Riemann mapping theorem

The (small ) Riemann mapping theorem is a statement of the theory of functions, named after Bernhard Riemann. Bernhard Riemann in 1851 sketched a proof in his thesis. In 1922, the statement was proved conclusively by Lipót Fejér and Frigyes Riesz. A now widespread evidence that uses the set of Montel, comes from Alexander Markovich Ostrowski from the year 1929. From Riemannian mapping theorem, there is a generalization that is referred to as a great Riemannian mapping theorem.

Riemann mapping theorem

Every simply connected domain can be mapped biholomorphic to the open unit disk.

To clarify the terms used in this sentence:

The open unit disk is defined as

"Genuine subset " states that the area must be unequal.

An open set in can be characterized by that each of its points surrounding a circular disk that is entirely in this set; in that it is in other words only the inner points.

A picture is biholomorphic if it is holomorphic and if its inverse map exists and this is also holomorphic. In particular, such mappings are homeomorphisms, ie in both directions continuously. From this and using the Riemann mapping theorem, one can conclude that all simply connected domains, the proper subsets are topologically equivalent.

For each point of the simply connected domain applies: There is exactly one of biholomorphic function on and with.

Large Riemannian mapping theorem

The big Riemannian mapping theorem, also known as uniformization, is a generalization of the above theorem. He says:

• The unit disk, or to equivalent thereto hyperbolic half-plane,
• The complex plane or
• Riemannian number ball

Note: It is comparatively easy to see that these three Riemann surfaces not biholomorphic equivalent in pairs: a biholomorphic map from to is by the theorem of Liouville not possible ( because holomorphic on and limited, ie constant) and the Riemann sphere is compact and is therefore only for purely topological reasons not homeomorphic and thus not biholomorphic equivalent to or.