Uniformizing Gromov hyperbolic spaces with Busemann functions

Given a complete Gromov hyperbolic space X that is roughly starlike from a point ω in its Gromov boundary ₆X, we use a Busemann function based at ω to construct an incomplete unbounded uniform metric space X_ whose boundary X_ can be canonically identified with the Gromov boundary _ωX of X relative to ω. This uniformization construction generalizes the procedure used to obtain the Euclidean upper half plane from the hyperbolic plane. Furthermore we show, for an arbitrary metric space Z, that there is a hyperbolic filling X of Z that can be uniformized in such a way that the boundary X_ has a biLipschitz identification with the completion Z of Z. We also prove that this uniformization procedure can be done at an exponent that is often optimal in the case of CAT (-1) spaces.