Rigidity and continuous extension for conformal maps of circle domains

We present sufficient conditions so that a conformal map between planar domains whose boundary components are Jordan curves or points has continuous homeomorphic extension to the closures of domains. Our involve notions cofat and C N E D CNED sets, i.e., countably negligible for extremal distances, recently introduced by author. use this result towards establishing rigidity class circle A domain is conformally rigid if every onto another restriction Möbius transformation. show point rigid. This strongest among all earlier works provides substantial evidence conjecture He–Schramm [Invent. Math. 115 (1994), no. 2, 297–310], relating problems removability.