The Dirichlet problem for a complex Hessian equation on compact Hermitian manifolds with boundary

TL;DR

This paper addresses the Dirichlet problem for complex Hessian equations on compact Hermitian manifolds with boundary, establishing existence of continuous solutions under certain conditions, advancing the understanding of complex geometric PDEs.

Contribution

It extends the solvability of the Dirichlet problem for complex Hessian equations from local Euclidean domains to compact Hermitian manifolds with boundary, under the presence of a subsolution.

Findings

Existence of continuous solutions on compact Hermitian manifolds with boundary.

Solution construction relies on pluripotential theory techniques.

Results apply to manifolds with locally conformal Kähler metrics.

Abstract

We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal K\"ahler metrics, provided a subsolution.