On rigidity of gradient K\"ahler-Ricci solitons with harmonic Bochner tensor

In this paper, we prove that complete gradient steady K\"ahler-Ricci solitons with harmonic Bochner tensor are necessarily K\"ahler-Ricci flat, i.e., Calabi-Yau, and that complete gradient shrinking (or expanding) K\"ahler-Ricci solitons with harmonic Bochner tensor must be isometric to a quotient of $N^k\times \mathbb{C}^{n-k}$, where $N$ is a K\"ahler-Einstein manifold with positive (or negative) scalar curvature.


Introduction
A complete Riemannian manifold (M, g ij ) is called a Ricci soliton, if there is a vector field X and a constant λ such that where λ > 0, λ = 0, or λ < 0 corresponds to shrinking, steady or expanding soliton respectively. Moreover, a Ricci soliton is called a gradient Ricci soliton if the vector field is a gradient vector field, i.e. X = ∇f for some smooth function f on M . In this case, the Ricci soliton equation above becomes Since R. Hamilton [16] introduced the concept of Ricci solitons in the mid 1980's, the study of Ricci solitons has attracted a lot of attention. Ricci solitons, as selfsimilar solutions to Hamilton's Ricci flow, are natural generalizations of Einstein metrics. Since they often arise as the dilation limits of the singularities of the Ricci flow, Ricci solitons play an important role in the singularity analysis of the Ricci flow.
Regarding steady solitons, it is well-known that compact ones (as well as compact expanding solitons) must be Einstein, see e.g., [9] for a proof. However, much less is known for noncompact steady Ricci solitons. In dimension n = 2, R. Hamilton showed that any 2-dimensional steady soliton is isometric to the Cigar soliton, 2010 Mathematics Subject Classification. Primary 53C44, 53C55. up to scaling. For n = 3, G. Perelman [21] conjectured that any 3-dimensional complete (κ-noncollapsed) steady soliton with positive sectional curvature is isometric to the Bryant soliton, which is the unique rotationally symmetric example on R 3 . Very recently, H.-D. Cao and the first author [7] made the first important progress on this problem. They showed that any n-dimensional, n ≥ 3, complete locally conformally flat steady Ricci soliton is either flat or isometric to the Bryant soliton (for n ≥ 4, Catino and Mantegazza [11] independently proved this result by using a different method). Subsequently, their work has been used by S. Brendle [1] in classifying 3-dimensional steady Ricci soliton satisfying certain asymptotic condition, and X. X. Chen and Y. Wang [13] in classifying 4-dimensional half-comformally flat steady solitons, respectively.
For more thorough discussions and results in Ricci solitons, the readers can refer to the survey [4] of H.-D. Cao and the references therein.
In this paper, we are interested in Kähler-Ricci solitons. R ij + ∇ i ∇jf = λg ij , for some constant λ ∈ R and such that ∇f is a holomorphic vector field, i.e. ∇ i ∇ j f = 0.
In [8], H.-D. Cao and R. Hamilton observed that complete noncompact gradient steady Kähler-Ricci solitons with positive Ricci curvature such that the scalar curvature attains the maximum must be Stein (and diffeomorphic to R 2n ). Later, under the same assumptions, A. Chau and L.-F. Tam [12], and R. Bryant [3] independently proved that such steady Kähler-Ricci solitons are actually biholomorphic to C n . Moreover, Chau and Tam showed that complete noncompact expanding Kähler-Ricci solitons with nonnegative Ricci curvature are also biholomorphic to C n .
To state our results, let us first recall that on Kähler manifolds there is a similar notion as the Weyl tensor, called the Bochner tensor. The Bochner tensor W ijkl is defined by where R ij = g kl R ijkl . We also denote the divergence of the Bochner tensor by Definition 1.2. A Kähler manifold M n is said to have harmonic Bochner tensor if C ijk = 0, i.e., Very recently, by using a similar argument as in the paper [7] of H.-D. Cao and the first author, Y. Su and K. Zhang [24] have shown that any complete noncompact gradient Kähler-Ricci soliton with vanishing Bochner tensor is necessarily Kähler-Einstein, and hence a quotient of the corresponding complex space form.
In this paper we investigate gradient Kähler-Ricci solitons with harmonic Bochner tensor, and extend the classification result of Su and Zhang. Our main results are: Theorem 1.1. Any complete gradient steady Kähler-Ricci soliton with harmonic Bochner tensor must be Kähler-Ricci flat (i.e., Calabi-Yau).
Theorem 1.2. Any complete gradient shrinking (or expanding) Kähler-Ricci soliton with harmonic Bochner tensor must be isometric to the quotient of N k × C n−k , where N k is a k-dimensional Kähler-Einstein manifold with positive (or negative) scalar curvature.
Remark 1.1. It is known that a compact Kähler manifold with vanishing Bochner tensor (also called Bochner-Kähler or Bochner-flat) is necessarily a compact quo- denote the complex space forms of constant holomorphic sectional curvature c and −c respectively (cf., e.g., corollary 4.17 in [2]). It follows immediately that any compact Kähler-Ricci soliton with vanishing Bochner tensor must be a quotient of complex space form.
Remark 1.2. In the Riemanian case, by using a rigidity result of Petersen and Wylie [22], Fernández-López and García-Río [15], and Munteanu and Sesum [18] proved that Ricci shrinkers with harmonic Weyl tensor must be rigid, i.e., a quotient of the product of an Einstein manifold and R k .
Acknowledgement The authors would like to thank their advisor, Professor Huai-Dong Cao, for suggesting this problem and for stimulating discussions. We are also grateful for his constant encouragement and support.

