SELF-SIMILAR SOLUTIONS TO THE MEAN CURVATURE FLOWS ON RIEMANNIAN CONE MANIFOLDS AND SPECIAL LAGRANGIANS ON TORIC CALABI YAU CONES

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1 Futaki, A., Hattori, K. and Yamamoto, H. Osaka J. Math. 51 (2014), SELF-SIMILAR SOLUTIONS TO THE MEAN CURVATURE FLOWS ON RIEMANNIAN CONE MANIFOLDS AND SPECIAL LAGRANGIANS ON TORIC CALABI YAU CONES AKITO FUTAKI, KOTA HATTORI and HIKARU YAMAMOTO (Received June 15, 2012, revised April 18, 2013) Abstract The self-similar solutions to the mean curvature flow have been defined and studied on the Euclidean space. In this paper we propose a general treatment of the selfsimilar solutions to the mean curvature flow on Riemannian cone manifolds. As a typical result we extend the well-known result of Huisken about the asymptotic behavior for the singularities of the mean curvature flows. We also extend results on special Lagrangian submanifolds on n to the toric Calabi Yau cones over Sasaki Einstein manifolds. 1. Introduction Let FÏ M [0, T ) V be a smooth family of immersions of an m-dimensional manifold M into a Riemannian manifold (V, Æg) of dimension m k. F is called a mean curvature flow if it satisfies (1) F t (p, t) H t(p) for all (p, t) ¾ M [0, T ) where H t is the mean curvature of the immersion F t Ï F(, t)ï M V. When V is the Euclidean space Ê mk there is a well-studied important class of solutions of (1), that is, self-similar solutions. They are immersions FÏ M Ê mk satisfying (2) H F where is a constant and F denotes the normal part of the position vector F. The solution of (2) is called shrinking, stationary (or minimal) or expanding depending on whether 0, 0 or 0. The purpose of this paper is to extend the definition of the self-similar solutions from the case when V is the Euclidean spaces to the case when V is a Riemannian cone manifold. Let (N, g) be an n-dimensional Riemannian manifold. We define the Riemannian cone manifold (C(N), Æg) over (N,g) by C(N) N Ê and Æg dr 2 r 2 g 2000 Mathematics Subject Classification. Primary 53C55; Secondary 53C21, 55N91.

2 1054 A. FUTAKI, K. HATTORI AND H. YAMAMOTO where r is the standard coordinate of Ê. If FÏ M C(N) is an immersion we define the position vector F of F at p ¾ M by (3) F (p) r(f(p)) r ¾ T F(p)C(N). Then the self-similar solution is defined as (4) H F where is a constant and F denotes the normal part of the position vector F. In this paper we propose a general treatment of the self-similar solutions to the mean curvature flows on Riemannian cone manifolds. As a typical result we extend the wellknown result of Huisken about the asymptotic behavior for the singularities of the mean curvature flows. In [9] Huisken introduced the rescaling technique and the monotonicity formula for the mean curvature flow of hypersurfaces in Euclidean space. Also in [9], using the monotonicity formula, Huisken proved that if the mean curvature flow has the type I singularity then there exists a smoothly convergent subsequence of the rescaling such that its limit satisfies the self-similar solution equation. In this paper we extend those techniques and consequences to Riemannian cone manifolds and an initial date manifold. We also give a construction of self-similar solutions on Riemannian cone manifolds. Let us recall the definition of type I singularity and its parabolic rescaling. Let M be a manifold and (V, Æg) a Riemannian manifold. Suppose FÏ M [0, T ) V is a mean curvature flow with maximal time T ½ of existence of the solution. One says that F develops a singularity of Type I as t T if there exists a constant C 0 such that supii t 2 C for all t ¾ [0, T ), M T t where II t is the second fundamental form with respect to the immersion F t Ï M V. Otherwise one says that F develops a singularity of Type II. Let M be a manifold and (C(N), Æg) the Riemannian cone manifold over a Riemannian manifold (N, g). Take a constant 0. For a map FÏ M [0, T )C(N), we define the parabolic rescaling of F of scale as follows; F Ï M [ 2 T, 0) C(N)Á F (p, s) N F p, T s2, r F p, T s2 where N Ï C(N) N Ê N is the standard projection. When the singularity does not occur at the apex of the cone one can show that the parabolic rescaling of type I singularity gives rise to a self-similar solution as shown

3 SELF-SIMILAR SOLUTIONS TO MEAN CURVATURE FLOW 1055 by Huisken. However when the singularity occurs at the apex we need some more conditions. Thus we are lead to the following definition of type I c singularity. DEFINITION 1.1. Let M be a manifold and (N, g) a Riemannian manifold. Suppose FÏ M [0, T ) C(N) is a mean curvature flow with T ½. We say that F develops a singularity of type I c if the following three conditions are satisfied: (a) F develops a singularity of type I as t T, (b) r(f t (p)) 0 for some p ¾ M as t T and (c) K 1 (T t) min M r 2 (F t ) K 2 (T t) for all t ¾ [0, T ) where K 1 and K 2 are positive constants. Examples of type I c singularities are given in Example 6.1. Theorem 1.2. Let M be an m-dimensional compact manifold and C(N) the Riemannian cone manifold over an n-dimensional Riemannian manifold (N, g). Let FÏ M [0, T )C(N) be a mean curvature flow, and assume that F develops a type I c singularity at T. Then, for any increasing sequence { i } i1 ½ of the scales of parabolic rescaling such that i ½ as i ½, there exist a subsequence { ik } ½ k1 and a sequence t ik T such that the sequence of rescaled mean curvature flow {F i k s ik } ½ k1 with s ik 2 i k (t ik T ) converges to a self-similar solution F ½ Ï M ½ C(N) to the mean curvature flow. The proof of this theorem is not substantially different from Huisken s original proof. But the merit of the idea to study on cones will be that we obtain examples of more non-trivial topology. In fact N {r 1} in C(N) is already a self-shrinker. Thus, any compact manifold can be a self-shrinker in some Riemannian cone manifold. It is also possible to study special Lagrangian submanifolds and Lagrangian self-similar solutions in Calabi Yau cones over Sasaki Einstein manifolds. A Sasaki manifold N is by definition an odd dimensional Riemannian manifold whose cone C(N) is a Kähler manifold. If the Kähler cone manifold is toric then the Sasaki manifold is said to be toric. It is proven in [4] and [2] that a Sasaki Einstein metric exists on a toric Sasaki manifold obtained from a toric diagram. A typical example is when N is the standard sphere of real dimension 2m 1. Then its cone is m1 {o}. It is natural to expect that we can extend results on special Lagrangian submanifolds or self-similar solutions on m1 to these toric Calabi Yau cones of height 1. In Theorem 7.5 we construct examples of complete special Lagrangian manifolds on toric Calabi Yau cones using the ideas of [6] and [11]. This construction includes the examples given in Theorem 3.1 in III.3 of Harvey Lawson [7]. Further construction of examples of special Lagrangians and Lagrangian self-similar solutions are given in the third author s subsequent paper [15], in which it is shown that, for any positive integer g, there are toric Calabi Yau 3- dimensional cones including Lagrangian self-shrinkers diffeomorphic to g S 1 where g is a compact orientable surface of genus g.

