Planar relative Schottky sets and quasisymmetric maps

TL;DR

This paper investigates quasisymmetric maps between measure-zero relative Schottky sets in planar domains, proving conformality, bi-Lipschitz properties, and establishing uniformization and rigidity results.

Contribution

It introduces new rigidity and uniformization theorems for quasisymmetric maps between relative Schottky sets in Jordan domains.

Findings

Quasisymmetric maps are conformal and locally bi-Lipschitz.

First derivatives of these maps are locally Lipschitz.

Established a uniformization theorem for relative Schottky sets.

Abstract

A relative Schottky set in a planar domain D is a subset of D obtained by removing from D open geometric discs whose closures are in D and are pairwise disjoint. In this paper we study quasisymmetric and related maps between relative Schottky sets of measure zero. We prove, in particular, that quasisymmetric maps between such sets in Jordan domains are conformal, locally bi-Lipschitz, and that their first derivatives are locally Lipschitz. We also provide a locally bi-Lipschitz uniformization result for relative Schottky sets in Jordan domains and establish rigidity with respect to local quasisymmetric maps for relative Schottky sets in the unit disc.