Green's functions and complex Monge-Amp\`ere equations

TL;DR

This paper establishes uniform lower bounds for Green's functions on compact Kähler manifolds, depending only on scalar curvature and volume form norms, using nonlinear Monge-Ampère equations to improve estimates for complex Monge-Ampère equations.

Contribution

It introduces new lower bounds for Green's functions on Kähler manifolds that depend solely on scalar curvature and volume form norms, not Ricci curvature.

Findings

Uniform lower bounds for Green's functions obtained

Bounds depend only on scalar curvature and volume form norms

Enhanced $C^1$ and $C^2$ estimates for Monge-Ampère equations

Abstract

Uniform $L^1$ and lower bounds are obtained for the Green's function on compact K\"ahler manifolds. Unlike in the classic theorem of Cheng-Li for Riemannian manifolds, the lower bounds do not depend directly on the Ricci curvature, but only on integral bounds for the volume form and certain of its derivatives. In particular, a uniform lower bound for the Green's function on K\"ahler manifolds is obtained which depends only on a lower bound for the scalar curvature and on an $L^q$ norm for the volume form for some $q>1$. The proof relies on auxiliary Monge-Amp\`ere equations, and is fundamentally non-linear. The lower bounds for the Green's function imply in turn $C^1$ and $C^2$ estimates for complex Monge-Amp\`ere equations with a sharper dependence on the function on the right hand side.