αijk ,R ⊥ ijαβ = ¯ R ijαβ + h αik h βjk − h αjk h βik . Suppose F : M × [0 , T ) → N is the mean curvature flow with initial value F : M → N . We have the following evolution equations. Lemma 2.1 ([20]) . Along the mean curvature flow we have (2.1) ∂∂t dµ t = −| H | dµ t , XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 5 ∂∂t h αij =∆ h αij + ¯ R αijk,k + ¯ R αkik,j − R lijk h αlk + 2 ¯ R αβjk h βik + 2 ¯ R αβik h βjk − ¯ R lkik h αlj − ¯ R lkjk h αli + ¯ R αkβk h βij − h αim ( h βjm h βll − h βkm h βjk ) − h αkm ( h βjm h βik − h βkm h βij ) − h βik ( h βjl h αkl − h βkl h αjl ) − h αjk h βij h βll + h β h e α , ¯ ∇ H e β i , (2.2) where ¯ R ABCD,E are the components of the first covariant derivative ¯ ∇ ¯ R of ¯ R . Throughout of the paper, we assume that the submanifold is connected, and theambient space N has bounded geometry. Recall that a Riemannian manifold is saidto have bounded geometry if (i) the sectional curvature is bounded; (ii) the injectiveradius is bounded from below by a positive constant. We always assume that N is a Riemannian manifold with bounded geometry satisfying − K ≤ K N ≤ K fornonnegative constant K , K , and the injective radius of N is bounded from belowby a positive constant i N .3. The extension of mean curvature flow
In this section, we prove the extension theorem for the mean curvature flow ofsubmanifolds in arbitrary codimension. The following Sobolev inequality can befound in [8].
Lemma 3.1 ([8]) . Let M n ⊂ N n + d be an n ( ≥ -dimensional closed submanifold ina Riemannian manifold N n + d with codimension d ≥ . Denote by i N the positivelower bound of the injective radius of N restricted on M . Assume the sectionalcurvature K N of N satisfies K N ≤ b . Let h be a non-negative C function on M .Then (cid:18)Z M h nn − dµ (cid:19) n − n ≤ C ( n, α ) Z M h |∇ h | + h | H | i dµ, provided b (1 − α ) − n ( ω − n V ol ( supp h )) n ≤ and ρ ≤ i N , where ρ = (cid:26) b − sin − b (1 − α ) − n ( ω − n V ol ( supp h )) n f or b real, (1 − α ) − n ( ω − n V ol ( supp h )) n f or b imaginary. Here α is a free parameter, < α < , and C ( n, α ) = 12 π · n − α − (1 − α ) − n nn − ω − n n . For b imaginary, we may omit the factor π in the definition of C ( n, α ) . Lemma 3.2.
Let M n ⊂ N n + d be an n ( ≥ -dimensional closed submanifold in aRiemannian manifold N n + d with codimension d ≥ . Assume K N ≤ K , where K is a nonnegative constant. Let f be a non-negative C function on M satisfying (3.1) K ( n + 1) n ( ω − n V ol ( supp f )) n ≤ , KEFENG LIU, HONGWEI XU, FEI YE, AND ENTAO ZHAO (3.2) 2 K − sin − K ( n + 1) n ( ω − n V ol ( supp f )) n ≤ i N . Then ||∇ f || ≥ ( n − n − (1 + s ) (cid:20) C ( n ) || f || nn − − H (cid:18) s (cid:19) || f || (cid:21) , where H = max x ∈ M | H | , C ( n ) = C ( n, nn +1 ) and s > is a free parameter.Proof. For all g ∈ C ( M ), g ≥ || g || nn − ≤ C ( n ) Z M ( |∇ g | + g | H | ) dµ. Substituting g = f n − n − into (3.3) gives (cid:18)Z M f nn − dµ (cid:19) n − n ≤ n − n − C ( n ) Z M f nn − |∇ f | dµ + C ( n ) Z M Hf n − n − dµ. By H¨older’s inequality, we get (cid:18) Z M f nn − dµ (cid:19) n − n ≤ C ( n ) (cid:20) n − n − (cid:18) Z M f nn − dµ (cid:19) (cid:18) Z M |∇ f | dµ (cid:19) + (cid:18) Z M H f dµ (cid:19) (cid:18) Z M f nn − dµ (cid:19) (cid:21) Then (cid:18) Z M f nn − dµ (cid:19) n − n ≤ C ( n ) (cid:20) n − n − (cid:18) Z M |∇ f | dµ (cid:19) + (cid:18) Z M H f dµ (cid:19) (cid:21) . This implies || f || nn − ≤ C ( n ) (cid:20) n − (1 + s )( n − ||∇ f || + H (cid:18) s (cid:19) || f || (cid:21) , which is desired. (cid:3) Now we establish an inequality involving the maximal value of the squared normof the mean curvature vector and its L n +2 -norm in the space-time. Proposition 3.3.
Suppose that F t : M n → N n + d ( n ≥ is the mean curvatureflow solution for t ∈ [0 , T ] , where N has bounded geometry. Then max ( x,t ) ∈ M × [ T ,T ] | H | ( x, t ) ≤ C Z T Z M t | H | n +2 dµ t dt ! n +2 , where C is a constant depending only on n , T , sup ( x,t ) ∈ M × [0 ,T ] | A | , K , K and i N .Proof. In the following proof, we always denote by C the constant depending onsome quantities. We make use of Moser iteration for parabolic equations. Here wefollow the computation in [6]. From the evolution equation of the second funda-mental form in Lemma 2.1, we have the following differential inequality.(3.4) ∂∂t | H | ≤ △| H | + β | H | , XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 7 where β is a positive constant depending only on n , sup ( x,t ) ∈ M × [0 ,T ] | A | , K and K . For 0 < R < R ′ < ∞ and x ∈ M , we set η = x ∈ B g (0) ( x, R ) ,η ∈ [0 , and |∇ η | g (0) ≤ R ′ − R x ∈ B g (0) ( x, R ′ ) \ B g (0) ( x, R ) , x ∈ M \ B g (0) ( x, R ′ ) . Since supp η ⊆ B g (0) ( x, R ′ ), we assume that R ′ is sufficiently small such that η satisfies (3.1) and (3.2) with respect to g (0). On the other hand, the area of somefixed subset in M is non-increasing along the mean curvature flow, hence η satisfies(3.1) and (3.2) with respect to each g ( t ) for t ∈ [0 , T ]. Putting f = | H | and B ( R ′ ) = B g (0) ( x, R ′ ), the inequality (3.4) implies that, for any q ≥ q ∂∂t Z B ( R ′ ) f q η dµ t ≤ Z B ( R ′ ) (cid:0) η f q − △ f dµ t + βf q η (cid:1) dµ t + Z B ( R ′ ) q f q η ∂∂t dµ t = Z B ( R ′ ) (cid:0) η f q − △ f dµ t + βf q η (cid:1) dµ t − Z B ( R ′ ) q f q +1 η dµ t ≤ Z B ( R ′ ) (cid:0) η f q − △ f dµ t + βf q η (cid:1) dµ t . (3.5)Here we have used the evolution equation of the volume form in Lemma 2.1. Inte-grating by parts we obtain Z B ( R ′ ) η f q − △ f dµ t = − q − q Z B ( R ′ ) |∇ ( f q η ) | dµ t + 4 q Z B ( R ′ ) |∇ η | f q dµ t + 4( q − q Z B ( R ′ ) h∇ ( f q η ) , f q ∇ η i dµ t ≤ − q Z B ( R ′ ) |∇ ( f q η ) | dµ t + 2 q Z B ( R ′ ) |∇ η | f q dµ t . (3.6)Thus by (3.5) and (3.6) we obtain1 q ∂∂t Z B ( R ′ ) f q η dµ t ≤ − q Z B ( R ′ ) |∇ ( f q η ) | dµ t + β Z B ( R ′ ) f q η dµ t + 2 q Z B ( R ′ ) |∇ η | f q dµ t . (3.7)This implies ∂∂t Z B ( R ′ ) f q η dµ t + Z B ( R ′ ) |∇ ( f q η ) | dµ t ≤ Z B ( R ′ ) |∇ η | f q dµ t + βq Z B ( R ′ ) f q η dµ t . (3.8)For any 0 < τ < τ ′ < T , define a function ψ on [0 , T ] by ψ ( t ) = ≤ t ≤ τ, t − ττ ′ − τ τ ≤ t ≤ τ ′ , τ ′ ≤ t ≤ T . KEFENG LIU, HONGWEI XU, FEI YE, AND ENTAO ZHAO
Then from (3.8) we get ∂∂t ψ Z B ( R ′ ) f q η dµ t ! dµ t + ψ Z B ( R ′ ) |∇ ( f q η ) | dµ t ≤ ψ Z B ( R ′ ) |∇ η | f q dµ t + ( βqψ + ψ ′ ) Z B ( R ′ ) f q η dµ t . (3.9)For any t ∈ [ τ ′ , T ], integrating both sides of (3.9) on [ τ, t ] implies Z B ( R ′ ) f q η dµ t + Z tτ ′ Z B ( R ′ ) |∇ ( f q η ) | dµ t dt ≤ Z T τ Z B ( R ′ ) |∇ η | f q dµ t dt + (cid:18) βq + 1 τ ′ − τ (cid:19) Z T τ Z B ( R ′ ) f q η dµ t dt. (3.10)By the Sobolev inequality in Lemma 3.2, we obtain Z B ( R ′ ) f qnn − η nn − dµ t ! n − n = || f q η || nn − ≤ n − (1 + s ) C ( n )( n − ||∇ ( f q η ) || + CC ( n ) (cid:16) s (cid:17) || f q η || , (3.11)where C depends on n and sup ( x,t ) ∈ M × [0 ,T ] | A | . Combining (3.10) and (3.11) im-plies that Z T τ ′ Z B ( R ′ ) f q (1+ n ) η n dµ t dt ≤ Z T τ ′ Z B ( R ′ ) f q η dµ t ! n Z B ( R ′ ) f nqn − η nn − µ t ! n − n dt ≤ max t ∈ [ τ ′ ,T ] Z B ( R ′ ) f q η dµ t ! n × Z T τ h n − (1 + s ) C ( n )( n − ||∇ ( f q η ) || + CC ( n ) (cid:16) s (cid:17) || f q η || i dt ≤ C max t ∈ [ τ ′ ,T ] Z B ( R ′ ) f q η dµ t ! n × Z T τ h ||∇ ( f q η ) || + || f q η || i dt ≤ C (cid:20) Z T τ Z B ( R ′ ) |∇ η | f q dµ t dt + (cid:16) βq + 1 τ ′ − τ (cid:17) Z T τ Z B ( R ′ ) f q η dµ t dt (cid:21) n , (3.12)where we have put s = 1 and C is a constant depending only on n and sup ( x,t ) ∈ M × [0 ,T ] | A | . XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 9
Note that |∇ η | g ( t ) ≤ |∇ η | g (0) e lt , where l = max ≤ t ≤ T || ∂g∂t || g ( t ) . Thus Z T τ Z B ( R ′ ) |∇ η | f q dµ t dt ≤ Z T τ Z B ( R ′ ) |∇ η | g (0) e lt f q dµ t dt ≤ e CT ( R ′ − R ) Z T τ Z B ( R ′ ) f q dµ t dt, for some positive constant C depending on n and sup ( x,t ) ∈ M × [0 ,T ] | A | . This to-gether with (3.12) implies that Z T τ Z B ( R ) f q (1+ n ) dµ t dt ≤ C (cid:18) βq + 1 τ ′ − τ + 2 e CT ( R ′ − R ) (cid:19) n × Z T τ Z B ( R ′ ) f q dµ t dt ! n , (3.13)where C is a positive constant depending on n and sup ( x,t ) ∈ M × [0 ,T ] | A | .Putting L ( q, t, R ) = R T t R B ( R ) f q dµ t dt , we have from (3.13)(3.14) L (cid:16) q (cid:16) n (cid:17) , τ ′ , R (cid:17) ≤ C (cid:18) βq + 1 τ ′ − τ + 2 e CT ( R ′ − R ) (cid:19) n L ( q, τ, R ′ ) n . We set µ = 1 + 2 n , q k = n + 22 µ k , τ k = (cid:16) − µ k +1 (cid:17) t, R k = R ′ µ k/ ) . Then it follows from (3.14) that L ( q k +1 , τ k +1 , R k +1 ) qk +1 ≤ C qk +1 h ( n + 2) β µ µ − · t + 4 e CT R ′ · µ ( √ µ − i qk × µ kqk L ( q k , τ k , R k ) qk . Hence L ( q m +1 , τ m +1 , R m +1 ) qm +1 ≤ C P mk =0 1 qk +1 h ( n + 2) β µ µ − · t + 4 e CT R ′ · µ ( √ µ − i P mk =0 1 qk × µ P mk =0 kqk L ( q , τ , R ) q . (3.15)As m → + ∞ , we conclude from (3.15) that(3.16) f ( x, t ) ≤ C nn +2 (cid:16) C + 1 t + e CT R ′ (cid:17)(cid:16) n (cid:17) n Z T Z M t f n +22 dµ t dt ! n +2 , for some positive constant C depending on n , sup M × [0 ,T ] , K and K .Note that we choose R ′ sufficient small such that(3.17) K ( n + 1) n ( ω − n V ol g (0) ( B ( R ′ )) n ≤ , and(3.18) 2 K − sin − K ( n + 1) n ( ω − n V ol g (0) ( B ( R ′ )) n ≤ i N . For g (0), there is a non-positive constant K depending on n , max x ∈ M | A | , K and K such that the sectional curvature of M is bounded from below by K . By theBishop-Gromov volume comparison theorem, we have V ol g (0) ( B ( R ′ )) ≤ V ol K ( B ( R ′ )) , where V ol K ( B ( R ′ )) is the volume of a ball with radius R ′ in the n -dimensionalcomplete simply connected space form with constant curvature K . Let R ′ be thelargest number such that K ( n + 1) n ( ω − n V ol K ( B ( R ′ )) n ≤ , and 2 K − sin − K ( n + 1) n ( ω − n V ol K ( B ( R ′ )) n ≤ i N . Then R ′ depends only on n , K , K , i N and sup ( x,t ) ∈ M × [0 ,T ] | A | , and V ol g (0) ( B ( R ′ ))satisfies (3.17) and (3.18). This together with (3.16) impliesmax ( x,t ) ∈ M × [ T ,T ] | H | ( x, t ) ≤ C Z T Z M t | H | n +2 dµ t dt ! n +2 , where C is a constant depending on n , T and sup ( x,t ) ∈ M × [0 ,T ] | A | , K , K and i N . (cid:3) Now we give a sufficient condition that assures the extension of the mean curva-ture flow of submanifolds in a Riemannian manifold.
