In this note we show that given two complete geodesic Gromov hyperbolic spaces that are roughly isometric and an arbitrary $\epsilon>0$ (not necessarily small), either the uniformization of both spaces with parameter $\epsilon$ results in uniform domains, or else neither uniformized space is a uniform domain. The terminology of "uniformization" is from [BHK], where it is shown that the uniformization, with parameter $\epsilon>0$, of a complete geodesic Gromov hyperbolic space results in a uniform domain provided $\epsilon$ is small enough.
@article{AFM_2021_46_1_a25,
author = {Jeff Lindquist and Nageswari Shanmugalingam},
title = {Rough isometry between {Gromov} hyperbolic spaces and uniformization},
journal = {Annales Fennici Mathematici},
pages = {449--464},
year = {2021},
volume = {46},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a25/}
}
TY - JOUR
AU - Jeff Lindquist
AU - Nageswari Shanmugalingam
TI - Rough isometry between Gromov hyperbolic spaces and uniformization
JO - Annales Fennici Mathematici
PY - 2021
SP - 449
EP - 464
VL - 46
IS - 1
UR - http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a25/
LA - en
ID - AFM_2021_46_1_a25
ER -
%0 Journal Article
%A Jeff Lindquist
%A Nageswari Shanmugalingam
%T Rough isometry between Gromov hyperbolic spaces and uniformization
%J Annales Fennici Mathematici
%D 2021
%P 449-464
%V 46
%N 1
%U http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a25/
%G en
%F AFM_2021_46_1_a25
Jeff Lindquist; Nageswari Shanmugalingam. Rough isometry between Gromov hyperbolic spaces and uniformization. Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 449-464. http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a25/
