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Rigidity theorems for circle domains (1809.05573v2)
Published 14 Sep 2018 in math.CV
Abstract: A circle domain $\Omega$ in the Riemann sphere is conformally rigid if every conformal map from $\Omega$ onto another circle domain is the restriction of a M\"{o}bius transformation. We show that circle domains satisfying a certain quasihyperbolic condition, which was considered by Jones and Smirnov, are conformally rigid. In particular, H\"{o}lder circle domains and John circle domains are all conformally rigid. This provides new evidence for a conjecture of He and Schramm relating rigidity and conformal removability.