Estimates for the complex Monge-Amp\`ere equation on Hermitian and balanced manifolds

TL;DR

This paper extends Yau's estimates for the complex Monge-Ampère equation to Hermitian and balanced manifolds, providing new a priori estimates and alternative proofs in non-Kähler settings.

Contribution

It generalizes classical estimates to non-Kähler manifolds, specifically Hermitian and balanced cases, and offers an alternative proof of Cherrier's theorem.

Findings

Established $C^{}$ a priori estimates for Hermitian and balanced backgrounds.

Extended Yau's estimates beyond Kähler manifolds.

Connected results to recent work by Guan-Li.

Abstract

We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove $C^{\infty}$ a priori estimates for a solution of the complex Monge-Ampere equation when the background metric is Hermitian (in complex dimension two) or balanced (in higher dimensions), giving an alternative proof of a theorem of Cherrier. We relate this to recent results of Guan-Li.