Removability, rigidity of circle domains and Koebe's conjecture

Abstract

A circle domain Ω in the Riemann sphere is conformally rigid if every conformal map of Ω onto another circle domain is the restriction of a Möbius transformation. We show that two rigidity conjectures of He and Schramm are in fact equivalent, at least for a large family of circle domains. The proof follows from a result on the removability of countable unions of certain conformally removable sets. We also introduce trans-quasiconformal deformation of Schottky groups to prove that a circle domain is conformally rigid if and only if it is quasiconformally rigid, thereby providing new evidence for the aforementioned conjectures.