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)
Manuscripts (2)
Removable Boundaries and Rigidity of Circle Domains
September 23, 2026 · source · cite as \cite{OAI:Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026}
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.
Koebe's Circle-Domain Conjecture
September 23, 2026 · source · cite as \cite{OAI:Koebes-Circle-Domain-Conjecture-September-23-2026}
We prove that every domain in the Riemann sphere is conformally equivalent to a circle domain, resolving Koebe's circle-domain conjecture positively.
Result 071 · Real and complex analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md