4 1056 A. FUTAKI, K. HATTORI AND H. YAMAMOTO In Section 8 we also study the infinitesimal deformations of special Lagrangian cone C() C(N) over a Legendrian submanifold in a Sasaki Einstein manifold N. We show that the parameter space H C() of those infinitesimal deformations is isomorphic to Ker(½ 2n) {³ ¾ C ½ ()Á ½ ³ 2n³}, see Theorem 8.6. This is also proved by Lemma 3.1 of [14], although the proof in this paper is different from [14]. This paper is organized as follows. In Section 2 we show fundamental formulas on mean curvature flows in Riemannian cone manifolds. In Section 3 we show the finite time blowup of the mean curvature from a compact manifold (Theorem 3.1). Section 4 is devoted to the proof of the monotonicity formula (Theorem 4.1). In Section 5 we see that the type I singularity is preserved under parabolic rescaling. In Section 6 we see that we obtain a self-similar solution by parabolic rescaling at a type I c singularity. In Section 7 we construct special Lagrangians in toric Calabi Yau cones. In Section 8 we study the infinitesimal deformations of special Lagrangian cones in Calabi Yau cones. 2. Self-similar solutions to the mean curvature flows on Riemannian cone manifolds Let FÏ M V be an immersion of an m-dimensional manifold M into an m k- dimensional Riemannian manifold (V, Æg). Thus the differential F x Ï T x M T F(x) V is injective for every x ¾ M, and we have a natural orthogonal decomposition of the vector bundle F (T V ) T M T M where T M M is the normal bundle. Denote by (resp. ) the projection Ï F (T V ) T M (resp. Ï F (T V ) T M). The second fundamental form II of the immersion FÏ M V is a section of the vector bundle T M Å (Å 2 T M) defined by II(X, Y ) (ÆÖ F (X)F (Y )) for X, Y ¾ ¼(T M). Here ÆÖ is the Levi-Civita connection of (V, Æg). The mean curvature vector field H of FÏ M V is a section of T M defined by H tr II, where the trace is taken with respect to the Riemannian metric g Ï F (Æg) on M. For the actual computations one often needs local expressions of the mean curvature vector. Let x 1,, x m and y 1,, y n be local coordinate charts around p ¾ U M and F(p) ¾ U ¼ V such that F U Ï U U ¼ is an embedding. Write F «(x 1,, x m ) y «(F(x 1,, x m )). Then we have the induced metric g i j F «x i F x j Æg «, where Æg Æg «dy «Ådy is the Riemannian metric on U ¼ V. Here we use the indices i, j, k, to denote the coordinates on M and «,,, to denote the coordinates

5 SELF-SIMILAR SOLUTIONS TO MEAN CURVATURE FLOW 1057 on V. The coefficients H «of the mean curvature vector field are given by the Gauß formula (5) H H «y «H «g i j 2 F «x i x j ¼i k F «j ¼ Æ «F F x k. x i x j Next we consider a smooth family of immersions FÏ M (a, b) V. Namely, for every time t in (a, b) Ê, F t Ï M V given by p F(p, t) is an immersion. We denote by g t the Riemannian metric F t (Æg) over M. For a fixed time t 0 in (a, b), the variation vector field ( Ft)(,t 0 ), considered as a section of F t 0 T V, is decomposed as F t (, t 0) Ú t 0 Ú t 0 where Ú t 0 (p) and Ú t 0 are respectively the sections of T M and T M. We denote by Ö t, div t, II t and H t respectively the Levi-Civita connection on (M,g t ), the divergence with respect to g t, the second fundamental form and the mean curvature vector field of the immersion F t Ï M V. Then following proposition is well-known as the first variation formula. Proposition 2.1. For every p in M, two tangent vectors X, Y at p and a compactly supported integrable function f on M, we have d dt g t (X, Y ) tt0 g t0 (Ö t 0 X Ú t 0, Y )g t0 (X, Ö t 0 Y Ú t 0 ) 2Æg(II t0 (X, Y ), Ú t 0 (p)), d dt tt0 M f dú gt M f (div t0 (Ú t 0 ) Æg(H t0, Ú t 0 )) dú gt0. Let FÏ M [0, T ) V be evolving by mean curvature flow with initial condition F 0 Ï M V : (6) F t (p, t) H t(p) for all (p, t) ¾ M [0, T ), F(p, 0) F 0 (p) for all p ¾ M. Applying the first variation formula in Proposition 2.1 to the mean curvature flows, we obtain following well-known properties for mean curvature flows.

6 1058 A. FUTAKI, K. HATTORI AND H. YAMAMOTO If FÏ M [0, T ) V is a mean curvature flow then the follow- Proposition 2.2. ing equation holds. (7) d Ô dt det((gt ) i j ) tt0 H t0 gô 2 det((gt0 ) i j ). If M is compact we also have d dt Vol gt (M) tt0 H t0 2 Æg dú g t0. M Proof. Because we consider the mean curvature flow, Ú t0 H t0 and therefore and Ú t 0 0 Ú t 0 (p) H t0 (p). It then follows from Proposition 2.1 that d dt (g t ) i j tt0 2Æg((II t0 ) i j, H t0 ). Then the first formula (7) follows from the well-known formula for the derivative of the determinant. To prove second formula, simply let f 1 on M in the first variation formula. Recall that, for an n-dimensional Riemannian manifold (N, g), we define the Riemannian cone manifold (C(N), Æg) over (N,g) by C(N) N Ê and Æg dr 2 r 2 g where r is the standard coordinate of Ê. Note that C(N) does not contain the apex. The most typical example of a cone is the case when N is the standard sphere S n in Ê n1. In this case the cone is Ê n1 {o}. For a map FÏ M Ê n1, one can consider the position vector of F(p) for p ¾ M, and using it, one can define selfsimilar solutions H F where is a constant. We can extend this idea to maps into Riemannian cone manifolds. Namely, for a smooth map FÏ M C(N) and p in M, we define the position vector F of F at p ¾ M by F (p) r(f(p)) r ¾ T F(p)C(N).

