Characterization of tangent quasicircles and quasiannuli

Abstract: 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.

• The paper characterizes when pairs of disjoint Jordan regions with at most one common boundary point admit quantitative quasiconformal maps to disks, using quasi-Möbius distortion of the relative hyperbolic metric.
• The relative hyperbolic metric is conformally natural and quasiconformally invariant up to controlled distortion, enabling unified treatment of tangent quasidisks, quasiannuli, and Schottky-set uniformization.
• The results classify straightenable Lipschitz graphs, show that all polynomial cusps are quasiconformally equivalent, and establish quantitative extension criteria for quasi-Möbius embeddings of tangent or separated circle pairs.

This paper by Ntalampekos develops a quantitative quasiconformal characterization of pairs of disjoint Jordan regions in the Riemann sphere $\widehat{\mathbb{C}}$ that can be mapped simultaneously to pairs of disks. The work completes a program initiated in the author's earlier uniformization theorem for Schottky sets, and it yields new results on Lipschitz graphs, polynomial cusps, and quasiconformal extensions of quasi-Möbius embeddings of pairs of circles (2602.17236).

Background and motivation

Ahlfors characterized quasicircles via the two-point condition that for every pair $z,w\in J$, the smaller arc $E$ between them satisfies $\diam(E)\le L|z-w|$; such curves bound quasidisks, and the dilatation constants are quantitatively related (2602.17236). For a pair of disjoint Jordan regions $U,V$ with quasicircular boundaries, boundary regularity alone is not sufficient for a uniform quasiconformal map sending both regions to disks: for disjoint open squares, the dilatation blows up as the relative distance $\Delta(U,V)=\dist(U,V)/\min\{\diam U,\diam V\}$ tends to $0$. Conversely, Herron's sufficient condition—uniform quasicircles together with $\Delta(\partial U,\partial V)\ge\delta>0$—is not necessary, since tangent disks fail the separation condition while being trivially equivalent to disks. Bonk's uniformization theorem and its extensions all retain a uniform relative separation hypothesis, while the author's Schottky-set theorem reduces arbitrary collections to the pairwise problem. The present paper supplies exactly this missing pairwise characterization.

The relative hyperbolic metric

The central new object is the relative hyperbolic metric. For disjoint Jordan regions $U,V$ whose closures intersect in at most one point, define

$$d_{V,U}(z,w)=\inf_\gamma \ell_{h_{U^*}}(\gamma),\qquad z,w\in\partial V\setminus\overline{U},$$

where the infimum runs over rectifiable curves in $z,w\in J$0 joining $z,w\in J$1 to $z,w\in J$2, and $z,w\in J$3 is the hyperbolic metric of $z,w\in J$4. When $z,w\in J$5 is a quasidisk this is a genuine metric; it is conformally invariant in the sense that conformal maps of the sphere preserving the regions preserve $z,w\in J$6. The paper computes the metric explicitly for half-planes (where $z,w\in J$7) and concentric circles, and establishes comparable estimates when boundaries lie near parallel lines or concentric circles within $z,w\in J$8-tubes, when a chord-arc curve separates the regions ("wormhole" configuration), and under positive relative separation.

Two structural facts underpin everything else. First, if $z,w\in J$9 is the domain bounded by $E$0 and a quasiconformal reflection of $E$1 along $E$2, then the identity from $E$3 to $E$4 is quantitatively quasisymmetric, where $E$5 is the quasihyperbolic metric; this reduces estimates of $E$6 to quasihyperbolic geometry. Second, a new result of independent interest states that a $E$7-quasiconformal map between domains is quantitatively quasisymmetric with respect to the quasihyperbolic metrics at all scales, refining the large-scale bi-Lipschitz comparison of Gehring–Osgood. Combining these yields the key invariance theorem: any quasiconformal map of the sphere induces a quantitatively quasisymmetric map $E$8.

Main characterization theorems

For tangent quasidisks—unbounded quasidisks $E$9 with disjoint closures—the paper proves that they admit a quasiconformal map to a pair of half-planes if and only if the identity $\diam(E)\le L|z-w|$0 is quasisymmetric, quantitatively. The proof proceeds through a normalization mapping $\diam(E)\le L|z-w|$1 to the real axis, Schwarz reflection across the real line, the observation that the induced boundary map is bi-Lipschitz from $\diam(E)\le L|z-w|$2 to the quasihyperbolic metric of the reflected domain, and finally the Beurling–Ahlfors extension theorem.

For quasiannuli—quasidisks with disjoint closures—the corresponding statement requires replacing quasisymmetry by quasi-Möbius distortion: the identity $\diam(E)\le L|z-w|$3 into the chordal metric must be quasi-Möbius. This switch is forced because quasiconformal maps of the sphere are not quantitatively quasisymmetric in general, only quasi-Möbius. Here the conformal model domain is an annulus rather than a strip, and the boundary map on $\diam(E)\le L|z-w|$4 is shown to be quasi-Möbiius before invoking the Beurling–Ahlfors machinery.

The two cases combine into a single statement: for quasidisks whose closures intersect in at most one point, a quasiconformal map to disks exists if and only if the identity on $\diam(E)\le L|z-w|$5 toward the chordal metric is quasi-Möbius, with dilatation depending only on the quasidisk constant and the distortion function. The author emphasizes that no comparably simple cross-ratio-type criterion should be expected for general pairs; even the special case treated below involves a condition as elaborate as quasisymmetry of an antiderivative.

