Finitely connected domains, Rational maps and Ahlfors functions

TL;DR

This paper explores the structure of rational maps associated with finitely connected domains, demonstrating that rational Ahlfors functions form a closed submanifold and establishing the existence of functions with non-positive residues.

Contribution

It introduces a new geometric framework for rational maps using Ahlfors functions and proves the existence of rational Ahlfors functions with non-positive residues.

Findings

Rational Ahlfors functions form a closed embedded submanifold.

A certain subset of rational maps forms a trivial bundle over the moduli space.

Existence of rational Ahlfors functions with non-positive residues is established.

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.