Uniformization and Embedding of Riemann Surfaces

In this chapter, we consider certain complex analytic characterizations of Riemann surfaces (some topological and $$\mathcal {C}^{\infty}$$ characterizations appear in Chap. 6 ). The first goal is the following Riemann surface analogue of the classical Riemann mapping theorem in the plane: Theorem 5.1 (Riemann mapping theorem) A simply connected Riemann surface is biholomorphic to the Riemann sphere ℙ1, to the complex plane ℂ, or to the unit disk Δ={z∈ℂ||z|<1}. The second goal of this chapter is the fact that every Riemann surface X may be obtained by holomorphic attachment of tubes at elements of a locally finite sequence of coordinate disks in a domain in ℙ1. In particular, for X compact, this allows one to form a canonical homology basis.