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Hyperbolic groups and spherical minimal surfaces

Antoine Song

Abstract

Let $M$ be a closed, oriented, negatively curved, $n$-dimensional manifold with fundamental group $Γ$. Let $S^\infty$ be the unit sphere in $\ell^2(Γ)$, on which $Γ$ acts by the regular representation. The spherical volume of $M$ is a topological invariant introduced by Besson-Courtois-Gallot. We show that it is equal to the area of an $n$-dimensional area-minimizing minimal surface inside the ultralimit of $S^\infty/Γ$, in the sense of Ambrosio-Kirchheim. Our proof combines the theory of metric currents with a study of limits of the regular representation of torsion-free hyperbolic groups.

Hyperbolic groups and spherical minimal surfaces

Abstract

Let be a closed, oriented, negatively curved, -dimensional manifold with fundamental group . Let be the unit sphere in , on which acts by the regular representation. The spherical volume of is a topological invariant introduced by Besson-Courtois-Gallot. We show that it is equal to the area of an -dimensional area-minimizing minimal surface inside the ultralimit of , in the sense of Ambrosio-Kirchheim. Our proof combines the theory of metric currents with a study of limits of the regular representation of torsion-free hyperbolic groups.
Paper Structure (25 sections, 32 theorems, 375 equations)

This paper contains 25 sections, 32 theorems, 375 equations.

Table of Contents

  1. Introduction
  2. Motivations and related works
  3. Strategy of proof
  4. Organization
  5. The spherical Plateau problem
  6. Setup
  7. Spherical Plateau solutions and ultralimits
  8. Approximation by currents with compact supports
  9. Pulled-tight sequence and no mass cancellation
  10. Minimality of the non-collapsed part
  11. Hyperbolic groups and non-collapsing
  12. Limits of the regular representation
  13. A Margulis type lemma
  14. The non-vanishing theorem
  15. Subspaces and subrepresentations for a separable subset
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $M$ be a closed, oriented, negatively curved $n$-manifold, with fundamental group $\Gamma$. Then, there is a mass-minimizing $n$-cycle $T$ inside the ultralimit $(S^\infty/\Gamma)_\omega$, whose mass is equal to $\mathop{\mathrm{SphereVol}}\nolimits(M)$. Moreover, the intersection of the support

Theorems & Definitions (67)

  • Theorem 1.1
  • Conjecture 1.2
  • Definition 2.1: Spherical volume
  • Definition 2.2: Spherical Plateau solution
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 57 more