Beurling’s free boundary value problem in conformal geometry

• Published in: Israel journal of mathematics Vol. 180; no. 1; pp. 223 - 253
• Main Authors: Bauer, Florian, Kraus, Daniela, Roth, Oliver, Wegert, Elias
• Format: Journal Article
• Language: English
• Published: Heidelberg The Hebrew University Magnes Press 01-12-2010
• Subjects: Algebra; Analysis; Applications of Mathematics; Group Theory and Generalizations; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Theoretical; Blaschke Product; Free Boundary; Bounded Function; Connected Domain; Free Boundary Problem
• ISSN: 0021-2172; 1565-8511
• DOI: 10.1007/s11856-010-0102-1

The subject of this paper is Beurling’s celebrated extension of the Riemann mapping theorem [
5
]. Our point of departure is the observation that the only known proof of the Beurling-Riemann mapping theorem (due to Beurling) 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-Poincaré 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.