The Dirichlet Problem for a Complex Monge-Ampere Type Equation on Hermitian Manifolds

TL;DR

This paper develops new techniques to solve a complex Monge-Ampère type equation on Hermitian manifolds, addressing challenges posed by torsion, curvature, and boundary shape, with implications for geometric problems.

Contribution

It introduces methods applicable to a broader class of fully nonlinear equations on complex manifolds, overcoming key a priori estimate difficulties.

Findings

Established a priori estimates for the equation on Hermitian manifolds.

Extended techniques to handle non-pseudoconvex boundary conditions.

Provided insights relevant to geometric problems involving complex Monge-Ampère equations.

Abstract

We are concerned with fully nonlinear elliptic equations on complex manifolds and search for technical tools to overcome difficulties in deriving a priori estimates which arise due to the nontrivial torsion and curvature, as well as the general (non-pseudoconvex) shape of the boundary. We present our methods, which work for more general equations, by considering a specific equation which resembles the complex Monge-Ampere equation in many ways but with crucial differences. Our work is motivated by recent increasing interests in fully nonlinear equations on complex manifolds from geometric problems.