Asymptotic behaviour of the Riemann mapping function at analytic cusps

Abstract

The well-known Riemann Mapping Theorem states the existence of a conformal map of a simply connected proper domain of the complex plane onto the upper half plane. One of the main topics in geometric function theory is to investigate the behaviour of the mapping functions at the boundary of such domains. In this work, we always assume that a piecewise analytic boundary is given. Hereby, we have to distinguish regular and singular boundary points. While the asymptotic behaviour for regular boundary points can be investigated by using the Schwarz Reflection at analytic arcs, the situation for singular boundary points is far more complicated. In the latter scenario two cases have to be differentiated: analytic corners and analytic cusps. The first part of the thesis deals with the asymptotic behaviour at analytic corners where the opening angle is greater than 0. The results of Lichtenstein and Warschawski on the asymptotic behaviour of the Riemann map and its derivatives at an analytic corner are presented as well as the much stronger result of Lehman that the mapping function can be developed in a certain generalised power series which in turn enables to examine the o-minimal content of the Riemann Mapping Theorem. To obtain a similar statement for domains with analytic cusps, it is necessary to investigate the asymptotic behaviour of a Riemann map at the cusp and based on this result to determine the asymptotic power series expansion. Therefore, the aim of the second part of this work is to investigate the asymptotic behaviour of a Riemann map at an analytic cusp. A simply connected domain has an analytic cusp if the boundary is locally given by two analytic arcs such that the interior angle vanishes. Besides the asymptotic behaviour of the mapping function, the behaviour of its derivatives, its inverse, and the derivatives of the inverse are analysed. Finally, we present a conjecture on the asymptotic power series expansion of the mapping function at an analytic cusp.