用平均曲率流形变某些紧型不可约埃尔米特对称空间的辛微分同胚

用平均曲率流形变某些紧型不可约埃尔米特对称空间的辛微分同胚 用平均曲率流形变某些紧型不可约埃尔米特 对称空间的辛微分同胚 卢广存 , 肖邦 北京师范大学数学科学学院,北京100875 摘要:本文推广Medos-Wang [1]中关于复投影空间上辛微分同胚的平均曲率流形变结论与证明 到一类包含复格罗斯曼流形及其紧全测地凯勒-爱因斯坦子流形的紧型不可约埃尔米特对称空 间上. 关键词:辛几何，凯勒-爱因斯坦流形，辛微分同胚，平均曲率流. 中图分类号: O186.16. Deforming symplectomorphism of certain irreducible Hermitian symmetric spaces of compact type by mean curvature ,ow Lu Guang-Cun , Xiao Bang School of Mathematical Sciences, Beijing Normal University, Beijing 100875 Abstract: In this paper, we generalize Medos-Wang’s arguments and results on the mean curvature ,ow deformations of symplectomorphisms of the complex projective spaces in [1] to a class of irreducible Hermitian symmetric spaces of compact type including complex Grassmann manifolds and their compact totally geodesic K?ahler-Einstein submanifolds. Key words: Symplectic geometry, Kahler-Einstein manifold, symplectomorphism, mean curvature ,ow. 0 Introduction Recall that a symplectic manifold (M, ω) is said to be K?ahler if there exists an integrable almost complex structure J on M such that the bilinear form g(X, Y ) = X, Y := ω(X, JY ) de,nes a Riemannian metric on M. The triple (ω, J, g) is called a K?ahler structure on M, g and ω are called a K?ahler metric and a K?ahler form, respectively. Such a K?ahler manifold is called a K?ahler-Einstein manifold if the Ricci form ρω ? ρg of g satis,es ρω = cω for some constant c ? R. Following [2] the curvature tensor R of a K?ahler manifold (M, ω, J, g) is de,ned by R(X, Y, Z, W ) = g(R(X, Y )W, Z) = g(R(Z, W )Y, X) 基 金 项 目: NNSF(10971014) , Research Fund for the Doctoral Program Higher Education of China (200800270003) and Program for Changjiang Scholars and Innovative Research Team in University of China. 作 者 简 介: Correspondence author: Lu Guangcun (1964-)， male， professor， major research direction: Symplectic Geometry and Nonlinear Analysis. -1- for X, Y, Z, W ? Γ(T M). Then the holomorphic sectional curvature in the direction X ? T M \ {0} is de,ned by H(X) = R(X, JX, X, JX)/[g(X, X)]2. (After extending g and R by C- ? linearity to T M ?R C, H(X) is equal to ?R(Z, Z, Z, Z)/[g(Z, Z)]2 for Z = (X ? ?1JX)/2 ? T (1,0)M). For a K?ahler manifold (M, J, g, ω) let Symp(M, ω) and Aut(M, J) denote the group of symplectomorphisms of the symplectic manifold (M, ω) and the group of biholomorphisms of the complex manifold (M, J), respectively. Their intersection is equal to the group of isometries of the K?ahler manifold (M, J, g, ω), I(M, J, g) := {φ ? Aut(M, J) | φ?g = g}. A K?ahler manifold (M, ω, J, g) is called homogeneous if I(M, J, g) acts transitively on M. Without special statements we always assume that M is closed (i.e. compact and bound- aryless) and connected throughout this paper. It is well-known that Symp(M, ω) is an in,nite dimensional Lie group whose Lie algebra is the space of symplectic vector ,elds. A lot of symplectic topology information of (M, ω) is contained in Symp(M, ω). (See beautiful books [3, 4, 5, 6] for detailed study). On the other hand I(M, J, g) is a ,nite dimensional Lie sub- group of Symp(M, ω). Hence in order to understand topology of Symp(M, ω), e.g. its homotopy groups, it is helpful to study the topology properties of the inclusion I(M, J, g) ? Symp(M, ω). To the author’s knowledge, the ,rst result in this direction was obtained by Smale [7], who (1) proved that there exists a continuous strong deformation retraction from Symp(S2, ωFS ) to (1) (n) (n) SO(3) = I(S2, i, gFS ). Hereafter gFS and ωFS denote, up to multiplying a positive number, the Fubini-Study metric and the associated K?ahler form on the complex projective spaces CP n respectively, and i is the standard complex structure on CP n. Recently, Jiayong Li and Jordan Alan Watts [8] strengthened this result. They constructed a strong deformation retraction from (1) Symp(S2, ωFS ) to SO(3) which is di,eologically smooth. In his famous paper [9] Gromov invented a powerful pseudo-holomorphic curve theory to study symplectic topology and got the following important results: ? For any two area forms ω1 and ω2 on CP 1 with CP 1 ω1 = CP 1 ω2, Symp(CP 1 ×CP 1, ω1 ?ω2) contracts onto (1) (1) I(CP 1 × CP 1, i × i, gFS ? gFS ) = Z/2Z extension of SO(3) × SO(3) ([9, ?2.4.A1]), and Symp(CP 1 × CP 1, ω1 ? ω2) cannot contract onto SO(3) × SO(3) if CP 1 ω1 = ω2 ([9, ?2.4.C2]). (A simple application of Moser theorem can reduce these to the case CP 1 (1) (1) ω1 = aωFS and ω2 = bωFS for nonzero a, b ? R). (2) (2) ? Symp(CP 2, ωFS ) contracts onto I(CP 2, i, gFS ) ([9, ?2.4.B3]). (1) Since Symp(S2, ωFS ) = Di, +(S2), Gromov’s results may be viewed as generalizations of (1) (1) (1) Smale’s theorem above in a direction. For Symp(S2 × S2, ωFS ? λωFS ) with S2 ωFS = 1 and λ = 1, so far some deep results were made by Abreu [10], Abreu and McDu, [11], Anjos and Granja [12] and others following an approach suggested by Gromov [9, ?2.4.C2]. A di,erent direction generalization of Smale’s theorem is to study the topology properties of the inclusion Symp(M, ω) ? Di,(M, ω). Using the parameterized Gromov-Witten invariant theory, Lˆe and Ono [13], and Seidel [14] got a few of interesting results in this direction. For example, (m) (n) with Pmn := CP m × CP n and CP 1 ωFS = CP 1 ωFS = 1, it was showed in [14] that the homomorphisms (m) (n) βk : πk(Symp(CP m × CP n, ωFS ? ωFS )) ? πk(Di,(Pmn)) -2- (m) (n) induced by the natural inclusion Symp(CP m ×CP n, ωFS ?ωFS ) ? Di,(Pmn) are not surjective for odd numbers k ? max{2m?1, 2n?1}. This gave the ,rst examples of symplectic manifolds of dimension > 4 for which the map π1(Symp) ? π1(Di,) is not surjective. In particular, this result implies that rank(cokerβ1) ? 2 for m = n = 1, which can also be derived from the above Gromov’s ,rst result. (See McDu,’s survey [15] for recent developments). In past ten years a new method (mean curvature ,ow (MCF) method) to the above question was developed by Mu-Tao Wang [1, 16, 17, 18, 19, 20, 21] and Smoczyk [22]. For compact Riemann surfaces they obtained the desired results (cf. [20, 21, 22]). Recently Ivana Medos and Mu-Tao Wang [1] applied the MCF to deform symplectomorphisms of CP n for each dimension n, and obtained a constant Λ0(n) ? (1, +?] only depending on n ? N, (see (9) for its de,nition), such that any Λ-pinched symplectomorphism of CP n with 1 2 1 1 1 1 n n ?1 + (1) 1 ? Λ ? Λ1(n) := Λ0(n) + Λ0(n) + 2 2 Λ0(n) Λ0(n) is symplectically isotopic to a biholomorphic isometry (cf.