A domain in the Riemann sphere is called a circle domain if every connected component of its boundary is either a round circle or a point. The famous Koebe uniformization conjecture states that every planar domain is conformally equivalent to a circle domain. This has only been proved in some special cases, such as domains with at most countably many boundary components, thanks to the major progress of He and Schramm in the 1990's.
In this talk, I will discuss uniqueness of the Koebe conformal map, which is closely related to the notion of conformal rigidity. More precisely, we say that a circle domain is (conformally) rigid if every conformal map of the domain onto another circle domain is the restriction of a Mobius transformation. It is well-known that some circle domains are rigid while some are not, but both sufficient and necessary conditions are yet to be found. I will survey recent results on a conjecture of He and Schramm relating rigidity to the notion of conformal removability. This is partly based on joint work with Dimitrios Ntalampekpos.
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