Stability and regularity of solutions of the Monge-Amp\`ere equation on Hermitian manifolds

TL;DR

This paper proves the stability and regularity of solutions to the complex Monge-Ampère equation on compact Hermitian manifolds, extending known results from Kähler to Hermitian settings and showing solutions are Hölder continuous.

Contribution

It establishes stability and Hölder regularity of Monge-Ampère solutions on Hermitian manifolds and extends Kähler results to the Hermitian case.

Findings

Solutions are stable under bounded variations in the right hand side.

Solutions are Hölder continuous.

Extension of Kähler-Einstein results to Hermitian manifolds.

Abstract

We prove stability of solutions of the complex Monge-Amp\`ere equation on compact Hermitian manifolds, when the right hand side varies in a bounded set in $L^p, p>1$ and it is bounded away from zero. Such solutions are shown to be H\"older continuous. As an application we extend a recent result of Sz\'ekelyhidi and Tosatti on K\"ahler-Einstein equation from K\"ahler to Hermitian manifolds.