Summary
Let Ω be a circle domain in the Riemann sphere
whose boundary has σ-finite linear measure. We show that Ω is rigid in the sense that any conformal homeomorphism of Ω onto any other circle domain is equal to the restriction of a Möbius transformation. Previously, Kaufman and Bishop have independently found examples of non-rigid circle, domains whose boundary is a Cantor set of (Hausdorff) dimension one.
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Oblatum 19-VI-1992 & 2-IV-1993
Supported by NSF and Sloan Foundation
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He, ZX., Schramm, O. Rigidity of circle domains whose boundary hasσ-finite linear measure. Invent Math 115, 297–310 (1994). https://doi.org/10.1007/BF01231761
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DOI: https://doi.org/10.1007/BF01231761