7 SELF-SIMILAR SOLUTIONS TO MEAN CURVATURE FLOW 1059 With respect to the bundle decomposition of F T F(p) C(N) T p M T p M, we decompose F (p) as F (p) F (p) F (p). Then we can define self-similar solutions by H F. For a Riemannian cone manifold (C(N), Æg) over an n-dimensional Riemannian manifold (N, g) and a point q in C(N), local coordinates (y «) n1 «1 around q are said to be associated with normal local coordinates of N when the part of coordinate (y «) n «1 becomes normal local coordinates of (N, g) around N (q) and y n1 is the standard coordinate of Ê, that is, y n1 r. Here, N is the projection of the cone manifold C(N) N Ê onto the first factor N. Note that under local coordinates associated with normal local coordinates of N, we have r Æ F r(f) F n1 for a given map FÏ M C(N). Let (x i ) i1 m be normal local coordinates centered at p of the Riemannian manifold (M, F (Æg)), and (y «) n1 «1 local coordinates of (C(N), Æg) associated with normal local coordinates centered at N (F(p)) of (N, g). Then calculating only (n1)-th coefficient H n1 (p) of mean curvature vector at p, namely, the coefficient of y ( n1 r), for the local expression of the mean curvature vector (5), we obtain the following local expression for H n1 (p); m n H n1 (p) m F «2 (8) (p). x i i1 2 r(f) x i 2 (p) r(f(p)) i1 «1 This easily follows from Ƽ n1 «r Æg «for 1 «, n. 3. Finite time singularity for mean curvature flows If the ambient space is the Euclidean space Ê mk and an initial date manifold M is compact, then the mean curvature flow does not have a long time solution. It is a well-known result of Huisken: Theorem 3.1 (Huisken [9]). Let F 0 Ï M Ê mk be an immersion of a compact m-dimensional manifold M. Then the maximal time T of existence of a solution FÏ M [0, T ) Ê mk of the mean curvature flow with initial immersion F 0 is finite.

8 1060 A. FUTAKI, K. HATTORI AND H. YAMAMOTO The proof follows by applying the parabolic maximum principle to the function f F 2 2mt which satisfies the evolution equation (ddt) f ½ f. One can show T (1(2m)) maxf 0 2, from which Theorem 3.1 follows. Using the position vector in a cone as defined in (3), we can extend this result when the ambient space is a Riemannian cone manifold as follows. Theorem 3.2. Let (C(N), Æg) be the Riemannian cone manifold over a Riemannian manifold (N, g) of dimension n, M a compact manifold of dimension m and FÏ M [0, T ) C(N) a mean curvature flow with initial condition F 0 Ï M C(N). Then the maximal time T of existence of the mean curvature flow is finite. Before the proof of this theorem, we want to prepare some lemmas. Lemma 3.3. Let (C(N), Æg) be a Riemannian cone manifold over a Riemannian manifold (N, g) of dimension n and FÏ M C(N) an immersion of a manifold M of dimension m. Then the following equation holds. where ½ is the Laplacian on (M, F (Æg)). Proof. ½(r 2 (F)) 2(Æg(H, F )m), Fix a point p in M. We take normal local coordinates (x i ) m i1 of (M, F (Æg)) centered at p and local coordinates (y «) n1 «1 of (C(N), Æg) associated with normal local coordinates of (N, g) centered at N (F(p)). Note that under these coordinates, y n1 r and F n1 r Æ F r(f). First of all, by the local expression of H n1 (p) in (8), we have the following equalities; (9) Æg(H(p), F (p)) H n1 (p)r(f(p)) r(f(p)) m i1 2 r(f) x i 2 (p) r(f(p))2 m n i1 «1 F «2 (p). x i Since (F Æg)(x i, x i ) 1 at p, we have m m (F Æg) x (p), i x (p) i i1 (10) n F r(f(p)) 2 «m i1 «1 x i (p) 2 m i1 Adding above two equations (9) and (10), we have (11) Æg(H(p), F (p))m r(f(p)) m i1 2 r(f) x i 2 r(f) x i 2 (p). (p) m r(f) x i i1 2 (p).

9 SELF-SIMILAR SOLUTIONS TO MEAN CURVATURE FLOW 1061 Since we took (x i ) m i1 as normal local coordinates of (M, F (Æg)) centered at p, the Laplacian ½ is È m i1 (xi ) 2, and thus we have at p (12) ½r 2 (F) m 2 i1 2 r 2 (F) r(f) x i 2 m i1 2 r(f) x i 2 m i1 r(f) x i 2. Thus from (11) and (12) we have shown that ½r 2 (F) 2(Æg(H, F )m). Lemma 3.4. Let (C(N), Æg) be a Riemannian cone manifold over an n-dimensional Riemannian manifold (N, g), M an m-dimensional manifold and FÏ M [0, T ) C(N) be a mean curvature flow with initial condition F 0 Ï M C(N). Then for any fixed time t in [0, T ) the following equality holds; (13) 2Æg(H t, F t ) t r 2 (F t ). Proof. Fix a point p in M. Take local coordinates (y «) n1 «1 of C(N) associated with normal local coordinates of N. Note that under these coordinates, y n1 r and Ft n1 r(f t ). Since F satisfies the mean curvature flow condition (6), the following equalities hold; from which (13) follows. Æg(H t (p), F t (p)) F Æg t (p, t), F t (p) r(f t (p)) t r(f t(p)) 1 2 Now we are in a position to prove Theorem 3.2. t r 2 (F t (p)), Proof of Theorem 3.2. Let f Ï M [0, T ) Ê be a function defined by f (p, t) r 2 (F t (p))2mt. For a fixed time t in [0, T ), by Lemma 3.3 and Lemma 3.4, where ½ t f t 2Æg(H t, F t )2m ½ t r 2 (F t ) ½ t f (, t) is the Laplacian with respect to the metric F t (Æg) on M. Since M is compact, there is a maximum of f (, 0) ( r 2 (F 0 )) on M, which we denote by C 0. By