Theorem 3.4.
Let F t : M n → N n + d ( n ≥ be the mean curvature flow solutionof closed submanifolds in a finite time interval [0 , T ) . If(i) there exist positive constants a and b such that | A | ≤ a | H | + b for t ∈ [0 , T ) ,(ii) || H || α,M × [0 ,T ) = (cid:16)R T R M t | H | α dµ t dt (cid:17) α < ∞ for some α ≥ n + 2 ,then this flow can be extended over time T .Proof. By H¨older’s inequality, it is sufficient to prove the theorem for α = n + 2.We will argue by contradiction.Suppose that the solution of the mean curvature flow can’t be extended over T .Then the second fundamental form becomes unbounded as t → T . From assumption( i ), | H | is unbounded either.First we choose an increasing time sequence t ( i ) , i = 1 , , · · · , such that t ( i ) → T as i → ∞ . Then we take a sequence of points x ( i ) ∈ M satisfying | H | ( x ( i ) , t ( i ) ) = max ( x,t ) ∈ M × [0 ,t ( i ) ] | H | ( x, t ) . Put Q ( i ) = | H | ( x ( i ) , t ( i ) ) , then Q ( i ) , i = 1 , , · · · is a nondecreasing sequence and lim i →∞ Q ( i ) = ∞ . Thistogether with lim i →∞ t ( i ) = T > i such that Q ( i ) t ( i ) ≥ Q ( i ) ≥ i ≥ i . Let h be the Riemannian metric on N . For i ≥ i and t ∈ [0 , F ( i ) ( t ) = F (cid:18) t − Q ( i ) + t ( i ) (cid:19) : ( M, g ( i ) ( t )) → ( N, Q ( i ) h ) , XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 11 where g ( i ) ( t ) = F ( i ) ( t ) ∗ ( Q ( i ) h ). Let H ( i ) and A ( i ) = h ( i ) jk be the mean curvaturevector and the second fundamental form of F ( i ) ( t ) respectively. Then we have(3.19) | H ( i ) | ( x, t ) ≤ on M × [0 , . From assumption ( i ) again, inequality (3.19) implies | A ( i ) | ≤ C , where C is aconstant independent of i . Since ( N, h ) has bounded geometry and Q ( i ) ≥ i ≥ i , ( N, Q ( i ) h ) also has bounded geometry for i ≥ i with the same boundingconstants as ( N, h ). It follows from Proposition 3.3 that for i ≥ i max ( x,t ) ∈ M ( i ) × [ , | H ( i ) | ( x, t ) ≤ C (cid:18)Z Z M t | H ( i ) | n +2 dµ g ( i ) ( t ) dt (cid:19) n +2 , where C is a constant independent of i .By [5], there is a subsequence of pointed mean curvature flow solutions F ( i ) ( t ) : ( M, g ( i ) ( t ) , x ( i ) ) → ( N, Q ( i ) h ) , t ∈ [0 , e F ( t ) : ( f M , e g ( t ) , e x ) → R n + d , t ∈ [0 , . Denote by e H the mean curvature vector of e F , t ∈ [0 , ( x,t ) ∈ f M × [ , | e H | ( x, t ) ≤ lim i →∞ C (cid:18)Z Z M t | H ( i ) | n +2 dµ g ( i ) ( t ) dt (cid:19) n +2 ≤ lim i →∞ C Z t ( i ) +( Q ( i ) ) − t ( i ) Z M t | H ( i ) | n +2 dµdt ! n +2 =0 . (3.20)The equality in (3.20) holds because R T R M | H | n +2 dµ t dt < + ∞ and ( Q ( i ) ) − → i → ∞ . On the other hand, according to the choice of the points, we have | e H | ( e x,
1) = lim i →∞ | H ( i ) | ( x ( i ) ,
1) = 1 . This is a contradiction. (cid:3)
Remark . When d = 1, Theorem 3.4 generalizes the theorems in [12, 25, 26]. Infact, for N n +1 = R n +1 , we have the following computations.(i) If h ij ≥ − C for ( x, t ) ∈ M × [0 , T ) with some C ≥
0, let λ i , i = 1 , · · · , n bethe principal curvatures. Then λ i + C ≥
0, which implies that X i ( λ i + C ) ≤ n ( X i ( λ i + C )) ≤ nH + 2 n C . On the other hand, X i ( λ i + C ) = | A | + 2 CH + nC ≥ | A | − H + ( n − C . Hence | A | ≤ (2 n + 1) H + (2 n − n + 1) C for t ∈ [0 , T ).(ii) If H > t = 0, then there exists a positive constant C such that | A | ≤ CH at t = 0. By [9], we know that H > t > ∂∂t (cid:18) | A | H (cid:19) = △ (cid:18) | A | H (cid:19) + 2 H (cid:28) ∇ H, ∇ (cid:18) | A | H (cid:19)(cid:29) − H | H ∇ i h jk − ∇ i H · h jk | . By the maximum principle we obtain that | A | /H is uniformly bounded fromabove by its initial data. Hence | A | ≤ CH for t ∈ [0 , T ).For general N n +1 with bounded geometry, we have similar computations. Henceour Theorem 3.4 is a generalization.At the end of this section, we would like to propose the following Open Problem 3.1.
Let F t : M → N be the mean curvature flow solution ofclosed submanifolds in a finite time interval [0 , T ) . Suppose || H || α,M × [0 ,T ) < ∞ for some α ≥ n + 2 . Is there a positive constant ω such that the solution exists in [0 , T + ω ) ? The convergence of mean curvature flow
In this section we obtain some convergence theorems for the mean curvatureflow. The extension theorem proved in section 3 will be used to give a positivelower bound on the existence time of the mean curvature flow.We need the following Sobolev inequality for submanifolds in the Euclideanspace.
Lemma 4.1.
Let M be an n ( ≥ -dimensional closed submanifold in R n + d . Thenfor all Lipschitz functions v on M , we have (cid:18) Z M v nn − dµ (cid:19) n − n ≤ C n (cid:18) Z M |∇ v | dµ + Z M | H | n +2 dµ Z M v dµ (cid:19) where C n is a positive constant depending only on n .Proof. The proof of the lemma for d = 1 was given in [12]. Using the same methodwe can prove the lemma for d > (cid:3) Now we begin to prove the following convergence theorem for the mean curvatureflow.
Theorem 4.2.
Let F : M n → R n + d ( n ≥ be a smooth closed submanifold. Thenfor any fixed p > , there is a positive constant C depending on n, p, V ol ( M ) and || A || n +2 , such that if || ˚ A || p < C , then the mean curvature flow with F as initial value has a unique solution F : M × [0 , T ) → R n + d in a finite maximal time interval, and F t converges uniformlyto a point x ∈ R n + d as t → T . The rescaled maps e F t = F t − x √ n ( T − t ) converge in C ∞ to a limiting embedding e F T such that e F T ( M ) is the unit n -sphere in some ( n + 1) -dimensional subspace of R n + d .Proof. We set Λ = || A || n +2 . Denote by T max the maximal existence time of themean curvature flow with F as initial value. It is easy to show that T max < + ∞ (see [22] for a proof).We split the proof to several steps. Step 1.
