Second order estimates for a class of complex Hessian equations on Hermitian manifolds

TL;DR

This paper establishes second order a priori estimates for solutions to complex Hessian equations on Hermitian manifolds, where the equations depend on the gradient, advancing the understanding of these nonlinear PDEs in complex geometry.

Contribution

It provides new second order estimates for complex Hessian equations with gradient dependence on compact Hermitian manifolds, extending previous results to a broader class of equations.

Findings

Derived second order a priori estimates for solutions

Extended estimates to equations with gradient dependence

Applicable to a class of complex Hessian equations on Hermitian manifolds

Abstract

In this paper, we derive an \emph{a priori} second order estimate for solutions which are in $\Gamma_{k+1}$ cone to a class of complex Hessian equations with both sides of the equation depending on the gradient on compact Hermitian manifolds.