Fully nonlinear elliptic equations on Hermitian manifolds for symmetric functions of partial Laplacians

TL;DR

This paper develops methods to solve fully nonlinear elliptic equations on Hermitian manifolds, extending previous work on plurisubharmonicity and solving the Dirichlet problem with smooth solutions.

Contribution

It introduces new interior estimates and existence results for fully nonlinear elliptic equations related to symmetric functions of partial Laplacians on Hermitian manifolds.

Findings

Established interior estimates for solutions.

Proved existence of smooth solutions for Dirichlet problems.

Extended previous results to more general Hermitian settings.

Abstract

We consider a class of fully nonlinear second order elliptic equations on Hermitian manifolds closely related to the general notion of $\bfG$-plurisubharmonicity of Harvey-Lawson and an equation treated by Sz\'ekelyhidi-Tosatti-Weinkove in the proof of Gauduchon conjecture. Under fairly general assumptions we derive interior estimates and establish the existence of smooth solutions for the Dirichlet problem as well as for equations on closed manifolds.