Conformal rigidity: progress and challenges

A domain in the Riemann sphere is called a $\mathit{\text{circle}}\mathit{\text{domain}}$ if every connected component of its boundary is either a round circle or a single point. A circle domain is $\mathit{\text{conformally}}\mathit{\text{rigid}}$ 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.