Uniformization of intrinsic Gromov hyperbolic spaces with Busemann functions

TL;DR

This paper demonstrates how to uniformize intrinsic Gromov hyperbolic spaces using Busemann functions, establishing a boundary correspondence and a Gehring-Hayman type theorem for conformal deformations.

Contribution

It introduces a method to uniformize hyperbolic spaces via Busemann functions and relates their boundaries through conformal deformations.

Findings

Established a Gehring-Hayman type theorem for conformally deformed spaces.

Proved that hyperbolic spaces can be uniformized by Busemann function densities.

Identified the Gromov boundary with the metric boundary of the deformed space.

Abstract

For any intrinsic Gromov hyperbolic space we establish a Gehring-Hayman type theorem for conformally deformed spaces. As an application, we prove that any complete intrinsic hyperbolic space with atleast two points in the Gromov boundary can be uniformized by densities induced by Busemann functions. Furthermore, we establish that there exists a natural identification of the Gromov boundary of $X$ with the metric boundary of the deformed space.