We give a necessary and sufficient condition so that a pair of disjoint Jordan regions in the sphere can be quasiconformally mapped to a pair of disks. As a consequence, we obtain a simple characterization that involves Lipschitz functions for the case that one of the Jordan regions is a half-plane. We apply these results to prove that all polynomial cusps are quasiconformally equivalent and that a quasisymmetric embedding of the union of two disjoint disks extends to a quasiconformal map of the sphere, quantitatively. Also, in combination with previous work of the author, we obtain a new characterization of compact sets that are quasiconformally equivalent to Schottky sets.
1. Introduction 1.1. Motivation. In this paper we study the question whether two disjoint Jordan regions in the sphere C = C ∪ {∞} can be mapped to a pair of disjoint disks with a quasiconformal homeomorphism of C. The closures of the regions are allowed to intersect each other. Our goal is to provide a characterization of such regions that is quantitative in nature. Namely the quasiconformal dilatation should depend only on the geometric features of the Jordan regions and not on their relative position.
Ahlfors [Ahl63] provided an excellent and deep characterization of curves that can be mapped to a circle with a quasiconformal homeomorphism of the plane. Specifically, a Jordan curve J in the plane can be quasiconformally mapped to a and in rigidity problems in complex dynamics [BLM16]. Moreover, it has been extended to non-planar carpets [MW13,Reh22].
Recent progress in uniformization problems in complex dynamics opened the way for uniformization results without the uniform relative separation assumption. Specifically, Luo and the author [LN24] characterized, under some mild conditions, Julia sets of rational maps that can be quasiconformally mapped to round gaskets; these are sets whose complementary components are Jordan regions that can potentially touch each other at the boundary. In particular, there is no separation between the Jordan regions. The methods of [LN24] were extended in [LMM26], where it is shown that certain Basilica Julia sets and limit sets of Kleinian groups can be quasiconformally mapped to round Basilicas.
While these results are purely dynamical and do not extend to arbitrary gaskets, the author [Nta25] was able to obtain a general characterization of sets in the sphere that can be quasiconformally mapped to Schottky sets; that is sets, whose complementary components are disks. The main result in [Nta25] (restated below in Theorem 1.4) asserts that if {U i } i∈I is a collection of disjoint Jordan regions in the sphere, then there exists a quasiconformal map of the sphere that maps each U i to a disk if and only if the collection {U i } i∈I is uniformly quasiconformally pairwise circularizable; that is, there exists K ≥ 1 such that for every pair U i , U j , i ̸ = j, there exists a K-quasiconformal map of the sphere that maps U i and U j to disks. For instance, this condition is satisfied under the assumption ∆(∂U i , ∂U j ) ≥ δ thanks to the result of Herron [Her87]. Therefore, Bonk’s theorem follows as a consequence.
Summarizing, the main result in [Nta25] allows the quasiconformal uniformization of a collection of disjoint Jordan regions by disks if every pair can be quasiconformally uniformized with controlled dilatation. Therefore, the last missing piece in this puzzle is to characterize pairs of disjoint Jordan regions that can be mapped quasiconformally to disks. This is precisely the main objective of the present work.
1.2. Main result. Our main result provides a quantitative characterization of pairs of disjoint quasidisks that can be quasiconformally mapped to pairs of disks. There are two fundamental cases that we treat separately: either the boundaries of the two quasidisks intersect at a point or the quasidisks have disjoint closures.
The main results are expressed in terms of the notion of the relative hyperbolic metric corresponding to a pair of quasidisks that we introduce in this paper. If U ⊂ C is an open set we denote by U * the complement of U . If ∂U contains at least two points, we denote by h U the hyperbolic metric on U .
Let U, V ⊂ C be disjoint Jordan regions such that U ∩ V is either empty or contains only one point; in particular, C \ (U ∪ V ) is connected. We define the relative hyperbolic metric corresponding to the pair (V, U ) as
where the infimum is taken over rectifiable curves γ : [a, b] → U * \ V such that γ(a) = z, γ(b) = w. It turns out that if V is a quasidisk, then d V,U is a metric on ∂V \ U . See Section 3.1 for details. This metric can be explicitly computed when U, V are disjoint half-planes or their boundaries are concentric circles; see Lemma 3.1. In Section 3.2 we estimate the metric in several other useful cases.
Our first main result provides a characterization in the case that the quasidisks U, V have a common point at the boundary. We may assume that this common point of “tangency” is at ∞. Thus, in the next statement we use the topology of C and | • | denotes the Euclidean metric.
Theorem 1.1 (Tangent quasidisks). Let U, V ⊂ C be unbounded quasidisks such that U ∩ V = ∅. There exists a quasiconformal map f : C → C that maps U and V to half-planes if and only if the identity map
The proof is given in Section 3.4. Next, we give the corresponding result for quasidisks with disjoint closures. Here we use the topology of the sphere and χ denotes the chordal metric.
Theorem 1.2 (Quasiannulus). Let U, V ⊂ C be quasidisks such that U ∩ V = ∅. There exists a quasiconformal map f : C → C that maps U and V to disks if and only if
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