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arXiv · 1408.6663

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

Julius Ross·David Witt Nyström
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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.

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Julius Ross, David Witt Nyström. 2014-08-28. Harmonic Discs of Solutions to the Complex Homogeneous Monge-Amp\`ere Equation. https://arxiv.org/abs/1408.6663

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