A circle domain in the Riemann sphere is a domain each of whose boundary components is a round circle or a point; it is conformally rigid if every conformal map from it onto another circle domain is the restriction of a Moebius transformation. He and Schramm (1994) conjectured that a circle domain is conformally rigid if and only if its boundary is conformally removable (removable for conformal homeomorphisms of the sphere), motivated by the role of rigidity in Koebe's Kreisnormierungsproblem, the conjecture that every planar domain is conformally equivalent to a circle domain. Younsi (2016) proved partial cases of the conjecture and posed, as the central open issue, the implication from removability to rigidity, together with concrete sub-questions on complements of Cantor sets. Rajala (2025) then disproved the converse implication by constructing a rigid circle domain with conformally non-removable boundary, so the conjecture now stands or falls with one open question: is every circle domain whose boundary is conformally removable conformally rigid? The question is open even when the boundary is a single Cantor set, where deciding it is tied to constructing Cantor sets that are conformally removable, do not have sigma-finite length, and yet admit non-Moebius conformal maps on their complement - beyond the classical Ahlfors-Beurling examples, whose complements are rigid.

Background

A planar domain Ω\Omega in the Riemann sphere C^\widehat{\mathbb{C}} is a circle domain if every connected component of its boundary ∂Ω\partial\Omega is either a round circle or a point. Koebe proved that every finitely connected domain is conformally equivalent to a circle domain, unique up to post-composition by a Moebius transformation (a conformal automorphism of C^\widehat{\mathbb{C}}), and in 1909 he conjectured the same for domains of arbitrary connectivity. This is Koebe's Kreisnormierungsproblem, still open for domains with uncountably many boundary components; He and Schramm settled the countably connected case in 1993, their proof leaning essentially on the uniqueness statement.

Uniqueness is equivalent to rigidity: a circle domain Ω\Omega is conformally rigid if every conformal map of Ω\Omega onto another circle domain is the restriction of a Moebius transformation. Finitely connected circle domains are rigid (Koebe), and He and Schramm proved rigidity for countably connected circle domains and, in 1994, for all circle domains whose boundary has σ\sigma-finite linear measure (finite or countable Hausdorff one-measure). On the other hand, circle domains with uncountably many boundary components need not be rigid: if EE is a Cantor set of positive area, then C^∖E\widehat{\mathbb{C}}\setminus E is a non-rigid circle domain.

The conjectured mechanism separating rigid from non-rigid domains is conformal removability. A compact set E⊂C^E\subset\widehat{\mathbb{C}} is conformally removable (also called CH-removable) if every homeomorphism of C^\widehat{\mathbb{C}} that is conformal on C^∖E\widehat{\mathbb{C}}\setminus E is a Moebius transformation. Quasicircles (images of circles under quasiconformal homeomorphisms, i.e. homeomorphisms distorting infinitesimal balls to uniformly bounded-eccentricity ellipses) and compact sets of σ\sigma-finite length are conformally removable, while sets of positive area are not. Removability must be distinguished from a weaker property: EE may fail to be removable for conformal maps on its complement, meaning there exists a conformal map on C^∖E\widehat{\mathbb{C}}\setminus E that is not the restriction of a Moebius map, even though every such map that extends to a homeomorphism of the sphere is Moebius. Ahlfors and Beurling (1950) constructed Cantor subsets of the unit circle having exactly this behavior: they proved that if EE is a Cantor subset of the unit circle T\mathbb{T} and the inner logarithmic capacity of T∖E\mathbb{T}\setminus E is less than one, then C^∖E\widehat{\mathbb{C}}\setminus E carries a non-Moebius conformal self-map, while such sets EE are conformally removable (they lie on the rectifiable curve T\mathbb{T}, so removability follows from Morera's theorem) and have finite length. Because these examples have σ\sigma-finite length, their complements are nevertheless rigid by the 1994 He-Schramm theorem.

