Harmonic Discs of Solutions to the Complex Homogeneous Monge-Ampère Equation

Abstract

We study regularity properties of solutions to the Dirichlet problem for the complex Homogeneous Monge-Ampère 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.

15 pages, 1 figure. v2 is a substantial rewrite with improved results reflected in a new title

References in corpus (1)

• Bergman kernels and equilibrium measures for ample line bundles