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We present sufficient conditions so that a conformal map between planar domains whose boundary components are Jordan curves or points has a continuous or homeomorphic extension to the closures of the domains. Our conditions involve the notions of cofat domains and C N E D CNED sets, i.e., countably negligible for extremal distances, recently introduced by the author. We use this result towards establishing conformal rigidity of a class of circle domains. A circle domain is conformally rigid if every conformal map onto another circle domain is the restriction of a Möbius transformation. We show that circle domains whose point boundary components are C N E D CNED are conformally rigid. This result is the strongest among all earlier works and provides substantial evidence towards the rigidity conjecture of He–Schramm [Invent. Math. 115 (1994), no. 2, 297–310], relating the problems of conformal rigidity and removability.
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This content will become publicly available on March 1, 2026
Point selections from Jordan domains in Riemannian surfaces
Using fiber bundle theory and conformal mappings, we continuously select a point from the interior of Jordan domains in Riemannian surfaces. This selection can be made equivariant under isometries, and take on prescribed values such as the center of mass when the domains are convex. Analogous results for conformal transformations are obtained as well. It follows that the space of Jordan domains in surfaces of constant curvature admits an isometrically equivariant strong deformation retraction onto the space of round disks. Finally we develop a canonical procedure for selecting points from planar Jordan domains.
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- Award ID(s):
- 2202337
- PAR ID:
- 10584388
- Publisher / Repository:
- America Mathematical Society
- Date Published:
- Journal Name:
- Transactions of the American Mathematical Society
- Volume:
- 378
- Issue:
- 1090
- ISSN:
- 0002-9947
- Page Range / eLocation ID:
- 1681 to 1696
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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