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, ϕ-uniformity and δ-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 ϕ-uniformity is invariant under quasimöbius mappings.

KEYWORDS:

• Roughly Apollonian bilipschitz mappings
• uniform domains
• Apollonian metric
• Gromov hyperbolicity

2000 MATHEMATICS SUBJECT CLASSICATIONS:

• Primary: 30C65
• 30F45
• Secondary:30C20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research was partly supported by NNSF of China (Nos. 11601529, 11671127, 11571216) and Hunan Provincial Natural Science Foundation of China (No. 2016JJ4122) and Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2017TP1017).