Characterization of tangent quasicircles and quasiannuli

TL;DR

This paper provides a comprehensive characterization of tangent quasicircles and quasiannuli, establishing conditions for quasiconformal mappings between Jordan regions and disks, with applications to polynomial cusps and Schottky sets.

Contribution

It introduces new necessary and sufficient conditions for quasiconformal equivalence of Jordan regions, including Lipschitz characterizations and extensions to complex sets.

Findings

Characterization of when disjoint Jordan regions can be quasiconformally mapped to disks.

Proof that all polynomial cusps are quasiconformally equivalent.

Extension of quasisymmetric embeddings to quasiconformal maps of the sphere.

Abstract

We give a necessary and sufficient condition so that a pair of disjoint Jordan regions in the sphere can be quasiconformally mapped to a pair of disks. As a consequence, we obtain a simple characterization that involves Lipschitz functions for the case that one of the Jordan regions is a half-plane. We apply these results to prove that all polynomial cusps are quasiconformally equivalent and that a quasisymmetric embedding of the union of two disjoint disks extends to a quasiconformal map of the sphere, quantitatively. Also, in combination with previous work of the author, we obtain a new characterization of compact sets that are quasiconformally equivalent to Schottky sets.