A UNIQUENESS THEOREM IN K¨AHLER GEOMETRY 1. Introduction ...

¨ A UNIQUENESS THEOREM IN KAHLER GEOMETRY PENGFEI GUAN, QUN LI, AND XI ZHANG Abstract. We consider compact K¨ ahler manifolds with their K¨ ahler 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 K¨ ahler-Einstein.

1. Introduction The study of extremal metrics is an important topic in K¨ahler geometry. K¨ahlerEinstein metrics, K¨ahler metrics with constant scalar curvature and other forms of elementary symmetric functions associated to the K¨ahler 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 K¨ahler manifold with nonnegative bisectional curvature and constant scalar curvature, then it is automatically a K¨ahler-Einstein manifold. In this paper, we consider K¨ahler metric g with its K¨ahler Ricci tensor Ric = (Ri¯j ) satisfying certain relation: (1.1)

F (Ric) = const,

for some elliptic F . We are interested to know when such metrics are K¨ahlerEinstein. We denote H = {all n × n Hermitian matrices}, Γn = {all positive n × n Hermitian matrices}, Γ1 = {A ∈ H| σ1 (A) > 0} Throughout the paper, we assume F is a function defined on a convex subset Γ of ¯ n ⊂ Γ and F is invariant under the unitary transformations. We will H such that Γ also assume F is monotone, that is, ∂F (1.2) ( ) > 0. ∂wi¯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. ahler manifold with K¨ ahler metric (gi¯j ) of Theorem 1. Let (M, g) be a compact K¨ quasi-positive bisectional curvature tensor. Suppose Rici¯j satisfies equation (1.1) with F as in (1.2). If either (1) Γ1 ⊃ 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.10771188. 1

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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 Xi¯j , (1.3)

n X

¯

¯

¯ γ s¯ ≥ 0, (F ij,γ s¯(A) + F γ j Ai¯s )Xi¯j X

i,j,γ,s=1

then g is a K¨ ahler-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 K¨ ahler surface with K¨ ahler metric (gi¯j ) of quasi-positive bisectional curvature tensor. If its K¨ ahler-Ricci tensor satisfies equation (1.1) with F as in (1.2), then g is a K¨ ahler-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 K¨ahler manifold with the K¨ahler metric gi¯j . In local coordinate, its K¨ahler form √ ω = −1gi¯j dz i ∧ d¯ zj is a closed real (1, 1)-form, this is the same as saying that ∂gi¯j ∂gk¯j ∂gi¯j ∂g ¯ = and = ijk k i ∂z ∂z ∂ z¯k ∂ z¯ for all i, j, k. The K¨ahler class of ω is the cohomology class [ω] in H 2 (M, R). By the Hodge theory, any other K¨ahler metrics in the same class is of the form √ ¯ ωφ = ω + −1∂ ∂φ for some real-valued function φ on M . The Christoffel symbols of the metric gi¯j are given by ¯ ¯ ∂g ¯ ¯ ∂g ¯ Γkij = g kl ijl and Γ¯ki¯j = g lk lji ∂z ∂ z¯ ¯ i¯ j −1 k where (g ) = (gi¯j ) . It is easy to see that Γij is symmetric in i and j and Γ¯ki¯j is symmetric in ¯i and ¯j. The curvature tensor of the metric gi¯j is defined as ∂ 2 gi¯j ∂gi¯n ∂gm¯j + g m¯n k . k l ∂z ∂ z¯ ∂z ∂ z¯l We see that Ri¯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 m Ri¯jk¯l = gm¯j Rik ¯ l =−

¯

¯

Ri¯jk¯l v i v j wk wl > 0 for all nonzero vectors v and w in the holomorphic tangent bundle of M .

¨ A UNIQUENESS THEOREM IN KAHLER GEOMETRY

3

The Ricci tensor of ω is obtained by taking the trace of the curvature tensor, ¯

Ri¯j = g kl Ri¯jk¯l = −∂i ∂¯j log det(gi¯j ).

(2.1) So its Ricci form is

Ric(ω) =

√

√ −1Ri¯j dz i ∧ d¯ z j = − −1∂ ∂¯ log det(gi¯j ).

