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