For any fixed positive number ε , we first show that if(4.1) || ˚ A || p < ε for some p >
1, then T max satisfies T max > T for some positive constant T depend-ing on n, p , Λ and independent of ε , and there hold || A ( t ) || n +2 < , || ˚ A ( t ) || p < ε for t ∈ [0 , T ] . XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 13
Put T = sup { t ∈ [0 , T max ) : || A ( t ) || n +2 < , || ˚ A ( t ) || p < ε } . We consider the mean curvature flow on the time interval [0 , T ).By the definition of T we have R M t | A | n +2 dµ t ≤ (2Λ) n +2 for t ∈ [0 , T ). FromLemma 4.1 we have for a Lipschitz function v ,(4.2) (cid:18) Z M v nn − dµ (cid:19) n − n ≤ C n (cid:18) Z M |∇ v | dµ + n n +22 (2Λ) n +2 Z M v dµ (cid:19) . From (2.2), we have ∂∂t | A | ≤ ∆ | A | + c | A | , for some positive constant c depending only on n . Putting u = | A | , we have ∂∂t u ≤ ∆ u + c u . (4.3)From (4.3) and (2.1) we have ∂∂t Z M t u n +22 dµ t = Z M t n + 22 u n +22 − ∂∂t udµ t + Z M t u n +22 ∂∂t dµ t = n + 22 Z M t u n +22 − (∆ u + cu ) dµ t − Z M t H u n +22 dµ t ≤ − nn + 2 Z M t |∇ u n +24 | dµ t + n + 22 c Z M t u n +22 +1 dµ t . (4.4)For the second term of the right hand side of (4.4), we have by H¨older’s inequality Z M t u n +22 +1 dµ t ≤ (cid:18) Z M t u n +22 dµ t (cid:19) n +2 · (cid:18) Z M t ( u n +22 ) n +2 n dµ t (cid:19) nn +2 ≤ (cid:18) Z M t u n +22 dµ t (cid:19) n +2 · (cid:18) Z M t u n +22 dµ t (cid:19) n +2 · (cid:18) Z M t ( u n +22 ) nn − dµ t (cid:19) n − n +2 ≤ (cid:18) Z M t u n +22 dµ t (cid:19) n +2 · (cid:18) Z M t u n +22 dµ t (cid:19) n +2 × (cid:20) C n (cid:18) Z M t |∇ u n +24 | dµ t + Z M t | H | n +2 dµ t Z M t u n +22 dµ t (cid:19)(cid:21) nn +2 ≤ (cid:18) Z M t u n +22 dµ t (cid:19) n +2 · (cid:20) C nn +2 n (cid:18) Z M t |∇ u n +24 | dµ t (cid:19) nn +2 + n n (2Λ) n C nn +2 n (cid:18) Z M t u n +22 dµ t (cid:19) nn +2 (cid:21) ≤ n n (2Λ) n C nn +2 n (cid:18) Z M t u n +22 dµ t (cid:19) + C nn +2 n · n + 2 ǫ n +22 (cid:18) Z M t u n +22 dµ t (cid:19) + C nn +2 n · nn + 2 ǫ − n +2 n Z M t |∇ u n +24 | dµ t , (4.5) for any ǫ >
0. Combining (4.4) and (4.5), we have ∂∂t Z M t u n +22 dµ t ≤ n + 22 c (cid:16) n n (2Λ) n C nn +2 n + C nn +2 n · n + 2 ǫ n +22 (cid:17)(cid:18) Z M t u n +22 dµ t (cid:19) + (cid:16) n c C nn +2 n ǫ − n +2 n − nn + 2 (cid:17) Z M t |∇ u n +24 | dµ t . (4.6)Picking ǫ = (cid:18) n ( n +2) c C nn +2 n (cid:19) nn +2 , inequality (4.6) reduces to ∂∂t Z M t | A | n +2 dµ t ≤ c (cid:18) Z M t | A | n +2 dµ t (cid:19) , (4.7)where c = n +22 c (cid:18) n n (2Λ) n C nn +2 n + C nn +2 n · n +2 (cid:18) n ( n +2) c C nn +2 n (cid:19) n (cid:19) .From (4.7), we see by the maximal principle that, for t ∈ [0 , min { T, T } ), where T = − ( ) n +2 c Λ n +2 , there holds(4.8) || A ( t ) || n +2 <
32 Λ . Now we consider the evolution equation of | ˚ A | . By a simple computation, wehave ∂∂t | ˚ A | ≤ ∆ | ˚ A | − |∇ ˚ A | + c | A | | ˚ A | , (4.9)where c ≥ c is a positive constant depending only on n .Define a tensor ˜˚ A by ˜˚ A αij = ˚ A αij + ση α δ ij , where η α = 1. Set h σ = | ˜˚ A | =( | ˚ A | + ndσ ) . Then from (4.9) we have(4.10) ∂∂t h σ ≤ ∆ h σ + c | A | h σ . For any r ≥ p >
1, we have1 r ∂∂t Z M t h rσ dµ t = Z M t h r − σ ∂∂t h σ dµ t + 1 r Z M t h pσ ∂∂t dµ t ≤ − r − r Z M t |∇ h r σ | dµ t + c Z M t | A | h rσ dµ t . (4.11)For the second term of the right hand side of (4.11), we have the followingestimate. XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 15 Z M t | A | h rσ dµ t ≤ (cid:18) Z M t | A | n +2 dµ t (cid:19) n +2 · (cid:18) Z M t h r · n +2 n σ dµ t (cid:19) nn +2 ≤ (2Λ) (cid:18) Z M t h rσ dµ t (cid:19) n +2 · (cid:18) Z M t ( h rσ ) nn − dµ t (cid:19) n − n · nn +2 ≤ (2Λ) (cid:18) Z M t h rσ dµ t (cid:19) n +2 · (cid:20) C n (cid:18) Z M |∇ h r σ | dµ t + n n +22 (2Λ) n +2 Z M h rσ dµ t (cid:19)(cid:21) nn +2 ≤ (2Λ) (cid:18) Z M t h rσ dµ t (cid:19) n +2 · (cid:20) C nn +2 n (cid:18) Z M |∇ h r σ | dµ t (cid:19) nn +2 + n n (2Λ) n C nn +2 n (cid:18) Z M h rσ dµ t (cid:19) nn +2 (cid:21) = n n (2Λ) n +2 C nn +2 n Z M h rσ dµ t + (2Λ) C nn +2 n (cid:18) Z M t h rσ dµ t (cid:19) n +2 · (cid:18) Z M |∇ h r σ | dµ t (cid:19) nn +2 ≤ n n (2Λ) n +2 C nn +2 n Z M h rσ dµ t + (2Λ) C nn +2 n · n + 2 µ n +22 Z M t h rσ dµ t + (2Λ) C nn +2 n · nn + 2 µ − n +2 n Z M |∇ h r σ | dµ t , (4.12)for any µ >
0. Therefore, combining (4.11) and (4.12) we have1 r ∂∂t Z M t h rσ dµ t ≤ (cid:18) c (2Λ) C nn +2 n · nn + 2 µ − n +2 n − r − r (cid:19) Z M t |∇ h r σ | dµ t + c (cid:18) n n (2Λ) n +2 C nn +2 n + (2Λ) C nn +2 n · n + 2 µ n +22 (cid:19) Z M h rσ dµ t . (4.13)Choose µ = (cid:16) c r p rp − p + r (cid:17) nn +2 , where c = c (2Λ) C nn +2 n · nn +2 . Then from (4.13), wehave(4.14) ∂∂t Z M t h rσ dµ t + (cid:16) − p (cid:17) Z M t |∇ h r σ | dµ t ≤ (cid:18) c + c (cid:16) r p rp − p + r (cid:17) n (cid:19) · r · Z M h rσ dµ t , where c = c (2Λ) C nn +2 n and c = c n n (2Λ) n +2 C nn +2 n · n +2 · c n .Let r = p . Then (4.17) reduces to(4.15) ∂∂t Z M t h pσ dµ t ≤ c Z M h pσ dµ t , where c = (cid:18) c + c (cid:16) p p − (cid:17) n (cid:19) · p .Letting σ →
