Uniformisation of Planar Domains

The aim in this final chapter is to show that the disk can act as a covering space for the vast majority of planar domains, in fact for any planar domain whose complement contains at least two points. Such domains are called hyperbolic. The rudiments of covering space theory are set out in the self-contained first section, including lifts of curves and a version of the Monodromy Theorem. The Modular Function, which is a covering of the twice punctured plane by the half-plane, is constructed by means of repeated Schwarz reflections. The generators of the automorphism group of the modular function are obtained and the action of this group on the half-plane described in some detail. The existence of the modular function is used to prove the Little Picard Theorem and Montel’s Second Theorem: the latter was a key ingredient in the proof of the Great Picard Theorem in Chap. 5 . In the final section, we work through a proof of the Uniformisation Theorem for planar domains, to the effect that every hyperbolic planar domain can be covered by the disk, the key point being that the disk is simply connected. In this sense, the Uniformisation Theorem is a generalisation of the Riemann Mapping Theorem.