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

Sébastien Alvarez

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.

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

Paper Structure

This paper contains 63 sections, 34 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Foliated Plateau problems
  3. Labourie's lamination and phase space
  4. Asymptotic counting of closed minimal surfaces
  5. Equidistribution and asymptotic counting of $k$-surfaces
  6. Outline of the paper
  7. Geodesics and geometric rigidity
  8. Setting
  9. The geodesic flow
  10. Boundary correspondence and Morse's lemma
  11. Hyperbolic properties and structural stability
  12. Gromov's geodesic rigidity
  13. Topological entropy and asymptotic counting
  14. Rigidity results
  15. Entropy and rigid inequality
  16. ...and 48 more sections

Key Result

Theorem 2.1

Quasi-geodesics in $X$ stay at bounded distance from geodesics.

Theorems & Definitions (45)

  • Theorem 2.1: Morse's lemma
  • Theorem 2.2: Structural stability
  • Theorem 2.3: Gromov's rigidity
  • Theorem 2.4: Besson-Courtois-Gallot, BCG
  • Theorem 2.5: Hamenstädt, HamenstadtMLS
  • Theorem 2.6: Guillemin-Kazhdan
  • proof : Sketch of proof
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3: Asymptotic Plateau problem
  • ...and 35 more