PENGFEI GUAN, QUN LI, AND XI ZHANG

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1 A UNIQUENESS THEOREM IN KÄHLER GEOMETRY PENGFEI GUAN, QUN LI, AND XI ZHANG Abstract. We consider compact Kähler manifolds with their Kähler Ricci tensor satisfying F (Ric) = constant. Under the nonnegative bisectional curvature assumption and certain conditions on F, we prove that such metrics are in fact Kähler-Einstein. 1. Introduction The study of extremal metrics is an important topic in Kähler geometry. Kähler- Einstein metrics, Kähler metrics with constant scalar curvature and other forms of elementary symmetric functions associated to the Kähler Ricci curvature tensor are examples of this type of metrics, we refer [5, 13, 6] and references therein. In [10], it was proved that if a compact Kähler manifold with nonnegative bisectional curvature and constant scalar curvature, then it is automatically a Kähler-Einstein manifold. In this paper, we consider Kähler metric g with its Kähler Ricci tensor Ric = (R i j) satisfying certain relation: (1.1) F (Ric) = const, for some elliptic F. Einstein. We denote We are interested to know when such metrics are Kähler- H = {all n n Hermitian matrices}, Γ n = {all positive n n Hermitian matrices}, Γ 1 = {A H σ 1 (A) > 0} Throughout the paper, we assume F is a function defined on a convex subset Γ of H such that Γ n Γ and F is invariant under the unitary transformations. We will also assume F is monotone, that is, (1.2) ( F ) > 0. w i j We will prove the following uniqueness theorem. Using the notation in [10, 12], we say a multi-linear form on a manifold is quasi-positive if it is nonnegative everywhere and strictly positive at least at one point. Theorem 1. Let (M, g) be a compact Kähler manifold with Kähler metric (g i j) of quasi-positive bisectional curvature tensor. Suppose Ric i j satisfies equation (1.1) with F as in (1.2). If either (1) Γ 1 Γ and F is locally concave in Γ, or The first author was supported in part by an NSERC Discovery grant, the third author was supported by NSF in China, No

2 2 PENGFEI GUAN, QUN LI, AND XI ZHANG (2) F satisfies the following condition that for every positive definite Hermitian matrix A and for every complex-valued vector X i j, (1.3) (F i j,γ s (A) + F γ j A i s )X X i j γ s 0, i,j,γ,s=1 then g is a Kähler-Einstein metric. If n = 2, we only need ellipticity condition (1.2) on F. More specifically, the result can be strengthened as follow. Theorem 2. Let (M, g) be a compact Kähler surface with Kähler metric (g i j) of quasi-positive bisectional curvature tensor. If its Kähler-Ricci tensor satisfies equation (1.1) with F as in (1.2), then g is a Kähler-Einstein metric. Our results can be interpreted as a nonlinear version of Wu s result for scalar curvature in [10]. The method we adopt to establish the main theorem is the constant theorem for fully nonlinear elliptic partial differential equations. This type of arguments was used in the real cases in recent papers [3, 4]. We remark that the curvature conditions imposed on (M, g) in Theorem 1 can be weakened. We refer the last section for the discussions. Acknowledgement: We would like to thank Lei Ni for bringing papers [10, 12] to our attention. 2. Preliminaries Let (M, g, J) be a compact Kähler manifold with the Kähler metric g i j. In local coordinate, its Kähler form ω = 1g i jdz i d z j is a closed real (1, 1)-form, this is the same as saying that g i j z k = g k j z i and z k = g i k z j for all i, j, k. The Kähler class of ω is the cohomology class [ω] in H 2 (M, R). By the Hodge theory, any other Kähler metrics in the same class is of the form g i j ω φ = ω + 1 φ for some real-valued function φ on M. The Christoffel symbols of the metric g i j are given by Γ k ij = g k l g i l z j and Γ kī j = gl k g lī z j where (g i j ) = (g i j) 1. It is easy to see that Γ k ij is symmetric in i and j and is Γ kī j symmetric in ī and j. The curvature tensor of the metric g i j is defined as R i jk l = g m jr m ik l = 2 g i j z k z l + gm n g i n g m j z k z l. We see that R i jk l is symmetric in i and k, in j and l and in the pairs {i j} and {k l}. We say (M, g, J) has positive holomorphic bisectional curvature if R i jk lv i v j w k w l > 0 for all nonzero vectors v and w in the holomorphic tangent bundle of M.

