Removability, rigidity of circle domains and Koebe's Conjecture

TL;DR

This paper explores the conditions under which circle domains are conformally rigid, establishing equivalences between different rigidity conjectures and linking removability with quasiconformal rigidity.

Contribution

It proves the equivalence of two major rigidity conjectures for a large class of circle domains and introduces trans-quasiconformal deformation of Schottky groups.

Findings

Two rigidity conjectures are equivalent for many circle domains.

Conformal rigidity is equivalent to quasiconformal rigidity in this context.

Removability of certain sets underpins the rigidity results.

Abstract

A circle domain $\Omega$ in the Riemann sphere is conformally rigid if every conformal map of $\Omega$ onto another circle domain is the restriction of a M\"{o}bius 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.