Deforming symplectomorphisms of complex projective spaces by the mean curvature flow
aa r X i v : . [ m a t h . DG ] J a n DEFORMING SYMPLECTOMORPHISMS OFCOMPLEX PROJECTIVE SPACESBY THE MEAN CURVATURE FLOW
Ivana Medoˇs and Mu-Tao Wang
Abstract
We apply the mean curvature flow to deform symplectomor-phisms of CP n . In particular, we prove that, for each dimension n , there exists a constant Λ, explicitly computable, such that anyΛ-pinched symplectomorphism of CP n is symplectically isotopicto a biholomorphic isometry.
1. Introduction
It was proposed in [ ] to use the mean curvature flow to studythe structure of the symplectomorphism group of a symplectic manifold( M, ω ). Consider the graph of a symplectomorphism f : M → M as anembedded submanifold Σ = { ( x, f ( x )) | x ∈ M } of the product mani-fold M × M . Σ can be viewed as a Lagrangian submanifold with respectto the symplectic structure π ∗ ω − π ∗ ω on M × M where π i is the projec-tion from M × M to the i -th factor, i = 1 ,
2. Suppose that M is endowedwith a compatible K¨ahler metric such that ω is the K¨ahler form. Thevolume of Σ with respect to the product metric naturally defines a func-tion on the symplectomorphism group of M which is symmetric withrespect to the inverse operation f f − . This provides a variationalapproach to study the topology of this infinite dimensional group. Thecritical point of the volume function corresponds to minimal Lagrangiansubmanifolds and the mean curvature flow is the negative gradient flow.By Smoczyk [ ], it is known that being Lagrangian is preserved by themean curvature flow when M is equipped with a K¨ahler-Einstein met-ric. Therefore, if Σ remains graphical along the mean curvature flow,the flow in turn gives a symplectic isotopy of f .In this article, we apply this idea to the complex projective space CP n with the Fubini-Study metric and prove that a pinched symplectomor-phism (see Definition 1) is symplectically isotopic to a biholomorphicisometry along the mean curvature flow. The authors are partially supported by National Science Foundation Grant DMS0605115 and DMS 0904281.Received July 15, 2009, this version January 25, 2011.
PROOF COPY 1 NOT FOR DISTRIBUTION IVANA MEDOˇS AND MU-TAO WANG
Denote by g and ω the Fubini-Study metric and the associated K¨ahlerform on CP n , respectively. Recall that a diffeomorphism f of CP n is asymplectomorphism if f ∗ ω = ω . Definition 1.
Let Λ be a constant ≥ . A symplectomorphism f of CP n is said to be Λ -pinched if (1.1) 1Λ g ≤ f ∗ g ≤ Λ g. The precise statement of the pinching theorem is the following.
Theorem 1.
For each positive integer n there exists a constant Λ( n ) > , such that, if f : CP n → CP n is a Λ -pinched symplectomorphism forsome < Λ < Λ( n ) , then:1) The mean curvature flow Σ t of the graph of f in CP n × CP n existssmoothly for all t ≥ .2) Σ t is the graph of a symplectomorphism f t for each t ≥ .3) f t converges smoothly to a biholomorphic isometry of CP n as t →∞ . The mean curvature flow forms a smooth one-parameter family ofsymplectomorphisms or a symplectic isotopy. Therefore the followingholds.
Corollary 1.
For each positive integer n , there exists a constant Λ( n ) , such that if f is a Λ -pinched symplectomorphism of CP n for some < Λ < Λ( n ) , then f is symplectically isotopic to a biholomorphicisometry. This theorem generalizes a previous theorem of the second author forRiemann surfaces in which no pinching condition is required.
Theorem 2. [ , ] Let (Σ , g , ω ) and (Σ , g , ω ) be two diffeo-morphic compact Riemann surfaces with Riemannian metrics g and g of the same constant curvature c . Suppose Σ is the graph of a sym-plectomorphism f : Σ → Σ and Σ t is the mean curvature flow in theproduct space Σ × Σ with initial surface Σ = Σ . Then Σ t remains thegraph of a symplectomorphism f t along the mean curvature flow. Theflow exists smoothly for all time and Σ t converges smoothly to a minimalLagrangian submanifold as t → ∞ . In Theorem 2, the long time existence for any c and the smoothconvergence for c > ]. The smooth convergencefor c ≤ ]. Using a differentmethod, Smoczyk [ ] proved the theorem when c ≤ ] (see also[ ]). In this case the symplectomorphism is indeed an area-preserving PROOF COPY NOT FOR DISTRIBUTIONEFORMING SYMPLECTOMORPHISMS 3 map. The boundary value problem for minimal area-preserving mapshas been studied by Wolfson [ ] and Brendle [ ].A theorem of Smale states that the isometry group SO (3) of S is acontinuous deformation retract of the oriented diffeomorphism group of S = CP , and Theorem 2 gives a new proof of this theorem. The defor-mation retract provided by the mean curvature flow is indeed smooth.We are informed by Prof. McDuff that it was proved by Gromov [ ] thatthe biholomorphic isometry group of CP is a deformation retract of itssymplectomorphism group. It seems that no similar result is known for CP n when n > § t remains the graph of a symplectomorphism f t as long as the flow exists smoothly. We study the evolution of theJacobian of the projection map π : Σ t → M (denoted by ∗ Ω) and provethat positivity is preserved by the maximum principle. This justifies theclaim by the implicit function theorem. (see § § t for each t >
0, and show that there is no finite timesingularity. (see § CP n ) and the pull-back metric f ∗ g isapproaching g as t → ∞ .Step 5. We prove that the second fundamental form of Σ t is uniformlybounded in t as t → ∞ . This gives the smooth convergence in thetheorem.Step 4 and 5 are done in §
2. Preliminaries2.1. Singular values of symplectic linear maps between vec-tor spaces.
Let (
V, g ) and ( ˜
V , ˜ g ) be 2 n -dimensional real inner productspaces, with almost complex structures J and ˜ J , respectively, compati-ble with the corresponding inner products. Then ω ( · , · ) = g ( J · , · ), and˜ ω = ˜ g ( ˜ J · , · ) are symplectic forms on V and ˜ V . Recall that a linear map PROOF COPY NOT FOR DISTRIBUTION IVANA MEDOˇS AND MU-TAO WANG L : ( V, ω ) → ( ˜ V , ˜ ω ) is said to be symplectic if:(2.1) ω ( u, v ) = ˜ ω ( L ( u ) , L ( v ))for any u, v ∈ V . In this context, the condition is equivalent to:(2.2) L ∗ ˜ J L = J, where L ∗ : ˜ V → V is the adjoint operator of L with respect to the innerproducts on ˜ V and V .For such L , we define E : V → ˜ V to be the map E = L [ L ∗ L ] − .Since L is an isomorphism, L ∗ L is a positive definite self-adjoint auto-morphism of V and the square root of L ∗ L is well-defined. Lemma 1. E is an isometry which intertwines with J and ˜ J , i.e. ˜ J E = EJ.
