Riemann's illustration theorem

The (small) Riemannian mapping theorem is a statement from the function theory , which was named after Bernhard Riemann . Bernhard Riemann sketched a proof in his dissertation in 1851. In 1922, the statement was finally proven by Lipót Fejér and Frigyes Riesz . A proof that is widespread today, using Montel's theorem , comes from Alexander Markowitsch Ostrowski from 1929. There is a generalization of Riemann's mapping theorem, which is referred to as the great Riemann's mapping theorem.

Riemann's illustration theorem

Every simply connected area can be mapped biholomorphically on the open unit disk . ${\displaystyle G\subsetneq \mathbb {C} }$ ${\displaystyle \mathbb {D} }$

To clarify the terms used in this sentence:

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

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

The notation means “real subset ” and states that the area must be unequal . ${\displaystyle G\subsetneq \mathbb {C} }$ ${\displaystyle G}$ ${\displaystyle \mathbb {C} }$

An open set in can be characterized by the fact that each of its points is surrounded by a circular disk that lies entirely within this set; in other words, that it consists only of internal points. ${\displaystyle \mathbb {C} }$

A mapping is biholomorphic if it is holomorphic and if its inverse mapping exists and this is also holomorphic. In particular, such mappings are homeomorphisms , i.e. continuous in both directions. From this and using Riemann's mapping theorem, one can conclude that all simply connected regions that are real subsets of are topologically equivalent. In fact, it is also topologically equivalent to these. ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$

For every point of the simply connected area the following applies: There is exactly one biholomorphic function of on with and . ${\displaystyle z}$ ${\displaystyle G}$ ${\displaystyle h}$ ${\displaystyle G}$ ${\displaystyle \mathbb {D} }$ ${\displaystyle h(z)=0}$ ${\displaystyle h'(z)>0}$

Large Riemannian set of illustrations

The large Riemannian figure theorem , also known as the uniformization theorem (proven by Paul Koebe , Henri Poincaré ), is a generalization of the above mentioned theorem. It says:

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

• the unit disk , or to the equivalent hyperbolic half-plane , ${\displaystyle \mathbb {D} }$ ${\displaystyle \mathbb {H} }$
• the complex number plane or ${\displaystyle \mathbb {C} }$
• the Riemann number ball ${\displaystyle \mathbb {P} ^{1}(\mathbb {C} ).}$

Note: It is comparatively easy to see that the three Riemann surfaces mentioned are pairwise not biholomorphically equivalent: According to Liouville's theorem, a biholomorphic mapping from to is not possible (because it is holomorphic to and restricted, i.e. constant) and the number sphere is compact and thus, for purely topological reasons, is not homeomorphic and therefore not biholomorphic equivalent to or . ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {D} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {D} }$ ${\displaystyle \mathbb {C} }$

Furthermore, the Riemann mapping theorem follows easily from the large Riemann mapping theorem by means of similar considerations. If a region is simply connected, it cannot be biholomorphic to the Riemann number sphere for reasons of compactness. If it is not the complex plane, then there is no restriction . But then there is a simply connected area in the dotted plane, then a branch of the square root exists . Hence it can not be biholomorphic . According to the great Riemann mapping theorem, must therefore be biholomorphic . That is the statement of the Riemann mapping theorem. ${\displaystyle G\subsetneq \mathbb {C} }$ ${\displaystyle G}$ ${\displaystyle 0\notin G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle G}$ ${\displaystyle \mathbb {D} }$

It must be said, however, that the former Riemann mapping theorem (or at least its proof ideas) are used to prove the large Riemann mapping theorem. In this way, no new derivation of the Riemann mapping theorem is obtained.

Individual evidence

1. ^ W. Fischer, I. Lieb: Funktionentheorie , Vieweg-Verlag 1980, ISBN 3-528-07247-4 , chapter IX, sentence 7.1
2. ↑ Otto Forster : Riemannsche surfaces , Heidelberger Taschenbücher Volume 184, Springer-Verlag, ISBN 3-540-08034-1 , sentence 27.9

literature

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