## On the volume growth of Kähler manifolds with nonnegative bisectional curvature.(English)Zbl 1348.53072

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, 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.

### MSC:

 53C55 Global differential geometry of Hermitian and Kählerian manifolds

### Keywords:

nonnegative bisectional curvature; volume growth

Zbl 1071.58020
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