Slim curves, limit sets and spherical CR uniformisations

TL;DR

This paper introduces a quantitative measure called slimness to analyze the geometry of curves in the boundary of complex hyperbolic space, leading to new insights into spherical CR uniformizations of 3-manifolds.

Contribution

It defines slimness for curves in the boundary sphere, studies deformations of R-circles, and applies these to classify spherical CR uniformizations of cusped 3-manifolds.

Findings

Slimness quantifies how close a path is to an R-circle.

Deformations of R-circles as slim curves are characterized.

Classifies certain spherical CR uniformizations of 3-manifolds.

Abstract

We consider here the $3$-sphere $\mathbf S^3$ seen as the boundary at infinity of the complex hyperbolic plane $\mathbf{H}^2_{\mathbf C}$. It comes equipped with a contact structure and two classes of special curves. First $\mathbf R$-circles are boundaries at infinity of totally real totally geodesic subspaces and are tangent to the contact distribution. Second, $\mathbf C$-circles, which 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 $\mathbf S^3$ is near to be an $\mathbf R$-circle. We analyze the classical foliation of the complement of an $\mathbf R$-circle by arcs of $\mathbf C$-circles. Next, we consider deformations of this situation where the $\mathbf R$-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 $\mathrm{PU}(2,1)$. As a consequence, we describe a class of spherical CR uniformizations of certain cusped $3$-manifolds.