Apollonian metric, uniformity and Gromov hyperbolicity

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, phi-uniformity and delta-hyperbolicity (in the sense of Gromov with respect to quasihyperbolic metric) of proper domains of 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 phi-uniformity is invariant under quasimobius mappings.