Quasiconformality and hyperbolic skew

TL;DR

This paper establishes a characterization of quasiconformal maps on hyperbolic spaces via bounded skew over equilateral hyperbolic triangles, and shows quasiconformal maps are quasisymmetric in the hyperbolic metric.

Contribution

It proves that bounded skew over hyperbolic triangles implies quasiconformality and vice versa, extending known Euclidean results to hyperbolic spaces and manifolds.

Findings

Bounded skew implies quasiconformality in hyperbolic spaces.

Quasiconformal maps are $ ext{eta}$-quasisymmetric in hyperbolic metric.

Results extend to hyperbolic manifolds and general metric spaces.

Abstract

We prove that if $f:\mathbb{B}^n \to \mathbb{B}^n$, for $n\geq 2$, is a homeomorphism with bounded skew over all equilateral hyperbolic triangles, then $f$ is in fact quasiconformal. Conversely, we show that if $f:\mathbb{B}^n \to \mathbb{B}^n$ is quasiconformal then $f$ is $\eta$-quasisymmetric in the hyperbolic metric, where $\eta$ depends only on $n$ and $K$. We obtain the same result for hyperbolic $n$-manifolds. Analogous results in $\mathbb{R}^n$, and metric spaces that behave like $\mathbb{R}^n$, are known, but as far as we are aware, these are the first such results in the hyperbolic setting, which is the natural metric to use on $\mathbb{B}^n$.