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arXiv · 1805.07481

Apollonian metric, uniformity and Gromov hyperbolicity

Yaxiang Li·Matti Vuorinen·Qingshan Zhou
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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.

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Yaxiang Li, Matti Vuorinen, Qingshan Zhou. 2018-05-19. Apollonian metric, uniformity and Gromov hyperbolicity. https://arxiv.org/abs/1805.07481

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