Uniformization problems in the plane: A survey

Abstract: In this survey we present the history and recent progress on several fundamental (quasi)conformal uniformization problems in the complex plane. Uniformization refers to the process of mapping a space to a canonical model by means of a well-behaved transformation that preserves the geometry and distorts shapes in a controlled fashion. A central problem in the area is Koebe's conjecture, which remains open after almost 120 years and predicts that each planar domain can be conformally mapped to a circle domain -- that is, a domain whose complementary components are points or closed disks. We trace the history of the conjecture, outline recent developments, and examine the associated uniqueness problem. We also discuss variants, with particular emphasis on the question whether a compact set can be mapped by a quasiconformal self-map of the plane to a Schottky set -- that is, a set in the plane whose complement is the union of disjoint open disks.

• The paper surveys conformal uniformization by circle domains, tracing results from the Riemann mapping theorem through He–Schramm’s countably connected case while highlighting that Koebe’s conjecture remains open for general domains.
• The paper explains how transboundary modulus, conformal rigidity, removability, and exhaustion methods shape current progress, including the discovery that rigidity does not generally imply boundary removability.
• The paper develops quasiconformal uniformization by Schottky sets, connecting pairwise circularizability and relative hyperbolic metrics to applications involving random carpets, Julia sets, and geometric group theory.

This survey by Ntalampekos presents the history and recent developments of uniformization problems in the complex plane, organized around two central themes: conformal uniformization of planar domains by circle domains (Koebe's conjecture) and quasiconformal uniformization of compact sets by Schottky sets. The unifying question is whether an arbitrary subset of the sphere can be transformed into a canonical object—a disk, a circle domain, a Schottky set—by maps whose distortion is controlled. The survey is notable for tracing how Schramm's transboundary modulus has become the principal tool underlying most modern progress in both directions.

Koebe's conjecture: history and partial results

A domain $\Omega \subset \widehat{\mathbb{C}}$ is a circle domain if every component of $\partial\Omega$ is a point or a circle. Koebe's conjecture (1908) asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain. The conjecture remains open after nearly 120 years.

The known cases follow a progression by connectivity. The Riemann mapping theorem settles the simply connected case; Koebe's uniformization theorem (1920) handles finitely connected domains; and the deepest result to date is the He–Schramm theorem (1993), which resolves the countably connected case with full uniqueness up to Möbius transformations. Beyond connectivity, the conjecture has been established under geometric hypotheses on complementary components:

Class of domains	Existence	Uniqueness
Simply connected	Riemann	yes
Finitely connected	Koebe	yes
Countably connected	He–Schramm	yes
Uniform domains	Herron–Koskela	Ntalampekos–Younsi
Cofat domains	Schramm	can fail
Cospread domains	Esmayli–Rajala	can fail
Gromov hyperbolic domains	Karafyllia–Ntalampekos	Ntalampekos–Younsi

Two features of this table deserve emphasis. First, the cofat and cospread conditions are strictly weaker than uniformity: cospread components may have vanishing area (the Julia set of $z^2+i$ is an example of a spread set), making Esmayli–Rajala's proof substantially more involved than Schramm's. Second, uniqueness fails in general for these weaker classes—if the boundary is a Cantor set of positive area, one constructs a homeomorphism of the plane that is conformal off the Cantor set yet not Möbius, via Uy's Lipschitz analytic functions. Thus existence and uniqueness genuinely decouple once connectivity becomes uncountable.

The Karafyllia–Ntalampekos result on Gromov hyperbolic domains carries particular structural weight: combined with Bonk–Heinonen–Koskela's earlier work, it yields that Gromov hyperbolicity (with respect to the quasihyperbolic metric) characterizes precisely the conformal images of uniform domains. This gives a conformally invariant class—which the Euclidean conditions of uniformity, cofatness, or spread are not—on which Koebe's conjecture holds with uniqueness.

The survey also records results "in the spirit" of Koebe's conjecture for other canonical targets: Hilbert and Grötzsch showed every domain is conformally equivalent to a slit domain, and Brandt–Harrington showed finitely connected domains admit conformal maps prescribing all non-degenerate complementary components up to homothety. Via a result of Schramm, Koebe's conjecture is equivalent to uniformization by square domains, and recent extremal problems of Bonk and Solynin–Vidanage for square/rectangular domains suggest a potentially productive route toward the general case.

