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PreprintCC BY 4.0arXiv

Koebe uniformization, John domains, and transboundary modulus

  • Christina Karafyllia
  • Dimitrios Ntalampekos
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Abstract

We prove Koebe's conjecture for John domains by showing that every planar John domain is conformally equivalent to a circle domain. Our proof is based on a geometric property of John domains, which we prove to be sufficient for uniformization by a circle domain. As a consequence, we obtain an alternative proof of Koebe's conjecture for inner uniform and Gromov hyperbolic domains, recovering results established in earlier work of the authors. Finally, we derive a necessary and sufficient condition for a domain to satisfy Koebe's conjecture by using Schramm's transboundary modulus and well-behaved exhaustions of circle domains constructed by the second author and Rajala.

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Cite

BibTeXRIS
BibTeX
@misc{karafyllia2026koebe,
  title = {{Koebe uniformization, John domains, and transboundary modulus}},
  author = {Christina Karafyllia and Dimitrios Ntalampekos},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31216},
  eprint = {2609.31216},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31216},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Koebe uniformization, John domains, and transboundary modulus
AU  - Christina Karafyllia
AU  - Dimitrios Ntalampekos
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31216
UR  - https://arxiv.org/abs/2609.31216
AB  - We prove Koebe's conjecture for John domains by showing that every planar John domain is conformally equivalent to a circle domain. Our proof is based on a geometric property of John domains, which we prove to be sufficient for uniformization by a circle domain. As a consequence, we obtain an alternative proof of Koebe's conjecture for inner uniform and Gromov hyperbolic domains, recovering results established in earlier work of the authors. Finally, we derive a necessary and sufficient condition for a domain to satisfy Koebe's conjecture by using Schramm's transboundary modulus and well-behaved exhaustions of circle domains constructed by the second author and Rajala.
ER  -