It is a real and closed (1, 1)-form. The scalar curvature of ω is ¯

R = g ij Ri¯j . Given any (1, 1) tensor Ti¯j ,its covariant derivatives are defined as ∂ T ¯ − Γlik Tl¯j ∂z k ij

Ti¯j,k =

(2.2) and

∂ ¯ T ¯ − Γ¯lj k¯ Ti¯l . ∂ z¯k ij We shall need the following commutation formulas for covariant derivatives

(2.3)

Ti¯j,k¯ =

Ti¯j,kl = Ti¯j,lk , Ti¯j,k¯¯l = Ti¯j,¯lk¯ , Ti¯j,k¯l = Ti¯j,¯lk + g m¯n Ri¯nk¯l Tm¯j − g m¯n Rm¯jk¯l Ti¯n . The second Bianchi identity in the K¨ahler case is Ri¯jk¯l,m = Ri¯jm¯l,k

and

Ri¯jk¯l,¯n = Ri¯jk¯n,¯l .

From (2.1), (2.2) and (2.3), one can easily check that Ri¯j,k = Rk¯j,i

and

Ri¯j,¯l = Ri¯l,¯j .

For simplicity, we choose the normal coordinate near the considered point. By directly calculation, we have (2.4)

Ri¯j,¯ll

= Ri¯l,¯jl = Ri¯l,l¯j − Riml ¯ ¯ ¯ j Rm¯ l + Rm¯ ll¯ j Rim ∂2R = ∂zi ∂ z¯j − Riml ¯ ¯ ¯ j Rm¯ j R im l + Rm¯

(2.5)

Ri¯j,l¯l = Rl¯j,i¯l = Rl¯j,¯li − Rm¯ji¯l Rlm ¯ + Rlmi j ¯ ¯ l Rm¯ ∂2R = ∂zi ∂ z¯j − Rm¯ji¯l Rlm ¯ + Rim ¯ Rm¯ j

From above we known ¯

¯

4Ric = g lk ∇l ∇k¯ Ric = g lk ∇k¯ ∇l Ric. ¯

Here 4 = 12 g lk (∇l ∇k¯ + ∇k¯ ∇l ) is the complex Laplacian. Definition 2.1 We say that the orthogonal bisectional curvature is nonnegative means that Ri¯ij¯j ≥ 0 for any i 6= 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 ⊂ Rn be a symmetric domain, set Γ = {A ∈ H : λ(A) ∈ Ψ}.

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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 2 2 ¯ ¯ ∂f f ∂F F us denote f˙k = ∂λ , f¨kl = ∂λ∂k ∂λ , F αβ = ∂A , and F αβ,γ s¯ = ∂A ∂¯ ∂A . It is well ¯ γs ¯ k l αβ αβ 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 F¨ (B, B) =

(2.6)

n X

f¨jk Bj¯j Bkk¯ + 2

j,k=1

X f˙j − f˙k |B ¯ |2 . λj − λk j k j 0, ∀i ≥ n − l + 1 (2.7) n n X X X f˙j − f˙k f˙i (A) f¨jk (A)Xj¯j Xkk¯ + 2 |Xj k¯ |2 + |Xik¯ |2 ≥ 0 λj − λk λk j,k=n−l+1

n−l+1≤j = ∇l {< ∇¯l Ric, Ric > + < Ric, ∇¯l Ric >} = 2 < ∇l ∇¯l Ric, Ric > + < ∇l Ric, ∇¯l Ric > + < ∇¯l Ric, ∇l Ric > = 2 < 4Ric, Ric > +|∇Ric|2 2 = |∇Ric|2 + 2{ ∂z∂i ∂Rz¯j Rj¯i + Rip¯Rp¯j Rj¯i − Rp¯ji¯l Rlp¯Rj¯i }.

Now rotate at a given point the coordinate system such that (Ri¯j ) is diagonal, and assume that {λi }n1 are the eigenvalues. Then, we have (3.2)

4|Ric|2

2

= |∇Ric|2 + 2 ∂z∂i ∂Rz¯i λi + 2λ3i − 2Ri¯jj¯i λi λj P 2 = |∇Ric|2 + 2 ∂z∂i ∂Rz¯i λi + i6=j Ri¯ij¯j (λi − λj )2 .

