Open Access
March 2016 On the volume growth of Kähler manifolds with nonnegative bisectional curvature
Gang Liu
J. Differential Geom. 102(3): 485-500 (March 2016). DOI: 10.4310/jdg/1456754016


Let $M$ be a complete Kähler 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 “A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature”, [J. Amer. Math. Soc. 17 (2004), 909–946, MR 2083471, Zbl 1071.58020]. There are two essential ingredients in the proof: the Cheeger–Colding theory on Gromov–Hausdorff convergence of manifolds, and the three-circle theorem for holomorphic functions in “Three circle theorems on Kähler manifolds and applications” by G. Liu [Arxiv: 1308.0710].


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Gang Liu. "On the volume growth of Kähler manifolds with nonnegative bisectional curvature." J. Differential Geom. 102 (3) 485 - 500, March 2016.


Published: March 2016
First available in Project Euclid: 29 February 2016

zbMATH: 1348.53072
MathSciNet: MR3466805
Digital Object Identifier: 10.4310/jdg/1456754016

Rights: Copyright © 2016 Lehigh University

Vol.102 • No. 3 • March 2016
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