The Hele-Shaw flow and moduli of holomorphic discs

Abstract

We present a new connection between the Hele-Shaw flow, also known as two-dimensional (2D) Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this we prove short time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also starting from a smooth Jordan domain. Applying the same ideas we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.

27 pages, 3 figures. v2 has improved exposition throughout, and some small correctifications (in particular to the proof of the short time existence with empty initial condition)

References in corpus (2)

• Planar elliptic growth
• Perturbation theorems for Hele-Shaw flows and their applications

Cited by in corpus (6)

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• On the separation distance of minimal Green energy points on compact Riemannian manifolds
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• Harmonic Discs of Solutions to the Complex Homogeneous Monge-Ampère Equation
• Geodesic Convexity of Small Neighborhood in the Space of Kähler Potentials