Since Ri¯ij¯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 Ri¯ij¯j > 0 at some point by assumption, we have λi = λj for any i 6= j. So the K¨ahler metric ω is of constant Ricci curvature, i.e. it is a K¨ahler-Einstein metric.

¨ A UNIQUENESS THEOREM IN KAHLER 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 (Rij ) = const., where Rij = g j l Ri¯l , then, 0

= 4F j

∂ ∂F ∂Ri ∂ ∂F j s¯ s,m ¯g ) ∂z m ( ∂Rj ∂ z¯m ) = ∂z m ( ∂Rj Ri¯ i i 2 ∂F F = ∂R + ∂R∂j ∂R j Ri¯ j,m ¯ Rk¯ j,mm ¯ l Ri¯ l,m i i P ∂F ∂ 2 R P k ∂2F = i ∂λi ∂zi ∂ z¯i + i,k,m ∂λi ∂λk ∇m ¯ ¯ Ri¯i ∇m Rkk P ∂F ∂F + 12 i,j Ri¯ij¯j ( ∂λ − ∂λ )(λi − λj ).

= (3.3)

i

j

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)

(

∂F ∂F − )(λi − λj ) ≤ 0, ∂λi ∂λj

for any i, j. Hence we conclude (3.5)

X ∂F ∂ 2 R ≥ 0. ∂λi ∂z i ∂ z¯i i

Since F is monotone, the strong maximum principle then yields R ≡ Const. It now follows Proposition 1 that g is K¨ahler-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 Cn in [11]. We adapt them to K¨ahler manifolds here. Let (4.1)

a = min rs (z), z∈M

where rs (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 .

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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 z0 ∈ M , pick an open neighborhood O of z0 , for any z ∈ O, let λ1 ≤ λ2 ≤ · · · ≤ λn be the eigenvalues of W at z. There is a positive constant C > 0 such that λn ≥ λn−1 ≥ · · · ≥ λn−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 ΛG = (λn−l+1 , · · · , λn ) be the ”good” eigenvalues of W at z, for simplicity, we also write G = ΛG 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 λ1 , . . . , λn , then the k-th elementary symmetric function of W is, X σk (W ) = λi1 · · · λik , where the sum is taken over all strictly increasing sequences i1 , . . . , ik of the indices from {1, . . . , n}. With this notation at hand, we set (4.3)

φ(z) = σl+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 c1 and c2 such that (h − k)(y) ≤ (c1 |∇φ| + c2 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 c1 and c2 depending only on n and C (independent of y and O). Finally, h ∼ k if h . k and k . h. Write W = (Wi¯j ), for each fixed point z ∈ O, we choose a normal complex coordinate system so that W is diagonal at z, and Wi¯i = λi , ∀ 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 à ! X X 0 ∼ φ(z) ∼ Wi¯i σl (G) ∼ Wi¯i , i∈B

i∈B

so Wi¯i ∼ 0, This relation yields that, for 1 ≤ m ≤ l, σm (W ) ∼ σm (G),

i ∈ B. ½

σm (W |j) ∼

σm (G|j), if j ∈ G; σm (G), if j ∈ B.

  σm (G|ij), if i, j ∈ G; σm (G|j), if i ∈ B, j ∈ G; σm (W |ij) ∼  σm (G), if i, j ∈ B, i 6= 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, X X 0 ∼ φα ∼ σl (G) Wi¯iα ∼ Wi¯iα i∈B

0 ∼ φα¯ ∼ σl (G)

X

i∈B

i∈B

Wi¯iα¯ ∼

X

i∈B

Wi¯iα¯

¨ A UNIQUENESS THEOREM IN KAHLER GEOMETRY

7

And by Proposition 2.2 in [9], ∂σl+1 (W ) ∼ ∂Wi¯j

½

σl (G), i = j ∈ B; 0, otherwise.

and for 1 ≤ m ≤ n,  if i = j, r = s, i 6= r;  σm−2 (W |ir), ∂ 2 σm (W ) −σm−2 (W |ij), if i = s, r = j, i 6= j; =  ∂Wi¯j ∂Wr¯s 0 otherwise. By the above relations, if l ≤ n − 2, we calculate the following, note that later on we’ll use notations Wi¯jr¯s to demonstrate the second order derivatives of the tensor Wi¯j with respect to zr and zs¯, similar for Wi¯jr and Wi¯j s¯.

