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Geometric Function Theory on Uniformly Quasiconformally Homogeneous Domains

Alastair Fletcher, Allyson Hahn

TL;DR

The paper extends geometric function theory to uniformly $K$-quasiconformally homogeneous domains using the quasihyperbolic metric $k_X$, establishing that quasiconformality and quasisymmetry coincide in this hyperbolic-like setting. It also analyzes normal quasiregular maps from such domains into $S^n$ or $\mathbb{R}^n$, showing they are uniformly continuous in $k_X$ and enjoy global Hölder control with exponent $\beta=(KM^2)^{1/(1-n)}$, with corresponding results for Euclidean targets. The results illuminate how transitive families of $K$-quasiconformal maps interact with hyperbolic-type geometry to yield robust distortion and continuity properties in higher dimensions. This framework broadens the reach of hyperbolic geometric function theory beyond the unit ball, potentially impacting uniformization and Teichmüller-type questions on more general domains.

Abstract

Uniformly quasiconformally homogeneous domains in $\mathbb{R}^n$ carry a transitive collection of $K$-quasiconformal maps for a fixed $K\geq 1.$ In this paper, we study two questions in this setting. The first is to show that quasiconformality and quasisymmetry with respect to the quasihyperbolic metric are equivalent. The second is to study normal quasiregular maps from such a domain into $S^n$ or $\mathbb{R}^n$ and show they enjoy geometric properties such as a uniform Hölder condition.

Geometric Function Theory on Uniformly Quasiconformally Homogeneous Domains

TL;DR

The paper extends geometric function theory to uniformly -quasiconformally homogeneous domains using the quasihyperbolic metric , establishing that quasiconformality and quasisymmetry coincide in this hyperbolic-like setting. It also analyzes normal quasiregular maps from such domains into or , showing they are uniformly continuous in and enjoy global Hölder control with exponent , with corresponding results for Euclidean targets. The results illuminate how transitive families of -quasiconformal maps interact with hyperbolic-type geometry to yield robust distortion and continuity properties in higher dimensions. This framework broadens the reach of hyperbolic geometric function theory beyond the unit ball, potentially impacting uniformization and Teichmüller-type questions on more general domains.

Abstract

Uniformly quasiconformally homogeneous domains in carry a transitive collection of -quasiconformal maps for a fixed In this paper, we study two questions in this setting. The first is to show that quasiconformality and quasisymmetry with respect to the quasihyperbolic metric are equivalent. The second is to study normal quasiregular maps from such a domain into or and show they enjoy geometric properties such as a uniform Hölder condition.
Paper Structure (11 sections, 19 theorems, 118 equations)

This paper contains 11 sections, 19 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Uniformly Quasiconformally Homogeneous Domains
  3. Statement of Results
  4. Preliminaries
  5. Quasihyperbolic Metric
  6. Quasiregular Mappings
  7. Quasisymmetry and Weak Quasisymmetry
  8. Families of Continuous Functions
  9. Normal Families
  10. Quasiconformal Implies Quasisymmetric on Uniformly K-QCH Domains
  11. Function Theory on Uniformly K-QCH Domains

Key Result

Theorem 1.2

Let $X,Y$ be proper subdomains of $\mathbb{R}^n$ and suppose $X$ is a uniformly $M$-QCH domain. Then, $f:(X,k_X)\rightarrow (Y,k_Y)$ is $K$-quasiconformal if and only if $f$ is $\eta$-quasisymmetric, in the respective quasihyperbolic metrics.

Theorems & Definitions (47)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 2.1: gehring1979uniform, Lemma 1
  • Proposition 2.2: gehring1976quasiconformally, Corollary 2.2
  • Definition 2.3
  • ...and 37 more