The Hele-Shaw flow and moduli of holomorphic discs

TL;DR

This paper establishes a novel connection between Hele-Shaw flow and holomorphic discs, proving short-term existence, uniqueness, and properties of related moduli spaces, advancing understanding in complex analysis and fluid dynamics.

Contribution

It introduces a new link between Hele-Shaw flow and holomorphic discs, proving existence, uniqueness, and describing the moduli space of quadrature domains.

Findings

Proved short time existence and uniqueness of Hele-Shaw flow with varying permeability.

Showed the moduli space of smooth quadrature domains is a smooth manifold.

Provided a local existence theorem for the inverse potential problem.

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.