0, (4.15) becomes ∂∂t Z M t | ˚ A | p dµ t ≤ c Z M | ˚ A | p dµ t . This implies by the maximal principle that, for t ∈ [0 , min { T, T } ), where T = ( n +2) ln c , there holds(4.16) || ˚ A ( t ) || p < ε. Set T = min { T , T } . We claim that
T > T . We prove this claim by contradic-tion. Suppose that T ≤ T . Then (4.8) and (4.16) hold on [0 , T ).If T < T max , from the smoothness of the mean curvature flow we see that thereexists a positive constant ϑ such that on [0 , T + ϑ ) we have || A ( t ) || n +2 <
53 Λ , || ˚ A ( t ) || p < ε. This contradicts to the definition of T .If T = T max , we will show that the mean curvature flow can be extended overtime T max .From (4.14), we have(4.17) ∂∂t Z M t h rσ dµ t + (cid:16) − p (cid:17) Z M t |∇ h r σ | dµ t ≤ c r n +1 · Z M h rσ dµ t , where c = max { c p n , c (3 p − n } .As in the proof of Proposition 3.3, for any τ, τ ′ such that 0 < τ < τ ′ < T max − θ ,and for any t ∈ [ τ ′ , T max − θ ], where θ is a small positive constant, we have from(4.17)(4.18) Z M t h rσ dµ t + (cid:16) − p (cid:17) Z tτ ′ Z M t |∇ h r σ | dµ t dt ≤ (cid:16) c r n +1 + 1 τ ′ − τ (cid:17) Z T max − θτ Z M t h rσ dµ t dt. As in (3.12), we have by (4.2) Z T max − θτ ′ Z M t h (1+ n ) σ dµ t dt ≤ Z T max − θτ ′ (cid:18)Z M t h rσ dµ t (cid:19) n · (cid:18)Z M t h nrn − σ dµ t (cid:19) n − n dt ≤ max t ∈ [ τ ′ ,T max − θ ] (cid:18)Z M t h rσ dµ t (cid:19) n · Z T max − θτ ′ (cid:18)Z M t h nrn − σ dµ t (cid:19) n − n dt ≤ C n − n n · max t ∈ [ τ ′ ,T max − θ ] (cid:18)Z M t h rσ dµ t (cid:19) n × Z T max − θτ ′ (cid:18)Z M t |∇ h r σ | dµ t + n n +22 (2Λ) n +2 Z M t h rσ dµ t (cid:19) dt. (4.19) XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 17
From (4.18) and (4.19), we have Z T max − θτ ′ Z M t h r (1+ n ) σ dµ t dt ≤ c (cid:18) c r n +1 + 1 τ ′ − τ (cid:19) n × (cid:18) Z T max − θτ Z M t h rσ dµ t dt (cid:19) n , (4.20)where c = C n − n n · max { , n n +22 (2Λ) n +2 T · pp − } .We put J ( r, t ) = Z T max − θt Z M t h rσ dµ t dt. Then from (4.20) we have(4.21) J (cid:16) r (cid:16) n (cid:17) , τ ′ (cid:17) ≤ c (cid:18) c r n +1 + 1 τ ′ − τ (cid:19) n J ( r, τ ) n . We let µ = 1 + 2 n , r k = pµ k , τ k = (cid:18) − µ k +1 (cid:19) t. Notice that µ >
1. From (4.21) we have J ( r k +1 , τ k +1 ) rk +1 ≤ c rk +1 (cid:18) c p n +1 + µ µ − · t (cid:19) rk µ krk · ( n +1) J ( r k , τ k ) rk . Hence J ( r m +1 , τ m +1 ) rm +1 ≤ c P mk =0 1 rk +1 (cid:18) c p n +1 + µ µ − · t (cid:19) P mk =0 1 rk · µ ( n +1) · P mk =0 krk J ( p, t ) p . As m → + ∞ , we conclude that(4.22) h σ ( x, t ) ≤ (cid:18)
1+ 2 n (cid:19) n ( n +1)( n +2)4 p c n p (cid:18) c p n +1 + ( n + 2) nt (cid:19) n +22 p (cid:18) Z T max − θ Z M t h pσ dµ t dt (cid:19) p . Now let σ → θ →
0. Then we have for t ∈ [ T max , T max ), | ˚ A | ( x, t ) ≤ C ( n, p, Λ , ε, T max ) < + ∞ . This implies that | A | ≤ a | H | + b on [0 , T max ) for some positive constants a and b independent of t . On the otherhand, we also have Z T max Z M t | H | n +2 dµ t dt < + ∞ , since T max < + ∞ . Now we apply Theorem 3.4 to conclude that the mean curvatureflow can be extended over time T max . This is a contradiction. This completes theproof of the claim.By the definition of T , for t ∈ [0 , T ], we also have(4.23) || A ( t ) || n +2 < , || ˚ A ( t ) || p < ε. This completes Step 1.
Step 2.
We denote by
V ol (Σ) the volume of a Riemannian manifold Σ, and set V = V ol ( M ). In this step we show that if we choose ε sufficiently small, then atsome time T ∈ [ T , T ], the mean curvature is bounded from bellow by a positiveconstant depending on n , p , V and Λ.Since the area of the submanifold is non-increasing along the mean curvatureflow, we see that for t ∈ [0 , T max ), there holds(4.24) V ol ( M t ) ≤ V. Since M t is a closed submanifold in the Euclidean space, by the total meancurvature inequality (for the proof see [3]), we have n n ω n ≤ Z M t | H | n dµ t ≤ | H | n max ( t ) V ol ( M t ) ≤ | H | n max ( t ) V. Here | H | max ( t ) = max M t | H | ( · , t ). This implies that for t ∈ [0 , T max ), there holds(4.25) | H | ( t ) ≥ n n ω n V − := c . On the other hand, by [18], there is a positive constant c depending only on n such that for t ∈ [0 , T max ), we have diam ( M t ) ≤ c Z M t | H | n − dµ t , where diam ( M t ) denotes the diameter of M t . This together with the H¨older in-equality, (4.23) and (4.24) implies that for t ∈ [0 , T max )(4.26) diam ( M t ) ≤ c n n − (2Λ) n − V n +2 := c . Since
T > T , we consider the mean curvature flow on [ T , T ].As (4.22), we have for t ∈ [ T , T ](4.27) | ˚ A | ≤ (cid:18) n (cid:19) n ( n +1)( n +2)4 p c n p (cid:18) c p n +1 + ( n + 2) nT (cid:19) n +22 p · T p · ε := c ε. Here c depends on n, p, V, Λ and is independent of ε .For u = | A | , since c ≤ c , we have by (4.3)(4.28) ∂∂t u ≤ ∆ u + c | A | u. Then by a standard Moser iteration process as for h σ in Step 1, we have for t ∈ [ T , T ](4.29) | A | ≤ (cid:18) n (cid:19) n ( n +1)2 c nn +2 (cid:18) c (cid:16) n + 22 (cid:17) n +1 + ( n + 2) nT (cid:19) · T n +2 ·
