aa r X i v : . [ m a t h . DG ] M a r MEAN CURVATURE FLOW IN SUBMANIFOLDS
HIROSHI NAKAHARA
Abstract.
We obtain explicit solutions of the mean curvature flow in some subman-ifolds of the Euclidean space. We give particularly an explicit solution of the flow ofa hypersurface in the Lagrangian self-expander L which is constructed in the article ofJoyce, Lee and Tsui and show that it converge to a minimal one.
1. IntroductionMean curvature flow evolves the submanifolds of the riemannian manifolds in thedirection of their mean curvature vectors. The short-time existence and uniqueness ofthe solution of the mean curvature flow equation was proved. The mean curvature vectorof a submanifold in a riemannian manifold is P j ( ∇ e j e j ) ⊥ where ∇ is the Levi-Civitaconnection, { e j } j is a orthonormal frame of the submanifold’s tangent bundle and ⊥ isthe orthogonal projection to the normal bundle of it. The mean curvature flow is animportant example of the changing. Sometimes the flow stops because of some singularpoints. Recently the finite time singularity is focused and mean curvature flow has beeninvestigated since it appeared from the study of annealing metals in physics in 1956.Many researchers study it and have found a lot of results. The mean curvature flow isthe steepest descent flow for the area functional and is described by a parabolic system ofpartial differential equations. The problem area of it is geometry but it implies questionsof partial differential equations. When M is a hypersurface in R n +1 and { M t } t ∈ [0 ,ǫ ) isthe solution of mean curvature flow, then, by the weak maximum principle of it (See also[1]), we can see that if the initial manifold M is in an open ball B (0 , r ) , where r > , then M t ⊂ B (0 , √ r − nt ) , for any t ∈ [0 , ǫ ) . Furthermore, other properties of the meancurvature flow in R N have been extensively studied. For example, Wang investigatesthe mean curvature flow of graphs in [6] and the author constructs explicit self-similarsolutions and translating solitons for the mean curvature flow in C n (= R n ) in his lastarticle [5]. Since the definition of the mean curvature vector is intricate as well as abstract,it is difficult or impossible to find the non-trivial and explicit solution of a given initialsubmanifold. However, we think of how explicit submanifolds of some concrete manifoldsmove by the mean curvature flow. The author noticed that if we consider the meancurvature flow of the hypersurfaces { s = constant } in the Lagrangian submanifold of theform of [3, Ansatz 3.1], they may change inside themselves. The prediction is correct with Mathematics Subject Classification. some restriction, to be showed in this paper. In this short article we deal with the meancurvature flow in the Lagrangian submanifold and give the explicit solutions. By the nextsection of this paper we get the following Theorem 1.1 and we treat a self-similar solutionfor the mean curvature flow in there. Note that we can learn self-similar solutions for themean curvature flow elementarily from [3] and [5], and the self-expander, which means aself-similar solution of expanding, in the articles is an explicit product manifold S n − × R in C n . Thus in the next steps, it is natural to study the mean curvature flow of the sphere S n − inside it and Theorem 1.1. Theorem . Let a > , E ≥ and α ≥ be constants. Define r : [0 , ∞ ) → R by r ( s ) = p /a + s and φ E : [0 , ∞ ) → R by φ E ( s ) = Z s t dt (1 /a + t ) p E (1 + at ) n e αt − . Set l s = { ( x r ( s ) e i φ E ( s ) , . . . , x n r ( s ) e i φ E ( s ) ); n X j =1 x j = 1 , x , . . . , x n ∈ R } , for s ∈ [0 , ∞ ) , and (1) L = [ s ∈ [0 , ∞ ) l s . ( Then, clearly, l s ⊂ L ⊂ C n . ) Fix s ∈ (0 , ∞ ) . Suppose that f is the solution of thefollowing initial-value problem (2) dfdt = − ( n − · E (1 + af ) n − e − αf Es (1 + af ) n f (0) = s Then { l f ( t ) } t is a mean curvature flow in L. In addition, when E = 1 , then L is theLagrangian self-expander in [3, Theorem C] and the domain of definition of f can beextended to [0 , ∞ ) , lim t →∞ f ( t ) = 0 and l is a minimal hypersurface in L.
2. Results and ProofsIn order to discuss the mean curvature flow in submanifolds, firstly, we consider thefollowing well known Proposition. Note that, in this article, when a manifold M is asubmanifold in a riemannian manifold N, then we denote A M,N the second fundamentalform of M in N and ∇ N , ∇ M the Levi-Civita connections on N and M respectively. Hence A M,N ∈ C ∞ ( M, ( T N/T M ) ⊗ T ∗ M ⊗ T ∗ M ) . Proposition . Let l, L be submanifolds in C n . Suppose that l is a submanifold in L. Put H to be the mean curvature vector of l in L, and ¯ H to be the mean curvature EAN CURVATURE FLOW IN SUBMANIFOLDS 3 vector of l in C n . Fix p ∈ l. Then H ( p ) = ¯ H ( p ) − X j A L, C n ( e j , e j ) where { e j } j is an orthonormal basis of T p l. Hence we can see that H ( p ) = π T p L ( ¯ H ( p )) , where π T p L ( ¯ H ( p )) is the orthogonal projection of ¯ H ( p ) to T p L. The reader can try to prove Proposition 2.1 or skip the proof below if it is already done.
