Foliated Plateau problems, geometric rigidity and equidistribution of closed $k$-surfaces

TL;DR

This paper surveys recent progress on the dynamics and rigidity of constant curvature surfaces in negatively curved 3-manifolds, extending thermodynamic concepts from geodesic flows to surface spaces.

Contribution

It introduces a new perspective on the dynamical properties of surface spaces, generalizing thermodynamic formalism and establishing geometric rigidity results.

Findings

Rigidity of hyperbolic marked area spectrum

Extension of thermodynamic formalism to surface spaces

Analysis of dynamical properties of constant curvature surfaces

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.