The Dirichlet Problem for the Complex Homogeneous Monge-Amp\`ere Equation

TL;DR

This paper surveys the Dirichlet problem for the complex Homogeneous Monge-Ampère Equation in complex domains and Kähler manifolds, linking it to Hele-Shaw flow and illustrating solution phenomena with examples.

Contribution

It provides a comprehensive survey and connects the problem to Hele-Shaw flow, including new examples of solution behaviors.

Findings

Connection between Monge-Ampère solutions and Hele-Shaw flow

Examples of diverse solution phenomena

Unified account of previous research

Abstract

We survey the Dirichlet problem for the complex Homogeneous Monge-Amp\`ere Equation, both in the case of domains in $\mathbb C^n$ and the case of compact K\"ahler manifolds parametrized by a Riemann surface with boundary. We then give a self-contained account of previous work of the authors that connects this with the Hele-Shaw flow, and give several concrete examples illustrating various phenomena that solutions to this problem can display.