[1, Corollay 5]). Here a symplecto- morphism ? of the K?ahler manifold (M, ω, J, g) is called Λ-pinched if 1 g ? ??g ? Λ2g Λ2 (cf. [1, Def.1]). The constant Λ0(n) was introduced above Remark 2 of [1, p.322], and it was shown that Λ0(1) = ? there. So [1] provides a new proof for the Smale’s result above. For n ? N we de,ne an increasing function [1, ?) Λ ? Λn by n 2n 1 1 1 1 Λ+ + Λ+ ? 1. (2) Λn := 2 Λ 2 Λ (This is obtained from [1, (3.11)] when Λ1 in [1, (3.10)] is replaced by Λ.) Then Λn = Λ0(n) if Λ = Λ1(n) by the proof of [1, Cor.5]. We here consider real 2n-dimensional compact K?ahler-Einstein manifolds (M, ω, J, g) sat- isfying the following three conditions. (A) The curvature tensor R is constant on subbundle {(X, JX, Y, JY ) | g(X, Y ) = 0, g(X, JY ) = 0, g(X, X) = 1 = g(Y, Y )}. Namely, for any p, q ? M and any unit orthogonal bases of (TpM, Jp, gp) and (TqM, Jq, gq), {a1, ? ? ? , a2n} and {a1, ? ? ? , a2n} with a2k = Jpa2k?1 and a2k = Jqa2k?1, k = 1, ? ? ? , n, it holds that R(ai, ak, ai, ak) = R(ai, ak, ai, ak) for all 1 ? i, k ? 2n. If (M, ω, J, g) is also homogeneous, this is equivalent to the following weaker (A’) For any p ? M and any unit orthogonal bases of (TpM, Jp, gp), {a1, ? ? ? , a2n} and {a1, ? ? ? , a2n} with a2k = Jpa2k?1 and a2k = Jqa2k?1, k = 1, ? ? ? , n, it holds that R(ai, ak, ai, ak) = R(ai, ak, ai, ak) for all 1 ? i, k ? 2n. (B) Re(R(X, Y , X, Y )) ? 0 for any X, Y ? T (1,0)M. (C) The holomorphic sectional curvature is positive, i.e. ? c0 > 0 such that ? ? ? ? u ? ?1Ju u + ?1Ju u ? ?1Ju u + ?1Ju R(u, Ju, u, Ju) = ?4R , , , ? c0 2 2 2 2 for any unit vector u ? T M. Here is our ,rst result. -3- ? Theorem 1. Let (M, ω, J, g) and (M, ω?, J, g?) be two real 2n-dimensional compact K?ahler- Einstein manifolds satisfying the above conditions (A) and (B). Then for any Λ-pinched sym- plectomorphism ? : (M, ω) ? (M, ω) with Λ ? [1, Λ1(n)] \ {?}, where Λ1(n) is given by (1), the following conclusions hold: (i) The mean curvature ,ow Σt of the graph of ? in M × M exists smoothly for all t > 0. (ii) Σt is the graph of a symplectomorphism ?t for each t > 0, and ?t is Λn-pinched along the mean curvature ,ow, where Λn is de,ned by (2). (iii) The ,ow converges to a totally geodesic Lagrangian submanifold of M × M and ?t con- verges smoothly to a biholomorphic isometry from M to M as t ? ? provided additionally ? that (M, ω, J, g) and (M, ω?, J, g?) satisfy the condition (C). Consequently, the symplecto- morphism ? : M ? M is symplectically isotopic to a biholomorphic isometry. By Cartan’s classi,cation, in addition to two exceptional spaces E6/(Spin(10)×SO(n+2)) and E7/(E6 × SO(2)), all irreducible Hermitian symmetric spaces of compact type have the following form of four types (in the terminology of [23, p. 518]): U(n + m)/U(n) × U(m), n, m ? 1, SO(2n)/U(n), n ? 2, Sp(n)/U(n) n ? 2, SO(n + 2)/SO(n) × SO(2), n ? 3. They are, respectively, holomorphically equivalent to (cf. [24]): ? GI(n, n + m) = G(n, n + m; C) the complex Grassmann manifold consisting of the (n ? 1)- dimensional, complex projective linear subspaces CP n?1 of the complex projective (n + m ? 1)- space CP n+m?1; ? GII(n, 2n) the complex analytic submanifold of G(n, 2n; C) consisting of all the (n ? 1)- dimensional, complex projective linear subspaces CP n?1 that lie in a non- singular quadric hypersurface Q2n?2(C) in CP 2n?1; ? GIII(n, 2n) the complex subvariety of G(n, 2n, C) consisting of all the (n ? 