Rigidity and continuous extension for conformal maps of circle domains

TL;DR

This paper establishes conditions under which conformal maps between certain planar domains extend continuously and proves that circle domains with CNED boundary components are conformally rigid, supporting the He-Schramm rigidity conjecture.

Contribution

It introduces new sufficient conditions involving cofat domains and CNED sets for continuous extension of conformal maps and proves the conformal rigidity of circle domains with CNED boundary components.

Findings

Conformal maps extend continuously under cofat and CNED conditions.

Circle domains with CNED boundary components are conformally rigid.

Supports the He-Schramm rigidity conjecture.

Abstract

We present sufficient conditions so that a conformal map between planar domains whose boundary components are Jordan curves or points has a continuous or homeomorphic extension to the closures of the domains. Our conditions involve the notions of cofat domains and CNED sets, i.e., countably negligible for extremal distance, recently introduced by the author. We use this result towards establishing conformal rigidity of a class of circle domains. A circle domain is conformally rigid if every conformal map onto another circle domain is the restriction of a M\"obius transformation. We show that circle domains whose point boundary components are CNED are conformally rigid. This result is the strongest among all earlier works and provides substantial evidence towards the rigidity conjecture of He-Schramm, relating the problems of conformal rigidity and removability.