On the volume growth of Kähler manifolds with nonnegative bisectional curvature (Q273219)

A complete Kähler manifold with nonnegative bisectional curvature admitting a holomorphic function of polynomial growth and having a maximal volume growth is biholomorphic and isometric to \(\mathbb{C}^n\). Later, the maximal growth condition was removed and based on this, \textit{L. Ni} [J. Am. Math. Soc. 17, No. 4, 909--946 (2004; Zbl 1071.58020)] conjectured that a complete Kähler manifold with nonnegative bisectional curvature, positive at one point, admitting a holomorphic function of polynomial growth has maximal volume growth. The present paper proves this conjecture. The proof is based on Cheeger-Colding results about Gromov-Hausdorff convergence and a previous result by the author on holomorphic functions of polynomial growth similar to the three circles theorem in complex analysis.