Local volume comparison for Kahler manifolds

TL;DR

This paper proves a sharp volume comparison theorem for small balls in Kahler manifolds with Ricci curvature bounds, assuming real analyticity, and shows limitations of Laplacian comparison in this setting.

Contribution

It establishes a new sharp volume comparison theorem for Kahler manifolds with Ricci bounds under real analyticity, and demonstrates the failure of Laplacian comparison in this context.

Findings

Sharp volume comparison theorem for small balls in Kahler manifolds.

Counterexample showing Laplacian comparison does not hold pointwise.

Highlights the importance of analyticity assumption in geometric comparisons.

Abstract

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