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

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 to a representative domain of the form Ar = {z : |z + 1/z| < 2r}, where r > 1 is a constant. The domain Ar 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.