Theorem

Uniformization Theorem

Topology

Every simply connected Riemann surface is conformally equivalent to one of exactly three model surfaces: the Riemann sphere, the complex plane, or the open unit disk. It generalizes the Riemann mapping theorem and gives a complete classification of Riemann surfaces via their universal covers.

Facts

Statement

The uniformization theorem states that every simply connected Riemann surface is conformally equivalent to exactly one of three model surfaces: the open unit disk, the complex plane, or the Riemann sphere. 1

Proof Year

1907 1

Classification

Statement Form

Classification Theorem 1

Connections

Sources

1. Uniformization theorem (Wikipedia)

Wikimedia Foundation

Uniformization theorem, lead section
the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.
Uniformization theorem, History section
The first rigorous proofs of the general uniformization theorem were given by Poincaré (1907) and Paul Koebe (1907a, 1907b, 1907c).
Lead section, statement-form reference
Since every Riemann surface has a universal cover which is a simply connected Riemann surface, the uniformization theorem leads to a classification of Riemann surfaces into three types: those that have the Riemann sphere as universal cover ("elliptic"), those with the plane as universal cover ("parabolic") and those with the unit disk as universal cover ("hyperbolic").

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Uniformization Theorem

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