2022/12/27 by Sébastien Alvarez, Alvarez, Sébastien, Ben Lowe +3
Mathematics · #53C12 #53C42 #57M50 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2212.13604
openalex publication_date 2022/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.