Result 071, Real and complex analysis

Koebe’s circle-domain conjecture

Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture.

Lean formalization Proof

The bigger picture

Why it matters

Can every region of the complex plane, however complicated its holes, be represented by one with only circular holes and missing points? The reported result says yes, using maps that preserve the local geometry of angles.

What changes?

The manuscripts report that every domain, a connected open region, on the Riemann sphere (the complex plane plus infinity) is conformally equivalent to a circle domain. This means a reversible analytic map preserves angles, while the model's complementary components are round disks or points. They also report conditional rigidity: if a circle domain's boundary is conformally removable, every conformal equivalence to another circle domain is a Möbius transformation. Neither claim restricts the number of complementary components.

What does that help mathematicians do?

Existence supplies circular models; rigidity controls their ambiguity under an additional assumption. A conformally removable boundary is one across which any sphere homeomorphism conformal elsewhere must also be conformal. For such circle domains, the reported rigidity theorem rules out equivalences beyond Möbius transformations, the sphere's conformal symmetries. This establishes the removability-to-rigidity direction of the He–Schramm conjecture, not its converse, and does not assert rigidity for every circle domain.

Are there practical applications?

The immediate value is foundational: researchers studying conformal geometry can represent arbitrary regions by models with geometrically simple complementary components, even when there are infinitely many. For removable boundaries, the rigidity claim further limits how those models can differ. The supplied abstracts describe structural theorems, not computational procedures for constructing the maps.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Removable Boundaries and Rigidity of Circle Domains

September 23, 2026 25 pages

We prove that a circle domain with conformally removable boundary is conformally rigid, with no restriction on the number of complementary components. This establishes the removability-to-rigidity direction of the He–Schramm Conjecture.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026,
  author = {{OpenAI}},
  title = {{Removable Boundaries and Rigidity of Circle Domains}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf}{OAI:Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026}},
  year = {2026}
}

Koebe's Circle-Domain Conjecture

September 23, 2026 64 pages

We prove that every domain in the Riemann sphere is conformally equivalent to a circle domain, resolving Koebe's circle-domain conjecture positively.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Koebes-Circle-Domain-Conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{Koebe's Circle-Domain Conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Koebes-Circle-Domain-Conjecture-September-23-2026/paper.pdf}{OAI:Koebes-Circle-Domain-Conjecture-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/071.md.

Koebe’s circle-domain conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves the removability-to-rigidity direction of the He–Schramm conjecture. If a circle domain has conformally removable boundary, then every conformal equivalence from it to another circle domain agrees on the source with a Möbius transformation. There is no bound on the number of complementary components.

The same Comparator file also states Koebe's circle-domain theorem, which gives a circle-domain representative for every domain in the Riemann sphere.

Koebe's circle-domain conjecture asks whether every domain in the Riemann sphere is conformally equivalent to a circle domain, whose complementary components are round disks or points. The formalization proves this for every domain. It also proves rigidity under a separate hypothesis: a conformal equivalence between circle domains is a Möbius transformation when the source boundary is conformally removable.

Comparator links

Result Comparator statement
Removable-boundary rigidity and the circle-domain theorem KoebeCircleDomains.lean
Koebe's circle-domain theorem and removable-boundary rigidity KoebeCircleDomains.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.