Conformal rigidity: progress and challenges

Abstract

A domain in the Riemann sphere is called a \textit{circle domain} if every connected component of its boundary is either a round circle or a single point. A circle domain is \textit{conformally rigid} if every conformal map onto another circle domain is the restriction of a Möbius transformation. Conformal rigidity is closely related to Koebe's conjecture, which asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain. In this article, we survey recent progress on conformal rigidity, with particular emphasis on its relationship with conformal removability.