Harmonic mappings of multiply connected domains

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In this paper the of Rad o-Kneser-Choquet is extended in two dierent 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 nitely 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 innity admits a unique harmonic mapping, with standard normalization at innity, 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. In 1926, T. Rad o[ 22] posed the problem to show that for any homeomorphism of the unit circle onto the boundary @ of a bounded convex domain ; the harmonic extension f maps the unit disk D univalently onto : In response, H. Kneser [14] supplied an elegant proof. Some 20 years later G. Choquet [4], apparently unaware that the was known, rediscovered it and gave another proof. Fortunately, the two proofs are dierent and even for simply connected domains they have dierent generalizations. The dichotomy between the two approaches of Kneser and Choquet comes into sharper focus as the is generalized to multiply connected domains. In presenting these generalizations, it will be expedient to distinguish between \Kneser’s theorem and \Choquet’s Kneser’s proof has little to do with convexity, while Choquet’s proof uses convexity in a more essential way. Indeed, Kneser’s proof applies (as he indicates in [14]) when is not convex, under the additional hypothesis that f(D) : We shall see that the main idea of his proof carries over to multiply connected domains. On other hand, by methods more akin to Choquet’s proof we will show that a nitely connected domain D can be mapped harmonically, with prescribed boundary values, onto a given convex domain with punctures at suitable points. Another result is that D can be mapped harmonically onto a punctured plane, and such a mapping is unique up to a normalization. Our proofs adapt an idea of Clunie and Sheil-Small [5], which gives yet another proof of the Rad o-Kneser-Choquet theorem.