Analytic Covering Maps

Abstract

In this chapter it will be shown that for every region Ω in the plane such that ℂ\Ω has at least two points, there is an analytic covering map τ:D →Ω. This is the essential part of what is called the Uniformization Theorem. The reader might want to review §9.7 before going much further. The reader will be assumed to be familiar with some basic topological notions such as the fundamental group and its properties. Some topological facts will be proved (especially in the first section) even though they may seem elementary and assumable to many.