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Conformal rigidity: progress and challenges

  • Dimitrios Ntalampekos
  • Malik Younsi
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Abstract

A domain in the Riemann sphere is called a \textit{circle domain} if every connected component of its boundary is either a round circle or a single point. A circle domain is \textit{conformally rigid} if every conformal map onto another circle domain is the restriction of a Möbius transformation. Conformal rigidity is closely related to Koebe's conjecture, which asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain. In this article, we survey recent progress on conformal rigidity, with particular emphasis on its relationship with conformal removability.

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Cite

BibTeXRIS
BibTeX
@misc{ntalampekos2026conformal,
  title = {{Conformal rigidity: progress and challenges}},
  author = {Dimitrios Ntalampekos and Malik Younsi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00560},
  eprint = {2610.00560},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00560},
}
RIS
TY  - UNPB
TI  - Conformal rigidity: progress and challenges
AU  - Dimitrios Ntalampekos
AU  - Malik Younsi
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00560
UR  - https://arxiv.org/abs/2610.00560
AB  - A domain in the Riemann sphere is called a \textit{circle domain} if every connected component of its boundary is either a round circle or a single point. A circle domain is \textit{conformally rigid} if every conformal map onto another circle domain is the restriction of a Möbius transformation. Conformal rigidity is closely related to Koebe's conjecture, which asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain. In this article, we survey recent progress on conformal rigidity, with particular emphasis on its relationship with conformal removability.
ER  -