Carathéodory's theorem (conformal mapping)

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematical complex analysis, Carathéodory's theorem, proved by Carathéodory (1913),[1] states that if U is a simply connected open subset of the complex plane C, whose boundary is a Jordan curve Γ then the Riemann map

f: UD

from U to the unit disk D extends continuously to the boundary, giving a homeomorphism

F : Γ → S1

from Γ to the unit circle S1.

Such a region is called a Jordan domain. Equivalently, this theorem states that for such sets U there is a homeomorphism

F : cl(U) → cl(D)

from the closure of U to the closed unit disk cl(D) whose restriction to the interior is a Riemann map, i.e. it is a bijective holomorphic conformal map.

Another standard formulation of Carathéodory's theorem states that for any pair of simply connected open sets U and V bounded by Jordan curves Γ1 and Γ2, a conformal map

f : UV

extends to a homeomorphism

F: Γ1 → Γ2.

This version can be derived from the one stated above by composing the inverse of one Riemann map with the other.

A more general version of the theorem is the following. Let

g : D\to U

be the inverse of the Riemann map, where DC is the unit disk, and UC is a simply connected domain. Then g extends continuously to

G : cl(D) → cl(U)

if and only if the boundary of U is locally connected. This result was first stated and proved by Marie Torhorst in her 1918 thesis,[2] under the supervision of Hans Hahn, using Carathéodory's theory of prime ends.

Context[edit]

Intuitively, Carathéodory's theorem says that compared to general simply connected open sets in the complex plane C, those bounded by Jordan curves are particularly well-behaved.

Carathéodory's theorem is a basic result in the study of boundary behavior of conformal maps, a classical part of complex analysis. In general it is very difficult to decide whether or not the Riemann map from an open set U to the unit disk D extends continuously to the boundary, and how and why it may fail to do so at certain points.

While having a Jordan curve boundary is sufficient for such an extension to exist, it is by no means necessary . For example, the map

f(z) = z2

from the upper half-plane H to the open set G that is the complement of the positive real axis is holomorphic and conformal, and it extends to a continuous map from the real line R to the positive real axis R+; however, the set G is not bounded by a Jordan curve.

References[edit]

  1. ^ Carathéodory, C. (1913), "Über die gegenseitige Beziehung der Ränder bei der konformen Abbildung des Inneren einer Jordanschen Kurve auf einen Kreis", Mathematische Annalen (Springer Berlin / Heidelberg) 73 (2): 305–320, doi:10.1007/BF01456720, ISSN 0025-5831, JFM 44.0757.01 
  2. ^ Torhorst, Marie (1921), "Über den Rand der einfach zusammenhängenden ebenen Gebiete", Mathematische Zeitschrift 9 (1-2): 44–65, doi:10.1007/BF01378335 

Further References[edit]