The Koebe Kreisnormierungsproblem (1908), one of the oldest open problems in geometric function theory, asks whether every domain in the Riemann sphere is conformally equivalent to a circle domain — a domain each of whose complementary components is a single point or a closed round disk. The conjecture is confirmed for finitely connected domains (Koebe), countably connected domains (He–Schramm, 1993), and for geometric classes including uniform, cofat, cospread, Gromov hyperbolic, and nondegenerate domains with bounded gap-ratio (Zhong, 2026). It remains open in full generality — most notably for domains with uncountably many complementary components subject to no geometric condition — and no counterexample is known. The problem is to prove the conjecture for all domains, or to exhibit a domain that is not conformally equivalent to any circle domain.

Background

Koebe's Kreisnormierungsproblem sits at the heart of geometric function theory, the branch of complex analysis that studies conformal maps — holomorphic bijections, i.e. angle- and orientation-preserving one-to-one analytic maps — between plane domains. A domain is a nonempty connected open set, and the natural ambient space is the Riemann sphere C^=C∪{∞}\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}, the extended complex plane. The Riemann mapping theorem (1851) gives every simply connected domain other than C^\hat{\mathbb{C}} itself a canonical model: it is conformally equivalent to the unit disk. For multiply connected domains the proposed canonical models are circle domains: domains in C^\hat{\mathbb{C}} each of whose complementary components (equivalently, boundary components) is either a single point or a closed round disk. Möbius transformations z↦(az+b)/(cz+d)z\mapsto(az+b)/(cz+d) map circles and points to circles and points, hence permute circle domains, which gives the model a built-in normalization freedom.

In 1908 Paul Koebe posed the Kreisnormierungsproblem ('circle normalization problem'): every domain in C^\hat{\mathbb{C}} should be conformally equivalent to a circle domain. He settled the finitely connected case (finitely many complementary components) by 1920. The conjecture has since been confirmed for progressively larger classes, each imposing either a cardinality or a quantitative geometric condition on the complement:

  • Countably connected domains (at most countably many complementary components): the landmark theorem of He and Schramm (1993), proved through a fixed-point argument on configurations of tangent circles (circle packings); the circle-domain model is unique up to post-composition with Möbius transformations.
  • Uniform domains (Herron–Koskela, 1990): domains without inward or outward cusps, in which any two points can be joined by a curve whose arc length and distance to the boundary are mutually comparable.
  • Cofat domains (Schramm, 1995): every complementary component occupies a definite area fraction of every Euclidean ball it meets but does not fill. Schramm's proof introduced transboundary extremal length, a generalization of the classical extremal length (a conformally invariant measure of the 'width' of a curve family) to the quotient space in which each complementary component is collapsed to a single point carrying counting measure; it has become the central tool of the field.
  • Cospread domains (Esmayli–Rajala, 2026): every complementary component contains, at every location and scale, a quasitripod — the image of a three-segment star under a quasisymmetric (relative-size-preserving) map — of comparable diameter; unlike cofat domains, the components may have zero area.
  • Gromov hyperbolic domains (Karafyllia–Ntalampekos, 2024; Duke Math. J., to appear): domains that are Gromov hyperbolic — every geodesic triangle is uniformly thin — when equipped with the quasihyperbolic metric, which weights infinitesimal distance by the reciprocal distance to the boundary. This is the first conformally invariant class beyond the countably connected case; it contains all finitely connected and all uniform domains, and the same theorem resolves a conjecture of Bonk–Heinonen–Koskela (2001) by showing these are precisely the conformal images of uniform domains.
  • Nondegenerate domains with bounded gap-ratio (Zhong, 2026): nondegenerate means every complementary component EE satisfies area⁡(E)≥κ diam⁡(E)2\operatorname{area}(E)\ge\kappa\,\operatorname{diam}(E)^2 for some uniform κ>0\kappa>0; bounded gap-ratio requires that whenever two complementary components come close, the ratio of the farthest to the nearest distance between them stays uniformly bounded, so nearby components cannot touch at one point while extending far apart.
  • Infinitely connected attracting Fatou domains of geometrically finite rational maps (Wang–Zhong, 2024): in complex dynamics, a Fatou domain is a maximal region on which the iterates of a rational map form a normal family.

None of these classes contains all domains, and the conjecture remains open for domains with uncountably many complementary components under no geometric hypothesis. The standard strategy approximates a domain by finitely connected subdomains, maps each onto a circle domain using Koebe's finitely connected theorem, normalizes, and passes to a locally uniform limit; the obstruction is the final step; under Carathéodory kernel convergence (domains converge when every compact subset of the limit is eventually contained in the approximants), a limit of circle domains need not be a circle domain — Rajala (2025) constructed explicit sequences exhibiting this failure. Moreover, uniqueness of the circle-domain model, which holds up to Möbius transformations in all countably connected cases, can fail for uncountably connected domains. On the other side, no counterexample and no structural obstruction is known: a companion rigidity program (He–Schramm 1994; Younsi 2016; Ntalampekos–Younsi 2020; Ntalampekos 2023) characterizes when the model is unique but does not produce existence. As of the March 2026 survey of Ntalampekos, the conjecture 'remains open after almost 120 years'.

Problem Statement

Prove or disprove Koebe's Kreisnormierungsproblem in its original, unrestricted form:

Every domain Ω⊂C^ is conformally equivalent to a circle domain.\text{Every domain }\Omega\subset\hat{\mathbb{C}}\text{ is conformally equivalent to a circle domain.}

That is, either:

(a) prove that for every nonempty connected open set Ω⊂C^\Omega\subset\hat{\mathbb{C}} there exists a univalent (injective) holomorphic map f:Ω→C^f:\Omega\to\hat{\mathbb{C}} whose image is a circle domain — a domain each of whose complementary components is a single point or a closed round disk; or

(b) exhibit a domain Ω⊂C^\Omega\subset\hat{\mathbb{C}} and prove rigorously that no conformal map from Ω\Omega onto any circle domain exists.

An affirmative answer must hold for arbitrary domains: no hypothesis on the number of complementary components (which may be uncountable), on their separation, size, fatness, gap-ratio, hyperbolicity, or any other geometric regularity of the boundary may be assumed. Results restricted to a proper subclass — countably connected, uniform, cofat, cospread, Gromov hyperbolic, nondegenerate with bounded gap-ratio, or any other — constitute progress but do not answer this question.

References

  1. Yi Zhong, Koebe uniformization of nondegenerate domains with bounded gap-ratios, Proceedings of the American Mathematical Society 154 (2026), no. 4, 1685–1699, DOI 10.1090/proc/17564, https://doi.org/10.1090/proc/17564; preprint arXiv:2408.03484, https://arxiv.org/abs/2408.03484
  2. Dimitrios Ntalampekos, Uniformization problems in the plane: A survey, preprint arXiv:2603.15098 (March 2026), https://arxiv.org/abs/2603.15098
  3. P. Koebe, Über die Uniformisierung beliebiger analytischen Kurven. III, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1908), 337–358, zbMATH 40.0467.01, https://zbmath.org/?q=an%3A40.0467.01
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  6. Oded Schramm, Transboundary extremal length, Journal d'Analyse Mathématique 66 (1995), 307–329, DOI 10.1007/BF02788827, https://doi.org/10.1007/BF02788827
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  9. Christina Karafyllia and Dimitrios Ntalampekos, Uniformization of Gromov hyperbolic domains by circle domains, preprint arXiv:2405.13782 (2024); Duke Mathematical Journal, to appear, https://arxiv.org/abs/2405.13782
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