  We introduce the notion of an asymptotically Poincar\'e family of surfaces in
an end of a quasi-Fuchsian manifold. We show that any such family gives a
foliation of an end by asymptotically parallel convex surfaces, and that the
asymptotic behavior of the first and second fundamental forms determines the
projective structure at infinity. As an application, we establish a conjecture
of Labourie from 1992 regarding constant Gaussian curvature surfaces. We also
derive consequences for constant mean curvature surfaces.