Applications

Schottky sets

Combining the pairwise characterization with the author's earlier uniformization theorem yields a characterization of compact sets quasiconformally equivalent to Schottky sets. For a collection of disjoint Jordan regions whose closures pairwise meet in at most one point, the following are quantitatively equivalent: (i) the identity maps on each $\diam(E)\le L|z-w|$6 are uniformly quasi-Möbius; (ii) each pair admits a uniformly quasiconformal circularization; (iii) there exists a quasiconformal map of the sphere sending every $\diam(E)\le L|z-w|$7 to a disk that is conformal ($\diam(E)\le L|z-w|$8-quasiconformal) on the complementary set $\diam(E)\le L|z-w|$9, and such a map restricted to $U,V$0 is unique up to post-composition by Möbius transformations.

Lipschitz graphs and cusps

A striking interplay between Lipschitz and quasiconformal geometry emerges. If $U,V$1 is Lipschitz, the graph of $U,V$2 and the real line can be mapped to parallel lines by a real-line-preserving quasiconformal homeomorphism of the plane if and only if an antiderivative of $U,V$3 is quasisymmetric, quantitatively. Concretely, graphs of $U,V$4 for $U,V$5 can be straightened, whereas the graph of $U,V$6 cannot—a sharp distinction governed solely by decay rates. A more general characterization for unbounded quasidisks $U,V$7 in the upper half-plane requires three conditions: $U,V$8 is a quasidisk, $U,V$9 lies in a two-sided $\Delta(U,V)=\dist(U,V)/\min\{\diam U,\diam V\}$0-Lipschitz squeeze of some $\Delta(U,V)=\dist(U,V)/\min\{\diam U,\diam V\}$1-Lipschitz function $\Delta(U,V)=\dist(U,V)/\min\{\diam U,\diam V\}$2, and an antiderivative of $\Delta(U,V)=\dist(U,V)/\min\{\diam U,\diam V\}$3 is quasisymmetric. Its necessity direction is technically the most involved part of the paper, constructing the auxiliary function $\Delta(U,V)=\dist(U,V)/\min\{\diam U,\diam V\}$4 from the quasisymmetric boundary values of a straightening map.

As a corollary, all polynomial cusps $\Delta(U,V)=\dist(U,V)/\min\{\diam U,\diam V\}$5, $\Delta(U,V)=\dist(U,V)/\min\{\diam U,\diam V\}$6, are quasiconformally equivalent. This contrasts with the fact that cusps cannot be straightened by quasiconformal maps (which quasi-preserve angles), and complements Chrontsios–Tyson's classification of polynomial spirals, where $\Delta(U,V)=\dist(U,V)/\min\{\diam U,\diam V\}$7 maps to $\Delta(U,V)=\dist(U,V)/\min\{\diam U,\diam V\}$8 precisely when $\Delta(U,V)=\dist(U,V)/\min\{\diam U,\diam V\}$9. Whether an analogous quantitative classification holds for cusps—with explicit dilatation bounds depending on $0$0—remains open.

Quasiconformal extension of embeddings of circle pairs

The paper resolves the extension problem for quasi-Möbius embeddings of unions of two circles. Every quasi-Möbius embedding of two tangent disk boundaries extends to a quasiconformal homeomorphism of the sphere, quantitatively, without assuming a priori that a homeomorphic extension exists (this follows automatically in the tangency case). For disks with disjoint closures, an extension exists provided either the moduli of separating curve families satisfy $0$1, or $0$2 for small modulus. The modulus condition cannot be dropped: the map fixing the unit circle and scaling $0$3 to $0$4 is uniformly quasi-Möbius but admits no uniformly controlled extension. Supporting these results, the paper supplies quantitative extension proofs—apparently absent from the literature despite frequent informal use in complex dynamics—for quasisymmetric homeomorphisms of pairs of parallel lines and of concentric circles, via a piecewise-linear triangulation argument and careful lift estimates for circle homeomorphisms.

Limitations and open questions

The characterization necessarily uses the metric $0$5, which the author concedes is not as simple as a cross-ratio condition; whether any genuinely simpler geometric criterion exists for general pairs is considered doubtful. The cusp corollary is qualitative: it asserts existence of quasiconformal equivalences among polynomial cusps but does not determine optimal dilatations, unlike the known spiral classification. In the annulus extension theorem, the modulus bound is necessary for quantitative control but not implied by quasi-Möbius distortion alone, so the stated hypotheses cannot be weakened. Finally, the strip- and annulus-extension results are established only for quasisymmetric boundary data with controlled local behavior; extending them to broader classes of boundary maps remains unaddressed.

Conclusion

The paper delivers the final pairwise ingredient for the quasiconformal uniformization theory of collections of Jordan regions by disks, formulated through the relative hyperbolic metric and with fully quantitative dependence of constants throughout. Beyond completing the Schottky-set characterization, the framework produces concrete classifications for Lipschitz graphs, proves the quasiconformal equivalence of all polynomial cusps, and settles the extension problem for quasi-Möbius embeddings of pairs of circles—including the non-uniform tangent case inaccessible to earlier uniform-domain methods.