In other words, E is a symplectic isometry.Proof. E is an isometry since:˜ g ( Eu, Ev ) = ˜ g ( L [ L ∗ L ] − u, L [ L ∗ L ] − v ) = g ( L ∗ L [ L ∗ L ] − u, [ L ∗ L ] − v )= g ([ L ∗ L ] u, [ L ∗ L ] − v )= g ([ L ∗ L ] − [ L ∗ L ] u, v )= g ( u, v )for any u, v ∈ V . Let P = [ L ∗ L ] , so that E = LP − . − J P − J and P are both positive definite ( − J P − J = J − P − J is positive definite since P − is and since J is an orthogonal operator), and, by the symplecticcondition (2.2), their squares are equal:( − J P − J ) = − J L − ( L ∗ ) − J = − L ∗ ˜ J ˜ J L = P . It follows that − J P − J = P . By using the symplectic condition L ∗ ˜ J L = J and the fact that P = L ∗ LP − , we obtain the desired result: − J P − J = P ⇒ − J P − J = L ∗ LP − ⇒ − ( L ∗ ) − J P − J = LP − ⇒ − ˜ J LP − J = LP − ⇒ − ˜ J EJ = E. Finally, the last equality implies E ∗ ˜ J E = J so E is in fact a sym-plectic isometry (condition (2.2)). q.e.d. PROOF COPY NOT FOR DISTRIBUTIONEFORMING SYMPLECTOMORPHISMS 5
Let ( v , . . . , v n ) be a basis of V that diagonalizes L ∗ L . L ∗ L is thepositive definite matrix: L ∗ L = λ . . . λ ... . . . λ n − . . . λ n with respect to this basis, for some λ i > i = 1 , . . . , n .Then, by construction, L ( v i ) = λ i E ( v i ); in other words: L = λ . . . λ ... . . . λ n − . . . λ n with respect to the bases ( v , . . . , v n ) and ( E ( v ) , . . . , E ( v n )), and thus λ i are the singular values of L . Lemma 2.
Let λ i be the singular values of L and v i be the associatedsingular vectors, i.e. L ( v i ) = λ i E ( v i ) . Then: ( λ i λ j − g ( J v i , v j ) = 0 . Proof.
By the symplectic condition (2.1) and Lemma 1: g ( J v i , v j ) =˜ g ( ˜ J L ( v i ) , L ( v j )) = λ i λ j ˜ g ( ˜ J E ( v i ) , E ( v j ))= λ i λ j ˜ g ( E ( J v i ) , E ( v j ))= λ i λ j g ( J v i , v j ) . q.e.d. Lemma 3. If α is a singular value of L , then so is α . Moreover, if V ( α ) denotes the subspace of singular vectors corresponding to a singu-lar value α , then dim V ( α ) = dim V (cid:18) α (cid:19) , and J restricts to an isomorphism between V ( α ) and V (cid:0) α (cid:1) .Proof. The first statement is a consequence of Lemma 2. Indeed,let ( v , . . . , v n ) be the basis described in the lemma. Then for each i ∈ { , . . . , n } there exists some j ∈ { , . . . , n } such that g ( J v i , v j ) = 0since J v i is a nonzero vector. Then, by the lemma, it follows that λ i λ j = 1.The second statement is trivial if α = 1. Assume that α = 1, and letdim V ( α ) = k , dim V (cid:0) α (cid:1) = l . By renumbering indexes, we may assumethat v , . . . , v k span V ( α ) (so that λ = . . . = λ k = α ). We claim that PROOF COPY NOT FOR DISTRIBUTION IVANA MEDOˇS AND MU-TAO WANG
J v , . . . , J v k belong to V (cid:0) α (cid:1) . Fix any 1 ≤ i ≤ k and consider J v i . Let V ′ be the orthogonal complement of V ( α ) such that V = V ( α ) ⊕ V ′ .Take any v m ∈ V ′ for 1 ≤ m ≤ n , thus we have Lv m = λ m v m for λ m = α . Lemma 2 implies g ( J v i , v m ) = 0 for any such v m , and therefore J v i is in the orthogonal complement of V ′ , or V ( α ) for each i = 1 , · · · k .Moreover, J v , . . . , J v k are linearly independent because v , . . . , v k are.It follows that k ≤ l . The same argument applies to V (cid:0) α (cid:1) and it followsthat k ≥ l .We conclude that k = l , and that J restricts to an isomorphism from V ( α ) to V (cid:0) α (cid:1) . q.e.d. Remark 1.
The preceding lemma implies that V splits into a directsum of singular subspaces of the following form: (2.3) V = V (1) k ⊕ V ( α ) k ⊕ V (cid:18) α (cid:19) k ⊕ . . . ⊕ V ( α s ) k s ⊕ V (cid:18) α s (cid:19) k s , where s is the total number of distinct singular values of L greater than1, α i are distinct singular values of L greater than 1, i = 1 , . . . , s , andthe superscripts represent dimension, k ≥ and k j > for j = 1 , . . . , s . Proposition 1.
Let L : ( V, ω ) → ( ˜ V , ˜ ω ) be a symplectic linear map,where V and ˜ V are real vector spaces of dimension n equipped withalmost complex structures J and ˜ J and inner products g and ˜ g com-patible with the respective complex structures; and where ω = g ( J · , · ) , ˜ ω = ˜ g ( ˜ J · , · ) . Then there exists an orthonormal basis of V with respectto which: (2.4) J = − . . .
01 0 . . . ... . . . . . . − . . . and: (2.5) L ∗ L = λ . . . λ ... . . . λ n − . . . λ n where λ i − λ i = 1 , for i = 1 , . . . , n .Proof. Lemma 3 and (2.3) imply that it is sufficient to find a basissatisfying (2.4) of the subspaces V ( α ) ⊕ V ( α ) for each singular value α = 1, as well as of V (1) if 1 is a singular value of L . PROOF COPY NOT FOR DISTRIBUTIONEFORMING SYMPLECTOMORPHISMS 7
Assume that there is a singular value α = 1, and let k = dim V ( α ).We choose an arbitrary basis u , . . . , u k of this space. Then J u , . . . , J u k is a basis of V ( α ). Putting these bases together provides a basis of V ( α ) ⊕ V ( α ) satisfying (2.4). Moreover, since u , . . . , u k are singularvectors of L with singular value α , and J u , . . . , J u k are singular valuesof L with singular value α , it follows that ( u , J u , u , J u , . . . , u k , J u k )is the desired basis.If a singular value is equal to 1 (i.e. if k > V (1) satisfying (2.4) suffices. q.e.d.Since the image of an orthonormal basis under an isometry is also anorthonormal basis, we obtain the following corollary. Corollary 2.
Let E : V → ˜ V be the isometry E = L [ L ∗ L ] − . If ( a , . . . , a n ) is a basis of V satisfying the properties of Proposition ,and if (˜ a , . . . , ˜ a n ) is the orthonormal basis ( E ( a ) , . . . , E ( a n )) of ˜ V ,then:(a) ˜ J = − . . .
