Harmonic mappings of multiply connected domains
The Radó-Kneser-Choquet theorem is extended in two different ways to multiply connected domains. One is a direct continuation of Kneser's idea and has nothing to do with convexity; while the other asserts that a finitely connected domain can be mapped harmonically with prescribed outer boundary correspondence onto a given convex domain with suitable punctures. It is also shown that a domain containing infinity admits a unique harmonic mapping, with standard normalization at infinity, onto a punctured plane. For domains of connectivity \(n\) the dilatation of the canonical mapping covers the unit disk exactly \(2n\) times. Furthermore, no other normalized harmonic mapping has the same dilatation.
scientific article; zbMATH DE number 950188