φαα¯

= =

n X

n X ∂ 2 σl+1 (W ) ∂σl+1 (W ) Wi¯jα Wr¯sα¯ + Wi¯jαα¯ ∂Wi¯j ∂Wr¯s ∂Wi¯j i,j,r,s=1 i,j=1 X X X σl−1 (W |ij)Wi¯iα Wj¯j α¯ − σl−1 (W |ij)Wi¯jα Wj¯iα¯ + σl (G)Wi¯iαα¯ i,j i6=j

=

(

X

+

i∈G j∈B

−(

X

X

+

i∈B j∈G

+

i∈G j∈B

X i∈B j∈G

X

+

i,j∈G i6=j

+

X

X

i

i,j i6=j

)σl−1 (W |ij)Wi¯iα Wj¯j α¯

i,j∈B i6=j

+

i,j∈G i6=j

X

)σl−1 (W |ij)Wi¯jα Wj¯iα¯ +

X

σl (G)Wi¯iαα¯

i

i,j∈B i6=j

We want to simplify the above expression, firstly X

σl−1 (W |ij)Wi¯iα Wj¯j α¯

∼

i∈G j∈B

X

σl−1 (G|i)Wi¯iα Wj¯j α¯

i∈G j∈B

=

X

σl−1 (G|i)Wi¯iα

i∈G

Similarly,

X

Wj¯j α¯ ∼ 0

j∈B

P

σl−1 (W |ij)Wi¯iα Wj¯j α¯ ∼ 0. P For any i ∈ B fixed, and ∀ α, −Wi¯iα¯ ∼ j∈B Wj¯j α¯ , so i∈B j∈G

j6=i

X i,j∈B i6=j

σl−1 (W |ij)Wi¯iα Wj¯j α¯ ∼ −

X i∈B

σl−1 (G)Wi¯iα Wi¯iα¯ = −

X

σl−1 (G)|Wi¯iα |2 .

i∈B

Finally, as |G| = l, |(G|ij)| = l − 2, so σl−1 (W |ij) ∼ σl−1 (G|ij) = 0, for all i, j ∈ G.

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PENGFEI GUAN, QUN LI, AND XI ZHANG

So, we have φαα¯

∼ −

X

σl−1 (G)|Wi¯iα |2 −

i∈B

−

X i∈G j∈B

X

σl−1 (G)|Wi¯jα |2 +

=

σl (G)Wi¯iαα¯ −

i=1

−

σl (G)Wi¯iαα¯ −

i∈B

−

σl (G)Wi¯iαα¯

X

σl−1 (G|i)(|Wi¯jα |2 + |Wj¯iα |2 )

σl−1 (G)|Wi¯jα |2

i,j∈B

∼

σl−1 (G|j)|Wi¯jα |2

i∈B j∈G

i∈G j∈B

X

X

X

X

i

i,j∈B i6=j

n X

σl−1 (G|i)|Wi¯jα |2 −

X

σl−1 (G|j)(|Wi¯jα |2 + |Wj¯iα |2 )

i∈B j∈G

X

σl−1 (G)|Wi¯jα |2

i,j∈B ¯

Since F αβ is diagonal at z, we do the contraction, n X

F αα¯ φαα¯

∼

α=1

n X α=1

−

X

F αα¯ σl (G)

n X

Wi¯iαα¯ −

F αα¯

α=1

X

F αα¯ σl−1 (G)

α=1

i∈B

X

n X

|Wi¯jα |2

i,j∈B

σl−1 (G|j)(|Wi¯jα |2 + |Wj¯iα |2 ).

i∈B j∈G

Taking second derivatives to the equation (1.1), we have that n X

¯

F αβ Wαβi ¯ = 0,

α,β=1

n X

¯

F αβ,γ s¯Wαβi ¯ Wγ s¯¯i +

α,β,γ,s=1

n X

¯

F αβ Wαβi ¯ ¯i = 0.

