Harmonic Discs of Solutions to the Complex Homogeneous Monge-Amp\`ere Equation

TL;DR

This paper investigates the regularity and harmonic discs of solutions to the complex homogeneous Monge-Ampère equation, revealing their connection to Hele-Shaw flows and demonstrating non-density of harmonic discs in certain cases.

Contribution

It establishes a link between solutions of the complex Monge-Ampère equation and Hele-Shaw flows, and analyzes the harmonic discs' density properties under various boundary conditions.

Findings

Harmonic discs are explicitly characterized for certain boundary data.

Examples show harmonic discs may not be dense in the product space.

Non-density of harmonic discs persists under small boundary data perturbations.

Abstract

We study regularity properties of solutions to the Dirichlet problem for the complex Homogeneous Monge-Amp\`ere equation. We show that for certain boundary data on $\mathbb P^1$ the solution $\Phi$ to this Dirichlet problem is connected via a Legendre transform to an associated flow in the complex plane called the Hele-Shaw flow. Using this we determine precisely the harmonic discs associated to $\Phi$. We then give examples for which these discs are not dense in the product, and also prove that this situation persists after small perturbations of the boundary data.