Summary: We consider the 3-sphere \(\mathbb{S}^3\) 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 \(\mathbb{S}^3\) 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 PU(2,1). As an application, we describe a class of spherical CR uniformisations of certain cusped 3-manifolds.