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

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 $Φ$ 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 $Φ$. 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.

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