10 1062 A. FUTAKI, K. HATTORI AND H. YAMAMOTO applying the maximum principle to the function f, it follows that f (p,t) r 2 (F t (p)) 2mt C 0 on M [0, T ). Therefore we obtain the following inequalities; t C 0 r 2 (F t (p)) 2m C 0 2m for all t in [0, T ). This means that the maximal time T is finite. 4. Monotonicity formula Next we turn to the monotonicity formula. For a fixed time T in Ê, we define the backward heat kernel T Ï Ê ( ½, T ) Ê as follows; T (y, t) 1 (4(T t)) m2 exp To simplify the notations, we use following abbreviation; F M t T Ï M M T t 2(T t) H T (r(f t (p)), t) dú gt, 2 Æg Ï M y 2 4(T t). F T (r(f t (p)), t (p) t) 2 2(T t) H t(p) dú gt. Then Huisken s monotonicity formula for a cone is the following. Theorem 4.1 (Monotonicity formula). Let M be a compact m-dimensional manifold without boundary, (C(N), Æg) the Riemannian cone manifold over an n-dimensional Riemannian manifold (N, g) and FÏ M [0, T ) C(N) the mean curvature flow with initial condition F 0 Ï M C(N). Then the following equation holds; (14) (15) d dt F T M t M T t 2(T t) H Proof. First we calculate the left term of (14) using (7). d dt M d dt M T (r(f t (p)), t) dú gt M 1 r 2 (4(T t)) exp (F t (p)) m2 4(T t) T (r(f t (p)), t) m r 2 (F t (p)) 2(T t) 4(T t) 2 2 Æg. Ôdet(gt,i j) dx 1 dx m r(f t (p))((t)r(f t (p))) 2(T t 0 ) H t (p) 2 Æg Æg dú gt.

11 SELF-SIMILAR SOLUTIONS TO MEAN CURVATURE FLOW 1063 It is clear that (16) F t (p) 2 Æg Æg r(f t (p)) r, r(f t(p)) r r 2 (F t (p)). Substituting (13) and (16) in (15), we have following formula; (17) d dt M M T (r(f t (p)), t) dú gt T (r(f t (p)), t) m 2(T t) F t (p) 2 Æg 4(T t) 2 Æg(H t (p), F t (p)) 2(T t) H t (p) 2 Æg dú gt. Let t and p be fixed. We take normal local coordinates (x i ) i1 m centered at p with respect to the Riemannian metric g t ( F t (Æg)) and local coordinates (y «) n1 «1 around F t (p) associated with normal local coordinates of (N, g). Under these coordinates, the Laplacian ½ t with respect to g t is 2 x 12 2 x m 2 at p. Under these coordinates we have following equations at the fixed t and p; (18) ½ t T (r(f t ), t) m 2 x i 2 xp T (r(f t ), t) i1 m x i i1 xp x i T (r(f t ), t) m x i T (r(f t ), t) xp r(ft )((x i )r(f t )) 2(T t) i1 r 2 (F t )((x i )r(f t )) 2 T (r(f t ), t) 4(T t) 2 ((x i )r(f t )) 2 r(f 2 t)(( x i 2 )r(f t )) 2(T t) 2(T t) Furthermore we want to express F t (p) under these coordinates. Now by our choice of the local coordinates of (x i ) i1 m, it is clear that (19) Æg F t (p), F x i t (p) x Æi j j..

12 1064 A. FUTAKI, K. HATTORI AND H. YAMAMOTO Note that y n1 r and Ft n1 r(f t ). The following equalities hold; F t (p) Æg m F t (p), F t (p) F x i t (p) x i i1 (20) Æg m F t (p), F t (p) i1 m Æg i1 r(f t (p)) r(f t (p)) r, n1 m i1 «1 x i F t «(p) x i r(f t (p)) F x i t (p) F t (p) x i F t (p) y «x i. x i Using (19) and (20), we can express the norm of F t (p) as follows; (21) F t (p) 2 Æg Æg F t (p), F t (p) r 2 (F t (p)) m i1 r(ft ) x i 2 (p). Applying (11) for F t and using (18) and (21), we have the following equality; (22) ½ t T (r(f t (p)), t) t) F t (p) 2 Æg m T (r(f t (p)), Æg(H t(p), F t (p)) 4(T t) 2 2(T t) 2(T t). In this equation (22) there are no local coordinates x i, so we have proven this equation (22) for all p in M globally. The equation (22) is equivalent to (23) m T (r(f t (p)), t) 2(T t) ½ t T (r(f t (p)), t) t) F t T (r(f t (p)), (p) 2 Æg Æg(H t(p), F t (p)) 4(T t) 2 2(T t).

13 SELF-SIMILAR SOLUTIONS TO MEAN CURVATURE FLOW 1065 Substituting (23) in (17), we have following equalities; d dt M T (r(f t (p)), t) dú gt M ½ t T (r(f t (p)), t) dú gt (24) M T (r(f t (p), t)) F t (p) 2 Æg 4(T t) 2 F t (p) 2 Æg 4(T t) 2 2 Æg(H t(p), F t (p)) 2(T t) H t0 (p) 2 Æg dú gt M M T (r(f t (p)), t) F t (p) 2 Æg 4(T t) 2 2 Æg(H t(p), F t (p)) 2(T t) H t (p) 2 Æg F T (r(f t (p)), t (p) t) 2 2(T t) H t(p) dú gt. This completes the proof of Theorem Singularities and the parabolic rescaling In this section we see that the property that a mean curvature flow develops type I singularities is preserved under parabolic rescaling. Proposition 5.1. Let M be an m-dimensional manifold and (C(N), Æg) the Riemannian cone manifold over an n-dimensional Riemannian manifold (N, g). If a map FÏ M [0, T ) C(N) is a mean curvature flow, then the parabolic rescaling of F of scale is also the mean curvature flow. Proof. (x i ) i1 m be local coordinates of M around p 0. Let (y «) n1 «1 be local coordinates of C(N) around F (p 0, s 0 ) associated with local coordinates N. Put (gs 0 ) i j (F s 0 Æg) x, i x j and Æg dú gt Fix (p 0, s 0 ) in M [ 2 T, 0). Let t T s 2 and t 0 T s 0 2. Let (g t0 ) i j (F t 0 Æg) x, i x j.

