Complete K\"ahler manifolds with nonnegative Ricci curvature

Abstract

We consider complete K\"ahler manifolds with nonnegative Ricci curvature. The main results are: 1. When the manifold has nonnegative bisectional curvature, we show that $\lim\limits_{r\to\infty}\frac{r^{2}}{vol(B(p, r))}\int_{B(p, r)}S$ exists. In other words, it depends only on the manifold. This solves a question of Ni. Also, we establish estimates among volume growth ratio, integral of scalar curvature, and the degree of polynomial growth holomorphic functions. The new point is that the estimates are sharp for any prescribed volume growth rate. 2. We discover a strong rigidity for complete Ricci flat K\"ahler metrics. Let $M^n (n\geq 2)$ be a complete K\"ahler manifold with nonnegative Ricci curvature and Euclidean volume growth. Assume either the curvature has quadratic decay, or the K\"ahler metric is $dd^c$-exact with quadratic decay of scalar curvature. If one tangent cone at infinity is Ricci flat, then $M$ is Ricci flat. In particular, the tangent cone is unique. In other words, we can test Ricci flatness of the manifold by checking one single tangent cone. This seems unexpected, since apriori, there is no equation on $M$ and the Bishop-Gromov volume comparison is not sharp on Ricci flat (nonflat) manifolds. Such result is in sharp contrast to the Riemannian setting: Colding and Naber showed that tangent cones are quite flexible when $Ric\geq 0$ and $|Rm|r^2<C$. This reveals subtle differences between Riemannian case and K\"ahler case. The result contains lots of examples, such as all noncompact Ricci flat K\"ahler surfaces of Euclidean volume growth (hyper-K\"ahler ALE 4-manifolds classified by Kronheimer), higher dimensional examples of Tian-Yau type, as well as an example with irregular cross section. It also covers Ricci flat K\"ahler metrics of Euclidean volume growth on Stein manifolds with $b_2 = 0$(e.g., $\mathbb{C}^n$).