Carathéodory's theorem (conformal mapping)

simply connected Julia set as an image of unit circle under Riemann map

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: U → D

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 : U→ V

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 D ⊂ C is the unit disk, and U ⊂ C 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

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

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

• Markushevich, A. I. (1977), Theory of functions of a complex variable. Vol. III, Chelsea Publishing Co., ISBN 0-8284-0296-5, MR 0444912
• Shields, Allen (1988), "Carathéodory and conformal mapping", The Mathematical Intelligencer 10 (1): 18–22, doi:10.1007/BF03023846, ISSN 0343-6993, MR 918659