The complex Monge-Amp\`{e}re equation on some compact Hermitian manifolds

TL;DR

This paper establishes Laplacian estimates and proves the existence and uniqueness of classical solutions for the complex Monge-Ampère equation on compact Hermitian manifolds, extending results to lower regularity functions.

Contribution

It provides new Laplacian estimates and existence results for the complex Monge-Ampère equation on Hermitian manifolds with less regular data.

Findings

Laplacian estimate holds for $F$ in $W^{1,q_0}$ with $q_0 > 2n$

Existence of a unique classical solution in $W^{3,q_0}$ when $F$ is in $W^{1,q_0}$

Results apply to both complex dimension two and higher dimensions

Abstract

We consider the complex Monge-Amp\`{e}re equation on compact manifolds when the background metric is a Hermitian metric (in complex dimension two) or a kind of Hermitian metric (in higher dimensions). We prove that the Laplacian estimate holds when $F$ is in $W^{1,q_{0}}$ for any $q_{0}>2n$. As an application, we show that, up to scaling, there exists a unique classical solution in $W^{3,q_{0}}$ for the complex Monge-Amp\`{e}re equation when $F$ is in $W^{1,q_{0}}$.