A Riemann mapping theorem for two-connected domains in the plane

TL;DR

This paper develops an explicit algebraic method to find conformal maps of two-connected planar domains, generalizing the Riemann mapping theorem and providing a practical approach for such mappings.

Contribution

It introduces a new explicit algebraic formula for conformal maps of two-connected domains using Ahlfors maps, extending classical Riemann mapping results.

Findings

Explicit algebraic conformal map formula for two-connected domains

Representation of the map via Ahlfors maps and extremal problems

Identification of a canonical representative domain analogous to the unit disc

Abstract

We show how to express a conformal map of a general two connected domain in the plane such that neither boundary component is a point to a representative domain which has the virtue of having an explicit algebraic Bergman kernel function. We shall explain why the representative domain is the best analogue of the unit disc in the two connected setting. The conformal map 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 can be found by solving the same extremal problem that determines a Riemann map in the simply connected case.