| # Koebe’s circle-domain conjecture | |
| The following describes the scope of the Lean formalization related to the following accompanying paper(s): | |
| - [Removable Boundaries and Rigidity of Circle Domains](../../preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026/paper.pdf) | |
| - [Koebe's Circle-Domain Conjecture](../../preprints/Koebes-Circle-Domain-Conjecture-September-23-2026/paper.pdf) | |
| ## 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](../ComparatorChallenges/KoebeCircleDomains.lean) | | |
| | Koebe's circle-domain theorem and removable-boundary rigidity | [KoebeCircleDomains.lean](../ComparatorChallenges/KoebeCircleDomains.lean) | | |