Koebe's circle domain conjecture (Kreisnormierungsproblem) and the equivalent Weyl-type realization of genus-zero hyperbolic surfaces as convex hull boundaries in hyperbolic 3-space

Koebe's circle domain conjecture (1908), also called the Kreisnormierungsproblem, asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain, i.e. a domain all of whose boundary components are points or round circles. It is proved for domains with at most countably many boundary components (Koebe 1920 for finitely many; He-Schramm 1993 for countably many) and for uncountably connected domains satisfying geometric hypotheses (uniform, cofat, cospread, or Gromov hyperbolic), but it remains open for arbitrary domains, as confirmed by a March 2026 survey (Ntalampekos, arXiv:2603.15098). Luo and Wu (Advances in Mathematics, 2024) proved the conjecture equivalent to a Weyl-type problem in hyperbolic geometry: every genus-zero complete hyperbolic surface is isometric to the boundary of the hyperbolic convex hull of a circle-type closed set in the sphere at infinity of hyperbolic 3-space. The companion uniqueness conjecture and the discrete-uniformization conjecture from the same paper were settled in the countable case by 2025-2026 rigidity theorems; the surviving open core is to prove or refute the conjecture for domains with uncountably many boundary components.

Background

A domain is a connected open subset of the Riemann sphere $\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ . Two domains are conformally equivalent if there is a bijective holomorphic map from one onto the other. A circle domain is a domain each of whose boundary components is either a single point or a round circle; the unit disk and the plane are the simplest examples. Moebius transformations, the maps $z\mapsto(az+b)/(cz+d)$ with $ad-bc\neq0$ , send circles and points to circles and points, so they map circle domains to circle domains and provide the natural normalization of any uniformizing map. The Riemann mapping theorem settles the simply connected case. Koebe proved the finitely connected case in 1920 and in 1908 posed the general question, now called the circle domain conjecture or Kreisnormierungsproblem: is every domain in the Riemann sphere conformally equivalent to a circle domain?

The strongest classical result is due to He and Schramm (1993): every countably connected domain is conformally equivalent to a circle domain, unique up to post-composition with a Moebius transformation. Their proof, and Schramm's alternative proof via transboundary extremal length (a variant of the classical conformal invariant extremal length, which measures curve families by reciprocal infima of conformal energies, adapted so that curves may cross boundary components), exhausts the domain by finitely connected subdomains and controls the limit map; the argument uses countability in an essential way. For domains with uncountably many boundary components only geometric sufficient conditions are known: Schramm's cofat domains (complementary components occupy a fixed area fraction of small disks), uniform domains (points joinable by twisted cones of uniformly bounded aperture), Esmayli-Rajala's cospread domains (complementary components contain uniform quasitripods at all locations and scales), and Karafyllia-Ntalampekos's Gromov hyperbolic domains (domains whose quasihyperbolic metric, the complete metric obtained from the Euclidean one by pointwise rescaling by the reciprocal of the distance to the domain boundary, is delta-hyperbolic, i.e. geodesic triangles are uniformly thin). A March 2026 survey tabulates these cases and notes that uniqueness of the uniformizing map genuinely fails in the uncountable case (e.g. complements of positive-area Cantor sets), so the conjecture is an existence statement.

Luo and Wu (Advances in Mathematics 458, 2024) discovered an equivalent formulation in hyperbolic geometry. Let $\mathbb{H}^3_P$ be the Poincaré ball model of hyperbolic 3-space, whose sphere at infinity is the unit sphere $S^2=\partial\mathbb{H}^3_P$ , and for a closed set $Y\subset S^2$ let $C_P(Y)$ denote the hyperbolic convex hull of $Y$ in $\mathbb{H}^3$ . A compact set $Y\subset S^2$ is of circle type if every connected component of $Y$ is either a round disk or a point, i.e. its complement is a circle domain. A theorem of Thurston states that for any closed $Y\subset S^2$ with more than two points, the boundary $\partial C_P(Y)$ , equipped with its intrinsic path metric (doubled across itself when the hull is 2-dimensional), is a genus-zero complete hyperbolic surface, meaning a complete Riemannian surface of constant Gaussian curvature $-1$ homeomorphic to a domain in the sphere; conversely every such surface is the Poincaré metric surface of a plane domain. Luo and Wu's Theorem 1.1 proves that the circle domain conjecture is equivalent to their Conjecture 1: every genus-zero complete hyperbolic surface is isometric to $\partial C_P(Y)$ for some circle-type closed set $Y\subset S^2$ . Combined with He-Schramm, this realizes every genus-zero complete hyperbolic surface with countably many topological ends (a topological end being a complementary component that persists as the surface is exhausted by larger and larger compact subsets). Their key quantitative tool, beyond Schramm's transboundary extremal length and a duality theorem for it, is an area bound: the Euclidean area of the boundary of any convex set in the Poincaré model is at most $16\pi$ (Theorem 4.1), a constant the authors believe can be improved to the optimal $8\pi$ .