proof of the main theorems
Let (M n , g ij , f ) be a gradient Kähler-Ricci soliton, i.e.
It is well-known that the following basic identities hold (see e.g. [9]).
Lemma 2.1. On a gradient Kähler-Ricci soliton, we have From now on, we assume (M n , g ij , f ) is a gradient Kähler-Ricci soliton with harmonic Bochner tensor so that Proof: On one hand, by differentiating (2.4), we obtain ∆R = ∇ k ∇kR = ∇ k R∇kf + R kl ∇k∇ l f.
Next, by taking the divergence on both sides of (2.6), we get That is, But, Hence, we have Therefore, formula (2.7) follows easily. Now, suppose that ∇f = 0 at some point p. Then we may choose an orthonormal frame {e 1 , e 2 , · · · , e n } of holomorphic vector fields at p such that e 1 is parallel to ∇f . Therefore, we have |∇ 1 f | = |∇f | and ∇ k f = 0 for k = 2, · · · , n. Lemma 2.3. Suppose ∇f = 0 at p. Then, under the frame {e 1 , e 2 , · · · , e n } chosen above, we have Proof: From (2.3) and (2.5), we have at p,

It follows that
In particular, for k ≥ 2, we have that However, on the other hand, it is easy to see that Therefore, R k1 = R 1k = 0 for k ≥ 2.
Lemma 2.3 tells us that ∇f is an eigenvector of the Ricci curvature tensor. Thus we may choose another orthonormal frame {w 1 = e 1 , w 2 , · · · , w n } at p such that |∇ 1 f | = |∇f | and the Ricci curvature is diagonalized at p, i.e.
Next, by setting i = 1 in (2.7) and dividing both sides of the equation by ∇ 1 f , we get (2.10). Proof: Since at point p, ∇f = 0, formula ( 2.10) implies that in a neighborhood of p we have Therefore, there are two possibilities I) R jī ∇ i f ∇jf = 0 at p, or II) R jī ∇ i f ∇jf = 0 at p. In this case, near p we have Taking covariant derivative on both sides gives us Evaluating the identity above at p under the orthonormal frame {w 1 , w 2 , · · · , w n } yields Thus, we have Rc(∇f, ∇f ) = λ n+4 |∇f | 2 whenever Rc(∇f, ∇f ) = 0. Now we are ready to prove the main theorems.
First, we may assume that f = Const, for otherwise we get that M is Kähler-Einstein from the soliton equation.
Therefore, we must have Rc(∇f, ∇f ) = 0 in A. Since f = Const in the interior of M \A, we have Rc(∇f, ∇f ) = 0 on the whole manifold M . It follows that ∇R = 0 on M . Then (2.5) implies that the Ricci curvature tensor is parallel on M . Therefore, by the de-Rahm decomposition theorem, the universal cover of M is isometric to N n−1 × C, where N is again an n − 1 dimensional Kähler-Ricci soliton with harmonic Bochner tensor. Thus by induction, we can finally get that M is isometric to a quotient of the product of a Kähler-Einstein manifold and the complex Euclidean space.
This finishes the proof.