14 1066 A. FUTAKI, K. HATTORI AND H. YAMAMOTO Then one can easily show that (25) (g s 0 ) i j 2 (g t0 ) i j. Thus the Christoffel symbols (¼ s 0 ) i jk with respect to g s 0 and (¼ t0 ) i jk with respect to g t 0 are related by (¼ s 0 ) i jk (¼ t0 ) i jk. One can also compute the Christoffel symbols of the Riemannian cone manifold C(N) as follows. If 1 «,, n, then Æ ¼ «(F s 0 (p 0 )) Æ ¼ «(F t 0 (p 0 )). If 1, n and «n 1 then Æ ¼ n1 (F s 0 (p 0 )) Æ ¼ n1 (F t 0 (p 0 )), and if 1 «, n and n 1 then Æ ¼ «n1 (F s 0 (p 0 )) (1)Æ ¼ «n1 (F t0 (p 0 )). By using these and the formula (5), one can show that the mean curvature vectors H t0 of F t0 and Hs 0 of Fs 0 are related by (26) (H s 0 (p 0 )) «1 2 (H t 0 (p 0 )) «, for 1 «n and (27) (H s 0 (p 0 )) n1 1 (H t 0 (p 0 )) n1. Now suppose that F is a mean curvature flow, so F satisfies Then F (p 0, s 0 ) s 1 2 n «1 F (p 0, t 0 ) t n «1 H s 0 (p 0 ). (H t0 (p 0 )) «Ht0 (p 0). y «(p 0) 1 (H t 0 (p 0 )) n1 y (p 0) n1 (Hs 0 (p 0 )) «y (p 0)(H «s 0 (p 0 )) n1 y (p 0) n1 is the mean curvature flow. This completes the proof of Propos- This means that F ition 5.1. Proposition 5.2. Let M be an m-dimensional manifold and C(N) the Riemannian cone over an n-dimensional Riemannian manifold (N,g). Let FÏ M [0,T )C(N) be a mean curvature flow. Then parabolic rescaling preserves the value of Ê M t T. This means that for all t in (0, T ) the following equation holds. M t T M s 0

15 SELF-SIMILAR SOLUTIONS TO MEAN CURVATURE FLOW 1067 where s Ê Ê 2 (t T ). Here we have used abbreviation for M t T and Ms T M t M M s 0 M T (r(f t (p)), t) dú gt, 0 (r(f s (p)), s) dú g t. 0 by and Proof. From the equation (25) in the proof of the Proposition 5.1, we get Õ det((g s ) i j) mô det((g t ) i j ) dú g s m dú gt. It follows that M s 0 M M M 1 r 2 (4( s)) exp (Fs (p)) m2 4(0 s) dú g s 1 2 (4( 2 (T t))) exp r 2 (F t (p)) m2 4 2 (T t) 1 r 2 (4(T t)) exp (F t (p)) m2 4(T t) m dú gt dú gt M t T. Proposition 5.3. Let M be an m-dimensional manifold and C(N) the Riemannian cone over an n-dimensional Riemannian manifold (N, g). Let F [0, T ) C(N) be a mean curvature flow. Then the parabolic rescaling preserves the condition that the mean curvature flow develops a Type I singularity. Proof. We have only to show that following two statements are equivalent. There exists some c 0 such that sup M t 2 c(t t) for all t ¾ [0, T ). There exists some c ¼ 0 such that sup M IIs 2 c ¼ s for all s ¾ [ 2 T, 0). Here II t and II s are the second fundamental form with respect to the immersion F t Ï M C(N) and Fs Ï M C(N) respectively. We can find a local expression of (II s )«i j and (II t ) i «j immediately by removing the inverse of Riemannian metric tensors (g s )i j ( (1 2 )(g t ) i j ) from equalities (26) and (II t ) n1 i j if (27). Hence, we find that (II s )«i j (II t ) i «j if 1 «n, and (II s )n1 i j «n 1, where s 2 (t T ). It then follows that (28) Hence we get (29) II s 2 (p) 1 2 II t 2 (p). (T t)ii t 2 s 2 2 II s 2 sii s 2.

16 1068 A. FUTAKI, K. HATTORI AND H. YAMAMOTO This mean that parabolic rescaling preserves the condition developing type I singularity. This completes the proof of Proposition Self-similar solutions This section is devoted to the proof of Theorem 1.2. Proof of Theorem 1.2. Take any increasing sequence { i } i1 ½ of the scales of the T, 0) C(N) parabolic rescaling such that i ½ as i ½. Let F i Ï M [ i 2 be the parabolic rescaling of the mean curvature flow FÏ M [0, T ) C(N). By Proposition 5.1, F i remains to be a mean curvature flow. Since F develops type I c singularity and in particular type I singularity, there exists a positive real number C 0 such that supii t 2 C M T t also develops type I singu- for all t in [0, T ), and by Proposition 5.3 the rescaled F i larity satisfying supii i s 2 C M s for all s in [ i 2, 0) with the same constant C 0 by (29). When s is restricted to the interval [a, b], we have the following bound (30) II i s 2 C b. Hence we have a uniform bound of the second fundamental form, and since F i satisfies the mean curvature flow, all the higher derivatives of the second fundamental form are uniformly bounded on [a, b] by [8]. On the other hand, by Theorem 4.1 the following monotonicity formula for holds. F i d 0 ds M i s M i s 0 F i 2s H i Integrating the both side of the above equation on any closed interval [a, b] ( ½, 0), we have (31) M i b 0 M i a 0 b a ds M i s 0 F i 2 Æg. 2s H i 2 Æg

17 SELF-SIMILAR SOLUTIONS TO MEAN CURVATURE FLOW 1069 where we take i sufficiently large so that [a, b] is contained in [ i 2 T, 0). By Proposition 5.2 we have M i a 0 M ui T where u i T a 2 i and M i b 0 M Úi T Ê M t T where Ú i T b 2 i. By the monotonicity formula, the derivative of the function such that is non-positive and Ê M t T 0, so for any increasing sequence {t i } ½ i1 t i T as i ½ the sequence Ê M ti T converges to a unique value. Now {u i } ½ i1 and {Ú i } ½ i1 are increasing sequences such that u i,ú i T as i ½. So Ê M i a 0 and Ê M i b converge to the same value as i ½. Therefore the left hand side of the equation (31) converges to 0 as i ½, and thus b (32) lim i½ a ds M i s 0 F i 2s H i 2 Æg 0. From this we can take a sequence s i ¾ [a, b] such that we have F i (33) 2s i H i 0 M i s i 0 as i ½. Suppose that p i attains min M r(f i s i ), and put Then p i also attains min M r(f ti ) and i Ï r 2 (F i (p i, s i )) 2 i r 2 (F(p i, t i )). (34) i 2 i r 2 (F(p i, t i )) s ir 2 (F(p i, t i )) T t i. It then follows from the condition (c) of Definition 1.1 that (35) bk 1 i ak 2. Thus, the image of F i (, s i ) uniformly stays away from the apex, and that F i (p i, s i ) stays in a compact region in C(N) for the minimum point (p i, s i ) for r(f i ). Put Ï bk 1. Let W be the manifold obtained from C(N) by cutting out the portion {r Ô 2}, and let V be the manifold obtained by gluing W and W smoothly 0

