Viscosity solutions to complex Hessian equations

TL;DR

This paper establishes existence and uniqueness of viscosity solutions for complex Hessian equations both locally in bounded domains and globally on compact hermitian manifolds, extending the theory to non-Kähler settings.

Contribution

It introduces new existence and uniqueness results for viscosity solutions to complex Hessian equations in both local and global contexts, including non-Kähler manifolds.

Findings

Unique viscosity solutions exist under suitable conditions.

Solutions are Hölder continuous if data are Hölder continuous.

Results extend to non-Kähler hermitian manifolds.

Abstract

We study viscosity solutions to complex hessian equations. In the local case, we consider $\Omega$ a bounded domain in $\mathbb{C}^n,$ $\beta$ the standard K\"{a}hler form in $\mathcal{C}^n$ and $1\leq m\leq n.$ Under some suitable conditions on $F, g$, we prove that the equation $(dd^c \varphi)^m\wedge\beta^{n-m}=F(x,\varphi)\beta^n,\ \f=g$ on $\pO$ admits a unique viscosity solution modulo the existence of subsolution and supersolution. If moreover, the datum are H\"{o}lder continuous then so is the solution. In the global case, let $(X,\omega)$ be a compact hermitian homogeneous manifold where $\omega$ is an invariant hermitian metric (not necessarily K\"{a}hler). We prove that the equation $(\omega+dd^c\varphi)^m\wedge\omega^{n-m}=F(x,\varphi)\omega^n$ has a unique viscosity solution under some natural conditions on $F.$