The Uniformization Theorem and Universal Covers

This paper will deal with the consequences of the Uniformization Theorem, which is a major result in complex analysis and differential geometry. We will proceed by stating the theorem, which is that for any simply connected Riemann surface, there exists a biholomorphic map to one (and only one) of the following three: the Riemann sphere, the open unit disk, and the complex plane. After the theorem is stated and the basic terms defined, we will go into its consequences by introducing the notions of covering spaces and universal covers. By utilizing another fundamental result in differential geometry, the Gauss-Bonnet Theorem, we will be able to use the notions of curvature and genus in order to come up with a simple method using only the topological properties of compact, orientable Riemann surfaces in order to decide which of the three model geometries (the Riemann sphere, the open unit disk, or the complex plane) is a universal cover for it.