Modulus of continuity for solutions to complex Monge-Amp\`ere equations on Hermitian manifolds

TL;DR

This paper proves that solutions to complex Monge-Ampère equations on compact Hermitian manifolds are uniformly log-continuous, extending previous results from the Kähler setting to a more general Hermitian context.

Contribution

It generalizes the uniform log-continuity result of solutions from Kähler to Hermitian manifolds, broadening the applicability of the theory.

Findings

Established uniform log-continuity of solutions on Hermitian manifolds

Extended previous Kähler case results to more general Hermitian manifolds

Provides a new proof technique for continuity estimates

Abstract

In this note, we give a proof of the uniform log-continuity of the solution to complex Monge-Amp\`ere equations on compact Hermitian manifolds, which is a generalization of the result of Guo-Phong-Tong-Wang in the K\"ahler case.