Second order estimates for fully nonlinear elliptic equations with gradient terms on Hermitian manifolds

TL;DR

This paper develops a priori second order estimates for a broad class of fully nonlinear elliptic equations with gradient dependence on Hermitian manifolds, motivated by complex geometry conjectures and recent research interests.

Contribution

It introduces new methods to obtain second order estimates for fully nonlinear elliptic equations involving gradient terms on Hermitian manifolds, extending previous results and addressing conjectures.

Findings

Global second order estimates derived for equations related to Gauduchon's conjecture

Methods applicable to a wide class of fully nonlinear PDEs in complex geometry

Addresses gradient dependence in elliptic equations on Hermitian manifolds

Abstract

We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a conjecture by Gauduchon which extends the Calabi conjecture; this was one of the original motivations to this work. We were also motivated by the fact that there had been increasing interests in fully nonlinear pde's from complex geometry in recent years, and aimed to develop general methods to cover as wide a class of equations as possible.