01 0 . . . ... . . . . . . − . . . with respect to (˜ a , . . . , ˜ a n ) ;and:(b) L is diagonalized with respect to these bases, with diagonal valuesordered in pairs whose product is 1: L = λ . . . λ ... . . . λ n − . . . λ n with λ i − λ i = 1 , for i = 1 , . . . , n .Proof. Part (a) follows from Proposition 1 and Lemma 1. Part (b)follows from the fact that L ( a i ) = λ i E ( a i ). q.e.d. Let Σ be thegraph of a symplectomorphism f : ( M, ω ) → ( ˜ M , ˜ ω ) between K¨ahler-Einstein manifolds ( M, g, ω ) and ( ˜
M , ˜ g, ˜ ω ) of real dimension 2 n and ofthe same scalar curvature. The product space ( M × ˜ M , G = g ⊕ ˜ g ) is thusa K¨ahler-Einstein manifold. We consider the evolution of Σ ⊂ M × ˜ M PROOF COPY NOT FOR DISTRIBUTION IVANA MEDOˇS AND MU-TAO WANG under the mean curvature flow. If J and ˜ J are almost complex structuresof M and ˜ M , respectively, then J = J ⊕ ( − ˜ J ) defines an almost complexstructure on M × ˜ M parallel with respect to G . Let Σ t be the meancurvature flow of Σ in M × ˜ M .Let Ω be the volume form of M extended to M × ˜ M naturally (moreprecisely, let Ω be the pullback of the volume form of M under theprojection π : M × ˜ M → M ). Denote by ∗ Ω the Hodge star of therestriction of Ω to Σ t . At any point q ∈ Σ t , ∗ Ω( q ) = Ω( e , . . . , e n )for any oriented orthonormal basis of T q Σ. ∗ Ω is the Jacobian of theprojection π from Σ t onto M . We shall show that ∗ Ω remains positivealong the mean curvature flow. By the implicit function theorem, thisimplies that Σ t is a graph over M .We apply the result in the previous section to choose a basis thatsimplifies the evolution equation of ∗ Ω. Suppose q ∈ Σ t is of the form q = ( p, f ( p )) for p ∈ M and f ( p ) ∈ ˜ M , and let ( a , . . . , a n ) be the basisof T p M satisfying the properties listed in Proposition 1, for L = Df p : T p M → T f ( p ) ˜ M , with the inner products understood to be the metrics g on M at p and ˜ g on ˜ M at f ( p ). Thus we have(2.6) a , a = J a , · · · , a n − , a n = J a n − on T p M . Define E : T p M → T f ( p ) ˜ M to be the isometry E = Df p [ Df ∗ p Df p ] − for p ∈ M . Let us also choose a basis of T f ( p ) ˜ M to be (˜ a , . . . , ˜ a n ) =( E ( a ) , . . . , E ( a n )), as per Corollary 2.Then(2.7) e i = 1 p | Df p ( a i ) | ( a i , Df p ( a i )) = 1 q λ i ( a i , λ i E ( a i ))and(2.8) e n + i = J ( p,f ( p )) e i = 1 q λ i ( J p a i , − ˜ J f ( p ) λ i E ( a i )) = 1 q λ i ( J p a i , − λ i E ( J p a i ))for i = 1 , . . . , n form an orthonormal basis of T q ( M × ˜ M ). By con-struction, e , . . . , e n span T q Σ, and e n +1 , . . . , e n span N q Σ. In termsof this basis at each point q ∈ Σ t : ∗ Ω = Ω( e , . . . , e n ) = 1 vuut n Y j =1 (1 + λ j ) . The second fundamental form of Σ t is, at each point q ∈ Σ t , character-ized by coefficients(2.9) h ijk = G ( ∇ M × ˜ Me i e j , J e k ) . PROOF COPY NOT FOR DISTRIBUTIONEFORMING SYMPLECTOMORPHISMS 9
Note that h ijk are completely symmetric with respect to i, j, and k .Before we prove Theorem 1, we remark that the long time existence ofthe flow can be proved under more relaxed ambient curvature conditions,but the convergence of the flow does require the more refined propertiesof the curvature of CP n .
3. Proof of Theorem 13.1. Evolution of ∗ Ω along the mean curvature flow. In the restof the paper we prove Theorem 1. We use the following conventionfor indexes: for any index i between 1 and 2 n , i ′ denote the index i + ( − i +1 . For example, 1 ′ = 2 and 2 ′ = 1. Unless otherwise ismentioned, all summation indexes range from 1 to 2 n . Proposition 2.
Let Σ be the graph of a symplectomorphism f :( M, ω ) → ( ˜ M , ˜ ω ) between K¨ahler-Einstein manifolds ( M, g, ω ) and ( ˜ M , ˜ g, ˜ ω ) of real dimension n and of the same scalar curvature. Suppose the meancurvature flow Σ t with Σ = Σ exists smoothly on [0 , t + ǫ ) for some ǫ > and each Σ t is the graph of a symplectomorphism f t : ( M, ω ) → ( ˜ M , ˜ ω ) .At each point q = ( p, f t ( p )) ∈ Σ t , ∗ Ω satisfies the following equation: ddt ∗ Ω =∆ ∗ Ω + ∗ Ω Q ( λ i , h ijk ) + X i,k λ i (1 + λ k )(1 + λ i ) ( R ikik − λ k ˜ R ikik ) , where Q ( λ i , h ijk ) = X i,j,k h ijk − X k X i We recall that Ω is a 2 n form. The notation Ω( e , . . . , J e p ( i ) , . . . , J e q ( j ) , . . . , e n )means that we replace J e p in the i -th position and J e q in the j -th po-sition and similarly in the rest of the paper.We denote A = ∗ Ω( X i,j,k h ijk ) − X p,q,k X i J a p = ( − p +1 a p ′ from (1). Fixing i < j , the termΩ( a , . . . , J a p ( i ) , . . . , J a q ( j ) , . . . , a n )is equal to ( − p +1 ( − q +1 Ω( a , . . . , a p ′ ( i ) , . . . , a q ′ ( j ) , . . . , a n )= ( − i + j ( δ pi ′ δ qj ′ − δ pj ′ δ qi ′ ) , as only those terms with p = i ′ and q = j ′ or p = j ′ and q = i ′ survive.On the other hand, we have q (1 + λ i ) q (1 + λ i ′ ) = λ i . Therefore, X p,q,k X i On the other hand, switching the last two arguments e k and e i in(3.2), using (3.3) again, and applying the skew-symmetry of curvaturetensor, we derive B = X p,k,i Ω( e . . . , J e p ( i ) , . . . , e n ) R ( J e p , e k , e i , e k )= ∗ Ω X p,k,i q λ i q λ p Ω( a . . . , J a p ( i ) , . . . , a n ) R ( J e p , e k , e i , e k )= ∗ Ω X k X i ( − i λ i R ( J e i ′ , e k , e i , e k ) , where we use J a p = ( − p +1 a p ′ and √ (1+ λ i ) q (1+ λ i ′ ) = λ i in the last equality.Denote by R and ˜ R the curvature tensors of M and ˜ M , respectively.We