2Λ := c . Here c = max { c n ( n +2) n , c n (3 n ) n } , and c = C n − n n · max { , n n +22 (2Λ) n +2 T · n +2 n } . Set G = (cid:16) t − T (cid:17) |∇ ˚ A | + | ˚ A | . We consider the evolution inequality of G on [ T , T ].As in [1], we have ∇ t ( ∇ ˚ A ) = ∇ ( ∇ t ˚ A ) + A ∗ A ∗ ∇ A. XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 19
Here ∇ is the connection on the spatial vector bundle, which for each t is agreewith the Levi-Civita connection of g ( t ). The evolution equation of ˚ A is ∇ t ˚ A = △ ˚ A + A ∗ A ∗ A. On the other hand, we have ∇ ( △ ˚ A ) = △ ( ∇ ˚ A ) + A ∗ A ∗ ∇ A. Hence ∇ t ( ∇ ˚ A ) = △ ( ∇ ˚ A ) + A ∗ A ∗ ∇ A. This implies(4.30) ∂∂t |∇ ˚ A | ≤ △|∇ ˚ A | + c | A | |∇ ˚ A | , where c is a positive constant depending only on n . Here we have used theinequality |∇ A | ≤ n n − |∇ ˚ A | , which was proved in [1].Combining (4.9) and (4.30) we have(4.31) ∂∂t G ≤ △ G + (cid:18)(cid:16) t − T (cid:17) c | A | − (cid:19) |∇ ˚ A | + c | A | | ˚ A | . From (4.27), (4.29) and (4.31), we have for t ∈ [ T , T ](4.32) ∂∂t G ≤ △ G + (cid:18)(cid:16) t − T (cid:17) c c − (cid:19) |∇ ˚ A | + c c c ε . Set T = min { T , T + c c } . Then T ≤ T ≤ T . For t ∈ [ T , T ], we have from(4.32) ∂∂t G ≤ △ G + c c c ε . By the maximal principle, this implies G ( t ) − G (cid:16) T (cid:17) ≤ c c c (cid:16) t − T (cid:17) ε for t ∈ [ T , T ]. Hence (cid:16) t − T (cid:17) |∇ ˚ A | ≤| ˚ A | (cid:16) T (cid:17) + c c c (cid:16) t − T (cid:17) ε ≤ c ε + c c c (cid:16) t − T (cid:17) ε . Then for t ∈ ( T , T ], there holds(4.33) |∇ ˚ A | ≤ c (cid:16) t − T (cid:17) ε + c c c ε . On the other hand, from [1], we know that |∇ H | ≤ n n − |∇ ˚ A | . Therefore, (4.33)implies that at t = T , we have(4.34) |∇ H | ≤ n n − · c (cid:16) T − T (cid:17) + c c c ! ε := c ε . Now we consider the submanifold M T at time T . Let x, y ∈ M T be two points suchthat | H | ( x, T ) = | H | min ( T ) := min M T | H | ( · , T ) and | H | ( y, T ) = | H | max ( T ) := max M T | H | ( · , T ). Let l : [0 , L ] → M T be the shortest geodesic such that l (0) = x and l ( L ) = y . Define a function η : [0 , L ] → R by η ( s ) = | H | ( l ( s ) , T ) for s ∈ [0 , L ].Then η (0) = | H | ( T ) and η ( L ) = | H | ( T ). By the definition of η , we have (cid:12)(cid:12)(cid:12)(cid:12) dds η ( s ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) dds | H | ( l ( s ) , T ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ( ∇| H | )( l ( s ) , T ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) | H ||∇ H | )( l ( s ) , T ) (cid:12)(cid:12)(cid:12)(cid:12) . This together with (4.29) and (4.34) implies(4.35) (cid:12)(cid:12)(cid:12)(cid:12) dds η ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ n c c ε. Then we have(4.36) η ( L ) − η (0) = Z L dds ηds ≤ diam ( M T ) · n c c ε. Combining (4.25), (4.26) and (4.36), we obtain(4.37) | H | ( T ) ≥ c − c ε, where c = 2 n c c c . We put ε = c c . Then if ε ≤ ε , (4.37) implies that(4.38) | H | ( T ) ≥ c . Step 3.
In this step, we finish the proof of Theorem 4.2.Consider the submanifold M T . Set ε = c [2 n ( n − c f or n ≥ , and ε = c √ c f or n = 3 . By (4.27) and (4.38), we see that if ε ≤ min { ε , ε } , then | A | ( T ) ≤ c ε + 1 n | H | ( T ) ≤ | H | ( T ) n − f or n ≥ , and | A | ( T ) ≤ | H | ( T ) f or n = 3 . We pick C = min { ε , ε } , which depends only on n, p, V and Λ. Then by theuniqueness of the mean curvature flow and the convergence theorem proved in [1],we conclude that the mean curvature flow with initial value F converges to a roundpoint in finite time. This completes the proof of Theorem 4.2. (cid:3) Corollary 4.3.
Let F : M n → R n + d ( n ≥ be a smooth closed submanifold.Suppose that the mean curvature is nowhere vanishing. Then for any fixed p > ,there is a positive constant C ′ depending on n, p , min M | H | and || A || n +2 , suchthat if || ˚ A || p < C ′ , then the mean curvature flow with F as initial value has a unique solution F : M × [0 , T ) → R n + d in a finite maximal time interval, and F t converges uniformlyto a point x ∈ R n + d as t → T . The rescaled maps e F t = F t − x √ n ( T − t ) converge in XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 21 C ∞ to a limiting embedding e F T such that e F T ( M ) is the unit n -sphere in some ( n + 1) -dimensional subspace of R n + d .Proof. It is easily to see that we can choose C in Theorem 4.2 such that C = C ( n, p, V, || A || n +2 ) depending on n, p , || A || n +2 and the upper bound V of thevolume of M . Since || A || n +2 ≥ n || H || n +2 ≥ n V ol ( M ) n +2 min M | H | , we have V ol ( M ) ≤ n − n +22 (min M | H | ) − ( n +2) || A || n +2 n +2 := V ′ . Then by Theorem 4.2, we can pick C ′ = C ( n, p, V ′ , || A || n +2 ), which depends on n, p , min M | H | and || A || n +2 . (cid:3) Theorem 4.4.
