Second order estimates for $\chi$-semi convex solutions of Hessian equations on Hermitian manifolds

TL;DR

This paper develops second order estimates for solutions to complex Hessian equations on Hermitian manifolds, extending concavity inequalities under semi-convexity assumptions and handling gradient-dependent terms.

Contribution

It introduces a modified concavity inequality for complex Hessian equations on Hermitian manifolds under semi-convexity, advancing second order estimate techniques.

Findings

Established a modified concavity inequality for complex Hessian equations.

Derived second order estimates for solutions with gradient-dependent terms.

Extended methods to compact Hermitian manifolds.

Abstract

In this paper, we establish the modified concavity inequality for complex Hessian equations under the semi-convexity assumption inspired by Lu \cite{Lu23} and Zhang \cite{Z24} for real case. Then second order estimates for admissible solutions of complex Hessian equations on compact Hermitian manifolds with both sides of equations depending on gradient terms are obtained by taking advantage of the crucial inequality.