Conformal rigidity versus removability

Uniqueness of the uniformizing map for a circle domain $D$ is equivalent to $D$ being conformally rigid. He and Schramm proved rigidity when the boundary has $\sigma$-finite length, and conjectured in 1994 that rigidity is equivalent to conformal removability of the boundary. Subsequent work strengthened the evidence: Ntalampekos–Younsi proved rigidity under an $L^2$ integrability condition on the hyperbolic distance, and Ntalampekos introduced CNED sets (countably negligible for extremal distance, generalizing Ahlfors–Beurling NED sets) proving rigidity whenever $\partial D$ is CNED—a class containing $\sigma$-finite-length sets and boundaries satisfying the Jones–Smirnov condition.

However, the He–Schramm conjecture was ultimately disproved by Rajala: there exists a conformally rigid circle domain whose boundary is not conformally removable. Any counterexample must have boundary containing both circles and points, since rigidity trivially implies removability for totally disconnected boundaries. Rajala's construction combines Ntalampekos's metric characterization of conformal maps with Wu's theorem that products of sufficiently thick Cantor sets are non-removable. Notably, only one direction of the original equivalence is refuted; whether removability implies rigidity remains open, and would follow if Ntalampekos's conjectured equivalence between removability and the CNED condition holds.

Exhaustions and approaches to the general case

The standard strategy for attacking Koebe's conjecture approximates $\Omega$ by finitely connected circle domains, applies Koebe's 1920 theorem, and passes to Carathéodory limits. The obstruction is that a sequence of circle domains need not converge to a circle domain—one can construct such sequences whose limit boundary contains line segments. External approximations suffice for countably connected domains (He–Schramm, Schramm); Rajala developed internal approximations (exhaustions by Jordan-curve-bounded domains) as an alternative proof. The sharp structural statement, due to Ntalampekos–Rajala, is that exhaustions work for exactly those domains satisfying the conjecture: if $\partial\Omega$0 is conformally equivalent to a circle domain, then some exhaustion produces locally uniformly convergent maps onto finitely connected circle domains with circle-domain limit. Consequently, to disprove Koebe's conjecture it suffices to exhibit a domain failing this exhaustion property, or the dual-type conditions of Rajala. This converts the conjecture into a more concrete, though still difficult, testable form.

Quasiconformal uniformization by Schottky sets

The second half of the survey treats a different regime: quasiconformal homeomorphisms of the entire sphere, so distortion control is global and boundary geometry becomes essential. A Schottky set is a compact set whose complement is a union of disjoint open disks.

For single Jordan curves, Ahlfors's characterization is definitive: a Jordan region is a quasidisk if and only if its boundary satisfies the quantitative turning condition $\partial\Omega$1 along connecting arcs. For pairs of curves, however, good individual geometry is not enough—the survey presents a normal-family argument showing that two squares sharing a vertex cannot be uniformly mapped to disks, since tangency formation contradicts local quasisymmetry. Herron's sufficient condition controls the relative distance $\partial\Omega$2.

Bonk's theorem (2013) extends this to arbitrary families of disjoint Jordan regions: if each region is an $\partial\Omega$3-quasidisk and pairwise relative distances are bounded below, then the complement is quasiconformally equivalent to a Schottky set, uniquely (up to Möbius postcomposition) when the set has area zero. This subsumes the earlier results of Herron–Koskela for uniform domains and McMullen for carpet limit sets of convex cocompact Kleinian groups. Its applications extend well beyond uniformization per se: Bonk–Merenkov used this framework to prove that every quasisymmetric self-map of the standard Sierpiński carpet is a Euclidean isometry, implying that the standard carpet is not quasisymmetric to any hyperbolic group limit set—a significant constraint in geometric group theory.