compute by Lemma 1, (2.7), and (2.8), R ( J e i ′ , e k , e i , e k )= R ( π ( J e i ′ ) , π ( e k ) , π ( e i ) , π ( e k )) + ˜ R ( π ( J e i ′ ) , π ( e k ) , π ( e i ) , π ( e k ))= 1(1 + λ k ) q (1 + λ i )(1 + λ i ′ ) [ R ( J a i ′ , a k , a i , a k ) − λ k λ i λ i ′ ˜ R ( ˜ J E ( a i ′ ) , E ( a k ) , E ( a i ) , E ( a k ))]= 1(1 + λ k ) q (1 + λ i )(1 + λ i ′ ) [ R ( J a i ′ , a k , a i , a k ) − λ k ˜ R ( E ( J a i ′ ) , E ( a k ) , E ( a i ) , E ( a k ))]= 1(1 + λ k ) q (1 + λ i )(1 + λ i ′ ) [( − i R ( a i , a k , a i , a k ) − ( − i λ k ˜ R ( E ( a i ) , E ( a k ) , E ( a i ) , E ( a k ))]= ( − i (1 + λ k ) q (1 + λ i )(1 + λ i ′ ) ( R ikik − λ k ˜ R ikik )= ( − i λ i (1 + λ k )(1 + λ i ) ( R ikik − λ k ˜ R ikik ) . q.e.d.The ambient curvature term B can be further simplified when M =˜ M = CP n . Corollary 3. Under the same assumption as in Proposition , if inaddition M and ˜ M are both CP n with the Fubini-Study metric, then: ddt ∗ Ω =∆ ∗ Ω + ∗ Ω " Q ( λ i , h ijk ) + X k odd (1 − λ k ) (1 + λ k ) . PROOF COPY NOT FOR DISTRIBUTION2 IVANA MEDOˇS AND MU-TAO WANG Proof. On CP n with the Fubini-Study metric h· , ·i , the sectional cur-vature is (see for example [ ]): K ( X, Y ) = ( || X ∧ Y || + 3 h J X, Y i ) | X | | Y | − h X, Y i . Therefore, with respect to the chosen orthonormal bases of T x M and T f ( x ) ˜ M , the sectional curvatures K and ˜ K of M and ˜ M are: K ( a i , a i ′ ) = 1 and K ( a r , a s ) = 14 for all other r, s ; and˜ K ( E ( a i ) , E ( a i ′ )) = 1 and ˜ K ( E ( a r ) , E ( a s )) = 14 for all other r, s. Therefore, R ikik = K ( a i , a k ) = 14 (1 + 3 δ ik ′ )and ˜ R ikik = ˜ K ( E ( a i ) , E ( a k )) = 14 (1 + 3 δ ik ′ )for any i, k with i = k .Plugging these into the expression for B , we obtain B = ∗ Ω4 X k X i = k λ i (1 − λ k )(1 + λ k )(1 + λ i ) (1 + 3 δ ik ′ )= ∗ Ω X k λ k ′ (1 − λ k )(1 + λ k )( λ k + λ k ′ ) + ∗ Ω4 X k − λ k λ k X i = k,k ′ λ i λ i + λ i ′ by dividing it into two summands with i = k ′ and i = k ′ . Using λ k λ k ′ =1 and X i = k,k ′ λ i λ i + λ i ′ = X i odd = k,k ′ λ i + λ ′ i λ i + λ i ′ = n − 1, we derive B = ∗ Ω X k − λ k (1 + λ k ) + ( n − ∗ Ω X k − λ k λ k . The second term vanishes as sums with odd k and even k cancel witheach other. Finally, we arrive at: B = ∗ Ω X k − λ k (1 + λ k ) = ∗ Ω X k odd (1 − λ k ) (1 + λ k ) . q.e.d.In this case B ≥ 0, with equality holding if and only if all the singu-lar values of f are equal (and thus necessarily equal to 1). Moreover, (1 − λ k ) (1+ λ k ) < 1, so B < n ( ∗ Ω) ≤ n n . PROOF COPY NOT FOR DISTRIBUTIONEFORMING SYMPLECTOMORPHISMS 13 We notice that Q ( λ i , h ijk ) is a quadratic form in h ijk which can berewritten as Q ( λ i , h ijk ) = X i,j,k h ijk − X k X i odd ( h iik h i ′ i ′ k − h ii ′ k ) − X k X i odd When each λ i = 1 , Q ((1 , . . . , , h ijk ) ≥ (3 − √ || h ijk || where || h ijk || = X i h iii + X i = j h ijj + X i See Appendix. q.e.d. Proposition 3. Let Q ( λ i , h jkl ) be the quadratic form defined in Propo-sition . In each dimension n , there exist Λ > such that Q ( λ i , h jkl ) is non-negative whenever ≤ λ i ≤ Λ for i = 1 , . . . , n . Moreover, forany ≤ Λ < Λ , there exists a δ > such that Q ( λ i , h jkl ) ≥ δ X i,j,k h ijk whenever ≤ λ i ≤ Λ for i = 1 , . . . , n .Proof. Since P i,j,k h ijk ≤ || h ijk || ≤ P i,j,k h ijk , by Lemma 4, Q ((1 , · · · , , h ijk ) ≥ − √ X i,j,k h ijk . Since being a positive definite matrix is an open condition, there is anopen neighborhood U of ( λ , . . . , λ n ) = (1 , · · · , 1) such that ( λ , . . . , λ n ) ∈ U implies Q ( λ i , h ijk ) is positive definite. Let δ ~λ be the smallest eigen-value of Q at ~λ ≡ ( λ , . . . , λ n ). Note that δ ~λ is a continuous functionin ~λ and set δ Λ = min { δ ~λ | ~λ = ( λ , . . . , λ n ) and 1Λ ≤ λ i ≤ Λ for i = 1 , . . . , n } . Λ defined by Λ ≡ sup { Λ | Λ ≥ δ Λ > } has the desired property. q.e.d. PROOF COPY NOT FOR DISTRIBUTION4 IVANA MEDOˇS AND MU-TAO WANG Remark 2. Λ is computable in each dimension n . In particular, Λ = ∞ when n = 1 , and Λ = √ 10 + √ when n = 2 . This canbe checked by dividing Q into smaller quadratic forms and compute theeigenvalues as in the Appendix. Corollary 4. Under the same assumption as in Proposition , sup-pose in addition that M and ˜ M are both CP n with the Fubini-Studymetric. There exist constants Λ > , depending only on n , such thatfor any Λ , ≤ Λ < Λ there exists a δ > with (3.5) (cid:18) ddt − ∆ (cid:19) ∗ Ω ≥ δ ∗ Ω | II | + ∗ Ω X k odd (1 − λ k ) (1 + λ k ) , whenever ≤ λ i ≤ Λ for every i . Here | II | is the norm of the secondfundamental form of Σ t . We recall the norm of the second fundamental form is | II | = s X i,j,k,l G ik G jl G (II( w i , w j ) , II( w k , w l ))= s X i,j,k,l,r,s G ik G jl G rs G ( ∇ M × ˜ Mw i w j , J w r ) G ( ∇ M × ˜ Mw k w l , J w s )with respect to an arbitrary basis w , . . . , w n of T q Σ with G ij = G ( w i , w j )and G ij = ( G ij ) − . By (2.9), | II | = sX i,j,k h ijk for the chosen basis (2.7). Proof. The result follows from Corollary 3 and Proposition 3. q.e.d. Short-time existence of the mean curvature flow in question is guaranteed bygeneral theory of quasilinear parabolic PDE. In order to establish long-time existence and convergence, we shall show that when an appropriatepinching holds initially, then f remains Λ -pinched along the flow, ∗ Ωsatisfies the differential inequality (3.5) along the flow, and min Σ t ∗ Ω isnon-decreasing in time. First we make several