3 A UNIQUENESS THEOREM IN KÄHLER GEOMETRY 3 (2.1) The Ricci tensor of ω is obtained by taking the trace of the curvature tensor, So its Ricci form is R i j = g k lr i jk l = i j log det(g i j). Ric(ω) = 1R i jdz i d z j = 1 log det(g i j). It is a real and closed (1, 1)-form. The scalar curvature of ω is R = g i j R i j. Given any (1, 1) tensor T i j,its covariant derivatives are defined as (2.2) T i j,k = z k T i j Γ l ikt l j and (2.3) T i j, k = z k T i j Γ l j k T i l. We shall need the following commutation formulas for covariant derivatives T i j,kl = T i j,lk, T i j, k l = T i j, l k, T i j,k l = T i j, lk + g m n R i nk lt m j g m n R m jk lt i n. The second Bianchi identity in the Kähler case is R i jk l,m = R i jm l,k and R i jk l, n = R i jk n, l. From (2.1), (2.2) and (2.3), one can easily check that R i j,k = R k j,i and R i j, l = R i l, j. For simplicity, we choose the normal coordinate near the considered point. By directly calculation, we have (2.4) R i j, ll = R i l, jl = R i l,l j R i ml jr m l + R m ll jr i m R i ml jr m l + R m jr i m = 2 R z i z j (2.5) R i j,l l From above we known = R l j,i l = R l j, li R m ji lr l m + R l mi lr m j R m ji lr l m + R i m R m j = 2 R z i z j Ric = g l k l kric = g l k k l Ric. Here = 1 2 gl k( l k + k l ) is the complex Laplacian. Definition 2.1 We say that the orthogonal bisectional curvature is nonnegative means that R iīj j 0 for any i j. We now list some properties related to functions defined on a set of Hermitian matrices. If F defined in the previous section is invariant under unitary transformations, then F is a symmetric function on eigenvalues of the matrices. On the other hand, let Ψ R n be a symmetric domain, set Γ = {A H : λ(a) Ψ}.

4 4 PENGFEI GUAN, QUN LI, AND XI ZHANG If f is symmetric in Ψ, we may extend f to F : Γ C, by F (A) = f(λ(a)). Let us denote f k = f λ k, f kl = 2 f λ k λ l, F α β = F A, and F α β,γ s = 2 F α β A α β A γ s. It is well known the second derivatives of F can be computed as follows, see [1]. Lemma 1. At any diagonal A Γ with distinct eigenvalues, let F (B, B) be the second derivative of F in direction B H, then (2.6) F n (B, B) = f jk B j jb k k + 2 f j f k B λ j λ j k 2. k j,k=1 j<k Corollary 1. (a). F satisfies condition (1.3) if and only if 0 l n, for all 0 λ 1 λ n and λ i > 0, i n l + 1 (2.7) j,k=n l+1 f jk (A)X j jx k k + 2 n l+1 j<k f j f k X λ j λ j k 2 + k i,k=n l+1 f i (A) λ k X i k 2 0 for every hermitian matrix X = (X j k) with X j k = 0 if j n l. (b). If F satisfies condition (1.3), then so is the function G(A) = F (A + ai) for any nonnegative constant a. Proof of part (a) of the corollary follows the same way as in the proof of Corollary 1 in [4]. Part (b) of the corollary follows part (a) directly. 3. Proof, Part 1 We start with the following proposition (a complex version of Berger s theorem, e.g., Theorem in [2]), which was implied in Lemma 1 in [10]. Since the proof is simple, we provide it here for completeness. Proposition 1. Let (M, ω) be a compact Kähler manifold with constant scalar curvature. Assume that the orthogonal bisectional curvature of ω is quasi-positive, then the Kähler metric ω must be Kähler-Einstein. Proof. By directly calculation, we have (3.1) Ric 2 = l l < Ric, Ric > = l {< lric, Ric > + < Ric, lric >} = 2 < l lric, Ric > + < l Ric, lric > + < lric, l Ric > = 2 < Ric, Ric > + Ric 2 = Ric 2 + 2{ 2 R z i z R j jī + R i p R p jr jī R p ji lr l p R jī }. Now rotate at a given point the coordinate system such that (R i j) is diagonal, and assume that {λ i } n 1 are the eigenvalues. Then, we have (3.2) Ric 2 = Ric R z i z i λ i + 2λ 3 i 2R i jjīλ i λ j = Ric R z i z i λ i + R iīj j(λ i λ j ) 2. Since R iīj j 0 and R C by assumption, the last term in the above equality is larger than or equal to zero. Hence we finally conclude that Ric 0. Then, the eigenvalue of Ricci curvature must be constantly. On the other hand, since R iīj j > 0 at some point by assumption, we have λ i = λ j for any i j. So the Kähler metric ω is of constant Ricci curvature, i.e. it is a Kähler-Einstein metric.

