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.

arXiv commentTo appear in Proceedings of the 2026 International Congress of Basic Science