Foliated Plateau problems, geometric rigidity and equidistribution of closed $k$-surfaces
Differential Geometry
2025-02-12 v1 Dynamical Systems
Abstract
In this note, we survey recent advances in the study of dynamical properties of the space of surfaces with constant curvature in three-dimensional manifolds of negative sectional curvature. We interpret this space as a two-dimensional analogue of the geodesic flow and explore the extent to which the thermodynamic properties of the latter can be generalized to the surface setting. Additionally, we apply this theory to derive geometric rigidity results, including the rigidity of the hyperbolic marked area spectrum.
Keywords
Cite
@article{arxiv.2502.07626,
title = {Foliated Plateau problems, geometric rigidity and equidistribution of closed $k$-surfaces},
author = {Sébastien Alvarez},
journal= {arXiv preprint arXiv:2502.07626},
year = {2025}
}
Comments
Survey paper. 29 pages. To appear in: Actes du s\'eminaire de Th\'eorie Spectrale et G\'eom\'etrie