5 A UNIQUENESS THEOREM IN KÄHLER GEOMETRY 5 Proof of the Main Theorem, Part 1. We adapt techniques of Ecker-Huisken in [7], where they treat Codazzi tensors on Riemannian manifolds. Since F (W ) = F (R j i ) = const., where Rj i = gj lr i l, then, (3.3) 0 = F = z m ( F R j i = F R j i R j i z m ) = R i j, mm + z ( F m R j i R i s, m g j s ) 2 F R j i Rl k R i j, mr k l,m = i F 2 R λ i z i z + i i,k,m 2 F λ i λ k m R iī m R k k F λ j )(λ i λ j ). i,j R iīj j( F λ i By the concavity assumption on F, the second term on the right hand side of above equality is non-positive. Again, the concavity of F yeilds (see [7]) (3.4) for any i, j. Hence we conclude (3.5) ( F λ i F λ j )(λ i λ j ) 0, i F 2 R λ i z i z i 0. Since F is monotone, the strong maximum principle then yields R Const. It now follows Proposition 1 that g is Kähler-Einstein. Remark 1. It s clear from the proof, Part 1 of Theorem 1 remains true if assumption on the bisectional curvature is replaced by the same assumption on the orthogonal bisectional curvature. 4. Proof, part 2 Our proof of Theorem 1 part 2 uses arguments of microscopic convexity principle for fully nonlinear equations of real variables in [4, 3]. This type of arguments were generalized to complex case to deal with plurisubharmonic solutions of fully nonlinear equations in complex domains in C n in [11]. We adapt them to Kähler manifolds here. Let (4.1) a = min z M r s(z), where r s (z) is the smallest eigenvalue of Ric at z. We set W = (Ric ag), so we have W 0 on M and W satisfies equation (4.2) F (W + ai) = constant. By Corollary 1, G(W ) = F (W + ai) still satisfies condition 1.3. Part 2 of Theorem 1 follows from the next proposition. Proposition 2. Suppose M and F as in Theorem 1, if W is a nonnegative definite symmetric tensor satisfying equation (1.1), and rank of W is less than n at some point, then W 0 on M.

