Uniformisation of intrinsic Gromov hyperbolic spaces with Busemann functions

Abstract In this paper, we investigate the extension of uniformisation results for Gromov hyperbolic spaces beyond the standard geodesic setting. By establishing a Gehring-Hayman type theorem for conformal deformations of any intrinsic Gromov hyperbolic space, we provide a framework for analysing spaces that do not necessarily admit geodesics. As a primary application, we prove that any complete intrinsic hyperbolic space with at least two points in the Gromov boundary can be uniformised by densities induced by Busemann functions. Furthermore, we establish that there exists a natural identification between the Gromov boundary of the original space and the metric boundary of the deformed space.