Local comparison theorems for Kähler manifolds

Gang Liu

Vol. 254 (2011), No. 2, 345–360

DOI: 10.2140/pjm.2011.254.345

Abstract

We establish a sharp relative volume comparison theorem for small balls on Kähler manifolds with lower bound on Ricci curvature, assuming real analyticity of the metric. The model spaces being compared to are complex space forms, that is, Kähler manifolds with constant holomorphic sectional curvature. Moreover, we give an example showing that on Kähler manifolds, the pointwise Laplacian comparison theorem does not hold when the Ricci curvature is bounded from below.

Keywords

volume comparison, Kähler manifolds

Mathematical Subject Classification 2010

Primary: 53BXX

Secondary: 53B35, 53B20

Milestones

Received: 14 March 2011

Accepted: 7 July 2011

Published: 27 February 2012