A Matrix Li-Yau-Hamilton estimate for the Green function on K\"ahler manifolds

TL;DR

This paper establishes a matrix Li-Yau-Hamilton inequality for the Green function on complete K"ahler manifolds with nonnegative holomorphic bisectional curvature, extending previous heat equation estimates to an elliptic setting.

Contribution

It introduces a novel elliptic matrix estimate for the Green function on K"ahler manifolds, generalizing prior heat equation results to a broader geometric context.

Findings

Proves a new matrix inequality for the Green function on K"ahler manifolds.

Extends Li-Yau-Hamilton estimates from heat equations to Green functions.

Provides a complex analogue of known Riemannian estimates.

Abstract

In this paper we prove a matrix Li-Yau-Hamilton inequality for the Green function on complete K\"ahler manifolds with nonnegative holomorphic bisectional curvature. This estimate can be seen as an elliptic analogue of the matrix estimate of Cao and Ni for the heat equation on K\"ahler manifolds, or the complex analogue of the estimate for Riemannian manifolds obtained previously by the author.