1)-dimensional, complex projective linear subspaces CP n?1 ? CP 2n?1, all of whose projective lines are con- tained in a general (i.e., nonsingular) linear complex. In terms of homogeneous coordinates in CP 2n?1, it is represented by the m-dimensional, complex vector subspaces of the vector space C2n, that are totally isotropic with respect to a nonsingular, alternating bilinear form on C2n; ? GIV(1, n + 1) the nonsingular, n-dimensional, complex quadric hypersurface Qn ? CP n+1, and real-analytically isomorphic to the real Grassmann manifold G+(2, n, R) of oriented, real projective lines in RP n+1. They have complex dimensions mn, n(n ? 1)/2, n(n + 1)/2, n, respectively. Let h and hI be the canonical K?ahler metrics on G(n, n + m; C) and GI(n, 2n), respectively. Both GII(n, 2n) and GIII(n, 2n) are totally geodesic submanifolds of (GI(n, 2n), hI). Denote by hII and hIII the induced metrics on GII(n, 2n) and GIII(n, 2n), respectively. Applying Theorem 1 to these manifolds we may get the following two results. Theorem 2. Let ω be the K?ahler form corresponding with the canonical metric h on G(n, n + m; C), g = Re(h) and J the standard complex structure. Then for every Λ-pinched symplecto- morphism ? ? Symp(G(n, n + m; C), ω) with Λ ? [1, Λ1(mn)] \ {?} the following holds: -4- (i) The mean curvature ,ow Σt of the graph of ? in G(n, n + m; C) × G(n, n + m; C) exists for all t > 0. (ii) Σt is the graph of a symplectomorphism ?t for each t > 0, and ?t is Λmn-pinched along the mean curvature ,ow, where Λmn is de,ned by (2). (iii) ?t converges smoothly to a biholomorphic isometry of (G(n, n + m; C), J, g) as t ? ?. Consequently, each such Λ-pinched symplectomorphism ? ? Symp(G(n, n + m; C), ω) is sym- plectically isotopic to a biholomorphic isometry of (G(n, n + m; C), J, g). Theorem 3. Let (M, ω, J, g) be a compact K?ahler-Einstein submanifold of (G(n, n + m; C), h) which is totally geodesic (e.g. (GII(n, 2n), hII) and (GIII(n, 2n), hIII) are such submanifolds of (GI(n, 2n), hI)). Set dim M = 2N. Then for every Λ-pinched symplectomorphism ? ? Symp(M, ω) with Λ ? [1, Λ1(N)] \ {?} the following holds: (i) The mean curvature ,ow Σt of the graph of ? in M × M exists for all t > 0. (ii) Σt is the graph of a symplectomorphism ?t for each t > 0, and ?t is ΛN-pinched along the mean curvature ,ow, where ΛN is de,ned by (2). (iii) ?t converges smoothly to a biholomorphic isometry of (M, J, g) as t ? ?. Consequently, each such Λ-pinched symplectomorphism ? : (M, ω) ? (M, ω) is symplectically isotopic to a biholomorphic isometry of (M, J, g). The paper is organized as follows. In Section 2 we review the evolution results along the mean curvature ,ow in [1]. The proof of Theorem 1 is given in Section 2. In Section 4 we prove Theorems 2 and 3. 1 Evolution along the mean curvature ,ow For convenience we review results in Section 2 of [1] in this section. A real 2N-dimensional Hermitian vector space is a real 2N-dimensional vector space V equipped with a Hermitian structure, i.e. a triple (ω, J, g) consisting of a symplectic bilinear form ω : V × V ? R, an inner product g and an complex structure J on V satisfying g = ω ? (Id × J). A Hermitian ? isomorphism from (V, ω, J, g) to another Hermitian vector space (V , ω?, J, g?) of real 2n dimension ? ? is a linear isomorphism L : V ? V satisfying: LJ = JL, L ω? = ω and L?g? = g. Proposition 1 and Corollary 2 in Section 2.1 of [1] can be summarized as follows. Proposition 4. For any linear symplectic isomorphism L from the real 2N-dimensional Her- ? mitian (V, ω, J, g) to (V , ω?, J, g?), let L : V ? V be the adjoint of L determined by g(L u?, v) = g?