On the volume growth of K\"ahler manifolds with nonnegative bisectional curvature

TL;DR

This paper proves that complete K"ahler manifolds with nonnegative bisectional curvature and certain conditions must exhibit maximal volume growth, confirming a conjecture by Ni.

Contribution

It establishes a new link between polynomial growth holomorphic functions and volume growth in K"ahler manifolds, using Cheeger-Colding theory and the three circle theorem.

Findings

Manifolds with the given conditions have maximal volume growth.

The proof combines Gromov-Hausdorff convergence with holomorphic function theory.

Confirms Ni's conjecture on volume growth in this setting.

Abstract

Let $M$ be a complete K\"ahler 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. There are two essential ingredients in the proof: The Cheeger-Colding theory on Gromov-Hausdorff convergence of manifolds; the three circle theorem for holomorphic functions.