Slim curves, limit sets and spherical CR uniformisations

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.