On uniform estimates for $(n-1)-$form fully nonlinear partial differential equations on compact Hermitian manifolds

TL;DR

This paper establishes uniform bounds for a broad class of fully nonlinear partial differential equations involving (n-1)-forms on compact Hermitian manifolds, extending previous methods with a novel elliptic operator approach.

Contribution

It introduces a new elliptic operator framework enabling a priori $L^ abla$ estimates for nonlinear PDEs on Hermitian manifolds, advancing the understanding of such equations.

Findings

Derived $L^ abla$ bounds for (n-1)-form PDEs

Extended comparison techniques to Hermitian manifolds

Provided a new method for maximum principle application

Abstract

We obtain a priori $L^\infty$ estimate for a general class of $(n-1)-$form fully nonlinear partial differential equations on compact Hermitian manifolds. Our method relies on the local version of comparison with auxiliary Monge-Amp\`ere equations, developed earlier by B. Guo and D. H. Phong. The key is to find the appropriate elliptic operator such that the maximum principle applies.