Busemann functions and uniformization of Gromov hyperbolic spaces

Abstract

The 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).