(u?, Lv). Then L L : V ? V is positive de,nite, and E := L(L L)?1/2 gives rise to a Her- ? mitian isomorphism from (V, ω, J, g) to (V , ω?, J, g?). Moreover, there exists an unitary basis {v1, ? ? ? , v2N} of (V, ω, J, g), i.e., and g(vi, vj) = δij Jv2k?1 = v2k, k = 1, ? ? ? , N, ? (and hence an unitary basis of (V , ω?, J, g?), {v?1, ? ? ? , v?2N} = {E(v1), ? ? ? , E(v2N)} ), such that -5- (i) The matrix representations of J and J? under them are all 0 1 1 0 .. (3) . J0 = 0 1 ?1 0 (ii) The map (L L)1/2 has the matrix representation under the basis {v1, ? ? ? , v2N}, (L L)1/2 = Diag(λ1, λ2, ? ? ? , λ2N?1, λ2N), where λ2i?1λ2i = 1 and λ2i?1 ? 1 ? λ2i, i = 1, ? ? ? , N. (iii) Under the bases {v1, ? ? ? , v2N} and {v?1, ? ? ? , v?2N} the map L has the matrix representation L = Diag(λ1, λ2, ? ? ? , λ2N?1, λ2N). Remark 5. From the arguments in [1] one can also choose the {v1, ? ? ? , v2N} such that λk, k = 1, ? ? ? , 2N in Proposition 4(ii) satisfy: λ2i ? 1 ? λ2i?1, i = 1, ? ? ? , N. ? Let (M, ω, J, g) and (M, ω?, J, g?) be two real 2N-dimensional K?ahler-Einstein manifolds, and let π1 : M ×M ? M and π2 : M ×M ? M be two natural projections. We have a product ? K?ahler manifold (M × M, π1?ω ? π2?ω?, J , G), where G = π1?g + π2?g? and J (u, v) = (Ju, ?Jv) for (u, v) ? T (M × M). For a symplectomorphism ? : (M, ω) ? (M, ω?) let Σ = Graph(?) = {(p, ?(p)) | p ? M}, and let Σt be the mean curvature ,ow of Σ in M × M. Denote by ? := π1?ωN, and by ?? the Hodge star of ?|Σt with respect to the induced metric on Σt by G. Then ?? is the Jacobian of the projection from Σt onto M, and ??(q) = ?(e1, ? ? ? , e2N) for q ? Σt and any oriented orthogonal basis {e1, ? ? ? , e2N} of TqΣt. The implicit function theorem implies that ??(q) > 0 if and only if Σt is locally a graph over M at q. Let q = (p, ?t(p)) ? Σt ? M × M. Set L := Dp?t : TpM ? T?t(p)M and E := 1 Dp?t[(Dp?t) Dp?t]? 2 : TpM ? T?t(p)M. Since L?L is a positive de,nite matrix, by the above arguments one can choose a holomorphic local coordinate system {z1, ? ? ? , zN} around p, zj = xj + iyj, j = 1, ? ? ? , N, such that (i) { ?x? 1 |p, ? ? ? , ?x?N |p, ?y? 1 |p, ? ? ? , ?y?N |p} is an orthogonal basis of the real 2N-dimensional vector space TpM, (ii) The complex structure Jp is given by the matrix J0 in (3) with respect to the base ? , ?y? 1 , ? ? ? , ?x?N , ?y?N . ?x1 (iii) L?L = Diag(λ21, λ22, ? ? ? , λ22N?1, λ22N) with respect to these basis, where λ2i?1λ2i = 1, λ2i?1 ? 