Problem Statement

Prove or disprove Koebe's circle domain conjecture in full generality: every domain (connected open set) in the Riemann sphere $\hat{\mathbb{C}}$ is conformally equivalent to a circle domain, i.e. a domain all of whose boundary components are single points or round circles.

By Luo-Wu's equivalence (Theorem 1.1 of Advances in Mathematics 458 (2024), article 109969), this is the same question as: every genus-zero complete hyperbolic surface is isometric, in its intrinsic path metric, to $\partial C_P(Y)$ , the boundary of the hyperbolic convex hull of a circle-type closed set $Y\subset S^2$ in the Poincaré ball model $\mathbb{H}^3_P$ of hyperbolic 3-space (with $\partial C_P(Y)$ read as the metric double when the hull is 2-dimensional).

The already-settled cases are not part of what remains: simply connected domains (Riemann mapping theorem), finitely and countably connected domains (Koebe 1920; He-Schramm 1993), and uncountably connected domains under the geometric hypotheses of uniformity, cofatness, cospreadness, or Gromov hyperbolicity. The open core is the case of domains with an arbitrary uncountable family of complementary components, equivalently genus-zero complete hyperbolic surfaces with uncountably many topological ends. No countability, geometric, measure-theoretic, or model-class restriction may be imposed on the domain.

References

Feng Luo, Tianqi Wu, The Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3-space, Advances in Mathematics 458 (2024), article 109969, DOI 10.1016/j.aim.2024.109969, https://doi.org/10.1016/j.aim.2024.109969 (preprint arXiv:1910.08001, https://arxiv.org/abs/1910.08001)
Zheng-Xu He, Oded Schramm, Fixed points, Koebe uniformization and circle packings, Annals of Mathematics (2) 137 (1993), no. 2, 369-406, DOI 10.2307/2946541, https://doi.org/10.2307/2946541
Dimitrios Ntalampekos, Uniformization problems in the plane: A survey, preprint arXiv:2603.15098 (2026), https://arxiv.org/abs/2603.15098
Christina Karafyllia, Dimitrios Ntalampekos, Uniformization of Gromov hyperbolic domains by circle domains, Duke Mathematical Journal (to appear), preprint arXiv:2405.13782, https://arxiv.org/abs/2405.13782
Behnam Esmayli, Kai Rajala, Conformal uniformization of domains bounded by quasitripods, Duke Mathematical Journal 175 (2026), no. 4, 555-605, DOI 10.1215/00127094-2025-0035, https://doi.org/10.1215/00127094-2025-0035 (preprint arXiv:2401.08485, https://arxiv.org/abs/2401.08485)
Oded Schramm, Transboundary extremal length, Journal d'Analyse Mathématique 66 (1995), 307-329, DOI 10.1007/bf02788827, https://doi.org/10.1007/bf02788827
Feng Luo, Yanwen Luo, Zhenghao Rao, A Rigidity Theorem for Convex Sets in Hyperbolic 3-Space, preprint arXiv:2602.23558 (2026), https://arxiv.org/abs/2602.23558
Xinrong Zhao, Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space, preprint arXiv:2606.15402 (2026), https://arxiv.org/abs/2606.15402
Feng Luo, Yanwen Luo, Zhenghao Rao, Xinrong Zhao, Discrete uniformization of polyhedral surfaces, preprint arXiv:2608.22491 (2026), https://arxiv.org/abs/2608.22491
Paul Koebe, Über die Uniformisierung beliebiger analytischen Kurven. Dritte Mitteilung, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1908), 337-358, JFM 40.0467.01 (zbMATH 2637577), https://zbmath.org/?q=an%3A2637577, digitized at EUDML: https://eudml.org/doc/58725
Paul Koebe, Abhandlungen zur Theorie der konformen Abbildung, Mathematische Zeitschrift 7 (1920), 235-301, DOI 10.1007/bf01199400, https://doi.org/10.1007/bf01199400