Fully nonlinear elliptic equations with gradient terms on compact almost Hermitian manifolds

TL;DR

This paper develops second order estimates for fully nonlinear elliptic equations with gradient terms on compact almost Hermitian manifolds, leading to existence results for complex Monge-Ampère and Hessian equations, and a priori estimates for deformed Hermitian-Yang-Mills equations.

Contribution

It introduces new second order estimates and existence proofs for complex nonlinear equations on almost Hermitian manifolds, extending classical results to this broader setting.

Findings

Established second order estimates for a class of fully nonlinear equations.

Proved existence of solutions for Monge-Ampère equations with gradient terms.

Provided $C^{ty}$ a priori estimates for deformed Hermitian-Yang-Mills equations.

Abstract

In this paper, we establish second order estimates for a general class of fully nonlinear equations with linear gradient terms on compact almost Hermitian manifolds. As an application, we first prove the existence of solutions for the Monge-Amp\`ere equation with linear gradient terms for $(n-1)$-plurisubharmonic functions, originated from Gaudochon conjecture, in the almost Hermitian setting. Second, we solve the Monge-Amp\`ere equation and Hessian equations with linear gradient terms. Third, we give the $C^{\infty}$ a priori estimates for the deformed Hermitian-Yang-Mills equation with supercritical phase. At last, we prove the existence of deformed Hermitian-Yang-Mills equation and complex Hessian quotient equations under supersolutions.