A Note on Wu-Zheng's Splitting Conjecture

Cao's splitting theorem says that for any complete K\"ahler-Ricci flow $(M,g(t))$ with $t\in [0,T)$, $M$ simply connected and nonnegative bounded holomorphic bisectional curvature, $(M,g(t))$ is holomorphically isometric to $\C^k\times (N,h(t))$ where $(N,h(t))$ is a Kahler-Ricci flow with positive Ricci curvature for $t>0$. In this article, we show that $k=n-r$ where $r$ is the Ricci rank of the initial metric. As a corollary, we also confirm a splitting conjecture of Wu-Zheng when curvature is assumed to be bounded.


Introduction
In [1], Cao proved the following splitting theorem. on a complex manifold M n and the initial metric g(0) has nonnegative and bounded holomorphic bisectional curvature. Then, rank R ij (x, t) is independent of x ∈ M and t ∈ (0, T ). Moreover, ker R ij (x, t) is independent t > 0 and is a parallel distribution for all t > 0. In particular, if M is simply connected, (M, g(t)) is holomorphically isometric to C n−k × (N k , h(t)) where h(t) with t ∈ [0, T ) is a complete solution to the Kähler-Ricci flow (1.1) on N with positive Ricci curvature for t > 0.
Let (M, g) be a complete Kähler manifold and r = max x∈M rank R ij (x). Then r is called the Ricci rank of the Kähler metric g. In this article, we shown that the number k in Cao's splitting theorem equals the Ricci rank of the initial metric g(0). Theorem 1.2. Let settings be the same as in Theorem 1.1 and r be the Ricci rank of the initial metric g(0). Then for any x ∈ M and t ∈ (0, T ).
In [6], Wu-Zheng presented the following splitting conjecture: Conjecture 1 (Wu-Zheng). Let M n be a complete Kähler manifold with nonnegative holomorphic bisectional curvature. Then, its universal covering manifoldM is holomorphically isometric to C n−r × N r , where r is the Ricci rank.
This is a noncompact generalization of the splitting theorem of Howard-Smyth-Wu [3] on compact Kähler manifolds with nonnegative holomorphic bisectional curvature which is also related to a conjecture proposed by Yau (Ref. [6,7]). In [6] and [7], Wu-Zheng showed that the conjecture is true for complete Kähler manifolds with nonnegative holomorphic bisectional curvature, real analytic Kähler metric and Ricci rank ≤ 2. The technique used by Zheng-Wu is delicate. They performed some careful analysis on the distribution defined by the kernel of the Ricci tensor. Because the distribution is not defined all over the manifold, they have to require that the metric is real analytic, so that the distribution can be paralleled extended to the whole manifold. As a direct corollary of Theorem 1.1 and Theorem 1.2, we can see that the conjecture is true if curvature is assumed to be bounded. Theorem 1.3. Let M n be a complete Kähler manifold with nonnegative and bounded holomorphic bisectional curvature. Then, its universal covering manifoldM is holomorphically isometric to C n−r × N r , where r is the Ricci rank.
In [4], Ni-Tam got splitting results of complete Kähler manifolds with nonnegative holomorphic bisectional curvature in a similar spirit of Cao's splitting theorem. However, Ni-Tam's splitting results are with respect to function theory on the Kähler manifold. For example, they got the following result. As a direct corollary of Theorem 1.3, we know that l equals the Ricci rank of the manifold if we further assume that the curvature of the manifold is bounded.

proof of Theorem 1.2
In this section, using a result of Ferus [2] on the completeness of the leaves of the nullity foliation of the curvature operator, we give a proof of Theorem 1.2.
Proof of Theorem 1.2. Let Ω = {x ∈ M |r = rank R ij (x, 0)} and L x = ker R ij (x, 0) for each x ∈ Ω. It is shown in [6] that L is a totally geodesic and flat foliation on Ω. In [2], Ferus showed that each leaf of the foliation L is complete. Let F x = ker R ij (x, t) for each x ∈ M and t > 0. It is shown by Cao's splitting theorem (Theorem 1.1) that F is independent of t and parallel for each t > 0. By continuity(Ref. [5]), we know that F ⊂ L on Ω and F is also parallel with respect to g(0). It suffices to show that L ⊂ F on Ω.
Since L is totally geodesic, flat and with complete leaves, we know that a leaf of L is a totally geodesic and isometric holomorphic immersion φ : C n−r → M. Note that on a Kähler-Ricci flow, where Ric = √ −1R ij dz ∧ dzj is the Ricci form. Therefore, for each t > 0, Note that φ * Ric(0) = 0 since φ : C n−r → M is a leaf of L. Hence Note that Ric(t) is nonnegative, so t 0 R(s)(φ)ds is a bounded plurisubharmonic function on C n−r . By Liouville property of pluri-subharmonic functions on complex Euclidean spaces (See [4, Theorem 0.2]), we know that t 0 R(s)(φ)ds is a constant function. Substituting this back to (2.4), we know that φ * Ric(t) = 0 for t > 0. Therefore L ⊂ F on Ω.
Remark 2.1. Instead of using paralleled translation to extend the Ricci kernel, here we use the Kähler-Ricci flow to extend the Ricci kernel.