preliminary observations.We consider vuut Y i (1 + λ i ) , for λ i > λ i λ i ′ = 1, where i ′ = i + ( − i +1 , i = 1 , . . . , n (in other words, λ k − λ k = 1 for k = 1 , . . . , n ). It can berewritten as: 1 sY i (1 + λ i ) = 1 Y i odd ( λ i + λ i ′ ) . PROOF COPY NOT FOR DISTRIBUTIONEFORMING SYMPLECTOMORPHISMS 15 This expression always has an upper bound: λ i λ i ′ = 1 implies that λ i + λ i ′ ≥ 2. Therefore,(3.6) 1 sY i (1 + λ i ) ≤ n , with equality if and only if λ i = 1 for all i .If λ i ’s are bounded, vuut Y i (1 + λ i ) also has a positive lower bound. Lemma 5. If ≤ λ i ≤ Λ for all i , where Λ > , then: n − ǫ ≤ sY i (1 + λ i ) , where ǫ = n − ) n > .Proof. The function x + x is increasing when x > 1. Therefore if ≤ λ i ≤ Λ for all i , then λ i + λ i ′ ≤ Λ + 1Λ . It follows that 12 n − ǫ ≤ sY i (1 + λ i ) ≤ n , where ǫ = n − ) n . q.e.d.On the other hand, a positive lower bound on vuut Y i (1 + λ i ) impliesa bound on each λ i . Lemma 6. If n − ǫ ≤ vuut Y i (1 + λ i ) , where < ǫ < n , then: ≤ λ i ≤ Λ for all i = 1 , . . . , n , where Λ = n n − ǫ + r(cid:16) n n − ǫ (cid:17) − > . PROOF COPY NOT FOR DISTRIBUTION6 IVANA MEDOˇS AND MU-TAO WANG Proof. If 12 n − ǫ ≤ sY i (1 + λ i ) = 1 Y i odd ( λ i + λ i ′ ) , then Y i odd ( λ i + λ i ′ ) ≤ n − n ǫ and λ i + λ i ′ ≤ n (1 − n ǫ ) Y j = i,j odd ( λ j + λ j ′ )for each i .Since λ j + λ j ′ ≥ j , the inequality implies λ i + λ i ′ ≤ n n − ǫ Since λ i λ i ′ = 1, it follows that:1Λ ≤ λ i ≤ Λwhere Λ = n n − ǫ + r(cid:16) n n − ǫ (cid:17) − 1. q.e.d.After these algebraic preliminaries, we return to the mean curvatureflow. Recall that f is Λ-pinched in the sense of Definition 1, if ≤ λ i ≤ Λ at each point p ∈ M in which λ i ’s are the singular values of Df p asin section 2.2. Proposition 4. Let Σ t be the mean curvature flow of the graph Σ of a symplectomorphism f : M → ˜ M where M = ˜ M = CP n with theFubini-Study metric. Suppose Σ t exists smoothly on [0 , T ) for some T > . Let ∗ Ω be the Jacobian of the projection π : Σ t → M . Let Λ be the constants characterized by Proposition .If ∗ Ω has the initial lower bound: n − ǫ ≤ ∗ Ω for ǫ = n (cid:18) − ′ + ′ (cid:19) for some < Λ ′ < Λ , then min Σ t ∗ Ω is nonde-creasing as a function in t . In particular, Σ t is the graph of a symplec-tomorphism f t : M → ˜ M . PROOF COPY NOT FOR DISTRIBUTIONEFORMING SYMPLECTOMORPHISMS 17 Proof. If initially n − ǫ ≤ ∗ Ω for ǫ = n (cid:18) − ′ + ′ (cid:19) . We computethat n n − ǫ = Λ ′ + ′ . Thus, by Lemma 6, f is Λ ′ -pinched. That in turnimplies that ∗ Ω initially satisfies inequality (3.5), and in particular,(3.7) (cid:18) ddt − ∆ (cid:19) ∗ Ω ≥ ∗ Ω X k odd (1 − λ k ) (1 + λ k ) . Thus ∗ Ω > n − ǫ for some [0 , T ′ ) with T ′ < T .Suppose at T ′ , ∗ Ω = n − ǫ for the first time after t = 0. But in [0 , T ′ ),we have ∗ Ω > n − ǫ and thus f is Λ ′ -pinched and inequality (3.7) issatisfied again. Since the right hand side of (3.7) is strictly positiveunless ∗ Ω = n , min Σ t ∗ Ω is non-decreasing in time by the maximumprinciple. q.e.d. Corollary 5. Under the same assumption as in Proposition , if theinitial symplectomorphism f is Λ -pinched, for Λ = (cid:20) (cid:18) Λ + 1Λ (cid:19)(cid:21) n + s(cid:20) (cid:18) Λ + 1Λ (cid:19)(cid:21) n − < Λ , then each f t is Λ -pinched along the mean curvature flow.Proof. The proof consists of only algebraic manipulation and thereis no need to apply the maximum principle again. We need a simplealgebraic formula which can be easily verified: for x > , y > x + p x − y if and only if x = y + y − . By the definition of Λ ,(3.9) 12 (cid:18) Λ + 1Λ (cid:19) = (cid:18) (cid:18) Λ + 1Λ (cid:19)(cid:19) n which is less than (cid:16) Λ + (cid:17) because Λ + > 2. Since Λ > > 1, it follows that Λ < Λ .Now suppose f is initially Λ -pinched, by Lemma 5, ∗ Ω has initiallower bound: 12 n − ǫ ≤ ∗ Ωfor(3.10) ǫ = 12 n − + ) n . PROOF COPY NOT FOR DISTRIBUTION8 IVANA MEDOˇS AND MU-TAO WANG Then, by Proposition 4, the lower bound of ∗ Ω remains true along theflow. Lemma 6 then implies that f is Λ ′ -pinched along the flow for(3.11) Λ ′ = n n − ǫ + vuut n n − ǫ ! − . We claim that with the given Λ and ǫ given by (3.10), Λ ′ is exactly Λ .In fact, from (3.11) and (3.8), we obtain n n − ǫ = 12 (cid:18) Λ ′ + 1Λ ′ (cid:19) . On the other hand from (3.10), we solve n n − ǫ = (cid:16) (cid:16) Λ + (cid:17)(cid:17) n = (cid:16) Λ + (cid:17) by (3.9). Therefore f is Λ pinched along the flow. q.e.d.We believed that the constant Λ can be further improved by consid-ering the evolution equation of λ i directly. In this article, we find thatthe evolution equation of ∗ Ω is sufficient to yield the desired constant,albeit not an optimal one.In Theorem 1, we choose a Λ that is slightly less than Λ in Corollary5, then f t will Λ ′ pinched along the flow for some Λ ′ < Λ and thus byCorollary 4, we have (3.5) all the way along the flow. We shall see thatthis is enough for the long time existence and convergence. We assume M = ˜ M = CP n . To prove long-time existence of the flow, we follow themethod in [ ]. We isometrically embed M × ˜ M into R N . The meancurvature flow equation in terms of the coordinate function F ( x, t ) ∈ R N is: ddt F ( x, t ) = H = ¯ H + V, where H ∈ T ( M × ˜ M ) /T Σ t is the mean curvature vector of Σ t in M , ¯ H ∈ T R N /T Σ t is the mean curvature vector of Σ t in R N , and V = − P a II M × ˜ M ( e a , e a ) where { e a } a =1 ··· n is an orthonormal basis of T Σ t . In the following