Foliated Plateau problems and asymptotic counting of surface subgroups

Abstract

In 2000, 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 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 Calegari–Marques–Neves in two key respects. Firstly, we work with all quasi-Fuchsian subgroups as opposed to merely asymptotically Fuchsian ones. Secondly, as their proof of rigidity breaks down in the present case, we require a different approach. Following ideas recently outlined by Labourie, 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.