α,β=1

While by the computation of the covariant derivatives on the K¨ahler manifold, we have that Wi¯iαα¯

= Wα¯iiα¯ = Wα¯iαi ¯ +

X s

Ws¯i Rα¯siα¯ −

X

Wα¯s Rs¯iiα¯

s

= Wααi ¯ Rααi ¯ ¯i + Wi¯i Ri¯iαα ¯ − Wαα ¯ ¯i .

¨ A UNIQUENESS THEOREM IN KAHLER GEOMETRY

Then, n X

F αα¯ φαα¯

∼

−σl (G)

α=1

X

F αα¯ Rααi ¯ ¯ ¯i Wαα

α∈G i∈B

−σl (G)

n X

X

¯

F αβ,γ s¯Wαβi ¯ Wγ s¯¯i

i∈B α,β,γ,s=1 n X X

F αα¯ |Wi¯jα |2

−σl−1 (G)

α=1 i,j∈B

−

n X X α=1

=

F αα¯ σl−1 (G|j)(|Wi¯jα |2 + |Wj¯iα |2 )

i∈B j∈G

−σl (G)

X

F αα¯ Rααi ¯ ¯ ¯i Wαα

α∈G i∈B

−σl (G) −σl (G)

n X X i∈B α,β=1 n X X

¯

¯ β F αα,β Wααi ¯ Wβ β¯¯i

¯

F αβ,β α¯ Wαβi ¯ Wβ α ¯¯i

i∈B α,β=1 n X X

−σl−1 (G)

F αα¯ |Wi¯jα |2

α=1 i,j∈B

−

n X X α=1

=

F αα¯ σl−1 (G|j)(|Wi¯jα |2 + |Wj¯iα |2 )

i∈B j∈G

−σl (G)

X

F αα¯ Rααi ¯ ¯ ¯i Wαα

α∈G i∈B

X X X X ¯ β¯ −σl (G) ( + +2 )F αα,β Wααi ¯ Wβ β¯¯i i∈B α,β∈B

α,β∈G

α∈B β∈G

X X X X X ¯ −σl (G) ( + + + )F αβ,β α¯ Wαβi ¯ Wβ α ¯¯i i∈B α,β∈B

−σl−1 (G) (4.4)

−(

X α∈G

n X X

α,β∈G

α∈G β∈B

α∈B β∈G

F αα¯ |Wi¯jα |2

α=1 i,j∈B

X X + ) F αα¯ σl−1 (G|j)(|Wi¯jα |2 + |Wj¯iα |2 ) α∈B

i∈B j∈G

So, n X X 1 X αα¯ F φαα¯ ∼ − F αα¯ Rααi IIi − III, ¯ −I − ¯ ¯i Wαα σl (G) α=1 α∈G i∈B

i∈B

9

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PENGFEI GUAN, QUN LI, AND XI ZHANG

where I

=

X X X X X X ¯ 2 ¯ β¯ ( + )F αβ,β α¯ |Wαβi +2 )F αα,β Wααi ( ¯ | ¯ Wβ β¯¯i + i∈B α,β∈B

i∈B α,β∈B

α∈B β∈G

α∈G β∈B

n X X 1 X X 1 ( ) F αα¯ |Wi¯jα |2 + F αα¯ |Wj¯iα |2 , λk α=1 λ j i,α∈B i,j∈B

k

j∈G

and IIi =

X

¯

¯

¯ β αβ,β α ¯ 2 (F αα,β Wααi |Wαβi ¯ | + ¯ Wβ β¯¯i + F

α,β∈G

F αα¯ 2 2 (|Wαβi ¯ | + |Wβ¯iα | ), λβ

finally, X

III =

¯ ¯

(F ij,j i +

i,α∈B j∈G

¯

F ii )|Wi¯jα |2 . λj

Using the nonnegativity of the hermitian matrix (Wαβ¯ ) ∈ C 2 , by a lemma in [3], ∀α, β |∇Wαβ¯ |2 ≤ C(Wαα¯ + Wβ β¯ ),

