Carathéodory's theorem (conformal mapping)

In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, first proved in Carathéodory (1913a), states that the conformal mapping sending the unit disk to the region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve. The result is one of Carathéodory's results on prime ends and the boundary behaviour of univalent holomorphic functions.

The first proof of Carathéodory's theorem presented here is a summary of the short self-contained account in Garnett & Marshall (2005), pp. 14–15; there are related proofs in Pomerenke (1992) and Krantz (2006).

Carathéodory's theorem. If f maps the open unit disk D conformally onto a bounded domain U in C, then f has a continuous one-to-one extension to the closed unit disk if and only if ∂D is a Jordan curve.

Clearly if f admits an extension to a homeomorphism, then ∂D must be a Jordan curve.

Conversely if ∂D is a Jordan curve, the first step is to prove f extends continuously to the closure of D. In fact this will hold if and only if f is uniformly continuous on D: for this is true if it has a continuous extension to the closure of D; and, if f is uniformly continuous, it is easy to check f has limits on the unit circle and the same inequalities for uniform continuity hold on the closure of D.

Suppose that f is not uniformly continuous. In this case there must be an ε > 0 and a point ζ on the unit circle and sequences z_n, w_n tending to ζ with |f(z_n) − f(w_n)| ≥ 2ε. This is shown below to lead to a contradiction, so that f must be uniformly continuous and hence has a continuous extension to the closure of D.

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