On the modulus of continuity of solutions to complex Monge-Amp\`ere equations

TL;DR

This paper establishes sharp continuity estimates for solutions to complex Monge-Ampère equations and applies these results to obtain diameter bounds for associated Kähler metrics, advancing understanding in complex geometry.

Contribution

It provides a new uniform modulus of continuity estimate for solutions, utilizing a PDE-based approach, and derives diameter bounds for Kähler metrics satisfying specific Monge-Ampère equations.

Findings

Sharp modulus of continuity estimates proved

Uniform diameter bounds for Kähler metrics established

PDE approach enhances previous supremum estimate methods

Abstract

In this paper, we prove a uniform and sharp estimate for the modulus of continuity of solutions to complex Monge-Amp\`ere equations, using the PDE-based approach developed by the first three authors in their approach to supremum estimates for fully non-linear equations in K\"ahler geometry. As an application, we derive a uniform diameter bound for K\"ahler metrics satisfying certain Monge-Amp\`ere equations.