Two relaxations of Bonk's hypotheses are documented. Ntalampekos's packing-quasiconformal maps—uniform limits of homeomorphisms, monotone but not injective—permit complete removal of the geometric assumptions, requiring only the summability condition $\partial\Omega$4. Since Rohde–Werness (with first public proof by Doherty–Miller) established that CLE carpets satisfy this summability almost surely, random carpets arising in the conformal loop ensemble theory of Sheffield–Werner are uniformizable by Schottky sets with probability one. On the dynamical side, Bonk–Lyubich–Merenkov uniformized Sierpiński-carpet Julia sets of postcritically finite rational maps, and Luo–Ntalampekos obtained an essentially complete characterization for gasket Julia sets: quasiconformal circularizability, fatness of the gasket, and parabolic multiplicity-3 contact points are equivalent, with strong uniqueness of the map even among arbitrary orientation-preserving homeomorphisms.

The characterization of Schottky sets and relative hyperbolic metrics

Ntalampekos's recent theorem provides a clean answer to the general quasiconformal Schottky problem at the level of pairs. Defining regions to be uniformly pairwise quasiconformally circularizable if every pair $\partial\Omega$5 admits a $\partial\Omega$6-quasiconformal map of the sphere sending both to disks, the theorem states that this condition is quantitatively equivalent to the existence of a quasiconformal map uniformizing $\partial\Omega$7 onto a Schottky set that is conformal ($\partial\Omega$8-quasiconformal) on $\partial\Omega$9, with unique restriction up to Möbius transformations. This strictly weakens Bonk's hypotheses and recovers his theorem as a corollary; its proof combines transboundary modulus estimates with the structure of Schottky reflection groups.

The reduction to pair-level analysis leaves exactly one missing piece, stated as Problem (annuli): find a necessary and sufficient quantitative condition for a pair of disjoint quasidisks to be quasiconformally mapped to a pair of disks. Ntalampekos answers this using the relative hyperbolic metric $z^2+i$0, defined on $z^2+i$1 by infimizing hyperbolic lengths of curves in $z^2+i$2. In model cases (concentric disks, parallel half-planes) this metric is comparable to Euclidean distance rescaled by boundary separation. The main results are sharp: two tangent quasidisks are quasiconformally straightenable to half-planes if and only if the identity from $z^2+i$3 to $z^2+i$4 is quasisymmetric; two quasidisks with disjoint closures are mappable to disks if and only if the corresponding identity is quasi-Möbius with respect to the chordal metric (the quasi-Möbius formulation being needed for genuine quantitative dependence). As a concrete application, for a Lipschitz function $z^2+i$5, the graph of $z^2+i$6 is quasiconformally straightened by a real-line-preserving map if and only if an antiderivative of $z^2+i$7 is quasisymmetric—yielding positive examples such as $z^2+i$8, $z^2+i$9, and negative ones such as $D$0.

Limitations and open problems

Several gaps remain explicit. Koebe's conjecture itself is unresolved for uncountably connected domains without geometric hypotheses, and the square-domain extremal approach of Bonk has not been extended beyond finite connectivity. The reverse direction of the He–Schramm rigidity conjecture (removability implies rigidity) survives Rajala's disproof and is tied to the conjectural equivalence between conformal removability and the CNED condition. The topological characterization of Schottky sets posed in the survey—if three listed consequences of circularizability suffice for topological circularizability—is known affirmatively only in the disjoint-closures case (Whyburn), and the touching case may involve unique homeomorphisms, as the Luo–Ntalampekos gasket theorem illustrates. Finally, the Luo–Ntalampekos gasket result depends intrinsically on complex-dynamical structure and does not generalize to arbitrary gaskets, leaving the general touching-regime problem dependent on a resolution of the annuli problem through the relative hyperbolic criterion.

Conclusion

The survey documents a coherent arc: from Koebe's century-old conjecture, through He–Schramm's countably connected theorem and Schramm's transboundary modulus, to a modern framework in which existence and uniqueness of canonical uniformizations are governed by quantitative, often pairwise, geometric conditions. The parallel development of conformal (circle domains) and quasiconformal (Schottky sets) uniformization reveals a recurring pattern—pairwise circularizability criteria, relative hyperbolic metrics, and summability hypotheses replacing rigid separation assumptions—and identifies precisely where the remaining obstacles lie: the general case of Koebe's conjecture, the surviving direction of the rigidity-removability equivalence, and the topological and geometric classification of touching configurations.