Some variational problems for the complex Monge--Amp{\`e}re operator

TL;DR

This paper investigates the Dirichlet problem for the complex Monge--Ampère equation on pseudoconvex Kähler manifolds, establishing existence of smooth solutions using flow-based methods under certain conditions.

Contribution

It extends previous work by Chou-Wang to prove the existence of smooth solutions for a class of variational problems involving the complex Monge--Ampère operator.

Findings

Existence of smooth solutions under specific conditions.

Flow-based methods effectively solve the Dirichlet problem.

Results applicable to strongly pseudoconvex Kähler manifolds.

Abstract

We consider the Dirichlet problem for the complex Monge--Amp\`ere equation on strongly pseudoconvex K\"ahler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, we establish existence of smooth solutions in a number of natural circumstances, following work of Chou-Wang.