A Riemann Mapping Theorem for Two-Connected Domains in the Plane

Abstract

We show how to express a conformal map Φ of a general two-connected domain in the plane, such that neither boundary component is a point, onto a representative domain of the form \(\mathcal{A}_r = \left\{ {z:\left| {z + 1/z} \right| < 2r} \right\}\), where r > 1 is a constant. The domain \(\mathcal{A}_r\) has the virtue of having an explicit algebraic Bergman kernel function, and we shall explain why it is the best analogue of the unit disc in the two-connected setting. The map Φ will be given as a simple and explicit algebraic function of an Ahlfors map of the domain associated to a specially chosen point. It will follow that the conformal map Φ can be found by solving the same extremal problem that determines a Riemann map in the simply connected case. In the last section, we show how these results can be used to give formulae for the Bergman kernel in two-connected domains.