Slim curves, limit sets and spherical CR uniformisations

Abstract

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