(4.5)

for some constant C independent of the indices. Therefore every term in I with at least two indices in B is under control since Wαα¯ ∼ 0 for α ∈ B. The only term in I needs to take care of is XX ¯ β¯ F αα,β Wααi 2 ¯ Wβ β¯¯i . i∈B ¯

α∈B β∈G

¯

¯ β We note that F αα,β ∼ F γ γ¯ ,β β for all α, γ ∈ B. This property can be obtained by our previous observation, σm (W |α) ∼ σm (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˜ (σ1 (W ), σ2 (W ), · · · , σn (W )). In turn, X X X ¯ ¯ β¯ F αα,β Wααi F γ γ¯ ,β β Wβ β¯¯i ( Wααi ¯ Wβ β¯¯i ∼ ¯ ) ∼ 0. α∈B

β∈G

α∈B β∈G

We have shown 0 . I. For the term IIi , we make use of condition (1.3). X ¯ β¯ (4.6) σl (G)IIi = σl (G) F αα,β Wααi ¯ Wβ β¯¯i +

X

α,β∈G α6=β

¯

2 (σl (G)F αβ,β α¯ + F αα¯ σl−1 (G|β))|Wαβi ¯ |

α,β∈G α6=β

(4.7)

+

X

F αα¯ σl−1 (G|β))|Wα¯iβ |2 ,

α,β∈G α6=β

The sum of the first two terms is nonnegative by the inequality (2.7).

¨ A UNIQUENESS THEOREM IN KAHLER GEOMETRY

11

To deal with III, XX

III =

α∈B

¯ ¯

(F ij,j i +

i∈B j∈G

¯

F ii )|Wi¯jα |2 . λj

For each α ∈ B, we may write X

¯ ¯

F ij,j i |Wi¯jα |2 =

n 1 X X i¯j,j¯i F |Xi¯j |2 , 2 i,j=1 α∈B

i∈B j∈G

with Xi¯j = Wi¯jα if i ∈ B, j ∈ G, and Xi¯j = 0 otherwise. Since Xi¯i = 0 for all i = 1, · · · , n, by Lemma 1, X i∈B j∈G

¯ ¯

¯

F ij,j i |Wi¯jα |2 =

¯

¯

¯

F j j − F ii F j j − F ii |Wi¯jα |2 = |Wi¯jα |2 + o(λi ), λj − λi λj

since λi ∼ 0 for i ∈ B. That yields 0 . III. In conclusion, we establish n X 1 X αα¯ F αα¯ Rααi F φαα¯ . − ¯. ¯ ¯i Wαα σl (G) α=1 α∈G

(4.8)

i∈B

Pn Then our assumption on the curvature yields that α=1 F αα¯ φαα¯ . 0, so φ ≡ 0 on M by the Strong Maximum Principle. By the assumption, there is a point z0 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 = minz∈M rs (z), where rs (z) is the smallest eigenvalue of Ric at z. W satisfies F˜ (W ) = constant,

(4.9)

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 W1¯1 ∈ G, W2¯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, IIi in (4.6) is γ|W1¯1,i |2 for some γ. From equation (4.9), we have X ¯ 11 F αα¯ Wαα,i 0= ¯ ∼ F W1¯ 1,i . α ¯

Since F 11 > 0, W1¯1,i ∼ 0. That is IIi ∼ 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

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 K¨ ahler manifold with K¨ ahler metric (gi¯j ) such that the orthogonal bisectional curvature and K¨ ahler-Ricci tensor of g are nonnegative. Suppose Rici¯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 K¨ ahler-Einstein. The geometric structure of K¨ahler manifolds with constant rank of K¨aher-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 1 part 2 in Theorem 1 are not mutually inclusive. First, it is well known that F = σkk 1 is concave in Γk and so is F = ( σσkl ) k−l for k > l. And convex combinations of these type of functions are in this category. We will see from the lemma below Pnthat F = σk satisfies condition (1.3). Therefore, any function of the form F = k=1 ak σklk for ak ≥ 0 and lk ≥ 1 also satisfies condition (1.3). Lemma 2. σk satisfies condition (1.3). Proof. As before, we may assume W > 0 is diagonal at the point of calculation. If α 6= β ∈ G, then ¯

= = = ≥ Therefore,

σn (W )F αβ,β α¯ + F αα¯ σn−1 (W |β) σn (W )(−σk−2 (W |αβ)) + σk−1 (W |α)σn−1 (W |β) σn−1 (W |β)σk−1 (W |α) − λβ σn−1 (W |β)σk−2 (W |αβ) σn−1 (W |β)σk−1 (W |αβ) 0. X

¯

[F αβ,β α¯ + F αα¯ σn−1 (W |β)]|Xαβ¯ |2 ≥ 0.