6 6 PENGFEI GUAN, QUN LI, AND XI ZHANG Proof. The proof is a complex version of the arguments in [4], see also [11]. Suppose W attains minimal rank l at z 0 M, pick an open neighborhood O of z 0, for any z O, let λ 1 λ 2 λ n be the eigenvalues of W at z. There is a positive constant C > 0 such that λ n λ n 1 λ n l+1 C. Let G = {n l + 1, n l + 2,, n} and B = {1, 2,, n l} be the good and bad sets of indices respectively. Let Λ G = (λ n l+1,, λ n ) be the good eigenvalues of W at z, for simplicity, we also write G = Λ G if there s no confusion. First we introduce a very useful notation, the so called k-th elementary symmetric functions. For a complex Hessian matrix W, write its eigenvalue as λ 1,..., λ n, then the k-th elementary symmetric function of W is, σ k (W ) = λ i1 λ ik, where the sum is taken over all strictly increasing sequences i 1,..., i k of the indices from {1,..., n}. With this notation at hand, we set (4.3) φ(z) = σ l+1 (W ). For two functions defined in an open set O M, y O, we say that h(y) k(y) provided there exist positive constants c 1 and c 2 such that (h k)(y) (c 1 φ + c 2 φ)(y). We also write h(y) k(y) if h(y) k(y) and k(y) h(y). Next, we write h k if the above inequality holds in O, with the constants c 1 and c 2 depending only on n and C (independent of y and O). Finally, h k if h k and k h. Write W = (W i j), for each fixed point z O, we choose a normal complex coordinate system so that W is diagonal at z, and W iī = λ i, i = 1,..., n. Following the computation in [9], we calculate φ and its derivatives up to second order. Since W is diagonal at z, we have that ( ) 0 φ(z) σ l (G) W iī, W iī so W iī 0, i B. This relation yields that, for 1 m l, σ m (W ) σ m (G), σ m (W ij) σ m (W j) { σm (G j), if j G; σ m (G), if j B. σ m (G ij), if i, j G; σ m (G j), if i B, j G; σ m (G), if i, j B, i j. where we denote (W i) to be the (n 1) (n 1) matrix with ith row and ith column deleted, and (W ij) the (n 2) (n 2) matrix with i,jth rows and i,jth columns deleted, and (G i) means the notations applied to the subset G. Also, 0 φ α σ l (G) W iīα W iīα 0 φᾱ σ l (G) W iīᾱ W iīᾱ

7 A UNIQUENESS THEOREM IN KÄHLER GEOMETRY 7 And by Proposition 2.2 in [9], σ l+1 (W ) W i j { σl (G), i = j B; 0, otherwise. and for 1 m n, 2 σ m (W ) W i j W r s = σ m 2 (W ir), if i = j, r = s, i r; σ m 2 (W ij), if i = s, r = j, i j; 0 otherwise. By the above relations, if l n 2, we calculate the following, note that later on we ll use notations W i jr s to demonstrate the second order derivatives of the tensor W i j with respect to z r and z s, similar for W i jr and W i j s. φ αᾱ = i,j,r,s=1 2 σ l+1 (W ) W i j W r s W i jαw r sᾱ + i,j=1 σ l+1 (W ) W W i jαᾱ i j = i,j = ( i G j B ( i G j B σ l 1 (W ij)w iīα W j jᾱ i,j i, + i, i,j B σ l 1 (W ij)w i jαw jīᾱ + i )σ l 1 (W ij)w iīα W j jᾱ + )σ l 1 (W ij)w i jαw jīᾱ + i i,j B σ l (G)W iīαᾱ σ l (G)W iīαᾱ We want to simplify the above expression, firstly i G j B σ l 1 (W ij)w iīα W j jᾱ i G j B σ l 1 (G i)w iīα W j jᾱ = σ l 1 (G i)w iīα W j jᾱ 0 i G j B Similarly, σ l 1 (W ij)w iīα W j jᾱ 0. For any i B fixed, and α, W iīᾱ j B W j i j jᾱ, so i,j B σ l 1 (W ij)w iīα W j jᾱ σ l 1 (G)W iīα W iīᾱ = σ l 1 (G) W iīα 2. Finally, as G = l, (G ij) = l 2, so σ l 1 (W ij) σ l 1 (G ij) = 0, for all i, j G.

8 8 PENGFEI GUAN, QUN LI, AND XI ZHANG So, we have φ αᾱ σ l 1 (G) W i jα 2 + i = i,j B σ l 1 (G) W iīα 2 i G j B σ l (G)W iīαᾱ i=1 i G j B σ l 1 (G) W i jα 2 i,j B σ l (G)W iīαᾱ σ l 1 (G) W i jα 2 i,j B σ l 1 (G i) W i jα 2 σ l (G)W iīαᾱ σ l 1 (G i)( W i jα 2 + W jīα 2 ) σ l 1 (G j)( W i jα 2 + W jīα 2 ) σ l 1 (G j) W i jα 2 Since F α β is diagonal at z, we do the contraction, F αᾱ φ αᾱ α=1 W iīαᾱ F αᾱ σ l 1 (G) F αᾱ σ l (G) α=1 α=1 σ l 1 (G j)( W i jα 2 + W jīα 2 ). α=1 F αᾱ i,j B W i jα 2 Taking second derivatives to the equation (1.1), we have that α,β=1 F α βw α βi = 0, α,β,γ,s=1 F α β,γ s W α βi W γ sī + α,β=1 F α βw α βiī = 0. While by the computation of the covariant derivatives on the Kähler manifold, we have that W iīαᾱ = W αīiᾱ = W αīᾱi + s W sī R α siᾱ s W α s R sīiᾱ = W αᾱiī + W iī R iīαᾱ W αᾱ R αᾱiī.