Let F : M n → R n + d ( n ≥ be a smooth closed submanifold. Thenfor any fixed p > n , there is a positive constant C depending on n, p, V ol ( M ) and || H || n +2 , such that if || ˚ A || p < C , then the mean curvature flow with F as initial value has a unique solution F : M × [0 , T ) → R n + d in a finite maximal time interval, and F t converges uniformlyto a point x ∈ R n + d as t → T . The rescaled maps e F t = F t − x √ n ( T − t ) converge in C ∞ to a limiting embedding e F T such that e F T ( M ) is the unit n -sphere in some ( n + 1) -dimensional subspace of R n + d .Proof. The idea to prove Theorem 4.4 is similar to the proof of Theorem 4.2. Weset Λ = || H || n +2 . Suppose(4.39) || ˚ A || p < ε for some fixed p > n and ε ∈ (0 , T ′ = sup { t ∈ [0 , T max ) : || H || n +2 < Λ , || ˚ A || p < ε } . As in the proof of Theorem 4.2, we consider the mean curvature flow on the timeinterval [0 , T ′ ).For | H | , we have the following inequality (see [1, 20] for the derivation) ∂∂t | H | ≤ ∆ | H | − |∇ H | + c | A | | H | , for some positive constant c depending only on n . Set w = | H | . Then(4.40) ∂∂t w ≤ ∆ w + c | ˚ A | w + c n w . From (4.40) we have for r > r ∂∂t Z M t w r dµ t ≤ − r − r Z M t |∇ w r | dµ t + c Z M t | ˚ A | w r dµ t + c n Z M t w r +1 dµ t . (4.41)Now we let r = n +22 . As in (4.5), we have Z M t w n +22 +1 dµ t ≤ C nn +2 n (cid:18) Z M t w n +22 dµ t (cid:19) + C nn +2 n · n + 2 ǫ n +22 (cid:18) Z M t w n +22 dµ t (cid:19) + C nn +2 n · nn + 2 ǫ − n +2 n Z M t |∇ w n +24 | dµ t , (4.42)for any ǫ > Z M t | ˚ A | w n +22 dµ t ≤ (200) C nn +2 n Z M w n +22 dµ t + (200) n +2 C nn +2 n · n + 2 µ n +22 Z M t w n +22 dµ t + (200) n +2 C nn +2 n · nn + 2 µ − n +2 n Z M |∇ w n +24 | dµ t , (4.43)for any µ > n + 2 · ∂∂t Z M t w n +22 dµ t ≤ (cid:18) c (200) n +2 C nn +2 n · nn + 2 µ − n +2 n + c n · C nn +2 n · nn + 2 ǫ − n +2 n − n ( n + 2) (cid:19) Z M t |∇ w r | dµ t + c (cid:18) (200) C nn +2 n + (200) n +2 C nn +2 n · n + 2 µ n +22 (cid:19) Z M t w n +22 dµ t + c n (cid:18) C nn +2 n + C nn +2 n · n + 2 ǫ n +22 (cid:19)(cid:18) Z M t w n +22 dµ t (cid:19) . (4.44)Now we pick µ = ǫ = (cid:18) c (200) n +2 C nn +2 n · nn +2 + c n · C nn +2 n · nn +26 n − n +2) (cid:19) nn +2 . Then from (4.44), we have(4.45) ∂∂t Z M t w n +22 dµ t ≤ c Z M t w n +22 dµ t + c (cid:18) Z M t w n +22 dµ t (cid:19) , where c and c are positive constants depending only on n .Let ρ ( t ) be the positive solution to the following Bernoulli equation ddt ρ = c ρ + c ρ ,ρ (0) =Λ n +2 . XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 23
Then ρ ( t ) = e c t n +2 + c c − c c e t , t ∈ (cid:20) , (cid:16) c c Λ n +2 + 1 (cid:17) c (cid:19) . Let T ′ > ρ ( t ) ≤ (2Λ) n +2 for t ∈ [0 , T ′ ]. Then by the maximal principle,we see that for t ∈ [0 , min { T ′ , T ′ } ), there holds Z M t w n +22 dµ t < (cid:16)
32 Λ (cid:17) n +2 , or equivalently,(4.46) || H ( t ) || n +2 <
32 Λ . Next, from (4.10) we have(4.47) ∂∂t h σ ≤ ∆ h σ + c | ˚ A | h σ + c n | H | h σ . From (4.47) we have for r > r ∂∂t Z M t h rσ dµ t ≤ − r − r Z M t |∇ h r σ | dµ t + c Z M t | ˚ A | h rσ dµ t + c n Z M t | H | h rσ dµ t . (4.48)As in (4.12), we have for r ≥ p > n , there holds Z M t | ˚ A | h rσ dµ t ≤ (200) C np n Z M h rσ dµ t + (200) p C np n · p − np ν pp − n Z M t h rσ dµ t + (200) p C np n · np ν − pn Z M |∇ h r σ | dµ t , (4.49)and Z M t | H | h rσ dµ t ≤ (200) C nn +2 n Z M h rσ dµ t + (200) n +2 C nn +2 n · n + 2 ̺ n +22 Z M t h rσ dµ t + (200) n +2 C nn +2 n · nn + 2 ̺ − n +2 n Z M |∇ h r σ | dµ t , (4.50)for any ν, ̺ > From (4.48), (4.49) and (4.50), we have1 r ∂∂t Z M t h rσ dµ t ≤ (cid:18) c (200) p C np n · np ν − pn + c n (200) n +2 C nn +2 n · nn + 2 ̺ − n +2 n − r − r (cid:19) Z M t |∇ h r σ | dµ t + (cid:18) c (200) C np n + c (200) p C np n · p − np ν pp − n + c n · (200) C nn +2 n + c n · (200) n +2 C nn +2 n · n + 2 ̺ n +22 (cid:19) Z M t h rσ dµ t . (4.51)Pick ν pn +2 = ̺ = (cid:18) c (200) p C np n · np + c n (200) n +2 C nn +2 n · nn +23 r − r (cid:19) nn +2 . Since r ≥ p > n , then ν pn +2 = ̺ ≤ (cid:18) c (200) p C np n · np + c n (200) n +2 C nn +2 n · nn +2 p − (cid:19) nn +2 · r nn +2 := c · r nn +2 . Then from (4.51), we have ∂∂t Z M t h rσ dµ t + Z M t |∇ h r σ | dµ t ≤ c r p + np − n Z M t h rσ dµ t , (4.52)where c = max n c (200) C np n + c n · (200) C nn +2 n , c n +2 p − n · c (200) p C np n · p − np ,c n +22 · c n · (200) n +2 C nn +2 n · n + 2 o . Letting r = p , we have from (4.52) ∂∂t Z M t h pσ dµ t ≤ c p p + np − n Z M t h pσ dµ t , (4.53)Now we apply the maximal principle and let σ →
0. Then for t ∈ [0 , min { T ′ , T ′ } ),where T ′ = c − p − np − n − , there holds || ˚ A ( t ) || p < ε. Set T ′ = min { T ′ , T ′ } . As in the Step 1 of the proof of Theorem 4.2, we canprove that T ′ > T ′ by contradiction. In fact, from the smoothness of the meancurvature flow we exclude the case where T ′ < T max . For the case where T ′ = T max ,since we have (4.52), which has similar form as (4.17), we can apply the standardMoser process to obtain the following estimate for small θ > h σ ( x, t ) ≤ (cid:18)
1+ 2 n (cid:19) n ( n +2)( p + n )4 p ( p − n ) c n p (cid:18) c p p + np − n + ( n + 2) nt (cid:19) n +22 p (cid:18) Z T max − θ Z M t h pσ dµ t dt (cid:19) p . XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 25
Here c = C n − n n · max { , (2Λ) n +2 T ′ } .Now we let σ → θ →
0. Then we have for t ∈ [ T max , T max ), | ˚ A | ( x, t ) ≤ C ′ ( n, p, Λ , ε, T max ) < + ∞ . This implies that | A | ≤ a ′ | H | + b ′ on [0 , T max ) for some positive constants a ′ and b ′ independent of t . On the otherhand we also have R T max R M t | H | n +2 dµ t dt < + ∞ . Applying Theorem 3.4 we con-clude that the mean curvature flow can be extended over time T max . This is acontradiction.We consider the mean curvature flow for t ∈ [ T ′ , T ′ ]. As (4.54), we have(4.55) | ˚ A | ( x, t ) ≤ (cid:18) n (cid:19) n ( n +2)( p + n )4 p ( p − n ) c n p (cid:18) c p p + np − n + ( n + 2) nT ′ (cid:19) n +22 p T ′ p · ε := c ε. By (4.40), we have ∂∂t w ≤ ∆ w + c | ˚ A | w + c n | H | w. Then similarly as (4.55), we get for t ∈ [ T ′ , T ′ ](4.56) | H | ( x, t ) ≤ (cid:18) n (cid:19) n ( n +1)2 c nn +2 (cid:18) c ( n + 2) n +1 + ( n + 2) nT ′ (cid:19) T ′ n +2 · (2Λ) := c . Here c = max n c (200) C nn +2 n + c n · (200) C nn +2 n , c ′ n +22 · c (200) n +2 C nn +2 n · n + 2 ,c ′ n +22 · c n · (200) n +2 C nn +2 n · n + 2 o .c = C n − n n · max { , (2Λ) n +2 T ′ } , and c ′ = (cid:18) c (200) n +2 C nn +2 n · nn +2 + c n (200) n +2 C nn +2 n · nn +2 n + 2 (cid:19) nn +2 . By (4.55) and (4.56), we have(4.57) | A | ( x, t ) ≤ c + c n := c , for t ∈ [ T ′ , T ′ ]. As in Step 2 of the proof of Theorem 4.2, we have for t ∈ [0 , T max ),there hold(4.58) | H | ( t ) ≥ n n ω n V − := c , and(4.59) diam ( M t ) ≤ c (2Λ) n − V n +2 := c , where V = V ol ( M ). Using a similar argument, for t ∈ [ T ′ , T ′ ], where T ′ = min { T ′ , T ′ + c c } , wehave(4.60) |∇ H | ≤ n n − · c (cid:16) t − T ′ (cid:17) + c c c ! ε := c ε . Combining (4.58), (4.59) and (4.60), we obtain that, at time T ′ , there is ε ′ = c n c c c , such that if ε ≤ ε ′ , then(4.61) | H | ( T ′ ) ≥ c . Set ε ′ = c [2 n ( n − c f or n ≥ , and ε ′ = c √ c f or n = 3 . By (4.27) and (4.38), we see that if ε ≤ min { ε ′ , ε ′ , } , then | A | ( T ′ ) ≤ c ε + 1 n | H | ( T ′ ) ≤ | H | ( T ′ ) n − f or n ≥ , and | A | ( T ′ ) ≤ | H | ( T ′ ) f or n = 3 . Then we can pick C = min { ε ′ , ε ′ , } , which depends only on n, p, V and Λ,and this completes the proof of Theorem 4.4. (cid:3) Using a similar argument as in the proof of Corollary 4.3, we have following
Corollary 4.5.
Let F : M n → R n + d ( n ≥ be a smooth closed submanifold.Suppose that the mean curvature is nowhere vanishing. Then for any fixed p > n ,there is a positive constant C ′ depending on n, p , min M | H | and || H || n +2 , suchthat if || ˚ A || p < C ′ , then the mean curvature flow with F as initial value has a unique solution F : M × [0 , T ) → R n + d in a finite maximal time interval, and F t converges uniformlyto a point x ∈ R n + d as t → T . The rescaled maps e F t = F t − x √ n ( T − t ) converge in C ∞ to a limiting embedding e F T such that e F T ( M ) is the unit n -sphere in some ( n + 1) -dimensional subspace of R n + d . Open Problems
In this section, we propose several open problems for the convergence of themean curvature flow of submanifolds. Denote by F n + d ( c ) the ( n + d )-dimensionalcomplete simply connected space form of constant sectional curvature c . Let M bean n -dimensional closed oriented submanifold in F n + d ( c ) with c ≥
0. Shiohama-Xu [15] showed that if | A | < α ( n, H, c ), then M is homeomorphic to a sphere for n ≥
4, or diffeomorphic to a spherical space form for n = 3. Here α ( n, H, c ) = nc + nH n − − n − n − p H + 4( n − cH . XTENSION AND CONVERGENCE OF MEAN CURVATURE FLOW 27
In [27], Xu-Zhao proved several differentiable sphere theorems for submanifoldssatisfying suitable pinching conditions in a Riemannian manifold. Recently, Xu-Gu [24] strengthened Shiohama-Xu’s topological sphere theorem for c = 0 to be adifferentiable sphere theorem. Motivated by these sphere theorems and the con-vergence theorem for the mean curvature flow due to Andrews and Baker [1], wepropose the following Open Problem 5.1.
Let M be an n -dimensional ( n ≥ smooth closed submani-fold in F n + d ( c ) with c > . Let M t be the solution of the mean curvature flow with M as initial submanifold. Suppose M satisfies | A | < α ( n, H, c ) . Then one of the following holds.a) The mean curvature flow has a smooth solution M t on a finite time interval ≤ t < T and the M t ’s converge uniformly to a round point as t → T .b) The mean curvature flow has a smooth solution M t for all ≤ t < ∞ and the M t ’s converge in the C ∞ -topology to a smooth totally geodesic submanifold M ∞ in F n + d ( c ) .In particular, M is diffeomorphic to the standard n -sphere. In [14], Shiohama-Xu obtained a topological sphere theorem for closed submani-folds satisfying || ˚ A || n < C ( n ) in F n + d ( c ) with c ≥ C ( n ) depending only on n . The following problems arise out of this topologicalsphere theorem and our convergence theorems. Open Problem 5.2.
Let M be an n -dimensional ( n ≥ smooth closed submani-fold in R n + d . Let M t be the solution of the mean curvature flow with M as initialsubmanifold. Then there exists an positive constant D ( n ) depending only on n ,such that if M satisfies || ˚ A || n < D ( n ) , then the mean curvature flow has a solution M t ’s on a finite time interval [0 , T ) and M t converges uniformly to a round point. In particular, M is diffeomorphic tothe standard n -sphere. For any 4-dimensional compact manifold M which is homeomorphic to a sphere,we hope to show that there exists an isometric embedding of the 4-sphere intoan Euclidean space such that || ˚ A || is small enough in the sense of Theorems 1.2or Open problem 5.2. In fact, Shiohama and the second author [14] proved thatfor any 4-dimensional compact submanifold M in an Euclidean space, we have || ˚ A || ≥ C (Σ i =1 β i ) / , where C is a universal positive constant and β i is the i -thBetti number of M , i = 1 , ,
3. Therefore it’s possible to isometrically embed atopological 4-sphere into an Euclidean space with small upper bound for || ˚ A || . Ifthis can be done, then we can deduce that M is diffeomorphic to a sphere. Thismay open a way to prove the smooth Poincar´e conjecture in dimension 4 which isnow one of the most challenging problems in geometry and topology.In general, for a homotopy sphere M , we can try to find its embedding in Eu-clidean spaces with small integral norm || ˚ A || n . Our results on mean curvature flowof arbitrary codimension reduce the problem of proving whether M is diffeomorphicto a sphere to the problem of finding the optimal embeddings of M into Euclideanspaces. Open Problem 5.3.
Let M be an n -dimensional ( n ≥ smooth closed submani-fold in F n + d ( c ) with c > . Let M t be the solution of the mean curvature flow with M as initial submanifold. Then there exists an positive constant E ( n ) dependingonly on n , such that if M satisfies || ˚ A || n < E ( n ) , then one of the following holds.a) The mean curvature flow has a smooth solution M t on a finite time interval ≤ t < T and the M t ’s converge uniformly to a round point as t → T .b) The mean curvature flow has a smooth solution M t for all ≤ t < ∞ and the M t ’s converge in the C ∞ -topology to a smooth totally geodesic submanifold M ∞ in F n + d ( c ) .In particular, M is diffeomorphic to the standard n -sphere. References [1] B. Andrews and C. Baker:
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E-mail address : [email protected] Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, PeoplesRepublic of China
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