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Let $M$ be a complete K\"ahler manifold with nonnegative bisectionalcurvature. Suppose the universal cover does not split and $M$ admits anonconstant holomorphic function with polynomial growth, we prove $M$ must beof maximal volume growth. This confirms a conjecture of Ni. There are twoessential ingredients in the proof: The Cheeger-Colding theory onGromov-Hausdorff convergence of manifolds; the three circle theorem forholomorphic functions.

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