A subsolution theorem for the Monge-Amp\`{e}re equation over an almost Hermitian manifold

TL;DR

This paper establishes a subsolution theorem for the Dirichlet problem of the complex Monge-Ampère equation on almost Hermitian manifolds, extending techniques from Hermitian geometry to a broader setting.

Contribution

It introduces a subsolution-based method to solve the Monge-Ampère equation on almost Hermitian manifolds, generalizing previous results from Hermitian to almost Hermitian contexts.

Findings

Existence of solutions given a smooth strictly J-psh subsolution.

Extension of subsolution techniques to almost Hermitian manifolds.

Framework for solving complex Monge-Ampère equations in broader geometric settings.

Abstract

Let $\Omega\subseteq M$ be a bounded domain with a smooth boundary $\partial\Omega$, where $(M,J,g)$ is a compact, almost Hermitian manifold. The main result of this paper is to consider the Dirichlet problem for a complex Monge-Amp\`{e}re equation on $\Omega$. Under the existence of a $C^{2}$-smooth strictly $J$-plurisubharmonic ($J$-psh for short) subsolution, we can solve this Dirichlet problem. Our method is based on the properties of subsolutions which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds.