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

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 $\mathrm{\Phi }$ 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 $\mathrm{\Phi }$. 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.