Rough isometry between Gromov hyperbolic spaces and unbounded uniformization

Abstract

In a recent paper, Zhou, Ponnusamy, and Rasila [Math. Nachr. (2025)] have established that the conformal deformations, with parameter $\hat{I}\mu >0$, of a Gromov hyperbolic space via Busemann functions are uniform spaces for sufficiently small $\hat{I}\mu$. In this paper, we demonstrate that if two proper, roughly starlike Gromov hyperbolic spaces are roughly isometric, then the uniformity of their conformal deformations is a simultaneous property; that is, either both are uniform spaces or neither is. Our results provide a counterpart to the work of Shanmugalingam and Lindquist [Ann. Fenn. Math. (2021)].