calculation, the index a is summed from 1 to 2 n , H = π M × ˜ MN Σ ( ∇ M × ˜ Me a e a ) = ∇ M × ˜ Me a e a − ∇ Σ e a e a = ∇ R N e a e a − π R N N ( M × ˜ M ) ( ∇ R N e a e a ) − ∇ Σ e a e a = ∇ R N e a e a − ∇ Σ e a e a + V = π R N N Σ ( ∇ Σ e a e a ) + V = ¯ H + V. Note that V is bounded since both M and ˜ M are compact. PROOF COPY NOT FOR DISTRIBUTIONEFORMING SYMPLECTOMORPHISMS 19 Following [ ], we assume that there is a singularity at space timepoint ( y , t ) ∈ R N × R . Consider the backward heat kernel of Huisken ρ y ,t at ( y , t ): ρ y ,t ( y, t ) = 14 π ( t − t ) n exp (cid:18) −| y − y | t − t ) (cid:19) . Let dµ t denote the volume form of Σ t . By Huisken’s monotonicityformula [ ], lim t → t Z ρ y ,t dµ t exists. Lemma 7. The limit lim t → t Z (1 − ∗ Ω) ρ y ,t dµ t exists and: ddt Z (1 − ∗ Ω) ρ y ,t dµ t ≤ C − δ Z ∗ Ω | II | ρ y ,t dµ t for some constant C > .Proof. By [ ]: ddt ρ y ,t = − ∆ ρ y ,t − ρ y ,t (cid:18) | F ⊥ | t − t ) + F ⊥ · ¯ Ht − t + F ⊥ · V t − t ) (cid:19) where F ⊥ ∈ T R N /T Σ t is the orthogonal component of F ∈ T R N .By [ ]: ddt dµ t = −| H | dµ t = − ¯ H · ( ¯ H + V ) dµ t . Combining these results, we obtain: ddt Z (1 − ∗ Ω) ρ y ,t dµ t ≤ Z [∆(1 − ∗ Ω) − δ ∗ Ω | II | ] ρ y ,t dµ t − Z (1 − ∗ Ω) (cid:20) ∆ ρ y ,t + ρ y ,t (cid:18) | F ⊥ | t − t ) + F ⊥ · ¯ Ht − t + F ⊥ · V t − t ) (cid:19)(cid:21) − Z (1 − ∗ Ω)[ ¯ H · ( ¯ H + V )] ρ y ,t dµ t = Z [∆(1 − ∗ Ω) ρ y ,t − (1 − ∗ Ω)∆ ρ y ,t ] dµ t − δ Z ∗ Ω | II | ρ y ,t dµ t − Z (1 − ∗ Ω) ρ y ,t (cid:20)(cid:18) | F ⊥ | t − t ) + F ⊥ · ¯ Ht − t + F ⊥ · V t − t ) (cid:19) + | ¯ H | + ¯ H · V (cid:21) dµ t = − δ Z ∗ Ω | II | ρ y ,t dµ t − Z (1 − ∗ Ω) ρ y ,t (cid:12)(cid:12)(cid:12)(cid:12) F ⊥ t − t ) + ¯ H + V (cid:12)(cid:12)(cid:12)(cid:12) dµ t + Z (1 − ∗ Ω) ρ y ,t (cid:12)(cid:12)(cid:12)(cid:12) V (cid:12)(cid:12)(cid:12)(cid:12) dµ t . PROOF COPY NOT FOR DISTRIBUTION0 IVANA MEDOˇS AND MU-TAO WANG Since V is bounded, and since R (1 − ∗ Ω) ρ y ,t dµ t ≤ R ρ ( y ,t ) dµ t < ∞ ,it follows that: ddt Z (1 − ∗ Ω) ρ y ,t dµ t ≤ C − δ Z ∗ Ω | II | ρ y ,t dµ t for some constant C . Now F ( t ) = R (1 − ∗ Ω) ρ y ,t dµ t is non-negativeand F ′ ( t ) ≤ C , or F ( t ) − Ct is non-increasing in t ∈ [0 , t ). From thisit follows that the limit as t → t exists. q.e.d.For ν > 1, the parabolic dilation D ν at ( y , t ) is defined by: D ν : R N × [0 , t ) → R N × [ − ν t , , ( y, t ) ( ν ( y − y ) , ν ( t − t )) . Let S ⊂ R N × [0 , t ) be the total space of the mean curvature flow,and let S ν ≡ D ν ( S ) ⊂ R N × [ − ν t , s denotes the new timeparameter, then t = t + sν .Let dµ νs be the induced volume form on Σ by F νs ≡ νF t + sν . Theimage of F νs is the s − slice of S ν , denoted Σ νs . Remark 3. Note that: Z (1 − ∗ Ω) ρ y ,t dµ t = Z (1 − ∗ Ω) ρ , dµ νs because ∗ Ω and ρ y ,t dµ t are invariant under parabolic dilation. Lemma 8. For any τ > : lim ν →∞ Z − − − τ Z ∗ Ω | II | ρ , dµ νs ds = 0 . Proof. From Remark 3: dds Z (1 − ∗ Ω) ρ , dµ νs = 1 ν ddt Z (1 − ∗ Ω) ρ y ,t dµ t . Then by Lemma 7: dds Z (1 − ∗ Ω) ρ , dµ νs ≤ Cν − δν Z ∗ Ω | II | ρ y ,t dµ t for some constant C . But ν R ∗ Ω | II | ρ y ,t dµ t = R ∗ Ω | II | ρ , dµ νs sincethe norm of the second fundamental form scales like the inverse of thedistance, so: dds Z (1 − ∗ Ω) ρ , dµ νs ≤ Cν − δ Z ∗ Ω | II | ρ , dµ νs . Integrating this inequality with respect to s from − − τ to − 1, weobtain: δ Z − − − τ Z ∗ Ω | II | ρ , dµ νs ds ≤ − Z (1 −∗ Ω) ρ , dµ ν − + Z (1 −∗ Ω) ρ , dµ ν − − τ + Cν . PROOF COPY NOT FOR DISTRIBUTIONEFORMING SYMPLECTOMORPHISMS 21 By Remark 3 and the fact that lim t → t Z (1 −∗ Ω) ρ y ,t dµ t exists (Lemma 7),the right-hand side of the inequality above approaches zero as ν → ∞ .q.e.d.We take a sequence ν j → ∞ . Then for a fixed τ : Z − − − τ Z ∗ Ω | II | ρ , dµ ν j s ds ≤ C ( j )where C ( j ) → τ j → C ( j ) τ j → 0, and s j ∈ [ − − τ j , − 1] so that(3.12) Z ∗ Ω | II | ρ , dµ ν j s j ≤ C ( j ) τ j . Observe that ρ , ( F ν j s j , s j ) = 1(4 π ( − s j ) ) n exp −| F ν j s j | − s j ) ! . When j is large enough, we may assume that τ j ≤ 1, and thus that s j ∈ [ − , − R > B R (0) ∈ R N ,we have: Z ∗ Ω | II | ρ , dµ ν j s j ≥ C ′ Z Σ νjsj ∩ B R (0) ∗ Ω | II | dµ ν j s j for a constant C ′ > 0, since s j are bounded and since | F ν j s j | ≤ R onΣ ν j s j ∩ B R (0).Then by inequality (3.12) and the fact that ∗ Ω has a positive lowerbound, we conclude the following result. Lemma 9. For any compact set K ⊂ R N : Z Σ νjsj ∩K | II | dµ ν j s j → as j → ∞ . Then, as shown in [ ], it follows thatlim t → t Z ρ y ,t dµ t ≤ . Finally, White’s theorem [ ] implies that ( y , t ) is a regular pointwhenever lim t → t Z ρ y ,t dµ t ≤ ǫ, contradicting the initial assumption that ( y , t ) is a singular point. PROOF COPY NOT FOR DISTRIBUTION2 IVANA MEDOˇS AND MU-TAO WANG In the precedingsections we have shown that the mean curvature flow Σ t of the graph ofsymplectomorphism f : CP n → CP n exists smoothly for all t > 0, andthat Σ t is a graph of symplectomorphisms for each t under the pinchingcondition. We conclude the proof of Theorem 1 by showing that Σ t converges to the graph of a biholomorphic isometry.By Proposition 2: (cid:18) ddt − ∆ (cid:19) ∗ Ω = ∗ Ω " Q ( λ i , h jkl ) + X k odd (1 − λ k ) (1 + λ k ) along the mean curvature flow, where Q ≥ ≤ λ i ≤ Λ .We use this result to derive the evolution equation of ln ∗ Ω, which wethen apply to show that lim t →∞ ∗ Ω = 12 n . Proposition 5. Under the same assumption as in