He and Schramm conjectured that rigidity of a circle domain is equivalent to conformal removability of its boundary. Younsi (2016) studied this conjecture systematically: he proved that removability of ∂Ω\partial\Omega is equivalent to removability of every Cantor set contained in ∂Ω\partial\Omega whenever ∂Ω\partial\Omega is a countable union of circles, Cantor sets and singletons; he proved a first non-trivial result on unions of removable sets (a union of quasicircles, Cantor sets and points is removable iff each Cantor set in it is); and, using David maps (solutions of the Beltrami equation with unbounded but exponentially integrable dilatation) to deform Schottky groups (groups generated by reflections in the boundary circles, used to extend conformal maps by reflection), he proved that a circle domain is conformally rigid if and only if it is quasiconformally rigid. Subsequent work strengthened the positive direction: Ntalampekos and Younsi (2020) proved rigidity for circle domains satisfying a quasihyperbolic-distance integrability condition (the quasihyperbolic distance is the path metric in which arc length is weighted by the reciprocal of the distance to the boundary; the condition of Jones and Smirnov, known to imply boundary removability, asks that this distance to a fixed base point be integrable, and it covers Hoelder and John circle domains - classical inner-connectivity classes in which, roughly, every point reaches a base point along an arc staying uniformly far from the boundary relative to the arc length traversed), and Ntalampekos (2023) proved rigidity whenever the boundary is a CNED set, i.e. countably negligible for extremal distance (the extremal distance, or modulus, of a curve family is a conformally invariant measure of how efficiently curves of the family can spread out; a set is negligible for extremal distance if removing it changes no such modulus) - a class containing all σ\sigma-finite length boundaries and all boundaries satisfying the Jones-Smirnov condition.

The landscape changed when Rajala (2025) disproved the conjecture in one direction: he constructed a conformally rigid circle domain whose boundary is conformally non-removable, combining a 1998 theorem of Wu (the product of two real Cantor sets is conformally non-removable whenever one factor is thick enough in the sense of logarithmic capacity) with Ntalampekos's metric characterization of conformal maps. Consequently, the conjecture now reduces to a single open implication: whether conformal removability of the boundary forces conformal rigidity. Ntalampekos has conjectured that conformal removability is equivalent to the CNED condition; if true, the open implication would follow from the 2023 rigidity theorem. The question is open even for complements of Cantor sets, which is precisely the special case Younsi (2016) analyzed: there, a counterexample would require a Cantor set that is conformally removable yet not removable for conformal maps on its complement and, unlike the Ahlfors-Beurling examples, does not have σ\sigma-finite length.

Problem Statement

Let Ω\Omega be a circle domain in the Riemann sphere C^\widehat{\mathbb{C}} (every boundary component is a round circle or a point) whose boundary ∂Ω\partial\Omega is conformally removable: every homeomorphism of C^\widehat{\mathbb{C}} that is conformal on C^∖∂Ω\widehat{\mathbb{C}}\setminus\partial\Omega is a Moebius transformation. Must Ω\Omega be conformally rigid, that is, must every conformal map of Ω\Omega onto another circle domain be the restriction of a Moebius transformation?

This is the remaining direction of the He-Schramm rigidity conjecture (1994); the converse implication was disproved by Rajala (2025), who constructed a rigid circle domain with conformally non-removable boundary. The question is open even in the special case Ω=C^∖E\Omega=\widehat{\mathbb{C}}\setminus E with EE a Cantor set, where rigidity of Ω\Omega is known to imply removability of EE but not conversely. Either every circle domain with conformally removable boundary is conformally rigid, or there is a circle domain with conformally removable, necessarily non-σ\sigma-finite-length boundary that admits a non-Moebius conformal map onto another circle domain.

Two concrete sub-questions posed by Younsi (2016) probe the Cantor-complement case and constitute decisive partial progress: (1) do there exist Cantor sets EE that do not have σ\sigma-finite length, are conformally removable, and are not removable for conformal maps on their complement (there exists a non-Moebius conformal map on C^∖E\widehat{\mathbb{C}}\setminus E), and are essentially different from the Ahlfors-Beurling Cantor subsets of a circle? (2) when a non-Moebius conformal map on C^∖E\widehat{\mathbb{C}}\setminus E stretches points of EE to nondegenerate continua, is it possible for all of these continua to be circles?