9 A UNIQUENESS THEOREM IN KÄHLER GEOMETRY 9 Then, F αᾱ R αᾱiī W αᾱ F αᾱ φ αᾱ σ l (G) α=1 α G σ l (G) F α β,γ s W α βi W γ sī α,β,γ,s=1 σ l 1 (G) F αᾱ W i jα 2 α=1 i,j B F αᾱ σ l 1 (G j)( W i jα 2 + W jīα 2 ) α=1 = σ l (G) F αᾱ R αᾱiī W αᾱ α G σ l (G) F αᾱ,β βw αᾱi W β βī α,β=1 σ l (G) F α β,βᾱ W α βi W βᾱī α,β=1 σ l 1 (G) F αᾱ W i jα 2 α=1 i,j B F αᾱ σ l 1 (G j)( W i jα 2 + W jīα 2 ) α=1 (4.4) = σ l (G) F αᾱ R αᾱiī W αᾱ α G σ l (G) ( + +2 )F αᾱ,β βw αᾱi W β βī α,β B α,β G α B β G σ l (G) ( )F α β,βᾱ W α βi W βᾱī α,β B α,β G α G α B β B β G σ l 1 (G) F αᾱ W i jα 2 α=1 i,j B ( + ) α G α B F αᾱ σ l 1 (G j)( W i jα 2 + W jīα 2 ) So, 1 σ l (G) α=1 F αᾱ φ αᾱ α G F αᾱ R αᾱiī W αᾱ I II i III,

10 10 PENGFEI GUAN, QUN LI, AND XI ZHANG where I = ( +2 )F αᾱ,β βw αᾱi W β βī + ( + )F α β,βᾱ W α βi 2 α,β B α B α,β B α G β G β B ( 1 ) F αᾱ W λ i jα F αᾱ W k λ jīα 2, j and II i = finally, k α,β G α=1 i,j B i,α B (F αᾱ,β βw αᾱi W β βī + F α β,βᾱ W α βi 2 + F αᾱ III = (F i j,jī + F iī ) W λ i jα 2. j i,α B λ β ( W α βi 2 + W βīα 2 ), Using the nonnegativity of the hermitian matrix (W α β) C 2, by a lemma in [3], α, β (4.5) W α β 2 C(W αᾱ + W β β), for some constant C independent of the indices. Therefore every term in I with at least two indices in B is under control since W αᾱ 0 for α B. The only term in I needs to take care of is F αᾱ,β βw αᾱi W β βī. 2 α B β G We note that F αᾱ,β β F γ γ,β β for all α, γ B. This property can be obtained by our previous observation, σ m (W α) σ m (G), if α B, and the fact discovered by Glaeser ([8]) that any symmetric function F (W ) acting on a matrix W can actually be written as F (W ) = F (σ 1 (W ), σ 2 (W ),, σ n (W )). In turn, F αᾱ,β βw αᾱi W β βī F γ γ,β βw β βī( W αᾱi ) 0. β G α B α B β G We have shown 0 I. For the term II i, we make use of condition (1.3). (4.6) σ l (G)II i = σ l (G) F αᾱ,β βw αᾱi W β βī (4.7) + α,β G + α,β G α,β G (σ l (G)F α β,βᾱ + F αᾱ σ l 1 (G β)) W α βi 2 F αᾱ σ l 1 (G β)) W αīβ 2, The sum of the first two terms is nonnegative by the inequality (2.7).