α6=β∈W

What is left in (1.3) is X F αα¯ X ¯ β¯ ¯ β β¯ + F αα,β Xαα¯ X R= |Xαα¯ |2 . λ α α,β∈W α6=β

α∈W

¨ A UNIQUENESS THEOREM IN KAHLER GEOMETRY

13

We now show that for F = σk , X X ¯ β¯ ¯ β β¯ + F αα¯ σn−1 (W |α)|Xαα¯ |2 ≥ 0. A= σn (W )F αα,β Xαα¯ X α∈W

α,β∈W α6=β

Since W is diagonal, it is equivalent to show (5.10) X X ¯ β β¯ + σn (W )σk−2 (W |αβ)Xαα¯ X σn−1 (W |α)σk−1 (W |α)|Xαα¯ |2 ≥ 0. α∈W

α,β∈W α6=β

We write Xαα¯ = γα . Since X X X 2 σk−1 (W |α)σk−1 (W |β)γα γ¯β + σk−1 (W |α)|γα |2 = | σk−1 (W |α)γα |2 , α∈W

α,β∈W α6=β

we have σk (W )A

≥ =

σk (W )A − σn (W )| X

X

α∈W

σk−1 (W |α)γα |2

α∈W

σn (W )[σk−2 (W |αβ)σk (W ) − σk−1 (W |α)σk−1 (W |β)]γα γ¯β

α,β∈W α6=β

+

X

2 (W |α)]|γα |2 [σn−1 (W |α)σk−1 (W |α)σk (W ) − σn (W )σk−1

α∈W

=

σn (W ) +

X

X

2 [σk (W |αβ)σk−2 (W |αβ) − σk−1 (W |αβ)]γα γ¯β

α,β∈W α6=β

σk−1 (W |α)σk (W |α)σn−1 (W |α)|γα |2

α∈W

≥

0,

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, 699–728. [2] A. Besse, Einstein manifolds, Ergeb.Math.Grenzgeb.Band 10, Springer-Verlag, Berlin and New York, 1987. [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), 1769-1791. [5] E. Calabi, Extremal K¨ ahler metrics, In Seminar on Differential Geometry , Ann. of Math. Stud., 102(1982), 259-290. Princeton Univ. Press, Princeton, N.J. ahler-Einstein surfaces, Invent. Math. 147(2002), [6] X.X. Chen and G. Tian, Ricci flow on K¨ 487-544. [7] K. Ecker and G. Huisken, Immersed hypersurfaces with constant Weingarten curvature, Math.Ann. 283(1989), 329-332. ees diff´ erentiables, Ann. Math., 77, no. 1, (1963), 193-209. [8] G. Glaeser, Fontions compos´ [9] P. Guan and X.N Ma, The Christoffel-Minkowski Problem I: Convexity of Solutions of a Hessian Equation, Invent.Math. 151, (2003), 553-577. [10] A. Howard, B. Smyth and H. Wu, On compact K¨ ahler manifolds of nonnegative bisectional curvature, I, Acta Math. 147, (1981), 51–56. [11] Q. Li, Ph.D. thesis, McGill University (2008).

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[12] H. Wu, On compact K¨ ahler manifolds of nonnegative bisectional curvature. II. Acta Math. 147, (1981), 57–70. [13] S.T. Yau, On the Ricci curvature of a compact K¨ ahler manifold and the complex MongeAmpere equation, Comm.Pure Appl. Math. 31(1978), 339-411. Department of Mathematics and Statistics, McGill University, Canada E-mail address: [email protected] Department of Mathematics and Statistics, McGill University, Canada E-mail address: [email protected] Department of Mathematics, Zhejiang University, P. R. China E-mail address: [email protected]