18 1070 A. FUTAKI, K. HATTORI AND H. YAMAMOTO along their boundaries. This V contains C(N) {r Ô } and the image of F i (M,si ) is included in that part. Since the higher derivatives of the second fundamental form are bounded as shown above, we can apply Theorem 1.2 in [3] (see also [1]) by taking (M k, p k ) to be (M, p k ), (N k, h k, x k ) to be (V, h, F k (p k, s k )) and F k to be F k, where the metric h is chosen so that h coincides with the cone metric on C(N) {r Ô }. Then we obtain a limit F ½ Ï M ½ N ½ which satisfies the equation of self-similar solution to the mean curvature flow by (33). But since x i F i (p i, s i ) stays in a compact region we have N ½ V. The limiting self similar solution then defines a flow in the cone C(N) satisfying the mean curvature equation. This completes the proof of Theorem 1.2. EXAMPLE 6.1 (Examples of type I c singularities). Here we show a simple example of the mean curvature flow developing the type I c singularity. For ½ a b ½, assume that there exists a mean curvature flow Ï M [a, b) N on (N, g), namely satisfies (s)(, s) Hs N, where H s N is the mean curvature vector with respect to the embedding (, s)ï M N. Then FÏ M [0, T (1 e 2m(b a) )) C(N) defined by F(p, t) Ï ((p, «(t)), (t)) ¾ N Ê, «(t) Ï a 1 2m log 1 t T (t) Ï Ô 2m(T t), becomes Ô a solution for mean curvature flow equation with initial data F 0 0 Ï M N { 2mT} C(N), where m dim M. The second fundamental form II C(N) t of the embedding F(, t)ï M C(N) is given by II C(N) t, II N «(t) r(f(p, t))g Mt Å r, where II«(t) N is the second fundamental form of the embedding M t (M, «(t)) N. Then we obtain II C(N) t 2 m Æg 1 1 2(T t) m p¾mii sup N 2 «(t) (p)2 g, since II N «(t) Æg r(f(p, t)) 1 II N «(t) g. Hence F develops a type I singularity at t T, if b ½ and sup II (p) s N g ½. p¾m,sa The condition (b) and (c) of Definition 1.1 are obviously satisfied since r(f(p, t)) Ô 2m(T t).

19 SELF-SIMILAR SOLUTIONS TO MEAN CURVATURE FLOW Special Lagrangian submanifolds in toric Calabi Yau cones In this section we construct special Lagrangian submanifolds in toric Calabi Yau cones. Let V be a Ricci-flat Kähler manifold with a Kähler form and of dim V n. Then the canonical line bundle K V is flat. V is said to be a Calabi Yau manifold if in addition K V is trivial and V admits a parallel holomorphic n-form Đ. This implies that, with a suitable normalization of Đ, we have n ĐĐ. n Ô 1 n! ( 1)n(n 1)2 2 Let L be a real oriented n-dimensional submanifold of V. Then L is called a special Lagrangian submanifold of V if L 0 and Im Đ L 0. Toric Calabi Yau cones are exactly the Kähler cones over Sasaki Einstein manifolds. They are described as toric Kähler cones obtained from toric diagram of height 1. This result was obtained in [4] and [2], which we outline below. DEFINITION 7.1 (Good rational polyhedral cones, cf. [12]). Let g be the dual of the Lie algebra g of an n-dimensional torus G. Let g be the integral lattice of g, that is the kernel of the exponential map expï g G. A subset C g is a rational polyhedral cone if there exists a finite set of vectors i ¾ g, 1 i d, such that C {y ¾ g y, i 0 for i 1,, d}. We assume that the set i is minimal in that for any j C {y ¾ g y, i 0 for all i j} and that each i is primitive, i.e. i is not of the form i a for an integer a 2 and ¾ g. (Thus d is the number of facets if C has non-empty interior.) Under these two assumptions a rational polyhedral cone C with nonempty interior is said to be good if the following condition holds. If {y ¾ C y, i j 0 for all j 1,, k} is a non-empty face of C for some {i 1,, i k } {1,, d}, then i1,, ik independent over and generates the subgroup {È k j1 a j i j a j ¾ Ê } g. are linearly DEFINITION 7.2 (Toric diagrams of height l, cf. [2]). An n-dimensional toric diagram with height l is a collection of i ¾ n g which define a good rational polyhedral cone and ¾ É n (É g ) such that (1) l is a positive integer such that l is a primitive element of the integer lattice n g.

20 1072 A. FUTAKI, K. HATTORI AND H. YAMAMOTO (2), i 1. We say that a good rational polyhedral cone C is associated with a toric diagram of height l if there exists a rational vector satisfying (1) and (2) above. The reason why we use the terminology height l is because using a transformation by an element of SL(n, ) we may assume that ¼ l 0 ½ and the first component of i is equal to l for each i. Theorem 7.3 ([4], [2]). Toric Sasaki Einstein manifolds are exactly those whose Kähler cones are obtained by the Delzant construction from toric diagram of fixed height and applying the volume minimization of Martelli Sparks Yau [13]. Equivalently, Toric Ricci-flat Kähler manifolds are exactly those obtained by the Delzant construction from toric diagram of fixed height and applying the volume minimization of Martelli Sparks Yau [13]. For a Ricci-flat toric Kähler cone V obtained from a toric diagram of height l, there exists a parallel holomorphic section of K Ål V. In particular if l 1 the Kähler cone manifold V is a Calabi Yau manifold. From now on we assume l 1. Then it is shown in [2] that the parallel holomorphic n-form is given in the form Đ e È n i1 i z i dz 1 dz n where z 1,, z n are holomorphic logarithmic coordinates. Since V is obtained from a toric diagram of height 1 we may assume t ( 1, 0,, 0). In this case we have Đ e z1 dz 1 dz n. We now apply a method used in [6] and [11]. Their method is summarized in [11] as follows. Proposition 7.4 ([11]). Let (V, J,, Đ) be a Calabi Yau manifold of complex dimension n, and H be a compact connected Lie group of real dimension n 1 acting effectively on V preserving the Calabi Yau structure. Suppose there exist a moment map Ï V h and a H-invariant (n 1)-form «such that for any X 1,, X n 1 ¾ h we have Im Đ(, X 1,, X n 1 ) d(«(x 1,, X n 1 ))

