In this paper the theorem of
Radó-Kneser-Choquet 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.
