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arXiv · 1809.05573

Rigidity theorems for circle domains

Dimitrios Ntalampekos·Malik Younsi
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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.

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Dimitrios Ntalampekos, Malik Younsi. 2018-09-14. Rigidity theorems for circle domains. https://doi.org/10.1007/s00222-019-00921-1

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