Koebe's 1908 Kreisnormierungsproblem predicts that every domain in the Riemann sphere is conformally homeomorphic to a circle domain, i.e. a domain whose complementary components are points or closed round disks; the conjecture remains open, though it is known for countably connected, cofat, cospread, and Gromov hyperbolic domains. This record concerns the conjecture restricted to a natural class of domains from complex dynamics: infinitely connected Fatou domains of rational maps. Wang and Zhong (2024) proved that an infinitely connected attracting Fatou domain of a geometrically finite rational map — one whose postcritical set meets the Julia set in finitely many points — is conformally homeomorphic to a circle domain, by showing that the quasicircle boundary components of the Cui–Peng holomorphic model are uniform and applying Schramm's cofat uniformization theorem. Both remaining dynamical sectors are open: (a) attracting Fatou domains of rational maps that are not geometrically finite, where the known quasicircle models lack uniformity and no conformally invariant sufficient condition is known to apply, and (b) parabolic Fatou domains, for which a holomorphic model was constructed only in 2025 and no uniformization is known. The problem asks whether every infinitely connected attracting or parabolic Fatou domain of an arbitrary rational map is conformally homeomorphic to a circle domain.

Background

A circle domain is a plane domain each of whose complementary components in the Riemann sphere C^\widehat{\mathbb{C}} is either a point or a closed round disk. In 1908 Paul Koebe conjectured (the Kreisnormierungsproblem) that every domain in C^\widehat{\mathbb{C}} is conformally homeomorphic to a circle domain; he settled the finitely connected case in 1920. He and Schramm confirmed the countably connected case in 1993, and in 1995 Schramm introduced transboundary extremal length — a modulus quantity for curve families that is allowed to run across boundary components, treating each complementary component as a single point of a quotient space — and used it to prove the conjecture for cofat domains, whose complementary components are uniformly "fat" in area. Later sufficient conditions include uniform domains (domains in which any two points can be joined by an arc whose length and distance from the boundary are each bounded by a constant multiple of the distance between the two points) and, very recently, cospread domains (a conformally invariant condition requiring the complementary components to be uniformly spread out), domains bounded by quasitripods (unions of three arcs of comparable diameters meeting at one common endpoint), nondegenerate domains with bounded gap-ratio (domains in which the gaps between complementary components are uniformly controlled relative to their sizes), and the conformally invariant class of Gromov hyperbolic domains, i.e. domains whose quasihyperbolic metric (the intrinsic path metric in which speed is weighted by the reciprocal of the Euclidean distance to the boundary) has geodesic triangles that are uniformly thin. The conjecture for arbitrary domains remains open.

Fatou domains of rational maps form a natural test class for the conjecture. For a rational map ff of degree at least two, the Fatou set is the largest open set on which the iterates of ff form a normal family (a family of holomorphic functions in which every sequence has a subsequence converging locally uniformly), and the Julia set is its complement; the Fatou set splits into connected components called Fatou domains. Sullivan's no-wandering-domains theorem implies every Fatou domain is eventually periodic, and a periodic domain is of one of four types: an attracting or superattracting basin (iterates converge to an attracting periodic cycle), a parabolic basin (iterates converge along attracting petals to a neutral periodic point whose multiplier — the complex derivative of the return map at the periodic point — is a root of unity), a Siegel disk (a Fatou domain on which an iterate of ff is conformally conjugate to an irrational rotation of a disk), or a Herman ring (a Fatou domain conformally conjugate to an irrational rotation of a round annulus). A Siegel disk is conformally a disk and a Herman ring is conformally a round annulus, so both are trivially circle domains; but an infinitely connected attracting or parabolic Fatou domain — one whose complement has infinitely many components, as happens for rational maps whose Julia set has infinitely many components — has uncountably many, highly irregular boundary components, so none of the classical uniformization theorems applies.

Two prior results structure the question. First, Cui and Peng (2008) proved that an infinitely connected fixed attracting Fatou domain UU of an arbitrary rational map is conformally conjugate to a completely invariant Fatou domain VV (a domain with g−1(V)=Vg^{-1}(V)=V) of a "simple attracting" rational map gg (the holomorphic model), whose boundary components are quasicircles or points; a quasicircle is a Jordan curve that is the image of a circle under a quasiconformal self-map of the plane (a homeomorphism whose infinitesimal distortion is uniformly bounded), equivalently a Jordan curve satisfying the bounded-turning condition: the smaller subarc between any two of its points has diameter at most a constant multiple of the distance between the points. However, the quasicircle constants of the Cui–Peng model are not uniform in general, while a domain whose boundary components are uniform quasicircles (quasicircles whose bounded-turning constants share one uniform bound) or points is cofat; hence Schramm's theorem cannot be applied directly. Second, Wang and Zhong (2024) closed the gap under a dynamical finiteness hypothesis: a rational map is geometrically finite if its postcritical set (the union of the forward orbits of its critical points) meets the Julia set in only finitely many points, and they proved that in that case the non-trivial boundary components of the model VV are uniform quasicircles — via bounded-degree and turning-distortion estimates on dynamical nested "puzzle pieces" (the nested closed disks obtained as components of preimages, under iterates of the map, of a fixed finite collection of Jordan domains covering a neighborhood of the Julia set) — so that VV, hence UU, is conformally homeomorphic to a circle domain. What their argument leaves open is precisely the two sectors stated as the problem below: the attracting case without geometric finiteness, and the parabolic case.

Problem Statement

Let ff be a rational map of degree at least two on the Riemann sphere, and let UU be an infinitely connected attracting (including superattracting) or parabolic Fatou domain of ff. Must UU be conformally homeomorphic to a circle domain?

The attracting case is known only under the additional hypothesis that ff is geometrically finite (Wang and Zhong, 2024). The problem is to settle, by proof or counterexample, both remaining sectors:

(a) Attracting sector: remove the geometric-finiteness hypothesis. Decide whether every infinitely connected attracting Fatou domain of an arbitrary rational map is conformally homeomorphic to a circle domain.

(b) Parabolic sector: decide the same question for infinitely connected parabolic Fatou domains of arbitrary rational maps, for which the recently constructed holomorphic model (Gao–Gao–Peng, 2025) currently yields only Jordan-curve boundary components.

A complete affirmative answer resolves both sectors. A single explicit rational map with an infinitely connected attracting or parabolic Fatou domain that is not conformally homeomorphic to any circle domain refutes the claim. The parent problem — Koebe's conjecture for arbitrary domains in C^\widehat{\mathbb{C}} — remains open but is not part of the required scope: this problem asks only for the two dynamical sectors above.

References

  1. Xiaoguang Wang and Yi Zhong, Koebe uniformization for infinitely connected attracting Fatou domains, arXiv:2406.13524 (2024). https://arxiv.org/abs/2406.13524
  2. Yi Zhong, Koebe uniformization of nondegenerate domains with bounded gap-ratio, Proceedings of the American Mathematical Society (published online January 2026), DOI 10.1090/proc/17564; preprint arXiv:2408.03484 (2024). https://arxiv.org/abs/2408.03484
  3. Christina Karafyllia and Dimitrios Ntalampekos, Uniformization of Gromov hyperbolic domains by circle domains, arXiv:2405.13782 (2024). https://arxiv.org/abs/2405.13782
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  12. Behnam Esmayli and Kai Rajala, Conformal Uniformization of Domains Bounded by Quasitripods, arXiv:2401.08485 (2024); Duke Mathematical Journal, to appear (2026). https://arxiv.org/abs/2401.08485