Abstract
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.
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Napier, T., Ramachandran, M. (2011). Uniformization and Embedding of Riemann Surfaces. In: An Introduction to Riemann Surfaces. Cornerstones. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4693-6_5
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