Riemann mapping theorem

Introduction

The (small) Riemann Mapping Theorem is a result in Complex Analysis named after Bernhard Riemann.

History

Bernhard Riemann outlined a proof in 1851 in his dissertation. In 1922, the theorem was finally proven by Lipót Fejér and Frigyes Riesz. A widely used proof today, which employs the Montel's theorem, was given by Alexander Ostrowski in 1929. The Riemann Mapping Theorem has a generalization known as the large Riemann Mapping Theorem.

Riemann Mapping Theorem

Every Connected space Domain (mathematical analysis) $G\subsetneq \mathbb {C}$ can be mapped biholomorphically onto the Unit circle $\mathbb {D}$.

Remark - Open Unit Disk

The open unit disk $\mathbb {D}$ is defined as

$\mathbb {D} :=\{z\in \mathbb {C} :|z|<1\}.$

Notation - Proper Subset

The notation $G\subsetneq \mathbb {C}$ means “proper Subset,” indicating that the domain $G$ is not equal to $\mathbb {C}$.

Characterization of Open Sets

An open set in $\mathbb {C}$ can be characterized by the property that every point in it is surrounded by a disk entirely contained within the set. In other words, it consists only of interior points.

Remark - Biholomorphic

A mapping is biholomorphic if it is holomorphic and its Inverse function also exists and is holomorphic. Such mappings are, in particular, Homeomorphism, i.e., continuous in both directions.

Simply Connected Domains

From this and using the Riemann Mapping Theorem, it can be concluded that all simply connected domains that are proper subsets of $\mathbb {C}$ are topologically equivalent. In fact, $\mathbb {C}$ itself is also topologically equivalent to these domains.

Existence of Biholomorphic Mappings

For any point $z$ in the simply connected domain $G$, the following holds:

There exists exactly one biholomorphic function $h$ from $G$ auf $\mathbb {D}$

mit $h(z)=0$ und $h'(z)>0$.

Large Riemann Mapping Theorem

The Large Riemann Mapping Theorem, also known as the Uniformization Theorem (proven by Paul Koebe and Henri Poincaré), generalizes the above theorem. It states:

Every simply connected Riemann surface is biholomorphically equivalent to exactly one of the following surfaces:

• The unit disk $\mathbb {D}$, or the hyperbolic half-plane $\mathbb {H}$ (which is equivalent to it),
• The complex plane $\mathbb {C}$, or
• The Riemann sphere $\mathbb {P} ^{1}(\mathbb {C} ).$.

Remark - Liouville's Theorem

It is relatively straightforward to see that the three mentioned Riemann surfaces are pairwise not biholomorphically equivalent. A biholomorphic mapping from $\mathbb {C}$ to $\mathbb {D}$ is not possible due to Liouville's Theorem (as a holomorphic function on $\mathbb {C}$ that is bounded must be constant), and the Riemann sphere is compact and therefore not homeomorphic, and hence not biholomorphically equivalent, to $\mathbb {D}$ or $\mathbb {C}$.

Dependencies of Proofs

It should be noted that the first Riemann Mapping Theorem (or at least its proof ideas) is used in the proof of the Large Riemann Mapping Theorem. Thus, this does not provide an independent derivation of the Riemann Mapping Theorem.

Literature

Eberhard Freitag & Rolf Busam: Funktionentheorie 1, Springer-Verlag, Berlin, ISBN 3-540-67641-4

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