References

  1. M. Younsi, Removability, rigidity of circle domains and Koebe's conjecture, Advances in Mathematics 303 (2016), 1300-1318. DOI: 10.1016/j.aim.2016.08.039, https://doi.org/10.1016/j.aim.2016.08.039 (preprint arXiv:1511.07348, https://arxiv.org/abs/1511.07348)
  2. Z.-X. He and O. Schramm, Fixed points, Koebe uniformization and circle packings, Annals of Mathematics (2) 137 (1993), 369-406. DOI: 10.2307/2946541, https://doi.org/10.2307/2946541
  3. Z.-X. He and O. Schramm, Rigidity of circle domains whose boundary has sigma-finite linear measure, Inventiones Mathematicae 115 (1994), 297-310. DOI: 10.1007/BF01231761, https://doi.org/10.1007/BF01231761
  4. L. Ahlfors and A. Beurling, Conformal invariants and function-theoretic null-sets, Acta Mathematica 83 (1950), 101-129. DOI: 10.1007/BF02392634, https://doi.org/10.1007/BF02392634
  5. D. Ntalampekos and M. Younsi, Rigidity theorems for circle domains, Inventiones Mathematicae 220 (2020), 129-183. DOI: 10.1007/s00222-019-00921-1, https://doi.org/10.1007/s00222-019-00921-1 (preprint arXiv:1809.05573, https://arxiv.org/abs/1809.05573)
  6. D. Ntalampekos, Rigidity and continuous extension for conformal maps of circle domains, Transactions of the American Mathematical Society 376 (2023), 5221-5239. DOI: 10.1090/tran/8923, https://doi.org/10.1090/tran/8923 (preprint arXiv:2210.04740, https://arxiv.org/abs/2210.04740)
  7. D. Ntalampekos, CNED sets: countably negligible for extremal distances, Selecta Mathematica (New Series) 30 (2024), Article 61. DOI: 10.1007/s00029-024-00951-5, https://doi.org/10.1007/s00029-024-00951-5
  8. K. Rajala, Rigid circle domains with non-removable boundaries, Proceedings of the London Mathematical Society (3) 131 (2025), Article e70081. DOI: 10.1112/plms.70081, https://doi.org/10.1112/plms.70081 (preprint arXiv:2409.19103, https://arxiv.org/abs/2409.19103)
  9. D. Ntalampekos and K. Rajala, Exhaustions of circle domains, International Mathematics Research Notices 2025 (2025), no. 19, Article rnaf296. DOI: 10.1093/imrn/rnaf296, https://doi.org/10.1093/imrn/rnaf296 (preprint arXiv:2312.06840, https://arxiv.org/abs/2312.06840)
  10. C. Karafyllia and D. Ntalampekos, Uniformization of Gromov hyperbolic domains by circle domains, Duke Mathematical Journal, to appear (2025). arXiv:2405.13782, https://arxiv.org/abs/2405.13782
  11. B. Esmayli and K. Rajala, Conformal uniformization of domains bounded by quasitripods, Duke Mathematical Journal 175 (2026), no. 4. DOI: 10.1215/00127094-2025-0035, https://doi.org/10.1215/00127094-2025-0035 (preprint arXiv:2401.08485, https://arxiv.org/abs/2401.08485)
  12. D. Ntalampekos, Uniformization problems in the plane: A survey, preprint (2026). arXiv:2603.15098, https://arxiv.org/abs/2603.15098
  13. C. J. Bishop, Conformal removability is hard, EMS Surveys in Mathematical Sciences, to appear (2026). Preprint (revised December 2025) available at https://www.math.stonybrook.edu/~bishop/papers/notborel.pdf
  14. J.-M. Wu, Removability of sets for quasiconformal mappings and Sobolev spaces, Complex Variables, Theory and Application 37 (1998), 491-506. DOI: 10.1080/17476939808815146, https://doi.org/10.1080/17476939808815146
  15. M. Younsi, On removable sets for holomorphic functions, EMS Surveys in Mathematical Sciences 2 (2015), 219-254. DOI: 10.4171/EMSS/12, https://doi.org/10.4171/EMSS/12 (preprint arXiv:1503.02582, https://arxiv.org/abs/1503.02582)
  16. P. W. Jones and S. K. Smirnov, Removability theorems for Sobolev functions and quasiconformal maps, Arkiv foer Matematik 38 (2000), 263-279. DOI: 10.1007/BF02384320, https://doi.org/10.1007/BF02384320