Beurling's free boundary value problem in conformal geometry

TL;DR

This paper provides a complete proof of Beurling's extension of the Riemann mapping theorem, introduces new analytic tools, and extends the theorem to maps with prescribed branching and boundary regularity.

Contribution

It combines geometric and analytic methods to fill gaps in the original proof and extends the theorem to broader classes of conformal maps.

Findings

Complete proof of Beurling-Riemann mapping theorem

Extension to maps with prescribed branching

Description of boundary regularity of solutions

Abstract

The subject of this paper is Beurling's celebrated extension of the Riemann mapping theorem \cite{Beu53}. Our point of departure is the observation that the only known proof of the Beurling-Riemann mapping theorem contains a number of gaps which seem inherent in Beurling's geometric and approximative approach. We provide a complete proof of the Beurling-Riemann mapping theorem by combining Beurling's geometric method with a number of new analytic tools, notably $H^p$-space techniques and methods from the theory of Riemann-Hilbert-Poincar\'e problems. One additional advantage of this approach is that it leads to an extension of the Beurling-Riemann mapping theorem for analytic maps with prescribed branching. Moreover, it allows a complete description of the boundary regularity of solutions in the (generalized) Beurling-Riemann mapping theorem extending earlier results that have been obtained by PDE techniques. We finally consider the question of uniqueness in the extended Beurling-Riemann mapping theorem.