Carathéodory's theorem (conformal mapping)

In mathematics, the set of Carathéodory treated the continuability angle -preserving mappings on the edge of its domain of definition.

Conformal mappings, Riemann mapping theorem

A conformal mapping is, by definition, a figure that receives angle. A mapping between two subsets of the complex plane if and only compliant if it is holomorphic or anti- holomorphic and the derivative vanishes nowhere.

The Riemann mapping theorem states that it is simply connected to each, open, proper subset conformal homeomorphism

Are on the unit disk. It was formulated in 1851 by Riemann, but not proved until 1912 by Carathéodory. The Riemann mapping theorem is used among other things for the geometrization of space.

The Riemann mapping theorem is remarkable, among other reasons, simply because cohesive, open subsets of the plane can be very complicated, for example, can be a nowhere differentiable its edge, fractal curve of infinite length or no continuous parameterized curve.

In general it is not true that the Riemann mapping to a continuous map

The edge can be continued on the unit circle. The set of Carathéodory means, however, that such a continuation exists when the edge of a Jordan curve, ie the image of a continuous, injective map is. This includes nichtdifferenzierbare, a fractal curve with, for example the Koch curve.

Set of Carathéodory

Sentence: It is a simply connected, open subset of the complex plane whose boundary is a Jordan curve. Then you can choose any conformal mapping

Continuously to a homeomorphism of Financial Statements

Continue on the closed disk. In particular, a homeomorphism

Conclusion: Each conformal mapping between two Jordan curves bounded simply connected open subsets of the plane can be continued to a homeomorphism.

Reversal

The following statements are equivalent for a bounded, simply connected domain:

• Each conformal mapping can be continued to a homeomorphism.
• The boundary of a Jordan curve.
• Each edge point is simple, that is, to each episode there with a curve whose image contains all.

From the equivalence follows: the boundary of a bounded convex region is a Jordan curve.

Higher-Dimensional generalizations

The steady continuability images on the edge of an open set is an extensive research topic in mathematics, see for example set of Korevaar - Schoen or Cannon- Thurston theory.