Complex Monge-Ampere equations on Hermitian manifolds

TL;DR

This paper extends classical results on complex Monge-Ampere equations from Kähler to Hermitian manifolds and applies these to generalize the Donaldson conjecture on geodesics in the space of Kähler metrics.

Contribution

It generalizes existence results of Monge-Ampere equations to Hermitian manifolds and extends the Donaldson conjecture to this broader setting.

Findings

Extended Monge-Ampere existence results to Hermitian manifolds

Generalized the Donaldson conjecture for Hermitian metrics

Connected classical PDE results with complex geometry applications

Abstract

We study complex Monge-Ampere equations on Hermitian manifolds, extending classical existence results of Yau and Aubin in the Kahler case, and those of Caffarelli, Kohn, Nirenberg and Spruck for the Dirichlet problem in $C^n$. As an application we generalize existing results on the Donaldson conjecture on geodesics in the space of Kahler metrics to the Hermitian setting.