Viscosity solution to complex Hessian equations on compact Hermitian manifolds

TL;DR

This paper establishes the existence and uniqueness of viscosity solutions to complex Hessian equations on compact Hermitian manifolds under certain conditions, advancing the understanding of nonlinear PDEs in complex geometry.

Contribution

It introduces a new approach to solving complex Hessian equations using viscosity solutions, including conditions for existence and uniqueness.

Findings

Existence of viscosity solutions under determinant domination

Uniqueness when the right hand side is strictly monotone

Reduction to solvability constant when right hand side is independent of the solution

Abstract

We prove the existence of viscosity solutions to complex Hessian equations on a compact Hermitian manifold that satisfy a determinant domination condition. This viscosity solution is shown to be unique when the right hand is strictly monotone increasing in terms of the solution. When the right hand side does not depend on the solution, we reduces it to the strict monotonicity of the solvability constant.