21 SELF-SIMILAR SOLUTIONS TO MEAN CURVATURE FLOW 1073 where X i ¾ h are identified with vector fields on V. Then for any c ¾ Z h, c ¼ ¾ Ê and any basis {Y 1,, Y n 1 } h, the set L c,c ¼ 1 (c)(«(y 1,, Y n 1 )) 1 (c ¼ ) is a H-invariant special Lagrangian submanifold of V. We refer the reader to [11] for the proof of Proposition 7.4. We now apply Proposition 7.4 to toric Calabi Yau manifold obtained from toric diagrams of height 1 with Đ e z1 dz 1 dz n, «Im(e z1 dz 2 dz n ), and with Y j 2 Im(z j ) and H the subtorus T n 1 generated by Y 1,, Y n 1. Then one easily finds that and Im Đ(, Y 1,, Y n 1 ) d(«(y 1,, Y n 1 )), «(Y 1,, Y n 1 ) 1 i n (ez1 ( 1) n e z 1 ). Thus the assumptions of Proposition 7.4 is satisfied, and we have proved the following. Theorem 7.5. Let V be a toric Calabi Yau manifold obtained from a toric diagram of height 1. Let Đ e z1 dz 1 dz n be the parallel holomorphic n-form described as above. Then there is a T n 1 -invariant special Lagrangian submanifold described as 1 (c) 1 i n (ez1 ( 1) n e z 1 ) c ¼ where T n 1 is a subtorus generated by Im(z 2 ),, Im(z n ) and Ï V h is a moment map. EXAMPLE 7.6. Take V to be the flat n, and let Û 1,, Û n be the standard holomorphic coordinates with Đ dû 1 dû n. The logarithmic holomorphic coordinates Ú 1,, Ú n are given by Û i e Úi. Thus, we have Đ e (Ú1 Ú n) dú 1 dú n.

22 1074 A. FUTAKI, K. HATTORI AND H. YAMAMOTO Taking t ( 1, 0,, 0) amounts to changing the coordinates z 1 Ú 1 Ú n, z 2 Ú 2,, z n Ú n. Then with the new coordinates we have Đ e z1 dz 1 dz n. In this situation the points in 1 (c) are described as Û 2 2 Û 1 2 c 2,, Û n 2 Û 1 2 c n. If n is even then (e z1 ( 1) n e z 1 )i n c ¼ if and only if (Û 1 Û n ) c ¼, and If n is odd then (e z1 ( 1) n e z 1 )i n c ¼ if and only if Im(Û 1 Û n ) c ¼. This is exactly the same as Theorem 3.1 in [7]. 8. Infinitesimal deformations of special Lagrangian cones In this section we consider the infinitesimal deformations of special Lagrangian cones embedded in the cone of Sasaki Einstein manifolds. DEFINITION 8.1. A Riemannian manifold (N, g) is called a Sasakian manifold if its Riemannian cone (C(N), Æg) is a Kähler manifold with respect to some integrable complex structure J over C(N). A Reeb vector field on the Sasakian manifold (N, g) is a Killing vector field on N given by Ï J(r(r)). For a Sasakian manifold (N, g), a contact form ¾ Đ 1 (N) on N is given by Ï g(, ). Then the Kähler form ¾ Đ 2 (C(N)) on C(N) is described as d(r 2 ). DEFINITION 8.2. For a smooth manifold N, a cone submanifold C of C(N) is a submanifold of C(N) which can be written as C C() for a submanifold N. For a Sasakian manifold (N, g, ), a cone submanifold C C(N) is a Lagrangian cone if it is a Lagrangian submanifold of (C(N), ). The following proposition is well-known but here we give a proof for readers convenience. Proposition 8.3. A submanifold N is Legendrian if and only if C() Ê C(N) is Lagrangian with respect to the Kähler form on C(N). Proof. Let N be a Legendrian submanifold. For any p ¾, open neighborhood U and u, Ú ¾ X (U), we have (u, Ú) d(u, Ú) ([u, Ú]) 0, u, g(u, ) (u) 0, r

23 SELF-SIMILAR SOLUTIONS TO MEAN CURVATURE FLOW 1075 since 0 and [u, Ú] ¾ X (U). Hence C() C(N) is Lagrangian. Conversely, let C() C(N) be Lagrangian and take u ¾ T p arbitrarily. Then (u) g(u, ) u, r 0, which implies that N is a Legendrian submanifold. Proposition 8.4. Let (V, J, ) be a Ricci-flat Kähler manifold of dim n with H 1 DR (V, Ê) 0, and assume that the canonical line bundle K V is holomorphically trivial. Then there exists a holomorphic n form Đ ¾ Đ (n,0) (V ) satisfying (36) n Ô 1 n! ( 1)n(n 1)2 2 n ĐĐ. Proof. From the assumption there is a nowhere vanishing holomorphic n form Đ 0 ¾ Đ (n,0) (V ) on V. Since Đ 0 is holomorphic, dđ 0 0. The Kähler form on V induces a hermitian metric on K V by h Ï h(đ 0, Đ 0 ) Ï n!( 1) n(n 1)2 Ô 1 2 n Đ0 Đ 0 n. Now we put Đ Ï h 1 2 e Ô 1 Đ 0 for ¾ C ½ (V, Ê), which satisfies the equation (36). Then it suffices to show that there exists ¾ C ½ (V,Ê) such that dđ0. From dđ 0 0, we have dđ Æ (h 12 eô 1 )Đ 0 h 12 eô h 1 Æ h Ô 1Æ Đ0 h 12 eô Æ log h Ô 1Æ Đ0. Thus the problem is reduced to show the existence of the function which satisfies Æ( (12) log h Ô 1) 0. Recall that is Ricci-flat Kähler form. Then the curvature form of the Hermitian connection on K V induced from h is equal to zero, we have dd c log h 0. Now we have assumed HDR 1 (V, Ê) 0, there exists ¾ Ç C½ (V, Ê) such that d c log h Ô ( 1 Ô 1)logh Æ dç (Æ )Ç. By comparing (0,1)-part, we have (logh Æ Ô 1Ç)0, consequently we obtain the assertion by putting Ç 2. From now on suppose (N, g, ) is a Sasaki Einstein manifold of dimension 2n 1, hence the Kähler structure on C(N) is Ricci-flat. Moreover we assume the canonical