1 ? λ2i for i = 1, ? ? ? , N. Obviously ?x? j = ?z? j + ?z? j , ?y? j = ?1?1( ?z? j ? ?z? j ). (iv) There exists a Hermitian vector space isomorphism E : (TpM, ωp, Jp, gp) ? (T?t(p)M, ω??t(p), J??t(p), g??t(p)) -6- such that under the orthogonal basis of (T?t(p)M, g??t(p)), ? ? ? ? E( |p), E( |p), ? ? ? , E( |p), ? ? ? , E( |p) , ?x1 ?y1 ?xN ?yN J??t(p) is also given by the matrix J0 in (3). By the choose of basis, we have ? ? ? ? ? ? ? ? g( |p, |p) = g( |p, |p) = δij, g( |p, |p) = g( |p, |p) = 0, ?i x?xj ?yi ?yj ?xi ?yj ?yi ?xj ? ? δld | , gld = g( , gld = gld = 0. d |p) = gdl = gdl = gld = l p ?z 2 ?z For j = 1, ? ? ? , N, set ? ? a2j?1 = and a2j = (4) |p |p. ?xj ?yj Then by (ii) above it holds that Jp(a2j?1) = a2j and Jp(a2j) = ?a2j?1, j = 1, ? ? ? , N. Let s = s + (?1)s+1, s = 1, ? ? ? , 2N, and let Jrs := g(Jas, ar). It follows that 0 if r = s , Js s = ?Jss and Jrs = (?1)s+1 if r = s . For i = 1, ? ? ? , 2N, let 1 1 ei = and e2N+i = (5) (ai, λiE(ai)) (Jpai, ?λiE(Jpai)). 1+ 1+ λ2i λ2i They form an orthogonal basis of Tq(M × M), and TqΣt = span({e1, ? ? ? , e2N)} and NqΣt = span({e2N+1, ? ? ? , e4N}) 2N and ?? = ?(e1, ? ? ? , e2N) = 1/ + λ2j). j=1(1 ? Proposition 6. ([1, Prop.2]) Let (M, g, J, ω) and (M, g?, J, ω) be two compact Ka?hler-Einstein manifolds of real dimension 2N, and let Σt be the mean curvature ,ow of the graph Σ of a symplectomorphism ? : (M, ω) ? (M, ω). Then ?? at each point q ? Σt satis,es the following equation: d λi(Rikik ? λ2kRikik) , (6) ? ? = ?? ? + ?? Q(λi, hjkl) + dt (1 + λ2k)(λi + λi ) k i=k where Q(λi, hjkl) = (?1)i+jλiλj(hi ikhj jk ? hi jkhj ik) h2ijk ? 2 i 0}. By Remark 2 and Lemma 4 in [1] (or the proof of [1, Prop.3]), Λ0(1) = ?, and ? ? 3 ? 5 3 ? 5 2 |II| = Q((1, ? ? ? , 1), hijk) ? h2ijk. 6 6 i,j,k Λ ? δΛ is nonincreasing. They imply Λ0(N) > 1. Clearly, δλ is continuous in λ, and [1, ?) Note that δΛ > 0 for every Λ ? [1, Λ0(N)). Indeed, by the de,nition of supremum we have a Λ ? (Λ , Λ0(N)) with δΛ > 0. So δΛ ? δΛ > 0. In addition, (7) and (8) imply 1 1 ? λi ? Λ ? λi ? Λ inf Q(λi, hjkl) = δΛ h2ijk = 1, hijk ? R, = inf δλ Λ Λ i,j,k for every Λ ? [1, Λ0(N)). Hence we get: Proposition 7. ([1, Prop. 3]) Let Q(λi, hjkl) be the the quadratic form de,ned in Proposition 6. Then for the constant Λ0(N) ? (1, +?] in (9), which only depends on 2N = dim M, Q(λi, hjkl) is nonnegative whenever Λ01(N) ? λi ? Λ0(N) for i = 1, ? ? ? , 2N. Moreover, for any Λ ? [1, Λ0(N)) it holds that Q(λi, hjkl) ? δΛ h2jkl ijk 1 whenever ? λi ? Λ for i = 1, ? ? ? , 2N. Λ 2 Proof of Theorem 1 Lemma 8. Let R be the curvature tensor of a K?ahler manifold (M, g, J, ω) of real dimension 2N. For any local holomorphic coordinate system (z1, ? ? ? , zn) on it, let Rrsrs = R ?z?r , ??z?s , ?z?r , ??z?s ? and zs = xs + ?1ys, s = 1, ? ? ? , n. Then ? ? ? ? ? ? ? ? ? R ?r, s. R , , , , , , = ?4Re(Rrsrs) s r s s r s ?y ?yr ?xr ?x ?x ?x ?x ?x In particular, we have ? ? ? ? R ?s, = ?4Rssss , , , ?xs ?ys ?xs ?ys ? that is, the holomorphic sectional curvature in the direction is given by ?xs ? 4Rss?ss? = ? H . s ?x [g( ?x? s , ?x? s )]2 -8- Proof. Since the only possible non-vanishing terms of the curvature components are of the form Ri?jk?l and those obtained from the universal symmetries of the curvature tensor, one easily prove ? ? ? ? R , , , ?xs ?yr ?xs ?yr ? ? ? ? ? ? ? ? ? ? ?1 ? ?1 ? = R + , , + , ?zs ?z?s ?zr ?z?r ?zs ?z?s ?zr ?z?r (10) = ?(Rrsrs + Rsrsr + Rrssr + Rsrrs), ? ? ? ? R , , , s r s ?xr ?x ?x ?x ? ? ? ? ? ? ? ? = R + + + + , , , ?zs ?z?r ?z?s ?zr ?z?r ?zs ?z?s ?zr (11) = Rrsrs + Rsrsr ? Rrssr ? Rsrrs. Note that Rrsrs = Rsrsr = Rsrsr. It follows from this and (10)-(11) that ? ? ? ? ? ? ? ? ? R R , , , , , , = ?2Rrsrs ? 