On the volume growth of Kähler manifolds with nonnegative bisectional curvature

Abstract

Let $M$ be a complete Kähler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and $M$ admits a nonconstant holomorphic function with polynomial growth; we prove $M$ must be of maximal volume growth. This confirms a conjecture of Ni in “A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature”, [J. Amer. Math. Soc. 17 (2004), 909–946, MR 2083471, Zbl 1071.58020]. There are two essential ingredients in the proof: the Cheeger–Colding theory on Gromov–Hausdorff convergence of manifolds, and the three-circle theorem for holomorphic functions in “Three circle theorems on Kähler manifolds and applications” by G. Liu [Arxiv: 1308.0710].