Holder continuity of solutions to the complex Monge-Ampere equations on compact Kahler manifolds

TL;DR

This paper investigates the conditions under which solutions to complex Monge-Ampère equations on compact Kähler manifolds are Hölder continuous, providing a partial converse to existing results about measure moderation.

Contribution

It establishes a partial converse to previous work, linking Hölder continuity of solutions to the measure's moderation on compact Kähler manifolds.

Findings

Hölder continuity implies measure moderation in solutions

Partial converse established between measure moderation and Hölder continuity

Advances understanding of regularity conditions for Monge-Ampère solutions

Abstract

We study H\"older continuity of solutions to the Monge-Amp\`{e}re equations on compact K\"ahler manifolds. In [DNS] the authors have shown that the measure $\omega_u^n$ is moderate if $u$ is H\"older continuous. We prove a theorem which is a partial converse to this result.