2Rsrsr = ?4Re(Rrsrs). s r s s r s ?y ?yr ?xr ?x ?x ?x ?x ?x The second equality may be derived from (10) directly. Lemma 8 is proved. Lemma 9. Under the assumptions of Lemma 8, if (M, g, J, ω) also satis,es the condition (B), then Re(Rrsrs) ? 0 for all 1 ? r, s ? n. n n Proof. Set X = ui ?z? i and Y = vj ?z? j with ui, vj ? C. Then i=1 j=1 n n ? ? ? ? R(X, Y , X, Y ) = = R ui , vj , vl uiukvjvlRijkl, j , uk i k ?z ?z ?z ?zl i,j,k,l=1 i,j,k,l=1 n n ? ? ? ? R(Y, X, Y, X) = = R vj , ui , uk vjvluiukRjilk. i , vl j l ?z ?z ?z ?zk i,j,k,l=1 i,j,k,l=1 Since Rjilk = Rijkl we get n R(X, Y , X, Y ) + R(Y, X, Y, X) = uiukvjvlRijkl + uiukvjvlRijkl i,j,k,l=1 = R(X, Y , X, Y ) + R(X, Y , X, Y ) = 2Re(R(X, Y , X, Y )). ? ? Taking X = , Y = , the desired results are obtained. ?zr ?zs Proposition 10. Let (M, ω, J, g) be a real 2n-dimensional compact K?ahler-Einstein manifold satisfying the conditions (A) and (B). Then for any symplectomorphism ? : M ? M, d (12) ? ? ? ? ? ? + ?? ? Q(λi, hjkl), dt along MCF Σt of the graph Σ of ?. Moreover, if ? is Λ-pinched for some Λ ? (1, Λ1(n)), then the symplectomorphism ?t : M ? M, whose graph is Σt, is Λn-pinched and d (13) ? ? ? ? ? ? + δΛ ? ??|II|2 dt along the mean curvature ,ow. In particular, minΣt ?? is nondecreasing as a function in t. -9- Proof. By the condition (A), Rikik = Rikik ?i, k. Hence the second term in the big bracket of (6) can be written as follows ( omitting |p in ?x? t |p and ?y? t |p), λi(Rikik ? λ2kRikik) λi(1 ? λ2k)Rikik = (14) (1 + λ2k)(λi + λi ) (1 + λ2k)(λi + λi ) k i=k k i=k λ2s?1(1 ? λ22r?1)R( ?x? s , ?x? r , ?x? s , ?x? r ) = (1 + λ22r?1)(λ2s?1 + λ2s) k=2r?1,i=2s?1,r=s λ2s(1 ? λ22r?1)R( ?y? s , ?x? r , ?y? s , ?x? r ) + (1 + λ22r?1)(λ2s?1 + λ2s) k=2r?1,i=2s λ2s?1(1 ? λ22r)R( ?x? s , ?y? r , ?x? s , ?y? r ) + (1 + λ22r)(λ2s?1 + λ2s) k=2r,i=2s?1 λ2s(1 ? λ22r)R( ?y? s , ?y? r , ?y? s , ?y? r ) + (1 + λ22r)(λ2s?1 + λ2s) k=2r,i=2s,r=s λ2s(1 ? λ22r) R( ?x? s , ?x? r , ?x? s , ?x? r ) λ2s?1(1 ? λ22r?1) = + (λ2s?1 + λ2s) (1 + λ22r?1) (1 + λ22r) r=s R( ?x? s , ?y? r , ?x? s , ?y? r )(λ22r ? 1)(λ2s ? λ2s?1) + (λ2s?1 + λ2s)(1 + λ22r) r,s ? ? ? ? ? ? ? ? (λ22r ? 1)(λ2s ? λ2s?1) = R( , , , ) ? R( , , , ) 2 s r s r s r s ?y ?y ?xr ?x ?x ?x ?x ?x (λ2s?1 + λ2s)(1 + λ2r) r,s ? ? ? ? ? ? ? ? (λ22r ? 1)(λ22s ? 1) ) ? R( = R( ) , , , , , , ?xs ?yr ?xs ?yr ?xs ?xr ?xs ?xr (1 + λ22s)(1 + λ22r) r,s ? ? ? ? ? ? ? ? (λ22r ? 1)(λ22s ? 1) ) ? R( = R( , , , , , , ) 2 2 s r s r s r s ?y ?y ?xr ?x ?x ?x ?x ?x (1 + λ2s)(1 + λ2r) r=s ? ? ? ? ? ? ? ? (λ22r ? 1)(λ22s ? 1) ) ? R( + R( ) , , , , , , ?xs ?yr ?xs ?yr ?xs ?xr ?xs ?xr (1 + λ22s)(1 + λ22r) r=s (λ22r ? 1)(λ22s ? 1) (λ22r ? 1)(λ22s ? 1) = [?4Re(Rrsrs)] + [?4Re(Rssss)] ? 0 2 2 (1 + λ22s)(1 + λ22r) (1 + λ2s)(1 + λ2r) r=s r=s because of Lemmas 8, 9 and our choice that λ2i?1 ? 1 ? λ2i, i = 1, ? ? ? , n. This leads to (12). Now if ? is Λ-pinched, then Λ1 ? λi(0) ? Λ for i = 1, ? ? ? , 2n. Since Λ1(n) < Λ0(n) in the 2 case Λ0(n) < ?, by Proposition 7 we get Q(λi(0), hjkl) ? δΛ ijk hjkl and hence d ? ? ? ? 