Apollonian metric, uniformity and Gromov hyperbolicity

TL;DR

This paper studies how certain geometric properties of domains in Euclidean space, like uniformity and hyperbolicity, are preserved under mappings that are roughly bilipschitz with respect to the Apollonian metric, with applications to quasiconformal and quasim"obius mappings.

Contribution

It establishes invariance of uniformity, $ ext{ extphi}$-uniformity, and Gromov hyperbolicity under roughly Apollonian bilipschitz mappings and characterizes such mappings for quasiconformal maps.

Findings

Uniformity and hyperbolicity are invariant under roughly Apollonian bilipschitz mappings.

Provides four equivalent conditions for quasiconformal maps to be roughly Apollonian bilipschitz.

$ ext{ extphi}$-uniformity is invariant under quasim"obius mappings.

Abstract

The main purpose of this paper is to investigate the properties of a mapping which is required to be roughly bilipschitz with respect to the Apollonian metric (roughly Apollonian bilipschitz) of its domain. We prove that under these mappings the uniformity, $\varphi$-uniformity and $\delta$-hyperbolicity (in the sense of Gromov with respect to quasihyperbolic metric) of proper domains of $\mathbb{R}^n$ are invariant. As applications, we give four equivalent conditions for a quasiconformal mapping which is defined on a uniform domain to be roughly Apollonian bilipschitz, and we conclude that $\varphi$-uniformity is invariant under quasim\"obius mappings.