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https://dx.doi.org/10.48550/ar...
Article . 2020
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Busemann functions and uniformization of Gromov hyperbolic spaces

Authors: Qingshan Zhou; Saminathan Ponnusamy; Antti Rasila;

Busemann functions and uniformization of Gromov hyperbolic spaces

Abstract

AbstractThe uniformization theory of Gromov hyperbolic spaces investigated by Bonk, Heinonen, and Koskela, generalizes the case where a classical Poincaré ball type model is used as the starting point. In this paper, we develop this approach in the case where the underlying domain is unbounded, corresponding to the classical Poincaré half‐space model. More precisely, we study conformal densities via Busemann functions on Gromov hyperbolic spaces and prove that the deformed spaces are unbounded uniform spaces. Furthermore, we show that there is a one‐to‐one correspondence between the bilipschitz classes of proper geodesic Gromov hyperbolic spaces that are roughly starlike with respect to a point on the Gromov boundary and the quasisimilarity classes of unbounded locally compact uniform spaces. Our result can be understood as an unbounded counterpart of the main result of Bonk, Heinonen, and Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque. 270 (2001).

Keywords

quasisymmetry, Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations, Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces, Mathematics - Complex Variables, Primary: 30L10, Secondary: 30L05, 30C65, Busemann function, FOS: Mathematics, Gromov hyperbolic space, Complex Variables (math.CV), uniformization

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
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