  In [17], Labourie initiated the study of the dynamical properties of the
space of $k$-surfaces, that is, suitably complete immersed surfaces of constant
extrinsic curvature in $3$-dimensional manifolds, which he presented as a
higher-dimensional analogue of the geodesic flow when the ambient manifold is
negatively curved. In this paper, following the recent work [5] of
Calegari--Marques--Neves, we study the asymptotic counting of surface subgroups
in terms of areas of $k$-surfaces. We determine a lower bound, and we prove
rigidity when this bound is achieved. Our work differs from that of [5] in two
key respects. Firstly, we work with all quasi-Fuchsian subgroups as opposed to
merely asymptotically Fuchsian ones. Secondly, as the proof of rigidity in [5]
breaks down in the present case, we require a different approach. Following
ideas outlined by Labourie in [19], we prove rigidity by solving a general
foliated Plateau problem in Cartan--Hadamard manifolds. To this end, we build
on Labourie's theory of $k$-surface dynamics, and propose a number of new
constructions, conjectures and questions.