11 A UNIQUENESS THEOREM IN KÄHLER GEOMETRY 11 To deal with III, For each α B, we may write III = (F i j,jī + F iī ) W λ i jα 2. j α B F i j,jī W i jα 2 = 1 2 α B i,j=1 F i j,jī X i j 2, with X i j = W i jα if i B, j G, and X i j = 0 otherwise. Since X iī = 0 for all i = 1,, n, by Lemma 1, F i j,jī W i jα 2 = F j j F iī W λ j λ i jα 2 = F j j F iī W i λ i jα 2 + o(λ i ), j since λ i 0 for i B. That yields 0 III. In conclusion, we establish (4.8) 1 σ l (G) α=1 F αᾱ φ αᾱ α G F αᾱ R αᾱiī W αᾱ. Then our assumption on the curvature yields that n α=1 F αᾱ φ αᾱ 0, so φ 0 on M by the Strong Maximum Principle. By the assumption, there is a point z 0 where the bisectional curvature is strictly positive. Since φ 0 and equation (4.8) is valid on M, the right hand side must also be vanishing identically. Therefore, G must be empty. That is W 0. Finally, we prove Theorem 2. Proof of Theorem 2. The proof follows the same steps as in the proof of part (2) of Theorem 1. We set W = g 1 (Ric ag), where a = min z M r s (z), where r s (z) is the smallest eigenvalue of Ric at z. W satisfies (4.9) F (W ) = constant, where F (A) = F (A + I), so F satisfies condition (1.2). Abusing the notation, we write F for F. We know the minimal rank l of W is less than or equal to 1. If l = 1, we can assume that W 1 1 G, W 2 2 B. All the steps in the proof of part (2) of Theorem 1 carry through without change, except in (4.6) we used condition (1.3). Since G = l = 1 in our case, II i in (4.6) is γ W 1 1,i 2 for some γ. From equation (4.9), we have 0 = α F αᾱ W αᾱ,i F 1 1 W 1 1,i. Since F 1 1 > 0, W 1 1,i 0. That is II i 0. If the minimal rank l = 0, then G = and (4.6) is trivial. In any case, (4.8) holds for n = 2 without condition (1.3).

12 12 PENGFEI GUAN, QUN LI, AND XI ZHANG 5. Discussions Assumptions in Theorem 1 can be weakened. In fact, in the proof of constant rank for W in Proposition 2, only the nonnegativity of orthogonal bisectional curvature and Ricci curvature are used. The quasi-positiveness of bisectional curvature assumption there is used to conclude W 0. Therefore, the proof in the last section may yield the following. Proposition 3. Let (M, g) be a compact Kähler manifold with Kähler metric (g i j) such that the orthogonal bisectional curvature and Kähler-Ricci tensor of g are nonnegative. Suppose Ric i j satisfies equation (1.1) and F satisfies (1.3), and let then W = Ric ag is of constant rank l < n, where a is defined in (4.1). In addition, if the orthogonal bisectional curvature is positive at some point, then (M, g) is Kähler-Einstein. The geometric structure of Kähler manifolds with constant rank of Käher-Ricci tensor in Proposition 3 have been studied in [12]. The proof of Proposition 3 follows the same lines of the proof of Theorem 1. The key differential inequality (4.8) still holds in this case for W = Ric ag. Therefore the maximum Principle implies W is of constant rank. We remark that in the second part of the proof, the constant rank result we derived is purely local, as a consequence, our main theorem is a local structure theorem. We discuss conditions imposed on F in Theorem 1. The conditions in part 1 and part 2 in Theorem 1 are not mutually inclusive. First, it is well known that F = σ 1 k k is concave in Γ k and so is F = ( σ k σ l ) 1 k l for k > l. And convex combinations of these type of functions are in this category. We will see from the lemma below that F = σ k satisfies condition (1.3). Therefore, any function of the form F = n k=1 a kσ l k k for a k 0 and l k 1 also satisfies condition (1.3). Lemma 2. σ k satisfies condition (1.3). Proof. As before, we may assume W > 0 is diagonal at the point of calculation. If α β G, then Therefore, σ n (W )F α β,βᾱ + F αᾱ σ n 1 (W β) = σ n (W )( σ k 2 (W αβ)) + σ k 1 (W α)σ n 1 (W β) = σ n 1 (W β)σ k 1 (W α) λ β σ n 1 (W β)σ k 2 (W αβ) = σ n 1 (W β)σ k 1 (W αβ) 0. W What is left in (1.3) is R = α,β W [F α β,βᾱ + F αᾱ σ n 1 (W β)] X α β 2 0. F αᾱ,β βx αᾱ Xβ β + α W F αᾱ λ α X αᾱ 2.