Proof.
From the definitions of the mean curvature vector and the second fundamentalform we have H ( p ) = X j A l,L ( e j , e j ) = X j ( ∇ Le j e j − ∇ le j e j ) = X j ( ∇ C n e j e j − A L, C n ( e j , e j ) − ∇ le j e j )= X j ( A l, C n ( e j , e j ) − A L, C n ( e j , e j )) = ¯ H ( p ) − X j A L, C n ( e j , e j ) . This finishes the proof. (cid:3)
In the following Theorem 2.2, from a direct calculation, the submanifolds L are La-grangian submanifold. Theorem . Let I be an interval of R and w : I → C \ { } be a smooth function.Suppose that ˙ w ( s ) = 0 , for any s ∈ I. Define submanifolds l s , for s ∈ I, in C n by l s = { ( x w ( s ) , . . . , x n w ( s )); n X j =1 x j = 1 , x , . . . , x n ∈ R } , and submanifold L in C n by L = [ s ∈ I l s . ( Clearly, l s ⊂ L ⊂ C n . ) Let H s be the mean curvature vector of l s in L. Then (3) H s ( x w ( s ) , . . . , x n w ( s )) = − ( n − w ( s ) ˙ w ( s )) | w ( s ) | | ˙ w ( s ) | · ∂∂s holds, where ∂/∂s = ( x ˙ w ( s ) , . . . , x n ˙ w ( s )) ∈ T ( x w ( s ) ,...,x n w ( s )) L. Thus, by the definitionof the mean curvature flow, if we suppose that f is a solution of the following ordinaldifferential equation df ( t ) dt = − ( n − w ( f ( t )) ˙ w ( f ( t ))) | w ( f ( t )) | | ˙ w ( f ( t )) | , then { l f ( t ) } t is a mean curvature flow in L. We notice that the following Lemma 2.3 holds and it is a lemma of Theorem 2.2.
H. NAKAHARA
Lemma . Let α ∈ C \ { } be a constant. Define a submanifold S in C n by S = { α ( x , . . . , x n ) ∈ C n ; n X j =1 x j = 1 , x , . . . , x n ∈ R } . Fix p ∈ S. Then H ( p ) = − n − | α | p, where H ( p ) is the mean curvature vector of S at p. Proof.
Let { e , . . . , e n − } be an orthonormal basis of T p S. Let V j be the plane which isgenerated by e j and −→ Op, where O = (0 , . . . , ∈ C n . Since the intersection of S and V j isa circle of radius | α | with center O, we can get curves c , . . . , c n − : R → S such that c j (0) = p, ˙ c j (0) = e j , ¨ c j (0) = − | α | p, for any j. We compute H ( p ) = n − X j =1 A S, C n ( e j , e j ) = n − X j =1 (cid:16) ∇ C n e j e j (cid:17) ⊥ = n − X j =1 (¨ c j (0)) ⊥ = n − X j =1 (cid:18) − | α | p (cid:19) ⊥ = − n − | α | p, where ⊥ is the orthogonal projection to T ⊥ p S. This completes the proof. (cid:3)
Now we prove Theorem 2.2. proof of Theorem 2.2.
We denote by ¯ H s the mean curvature vector of l s in C . Fix p = ( x w ( s ) , . . . , x n w ( s )) ∈ l s . By Lemma 2.3,¯ H s ( p ) = − n − | w ( s ) | ( p ) . By Proposition 2.1, we have H ( p ) = π T p L ( ¯ H ( p )) = − n − | w ( s ) | · π T p L ( p )From a direct calculation, we can see ∂/∂s ⊥ T p l s . Hence we obtain H ( p ) = − n − | w ( s ) | · p · ∂/∂s∂/∂s · ∂/∂s · ∂∂s = − ( n − w ( s ) ˙ w ( s )) | w ( s ) | | ˙ w ( s ) | · ∂∂s . This finishes the proof. (cid:3)
Next we consider the following Remark 4 and Figure 1. If we put w = · · · = w in theconstruction of the Lagrangian self-expander given by Joyce, Lee and Tsui [3, ThoremC], then we can find a minimal hypersurface in the self-expander. Remark . Let a > α ≥ r : R → R by r ( s ) = p /a + s and φ : R → R by φ ( s ) = Z s | t | dt (1 /a + t ) p (1 + at ) n e αt − . EAN CURVATURE FLOW IN SUBMANIFOLDS 5 L l s l C (minimal) H sn Figure 1.
