Uniformisation des surfaces de Riemann

TL;DR

This paper presents an elementary proof of the uniformization theorem for Riemann surfaces, avoiding advanced concepts like paracompacity, by leveraging topological and holomorphic properties.

Contribution

It offers a novel proof of the uniformization theorem using elementary holomorphic function properties and topological methods, bypassing complex analytic prerequisites.

Findings

Proof relies solely on elementary holomorphic properties

Utilizes a topological characterization of the sphere

Generalizes the double construction of Riemann surfaces

Abstract

A proof of the uniformization theorem of Riemann surface is given with only elementary properties of holomorphic functions and not using the paracompacity of the surface. This proof leans on an holomorphic version of the topological characterization, due to Brown, of the sphere as variety covered by two discs, a generalization of the construction of double of a Riemann surface with boundary and the arithmetic, due to Jordan, of separation in surfaces