Busemann functions and uniformization of Gromov hyperbolic spaces

TL;DR

This paper extends the uniformization theory of Gromov hyperbolic spaces to unbounded models using Busemann functions, establishing a correspondence between hyperbolic spaces and uniform spaces.

Contribution

It develops an unbounded uniformization approach for Gromov hyperbolic spaces using Busemann functions, generalizing previous bounded models.

Findings

Established a one-to-one correspondence between hyperbolic and uniform spaces.

Proved that deformed spaces are unbounded uniform spaces.

Extended the uniformization theory to unbounded models.

Abstract

Uniformization theory of Gromov hypebolic spaces investigated by Bonk, Heinonen and Koskela, generalizes the case where a classical Poincar\'e 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 Poincar\'e 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 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 in "Uniformizing Gromov Hyperbolic Spaces", Ast\'erisque 270 (2001).