Proposition , ateach point q ∈ Σ t , ln ∗ Ω satisfies the following equation: ddt ln ∗ Ω =∆ ln ∗ Ω + Q ( λ i , h jkl ) + X k X i = k λ i (1 + λ k )( λ i + λ i ′ ) ( R ikik − λ k ˜ R ikik ) , where R ijkl and ˜ R ijkl are the coefficients of the curvature tensors of M and ˜ M with respect to the chosen bases ( ) and ( ), i ′ = i + ( − i +1 ,and (3.13) Q ( λ i , h jkl ) = Q ( λ i , h jkl ) + X k " X i odd ( λ i − λ i ′ ) h ii ′ k with Q ( λ i , h jkl ) given by Proposition and equation ( ).Proof. We compute ddt ln ∗ Ω = 1 ∗ Ω ddt ∗ Ω and ∆(ln ∗ Ω) = ∗ Ω∆( ∗ Ω) − |∇ ∗ Ω | ( ∗ Ω) . By Proposition 2, it follows that (cid:18) ddt − ∆ (cid:19) ln ∗ Ω = Q ( λ i , h jkl )+ X k X i = k λ i (1 + λ k )( λ i + λ i ′ ) ( R ikik − λ k ˜ R ikik )+ |∇ ∗ Ω | ( ∗ Ω) . We compute( ∗ Ω) k = X i Ω( e , . . . , ( ∇ M × ˜ Me k − ∇ Σ e k ) e i , . . . , e n )= X i Ω( e , . . . , h∇ M × ˜ Me k e i , J e p iJ e p , . . . , e n )= X p,i Ω( e , . . . , J e p , . . . , e n ) h pik PROOF COPY NOT FOR DISTRIBUTIONEFORMING SYMPLECTOMORPHISMS 23 As the simplification of the expression A in the proof of Proposition2, we obtain( ∗ Ω) k = ∗ Ω X i ( − i λ i h ii ′ k = − ∗ Ω X i odd ( λ i − λ i ′ ) h ii ′ k . It follows that: |∇ ∗ Ω | ( ∗ Ω) = X k " X i odd ( λ i − λ i ′ ) h ii ′ k , and thus (cid:18) ddt − ∆ (cid:19) ln ∗ Ω = Q ( λ i , h jkl )+ X k X i = k λ i (1 + λ k )( λ i + λ i ′ ) ( R ikik − λ k ˜ R ikik ) , where Q ( λ i , h jkl ) = Q ( λ i , h jkl ) + X k " X i odd ( λ i − λ i ′ ) h ii ′ k is a new qua-dratic form in h ijk , with coefficients depending on the singular valuesof f . q.e.d. Corollary 6. Under the same assumption as in Proposition , sup-pose in addition that M and ˜ M are both CP n with the Fubini-Studymetric, then: ddt ln ∗ Ω =∆ ln ∗ Ω + Q ( λ i , h ijk ) + X k odd (1 − λ k ) (1 + λ k ) . Proof. This is a direct consequence of Proposition 5 and Corollary 3.q.e.d. Remark 4. Q is a positive definite quadratic form of h ijk whenever Q is, and in fact it allows for an improvement of the pinching constant. We use the evolution equation of ln ∗ Ω to show that lim t →∞ ∗ Ω = 12 n .Fix a k and notice that(1 − λ k ) (1 + λ k ) = ( λ k − λ k ′ ) ( λ k + λ k ′ ) = x − x , where x = ( λ k + λ k ′ ) .Since λ k λ k ′ = 1, it follows that λ k + λ k ′ ≥ 2, and thus x ≥ x ≤ (cid:16) Λ + (cid:17) .We claim x − x ≥ c (cid:18) 12 ln x − ln 2 (cid:19) for c = + ) . PROOF COPY NOT FOR DISTRIBUTION4 IVANA MEDOˇS AND MU-TAO WANG To see this, let f ( x ) = x − x , g ( x ) = c (cid:0) ln x − ln 2 (cid:1) and notice that f (4) = g (4) = 0. We compute f ′ ( x ) = x − x + 4 x = 4 x and g ′ ( x ) = c x . Thus f ′ ( x ) g ′ ( x ) = 4 x xc = 8 cx ≥ . The last inequality follows from the choice of c and the fact that x ≤ (cid:16) Λ + (cid:17) . Now since f (4) = g (4) and f ′ ( x ) ≥ g ′ ( x ) for 4 ≤ x ≤ (cid:16) Λ + (cid:17) , it follows that f ( x ) ≥ g ( x ).Substituting back, we obtain( λ k − λ k ′ ) ( λ k + λ k ′ ) ≥ c (ln( λ k + λ k ′ ) − ln 2) , and thus X k odd (1 − λ k ) (1 + λ k ) = X k odd ( λ k − λ k ′ ) ( λ k + λ k ′ ) ≥ c − ln Y k odd λ k + λ k ′ − n ln 2 ! = − c (cid:18) ln ∗ Ω − ln 12 n (cid:19) . Therefore under the pinching condition: (cid:18) ddt − ∆ (cid:19) (cid:18) ln ∗ Ω − ln 12 n (cid:19) ≥ − c (cid:18) ln ∗ Ω − ln 12 n (cid:19) . The pinching condition holds along the mean curvature flow, so thisholds for all times. By the comparison principle for parabolic equations,lim t →∞ min Σ t ln ∗ Ω − ln 12 n = 0, and thus lim t →∞ min Σ t ∗ Ω = 12 n . This in turnimplies, by Lemma 6, that λ i → t → ∞ for all i .For the rest of the proof, we modify the method from [ ] to showthe second fundamental form is uniformly bounded in time. Let ǫ > η ǫ = ∗ Ω − n + ǫ . Note that min Σ t η ǫ is nondecreasing, and η ǫ → ǫ when t → ∞ . Let T ǫ ≥ η ǫ | T ǫ > t ≥ T ǫ : η ǫ > p ∈ M , and all t > T ǫ : ddt η ǫ = ∆ η ǫ + ∗ Ω( Q + B ) ≥ ∆ η ǫ + δ ∗ Ω | II | = ∆ η ǫ + δη ǫ η ǫ ∗ Ω | II | . PROOF COPY NOT FOR DISTRIBUTIONEFORMING SYMPLECTOMORPHISMS 25 On the other hand, from [ ], | II | satisfies the following equationalong the mean curvature flow: ddt | II | = ∆ | II | − |∇ II | + [( ∇ M∂ k ) R ( J e p , e i , e j , e k ) + ( ∇ M∂ j R )( J e p , e k , e i , e k )] h pij − R ( e l , e i , e j , e k ) h plk h pij + 4 R ( J e p , J e q , e j , e k ) h qik h pij − R ( e l , e k , e i , e k ) h plj h pij + R ( J e p , e k , J e q , e k ) h qij h pij + X p,r,i,m ( X k h pik h rmk − h pmk h rik ) + X i,j,m,k ( X p h pij h pmk ) . Since M × ˜ M is a symmetric space, the curvature tensor R of M × ˜ M is parallel, and thus | II | satisfies: ddt | II | ≤ ∆ | II | − |∇ II | + K | II | + K | II | for positive constants K and K that depend only on n .Therefore: ddt ( η − ǫ | II | ) ≤ − η − ǫ | II | (∆ η ǫ + δ ∗ Ω | II | ) + η − ǫ (∆ | II | − |∇ II | + K | II | + K | II | )= − η − ǫ ∆ η ǫ | II | + η − ǫ ∆ | II | − η − ǫ |∇ II | + η − ǫ ( η ǫ K − δ ∗ Ω) | II | + η − ǫ K | II | = ∆( η − ǫ ) | II | − η − ǫ |∇ η ǫ | | II | + η − ǫ ∆ | II | − η − ǫ |∇ II | + η − ǫ ( η ǫ K − δ ∗ Ω) | II | + η − ǫ K | A | = ∆( η − ǫ ) | II | − η ǫ |∇ ( η − ǫ ) | | II | + η − ǫ ∆ | II | − η − ǫ |∇ II | + η − ǫ ( η ǫ K − δ ∗ Ω) | II | + η − ǫ K | II | = ∆( η − ǫ | II | ) − ∇ ( η − ǫ ) · ∇ ( | II | ) − η ǫ |∇ ( η − ǫ ) | | II | − η − ǫ |∇ II | + η − ǫ ( η ǫ K − δ ∗ Ω) | II | + η − ǫ K | II | . We apply the relation that − ∇ ( η − ǫ ) · ∇ ( | II | ) − η ǫ |∇ ( η − ǫ ) | | II | = − η ǫ ∇ ( η − ǫ ) · ∇ ( η − ǫ | II ) . Therefore the function ψ = η − ǫ | II | satisfies: ddt ψ ≤ ∆ ψ − η ǫ ∇ η − ǫ · ∇ ψ + ( η ǫ K − δ ∗ Ω) ψ + K ψ ≤ ∆ ψ − η ǫ ∇ η − ǫ · ∇ ψ + ( ǫK − δC ) ψ + K ψ, where C = min Σ ∗ Ω, since min Σ t ∗ Ω is nondecreasing and η ǫ ≤ ǫ . ǫ can bechosen small enough so that ǫK − δC < 0. Then by the comparisonprinciple for parabolic PDE, ψ ≤ y ( t ) for all t ≥ T ǫ , where y ( t ) is thesolution of the ODE ddt y = − ( δC − ǫK ) y + K y PROOF COPY NOT FOR DISTRIBUTION6 IVANA MEDOˇS AND MU-TAO WANG satisfying the initial condition y ( T ǫ ) = max Σ Tǫ ψ . y ( t ) can be solved ex-plicitly: y ( t ) = K δC − ǫK , if max Σ Tǫ ψ = K δC − ǫK K Ke K t ( δC − ǫK ) Ke K t − , otherwise , where K is a constant satisfying K > Σ Tǫ ψ > K δC − ǫK , and K < Σ Tǫ ψ < K δC − ǫK . Thus | II | ≤ η ǫ y ( t ) ≤ ǫy ( t )for all t ≥ T ǫ .Sending t → ∞ and ǫ → 0, we conclude that max Σ t | II | → t → ∞ .Finally, the induced metric and the volume functional both have analyticdependence on F , so by Simon’s theorem [ ] the flow converges to aunique limit at infinity.Since λ i → i as t → ∞ , the limit map is an isometry. Denoteit by f ∞ . Being symplectic is a closed property, so f ∞ is symplectic.Then at every p ∈ M : Df ∞ J = ˜ J Df ∞ The same is true for the inverse of f ∞ , and thus the map f ∞ is biholo-morphic. 4. Appendix4.1. Proof of Lemma 4. We recall that h ijk is symmetric in all threeindexes, that all indexes range from 1 to 2 n unless otherwise (such as i odd) is mentioned, and that i ′ = i + ( − i +1 . The object of study isthe quadratic form ˜ Q ( h ijk ) given by X i,j,k h ijk − X k X i odd ( h iik h i ′ i ′ k − h ii ′ k ) + 4 X k X i odd Lemma 10. The three summands of ˜ Q in ( ) can be rewritten inthe following way: A = X i h iii + 3 X i odd ( h ii ′ i ′ + h i ′ ii )+ 3 X i odd For each odd i , the expression in ˜ Q can be further dividedinto two identical quadratic forms of two variables, each has smallesteigenvalue 3 − √ 5. For each index ( i, j ) with i odd < j odd, the expres-sion in ˜ Q can be further divided into four identical quadratic forms ofthree variables, each has smallest eigenvalue 2. For each index ( i, j, k )with i odd < j odd < k odd, the expression in ˜ Q can be further dividedinto two identical quadratic forms of four variables, each has smallesteigenvalue 4. q.e.d.First of all,(4.2) A = X i h iii + 3 X i In the first summand, it is possible that j equals i ′ , thus(4.3) X i Therefore, X i Minimal Lagrangian diffeomorphisms between domains in the hyper-bolic plane. J. Differential Geom. (2008), 1–20, MR2434257, Zbl 1154.53034.[2] M. Gromov, Pseudoholomorphic curves in symplectic manifolds. Invent. Math. (1985), 307–347, MR0809718, Zbl 0592.53025.[3] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow. J.Differential Geom. (1990), no. 1, 285–299, MR1030675, Zbl 0694.53005.[4] H. B. Lawson, J. Simons, On stable curves and their application to global prob-lems in real and complex geometry Ann. Math. (1973), no. 3, 427–450,MR0324529, Zbl 0283.53049.[5] Y.-I. Lee, Lagrangian minimal surfaces in K¨ahler-Einstein surfaces of negativescalar curvature. Comm. Anal. Geom. (1994), no. 4, 579–592, MR1336896, Zbl0843.58024.[6] D. McDuff, D. Salomon, Introduction to symplectic topology. Oxford Mathemati-cal Monographs. Oxford Science Publications. The Clarendon Press, Oxford Uni-versity Press, New York, 1995. viii+425 pp. ISBN: 0-19-851177-9, MR1373431,Zbl 0844.58029.[7] R. Schoen, The role of harmonic mappings in rigidity and deformation problems. Complex geometry (Osaka, 1990), 179–200, Lecture Notes in Pure and Appl.Math., 143, Dekker, New York, 1993, MR1201611, Zbl 0806.58013.[8] L. Simon, Asymptotics for a class of nonlinear evolution equations, with appli-cations to geometric problems. Ann. of Math. (2) (1983), no. 3, 525–571,MR0727703, Zbl 0549.35071.[9] S. Smale, Diffeomorphisms of the 2-sphere. Proc. Amer. Math. Soc. (1959),621–626, MR0112149, Zbl 0118.3910. PROOF COPY NOT FOR DISTRIBUTION2 IVANA MEDOˇS AND MU-TAO WANG [10] K. Smoczyk, A canonical way to deform a Lagrangian submanifold. preprint,dg-ga/9605005.[11] K. Smoczyk, Angle theorems for the Lagrangian mean curvature flow. Math. Z. (2002), no. 4, 849–883, MR1922733, Zbl 1020.53045.[12] M.-T. Wang, Deforming area preserving diffeomorphism of surfaces by meancurvature flow. Math. Res. Lett. (2001), no.5-6, 651-662, MR1879809, Zbl1081.53056.[13] M.-T. Wang, Long-time existence and convergence of graphic mean curva-ture flow in arbitrary codimension. Invent. Math. (2002), no. 3, 525–543,MR1908059, Zbl 1039.53072.[14] M.-T. Wang, Mean curvature flow in higher codimenison. Second InternationalCongress of Chinese Mathematicians, 275–283, New Stud. Adv. Math., 4, Int.Press, Somerville, MA, 2004, MR2497990.[15] M.-T. Wang, Mean curvature flow of surfaces in Einstein four-manifolds. J.Differential Geom. (2001), no. 2, 301–338, MR1879229, Zbl 1035.53094.[16] M.-T. Wang, A convergence result of the Lagrangian mean curvature flow. Third International Congress of Chinese Mathematicians. Part 1, 2, 291–295,AMS/IP Stud. Adv. Math., 42, pt. 1, 2, Amer. Math. Soc., Providence, RI,2008, MR2409639, Zbl 1170.53046.[17] B. White, A local regularity theorem for classical mean curvature flow. Ann. ofMath. (2) (2005), no. 3, 1487–1519, MR2180405, Zbl 1091.53045.[18] J. Wolfson, Minimal Lagrangian diffeomorphisms and the Monge-Amp`ere equa-tion. J. Differential Geom. (1997), 335–373, MR1484047, Zbl 0926.53032.(1997), 335–373, MR1484047, Zbl 0926.53032.