# Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem

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*Henri Paul de Saint-Gervais*

Translated from the French by Robert Burns, York University, Toronto

A publication of the European Mathematical Society

In 1907, Paul Koebe and Henri Poincaré almost simultaneously proved
the uniformization theorem: Every simply connected Riemann surface is
isomorphic to the plane, the open unit disc, or the sphere.

It took a whole century to get to the point of stating this theorem and
providing a convincing proof of it, relying as it did on prior work of
Gauss, Riemann, Schwarz, Klein, Poincaré , and Koebe, among others. The
present book offers an overview of the maturation process of this
theorem.

The evolution of the uniformization theorem took place in parallel with
the emergence of modern algebraic geometry, the creation of complex
analysis, the first stirrings of functional analysis, and with the
flowering of the theory of differential equations and the birth of
topology. The uniformization theorem was, thus, one of the lightning rods
of 19th century mathematics. Rather than describe the history of a
single theorem, the book aims to return to the original proofs, to look at
these through the eyes of modern mathematicians, to inquire as to their
correctness, and to attempt to make them rigorous while respecting,
as much as possible, the state of mathematical knowledge at the time, or,
if this should prove impossible, then to use modern mathematical tools
that were not available to the authors of the original proofs.

This book will be useful to mathematicians wishing to cast a
glance back at the history of their discipline. It should also provide
graduate students with a non-standard approach to concepts of great
importance for modern research.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in Riemann surfaces.