  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.
