Fully non-linear elliptic equations on compact Hermitian manifolds

TL;DR

This paper develops a unified approach to derive a priori estimates for fully non-linear elliptic equations on compact Hermitian manifolds, enabling solutions to complex Hessian and quotient equations.

Contribution

It introduces a general method for a priori estimates applicable to various fully non-linear equations on Hermitian and Riemannian manifolds, extending previous specific cases.

Findings

Established a priori estimates for a broad class of non-linear equations

Solved Hessian quotient equations on Kähler manifolds under subsolution assumptions

Method applicable to both Hermitian and Riemannian settings

Abstract

We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge-Amp\`ere, Hessian and inverse Hessian equations. As an application we solve a class of Hessian quotient equations on K\"ahler manifolds assuming the existence of a suitable subsolution. The method also applies to analogous equations on compact Riemannian manifolds.