Some comparison theorems for Kahler manifolds

TL;DR

This paper establishes comparison theorems for Kahler manifolds, including complex Hessian, eigenvalue, and volume comparisons, extending previous results by Li and Wang.

Contribution

It introduces new comparison theorems for Kahler manifolds involving complex Hessian, eigenvalues, and volume, under curvature bounds.

Findings

Complex Hessian comparison for distance functions

Eigenvalue comparison results

Volume comparison based on scalar curvature

Abstract

In this work, we will verify some comparison results on Kahler manifolds. They are complex Hessian comparison for the distance function from a closed complex submanifold of a Kahler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scalar curvature. This work is motivated by comparison results of Li and Wang .