Characterization of tangent quasicircles and quasiannuli

TL;DR

We develop a quantitative criterion for when two disjoint Jordan regions $U,V$ in $\\widehat{\\mathbb C}$ can be quasiconformally mapped to disks, using the relative hyperbolic metric $d_{V,U}$ as the central tool. The main results separate into the tangent-quasicircle case and the quasiannulus case, with a precise, data-dependent condition on the identity map $(\partial V,d_{V,U})\to(\partial V,|\cdot|)$ being quasisymmetric in the tangent scenario or quasi-Möbius in the disjoint-closures scenario. These criteria yield a quantitative uniformization principle that extends to Schottky sets and has concrete applications to graphs, polynomial cusps, and the extension of circle-pair embeddings, all with explicit distortion dependencies. The work unifies hyperbolic geometry, uniform-domain theory, and quasiconformal/quasi-Möbius analysis to provide robust, geometry-driven criteria and extension mechanisms for mapping gasket-type configurations to idealized disk configurations.

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.

Figures (8)

• Figure 1: Two unbounded quasidisks $U, V$ whose boundaries are close to parallel lines, as in Lemma \ref{['lemma:quasicircle_parallel']}. In this case, $d_{V,U}(z,w)\simeq_K \mathop{\mathrm{dist}}\nolimits_e(\partial U,\partial V)^{-1}|z-w|$.
• Figure 2: Two unbounded quasidisks $U,V$ and a chord-arc curve $J$ whose distance and Hausdorff distance to each of $\partial U,\partial V$ are comparable to $1$, as in Lemma \ref{['lemma:wormhole']}. In this case, $d_{V,U}(z,w)\simeq_K |z-w|$.
• Figure 3: Two quasidisks $U,V$ whose boundaries are close to concentric circles, as in Lemma \ref{['lemma:quasicircle_concentric']}. In this case, $d_{V,U}(z,w)\simeq_K {\mathop{\mathrm{dist}}\nolimits_e(\partial U,\partial V)}^{-1}{|z-w|}$.
• Figure 4: Shown in gray is the region $Z$ bounded by $\partial U$ and a quasiconformal reflection $g(\partial U)$ of $\partial U$ along $\partial V$ when $\overline U\cap \overline V=\emptyset$ (left) and when $\overline U\cap \overline V$ is a singleton (right).
• Figure 5: The configuration in the proof of Lemma \ref{['lemma:reflection_distance_chi']}\ref{['lemma:reflection_distance:1_chi']}.

Theorems & Definitions (86)

• Theorem 1.1: Tangent quasidisks
• Theorem 1.2: Quasiannulus
• Theorem 1.3
• Theorem 1.4: Schottky sets
• Theorem 1.5: Graphs
• Theorem 1.6
• Corollary 1.7: Cusps
• Theorem 1.8: Tangent disks
• Theorem 1.9: Annulus
• Lemma 2.1