24 1076 A. FUTAKI, K. HATTORI AND H. YAMAMOTO bundle K C(N) is trivial. Since (N, g) is an Einstein manifold with positive Ricci curvature, then H 1 (C(N), Ê) H 1 (N, Ê) 0. Therefore we have a holomorphic n-form Đ on C(N) satisfying (36). Now we denote by H É and H the mean curvature vector of C() C(N) and N, respectively. Then the direct calculation gives H É r 2 H, therefore C() is minimal if and only if is minimal. It is well known that the mean curvature of a Lagrangian submanifold embedded in a Calabi Yau manifold is equal to d under the identification of vector fields and 1-forms by the symplectic form, where is the Lagrangian angle. Then the Lagrangian submanifold embedded in the Calabi Yau manifold is minimal if and only if the Lagrangian angle is constant. In particular it is special Lagrangian if the Lagrangian angle is equal to zero. Hence N is minimal Legendrian if and only if C() C(N) is Lagrangian with constant Lagrangian angle. In [14], the infinitesimal deformation spaces of minimal Legendrian submanifolds embedded in -Sasaki Einstein manifolds are studied. Here we observe the infinitesimal deformation spaces of special Lagrangian cones in C(N), using the results obtained in [5]. Let C() be a special Lagrangian submanifold in C(N), and we have orthogonal decompositions T C(N) C() T C() NC() and T N T N, where N, NC() are normal bundles. Then for any (x, r) ¾ C() we have the natural identification N (x,r) C() N x. The infinitesimal deformations of cone submanifolds of C(N) is generated by the smooth 1-parameter families of cone submanifolds {C( t ) 1 N ( t )Á t }, where { t Á t } is the smooth families of submanifolds of N which satisfies 0, and N Ï N Ê N is the projection onto the first component. Since the infinitesimal deformations of N are parameterized by smooth sections of N, the infinitesimal deformations of cone submanifolds are parameterized by A C() Ï {«N «0 ¾ ¼(NC())Á «0 ¾ ¼(N )}. Then «(x,r) ¾ N (x,r) C() N x is independent of r for each «¾ A C(). Since C() is Lagrangian, NC() is identified with the cotangent bundle T C() by the bundle isomorphism ÇÏ NC() T C() defined by Ç(Ú) Ï Ú Ç Ç(Ú, ). By the results in [5], the infinitesimal deformations of special Lagrangian submanifolds of C() are parameterized by harmonic 1-forms on C(). Thus the infinitesimal deformations of special Lagrangian cones of C() are parameterized by H C() Ï {Ç(«) ¾ Đ 1 (C())Á «¾ A C(), d Ç(«) d Ç(«) 0}, where Ç is the isomorphism induced by, and is the Hodge star with respect to the induced metric Æg C(). To study the vector space H C(), we need the next lemma.

25 SELF-SIMILAR SOLUTIONS TO MEAN CURVATURE FLOW 1077 Lemma 8.5. Under the natural identification T (x,r) C() T x T r Ê, we have Ç(A C() ) { (x,r) r³(x) dr r 2 x ¾ Đ 1 (C())Á ³ ¾ C ½ (), ¾ Đ 1 ()}. Proof. Define a diffeomorphism m a exp(ar(r))ï C(N) C(N) by m a (p,r)(p,ar) for a 0. First of all we show that m a is a biholomorphism. Since (dda)(m a ) J (m a ) L r(r) J, it suffices to show L r(r) J 0. Now we may write r(r) J, then for any x ¾ C(N) and open neighborhood x ¾ U C(N) and Ú ¾ X (C(N)), (L J J)(Ú) L J (J Ú) J(L J Ú) [J, J Ú] J([J, Ú]) N J (, Ú) J 2 [, Ú] J[, J Ú] N J (, Ú) J{(L J)(Ú)}, where N J is the Nijenhuis tensor. Thus we have L J J 0 since J is integrable and L J 0, hence m a is a biholomorphism. Next we show that Ç(A C() ) { ¾ Đ 1 (C())Á m a a 2 for all a ¾ Ê }. Since m a satisfies m a Æg m a (dr 2 r 2 g)d(ar) 2 (ar) 2 g a 2 Æg, we obtain m a a 2. By the definition of A C(), we may write A C() {«¾ Đ 1 (C())Á (m a ) ««for all a ¾ Ê }. For any «¾ ¼(NC()), we have m a (Ç(«)) m a ( «) (ma ) 1 «m a a2 Ç((m a ) 1 «) a 2 Ç(«)a 2 Ç((m a ) 1 ««). Therefore the equation m a (Ç(«))a2 Ç(«) holds for all a ¾ Ê if and only if «¾A C(). Now we take ¾ Đ 1 (C()) and decompose it as (x,r) (x, r)(x, r) dr such that (x, r) ¾ T x and ¾ C½ (C()). then m a a 2 is equivalent to m a m a m a a dr, (x, ar) a 2 (x, r), (x, ar) a(x, r). Thus we may put r 2 and r³ for some ¾ Đ 1 () and ³ ¾ C ½ ().

26 1078 A. FUTAKI, K. HATTORI AND H. YAMAMOTO Theorem 8.6. The vector space H C() is isomorphic to where ½ d d and d is a formal adjoint operator of d with respect to the metric Æg. Ker(½ 2n) {³ ¾ C ½ ()Á ½ ³ 2n³}, Proof. From Lemma 8.5, all ¾ Ç(A C() ) can be written as r³ dr r 2. Then we have d r dr (2 d³)r 2 d, from which it follows that d 0 is equivalent to 2 d³. Next we calculate d. Denote by vol the volume form of g. Since the volume forms of Æg C() is given by r n 1 dr vol, we can deduce r n 3 dr, dr r n 1 vol, where is the Hodge star operator with respect to g. Consequently, we obtain Hence d d 0 is equivalent to d r n 1 dr (d n³ vol ). 1 2 d³, n³ vol 1 2 d d³ 0, and the latter equation is equivalent to d d³ 2n³. In [14], the infinitesimal deformation spaces of minimal Legendrian submanifolds in Sasaki Einstein manifolds are studied. Proposition 8.6 is also obtained from the case of -Ricci constant A is equal to 2n 2 in [14]. Here we should pay attention that the dimension of infinitesimal deformation spaces obtained in [14] is equal to 1 dim Ker(½ 2n), since the deformations of C() generated by Reeb vector field is not special Lagrangian cone, but minimal Lagrangian cone whose Lagrangian angle is not equal to zero. Actually, if we put «, then Ç(«) r dr and d d(r n vol ) nr n 1 dr vol 0, accordingly this «does not generate deformations of special Lagrangian cones.

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