Beurlings free boundary value problem in conformal geometry

The subject of this paper is Beurlings 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 Beurlings geometric and approximative approach. We provide a complete proof of the Beurling-Riemann mapping theorem by combining Beurlings geometric method with a number of new analytic tools, notably $H^p$-space techniques and methods from the theory of Riemann-Hilbert-Poincare 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.