Koebe’s circle-domain conjecture

What OpenAI claims

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)

Papers (2)

1. Removable Boundaries and Rigidity of Circle Domains

   23 September 2026 · Preprint, not peer reviewed · PDF

   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.

2. Koebe's Circle-Domain Conjecture

   23 September 2026 · Preprint, not peer reviewed · PDF

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

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.