Koebe's Conjecture on Circle Domain Uniformization

Determine whether for every domain Ω in the Riemann sphere there exists a conformal map from Ω onto a circle domain, where a circle domain is a domain whose complementary components are closed disks or points (Koebe's conjecture).

Background

The paper studies quasiconformal uniformization of compact subsets of the Riemann sphere to Schottky sets and discusses related uniformization problems. In this context, the transboundary modulus tool of Schramm has been central in advancing uniformization and rigidity results and is closely tied to Koebe's conjecture.

Koebe's conjecture concerns conformal uniformization of arbitrary planar domains to circle domains. Although the present work focuses on quasiconformal homeomorphisms of the entire sphere, the conjecture remains a foundational open question in complex analysis and serves as a motivating backdrop for uniformization results, including those for Sierpiński carpets.

References

Koebe's conjecture asserts that for every domain $\Omega\subset \widehat $ there exists a conformal map from $\Omega$ onto a circle domain.

— Quasiconformal characterization of Schottky sets  (2507.22658 - Ntalampekos, 30 Jul 2025) in Section 1.1 Background

This remains the most general result known to date concerning Koebe's conjecture, while the conjecture in full generality is still open.

— Koebe uniformization, John domains, and transboundary modulus  (2609.31216 - Karafyllia et al., 25 Sep 2026) in Section 1, subsection “Background”