Finitely connected domains, Rational maps and Ahlfors functions

2013/09/11 by Maxime Fortier Bourque, Bourque, Maxime Fortier, Malik Younsi +1
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:26C15 #msc:32G15 #primary 32G15 #secondary 26C15

paper · pdf · doi:10.48550/arxiv.1309.2885

20 pages

arxiv created 2013/09/11 · arxiv updated 2013/09/12

Abstract

Using Ahlfors functions, Grunsky maps and the Bell representation theorem, we show that a certain subset of the rational maps of degree n forms a trivial bundle over the moduli space of non-degenerate n-connected domains with one marked tangent vector with fiber the n-fold symmetric product of the circle. A consequence is that the set of rational Ahlfors functions of degree n forms a closed embedded submanifold inside the space of rational maps of degree n. As an application, we show the existence of rational Ahlfors functions with non-positive residues, resolving a question left open in a previous paper by the authors.

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