Remark 4In the situation of Theorem 2.2, if we put I = R and w ( s ) = r ( s ) e iφ ( s ) , then L is theLagrangian self-expander constructed in [3, Theorem C]. Then we computeRe ( ¯ w ( s ) ˙ w ( s )) | w ( s ) | | ˙ w ( s ) | = Re (cid:16) r ( s ) e − iφ ( s ) ( ˙ r ( s ) e iφ ( s ) + ir ˙ φ ( s ) e iφ ( s ) ) (cid:17) r ( s ) · | ˙ r ( s ) e iφ ( s ) + ir ( s ) ˙ φ ( s ) e iφ ( s ) | = r ( s ) ˙ r ( s ) r ( s ) · | ˙ r ( s ) + ir ( s ) ˙ φ ( s ) | = r ( s ) ˙ r ( s ) r ( s ) ˙ r ( s ) + r ( s ) ˙ φ ( s ) = ss + s / ((1 + at ) n e αs − s + s/ ((1 + at ) n e αs − s (1 + as ) n e αs / ((1 + as ) n e αs − as ) n e αs − s (1 + as ) n e αs = (1 + as ) n − e − αs s (1 + as ) n . (4) H. NAKAHARA
By the equation (3), the last computation and L’Hˆopital’s rule, we obtain H ( x w (0) , . . . , x n w (0)) = − ( n − · Re ( ¯ w (0) ˙ w (0)) | w (0) | | ˙ w (0) | · ∂∂s = − ( n − · lim s → (1 + as ) n − e − αs s (1 + as ) n · ∂∂s = − ( n − · lim s → n (1 + as ) n − · as + 2 αse − αs (1 + as ) n + s · n (1 + as ) n − · as · ∂∂s = ~ . (5)Therefore, in this case, l is minimal in L. Secondly, the author is going to prove a generalversion of the fact l is volume-minimizing as well as minimal in his next paper. Seealso it. A bit of information is below. We can see π ∗ (vol l ) is a calibration of L thatis described in the section 4 of [2] and that is a little difficult to prove, and l is thecalibrated submanifold, where π is a projection from L to l defined by π ( x w ( s ) , . . . , x n w n ( s )) = ( x w (0) , . . . , x n w n (0))and vol l is the volume form of l with respect to the induced metric of the Euclideanmetric in C n . Note that π is well-defined. This implies that l is volume-minimizing aswell as minimal. In addition, without the restriction w = · · · = w n , the submanifold { y = 0 } is a calibrated manifold in the self-expander L [3, Thorem C].Further we have to research the following situation. Remark . Let a > , E > α ≥ r : (0 , ∞ ) → R by r ( s ) = p /a + s and φ E : (0 , ∞ ) → R by φ E ( s ) = Z s t dt (1 /a + t ) p E (1 + at ) n e αt − . In the situation of Theorem 2.2, if we put I = (0 , ∞ ) and w ( s ) = r ( s ) e iφ E ( s ) , then L isthe Lagrangian self-similar solution constructed in [5, Theorem 1.3]. Now we write ∂∂s =( x ˙ w . . . , x n ˙ w )= ˙ w · ( x , . . . , x n )=( ˙ re iφ E + r i ˙ φ E e iφ E ) · ( x , . . . , x n )=( ˙ r + r i ˙ φ E ) · e iφ E · ( x , . . . , x n ) . (6)From (4) and (6), we obtain EAN CURVATURE FLOW IN SUBMANIFOLDS 7 H s ( x w ( s ) , . . . , x n w ( s )) = − ( n −
1) Re ( ¯ w ( s ) ˙ w ( s )) | w ( s ) | | ˙ w ( s ) | · ∂∂s = − ( n − r ˙ rr · | ˙ r + ir ˙ φ E | · ( ˙ r + r i ˙ φ E ) · e iφ E · ( x , . . . , x n )= − ( n − r ˙ rr · ( ˙ r − ir ˙ φ E ) · e iφ E · ( x , . . . , x n )= − ( n − s /a + s · s p /a + s − i s p (1 /a + s ) { E (1 + as ) n e αs − } ! − · e iφ E · ( x , . . . , x n )= − ( n − /a + s · p /a + s − i p (1 /a + s ) { E (1 + as ) n e αs − } ! − · e iφ E · ( x , . . . , x n ) . Thereforelim s → +0 H s ( x w ( s ) , . . . , x n w ( s )) = − ( n − a · (cid:18) √ a − i r aE − (cid:19) − · ( x , . . . , x n )= − ( n − a · √ a + i p a/ ( E − a + a/ ( E − · ( x , . . . , x n )= − ( n − · √ a + i p a/ ( E − / ( E − · ( x , . . . , x n )= − ( n − √ a · i p / ( E − / ( E − · ( x , . . . , x n )= − ( n − √ a · E − i √ E − E · ( x , . . . , x n ) . (7)By (7), we can check lim E → (cid:18) lim s → +0 H s ( x w ( s ) , . . . , x n w ( s )) (cid:19) = ~ References [1]
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Department of MathematicsTokyo Institute of Technology2-21-1 O-okayama, Meguro, TokyoJapan
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