Rough isometry between Gromov hyperbolic spaces and unbounded uniformization
Vasudevarao Allu, Alan P Jose
TL;DR
The paper proves that uniformity of conformal deformations X_ε built from Busemann functions on a proper, δ-hyperbolic, roughly starlike space is preserved under λ-rough isometries: if X_ε is uniform for some ε>0, then the roughly isometric Y has Y_ε also uniform for the same ε, provided ε is sufficiently small. The approach hinges on precise comparisons of Busemann functions and conformal densities under rough isometries, a discretization technique for path lengths, and a Gehring–Hayman-type control that ties δ_ε to the density ρ_ε. This extends the intrinsic uniformization program of Bonk–Heinonen–Koskela and aligns with prior results by Lindquist and Shanmugalingam while emphasizing the role of Busemann-based densities. The counterexample for rough similarities highlights the sharpness of the rough isometry condition in preserving uniformity under conformal deformation.
Abstract
In a recent paper, Zhou, Ponnusamy, and Rasila [Math. Nachr. (2025)] have established that the conformal deformations, with parameter $ε>0$, of a Gromov hyperbolic space via Busemann functions are uniform spaces for sufficiently small $ε$. 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)].