Regularity of solutions to Monge--Amp\`ere equations on compact Hermitian manifolds

TL;DR

This paper investigates the stability and regularity of solutions to complex Monge--Ampère equations on compact Hermitian manifolds, establishing continuity, stability, and geometric bounds relevant to complex differential geometry.

Contribution

It provides new regularity results for degenerate Monge--Ampère equations on Hermitian manifolds, including global continuity and diameter bounds for related flows.

Findings

Solutions are H"older continuous under certain conditions.

Global continuity is achieved when the reference form is a pullback of a Hermitian metric.

A uniform diameter bound for the twisted Chern--Ricci flow is established.

Abstract

We study the stability and H\"older continuity of solutions to degenerate complex Monge--Amp\`ere equations associated with a (non-closed) big form on compact Hermitian manifolds. We also show that the solution is globally continuous when the reference form is the pullback of a Hermitian metric. As a consequence, we establish a uniform diameter bound for the twisted Chern--Ricci flow.