Journal of Differential Geometry

On the volume growth of Kähler manifolds with nonnegative bisectional curvature

Gang Liu

Abstract

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].

Article information

Source
J. Differential Geom., Volume 102, Number 3 (2016), 485-500.

Dates
First available in Project Euclid: 29 February 2016

https://projecteuclid.org/euclid.jdg/1456754016

Digital Object Identifier
doi:10.4310/jdg/1456754016

Mathematical Reviews number (MathSciNet)
MR3466805

Zentralblatt MATH identifier
1348.53072

Citation

Liu, Gang. On the volume growth of Kähler manifolds with nonnegative bisectional curvature. J. Differential Geom. 102 (2016), no. 3, 485--500. doi:10.4310/jdg/1456754016. https://projecteuclid.org/euclid.jdg/1456754016