On the volume growth of kähler manifolds with nonnegative bisectional curvature

Let M be a complete Kahler 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 [17]. There are two essential ingredients in the proof: the Cheeger Colding theory [2] [5] on Gromov Hausdorff convergence of manifolds and the three circle theorem for holomorphic functions in [14].