Apollonian metric, uniformity and Gromov hyperbolicity



: Li YX, Vuorinen M, Zhou QS

Publisher: TAYLOR & FRANCIS LTD

: 2019

: Complex Variables and Elliptic Equations

: COMPLEX VARIABLES AND ELLIPTIC EQUATIONS

: COMPLEX VAR ELLIPTIC

: 14

: 1747-6933

: 1747-6941

DOI: https://doi.org/10.1080/17476933.2019.1579203

: https://research.utu.fi/converis/portal/detail/Publication/40257516


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.

Last updated on 2024-26-11 at 11:07