0 at t = 0. dt 1 1 1 Note that Lemma 5 of [1] implies that 2n ? (n, Λ) ? ?? at t = 0, where (n, Λ) = 2n ? (Λ+1Λ )n . Then repeating the proof of Proposition 4 and Corollary 5 in [1] we may get (13). Using this proposition we may prove Theorem 1(i) and (ii) as in [1, ?3.3]. In order to prove Theorem 1(iii) we only need to prove the following proposition. - 10 - Proposition 11. Under the assumptions of Proposition 10, suppose further that (M, ω, J, g) also satis,es the condition (C). Then along the mean curvature ,ow it holds that (1 ? λ2k)2 d . (15) ? ? ? ? ? ? + δΛ ? ??|II|2 + c0 ? ?? dt (1 + λ2k)2 k odd Proof. Under the further assumption, by (14) we have λi(Rikik ? λ2kRikik) (1 + λ2k)(λi + λi ) k i=k (λ22r ? 1)(λ22s ? 1) (λ22r ? 1)(λ22s ? 1) = [?4Re(Rrsrs)] + [?4Re(Rssss)] (1 + λ(1 + λ22s)(1 + λ22r) 22s)(1 + λ22r) r=s r=s (λ22r ? 1)(λ22s ? 1) . ? c0 (1 + λ22s)(1 + λ22r) r=s This and Propositions 6 and 7 give (15). As in [1], using this we may prove that λi ? 1 and maxΣt |II|2 ? 0 as t ? ?, and hence that the ,ow converges to a totally geodesic Lagrangian submanifold of M × M as t ? ? and that ?t converges smoothly to a biholomorphic isometry ?? : M ? M. Theorem 1 is proved. 3 Proofs of Theorems 2, 3 Following [25] let us recall the matrix representations of the Hermitian symmetric spaces GI(n, n + m) = G(n, n + m; C), GII(n, 2n) and GIII(n, 2n). Let M(n + m, n; C) = {A ? Cn×(n+m) | rankA = n }. Then GL(n; C) := {Q ? Cn×n | detQ = 0} acts freely on M(n, n; C) from the left by matrix multiplication. The complex Grassmann manifold G(n, n + m; C) may be de,ned as the quotient GI(n, n + m) = G(n, n + m; C) := M(n, n + m; C)/GL(n; C). For A ? M(n, n + m; C), denote by [A] ? G(n, n + m; C) the GL(n; C)-orbit of A in M(n, n + m; C). Any representative matrix B of [A] is called a homogeneous coordinate of the point [A]. For integers 1 ? α1 < ? ? ? < αn ? n + m let {αn+1, ? ? ? , αn+m} be the complement of {α1, ? ? ? , αn} in the set {1, 2, . . . , n + m}. Write A ? M(n, n + m; C) as A = (A1, ? ? ? , An+m) and Aα1???αn = (Aα1, ? ? ? , Aαn) ? Cn×n, Aαn+1???αn+m = (Aαn+1, ? ? ? , Aαn+m) ? Cn×m, where A1, ? ? ? , An+m are n × 1 matrices. De,ne Uα1,??? ,αn = {[A] ? G(n, n + m; C) | detAα1???αn = 0 }, [A] ? Z = (Aα1???αn)?1Aαn+1???αn+m. Θα1???αn : Uα1???αn ? Cn×m ? Cnm, We call Z the local coordinate of [A] ? G(n, n + m; C), and 1 ? α1 < ? ? ? < αn ? n Uα1???αn, Θα1???αn the canonical atlas on G(n, n + m; C) ([25, 26, 27]). There exists a unique (up to multiplying a nonzero real constant) metric h on G(n, n + m; C) which is invariant under the action of the - 11 - group of motions in G(n, n + m; C) (cf.[26]). It is well-known (e.g. [25] ) that this metric in the local chart (U1???n, Z = Θ1???n) on G(n, n + m; C) is given by h = Tr[(In + ZZ )?1dZ(Im + Z Z)?1dZ ] = ??? log det(I + ZZ ), (16) where Z and dZ are the conjugate transposes of Z and dZ respectively, Tr denotes the trace, ? ? iα iα and ? = and ?? = . It is K?ahler-Einstein. i,α dZ i,α dZ ?Ziα ?Ziα Correspondingly, we have also matrix representations of the Hermitian symmetric spaces II G (n, 2n) and GIII(n, 2n) (cf. [25]), 0 In GII(n, 2n) = A =0 , [A] ? G(n, 2n) ?A ? [A] such that A In 0 0 In GIII(n, 2n) = [A] ? G(n, 2n) ?A ? [A] such that A A =0 . ?In 0 Let hI be the canonical K?ahler metric on GI(n, 2n), which in the coordinate chart Uα1???αn is given by ??? ln det(I + ZZ ) as in (16). It induces a K?ahler metric hII on GII(n, 2n) given by hII = ??? ln det(I ? ZZ) (17) [A] ? Z kl([A]) in the induced coordinate system GII(n, 2n) ? Uα1???αn with Z ? Cn×n k