We consider the 3-sphere S3 seen as the boundary at infinity of the complex hyperbolic plane HC2. It comes equipped with a contact structure and two classes of special curves. First, R-circles are boundaries at infinity of totally real totally geodesic subspaces and are tangent to the contact distribution. Second, C-circles are boundaries of complex totally geodesic subspaces and are transverse to the contact distribution. We define a quantitative notion, called slimness, that measures to what extent a continuous path in the sphere S3 is near to being an R-circle. We analyse the classical foliation of the complement of an R-circle by arcs of C-circles. Next, we consider deformations of this situation where the R-circle becomes a slim curve. We apply these concepts to the particular case where the slim curve is the limit set of a quasi-Fuchsian subgroup of PU(2,1). As an application, we describe a class of spherical CR uniformisations of certain cusped 3-manifolds.
@article{10_4171_ggd_789,
author = {Elisha Falbel and Antonin Guilloux and Pierre Will},
title = {Slim curves, limit sets and spherical {CR} uniformisations},
journal = {Groups, geometry, and dynamics},
pages = {1507--1557},
year = {2024},
volume = {18},
number = {4},
doi = {10.4171/ggd/789},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/789/}
}
TY - JOUR
AU - Elisha Falbel
AU - Antonin Guilloux
AU - Pierre Will
TI - Slim curves, limit sets and spherical CR uniformisations
JO - Groups, geometry, and dynamics
PY - 2024
SP - 1507
EP - 1557
VL - 18
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/789/
DO - 10.4171/ggd/789
ID - 10_4171_ggd_789
ER -
%0 Journal Article
%A Elisha Falbel
%A Antonin Guilloux
%A Pierre Will
%T Slim curves, limit sets and spherical CR uniformisations
%J Groups, geometry, and dynamics
%D 2024
%P 1507-1557
%V 18
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/789/
%R 10.4171/ggd/789
%F 10_4171_ggd_789