13 A UNIQUENESS THEOREM IN KÄHLER GEOMETRY 13 We now show that for F = σ k, A = σ n (W )F αᾱ,β βx αᾱ Xβ β + F αᾱ σ n 1 (W α) X αᾱ 2 0. α W α,β W Since W is diagonal, it is equivalent to show (5.10) σ n (W )σ k 2 (W αβ)x αᾱ Xβ β + σ n 1 (W α)σ k 1 (W α) X αᾱ 2 0. α W α,β W We write X αᾱ = γ α. Since σ k 1 (W α)σ k 1 (W β)γ α γ β + σk 1(W 2 α) γ α 2 = σ k 1 (W α)γ α 2, α W α W α,β W we have σ k (W )A σ k (W )A σ n (W ) σ k 1 (W α)γ α 2 α W = σ n (W )[σ k 2 (W αβ)σ k (W ) σ k 1 (W α)σ k 1 (W β)]γ α γ β α,β W + [σ n 1 (W α)σ k 1 (W α)σ k (W ) σ n (W )σk 1(W 2 α)] γ α 2 α W = σ n (W ) 0, + α W α,β W [σ k (W αβ)σ k 2 (W αβ) σ 2 k 1(W αβ)]γ α γ β σ k 1 (W α)σ k (W α)σ n 1 (W α) γ α 2 the last inequality follows from Lemma 2.4 in [9]. References [1] J.M. Ball, Differentiablity properties of symmetric and isotropic functions, Duke Math. J., 51, 1984, [2] A. Besse, Einstein manifolds, Ergeb.Math.Grenzgeb.Band 10, Springer-Verlag, Berlin and New York, [3] B. Bian and P. Guan, A Microscopic Convexity Principle for Fully Nonlinear Elliptic Equations, to appear in Inventiones Mathematicae (2009). [4] L. Caffarelli, P. Guan and X.N. Ma, A Constant Rank Theorem for Solutions of Fully Nonlinear Elliptic Equations, Comunications on Pure and Applied Mathematics, 60, (2007), [5] E. Calabi, Extremal Kähler metrics, In Seminar on Differential Geometry, Ann. of Math. Stud., 102(1982), Princeton Univ. Press, Princeton, N.J. [6] X.X. Chen and G. Tian, Ricci flow on Kähler-Einstein surfaces, Invent. Math. 147(2002), [7] K. Ecker and G. Huisken, Immersed hypersurfaces with constant Weingarten curvature, Math.Ann. 283(1989), [8] G. Glaeser, Fontions composées différentiables, Ann. Math., 77, no. 1, (1963), [9] P. Guan and X.N Ma, The Christoffel-Minkowski Problem I: Convexity of Solutions of a Hessian Equation, Invent.Math. 151, (2003), [10] A. Howard, B. Smyth and H. Wu, On compact Kähler manifolds of nonnegative bisectional curvature, I, Acta Math. 147, (1981), [11] Q. Li, Ph.D. thesis, McGill University (2008).

14 14 PENGFEI GUAN, QUN LI, AND XI ZHANG [12] H. Wu, On compact Kähler manifolds of nonnegative bisectional curvature. II. Acta Math. 147, (1981), [13] S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge- Ampere equation, Comm.Pure Appl. Math. 31(1978), Department of Mathematics and Statistics, McGill University, Canada address: guan@math.mcgill.ca Department of Mathematics and Statistics, McGill University, Canada address: qunli@math.mcgill.ca Department of Mathematics, Zhejiang University, P. R. China address: xizhang@zju.edu.cn