Carathéodory's theorem (conformal mapping)

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See also Carathéodory's theorem for other meanings.

In mathematics, Carathéodory's theorem in complex analysis 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.

[edit] 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.