Rigidity theorems for circle domains

TL;DR

This paper proves that certain classes of circle domains, including Holder and John domains, are conformally rigid under specific quasihyperbolic conditions, supporting a conjecture linking rigidity and removability.

Contribution

It establishes conformal rigidity for circle domains satisfying a quasihyperbolic condition, extending known results to Holder and John domains.

Findings

Holder circle domains are conformally rigid.

John circle domains are conformally rigid.

Supports conjecture linking rigidity and conformal removability.

Abstract

A circle domain $\Omega$ in the Riemann sphere is conformally rigid if every conformal map from $\Omega$ onto another circle domain is the restriction of a M\"{o}bius transformation. We show that circle domains satisfying a certain quasihyperbolic condition, which was considered by Jones and Smirnov, are conformally rigid. In particular, H\"{o}lder circle domains and John circle domains are all conformally rigid. This provides new evidence for a conjecture of He and Schramm relating rigidity and conformal removability.