Rigidity theorems for circle domains

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 Möbius transformation. We show that circle domains satisfying a certain quasihyperbolic condition, which was considered by Jones and Smirnov, are conformally rigid. In particular, Hölder 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.

arXiv comment43 pages, 1 figure. Added more details and topological preliminaries in Section 3. Results unchanged

References in corpus (1)

• Planar relative Schottky sets and quasisymmetric maps

Cited by in corpus (5)

• Polyhedral approximation of metric surfaces and applications to uniformization
• Rigidity and continuous extension for conformal maps of circle domains
• Metric definition of quasiconformality and exceptional sets
• CNED sets: countably negligible for extremal distances
• Conformal uniformization of planar packings by disk packings