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Spherical volume and spherical Plateau problem

Antoine Song

TL;DR

This work introduces the spherical Plateau problem as an infinite‑dimensional variational model for a topological invariant called the spherical volume, defined via integral currents on the spherical quotient $S^\infty/\lambda_\Gamma(\Gamma)$. Central technical tools include Ambrosio–Kirchheim metric currents, Wenger’s intrinsic flat compactness, and a barycenter map with a sharp Jacobian bound that underpins intrinsic uniqueness results. The authors prove intrinsic uniqueness of spherical Plateau solutions for closed hyperbolic manifolds (and extend the paradigm to 3‑manifolds), establishing an intrinsic rigidity that ties the spherical Plateau solution to canonical geometric structures scaled by $\frac{(n-1)^2}{4n}g_0$ or $\frac13 g_{\mathrm{hyp}}$ in dimension 3. They also develop an analogue of Dehn fillings via Plateau‑level constructions, showing asymptotic rigidity: as the filling tori injectivity radii blow up, the spherical Plateau solutions converge to the underlying hyperbolic core, thereby providing a higher‑dimensional counterpart to Thurston’s Dehn filling phenomenon. Together, these results connect a variational, current‑theoretic framework with classical rigidity, entropy, and geometric decomposition phenomena, and open several questions about well‑ordering and extrinsic rigidity in broader settings.

Abstract

Given a closed oriented manifold or more generally a group homology class, we introduce the spherical Plateau problem, which is a variational problem corresponding to a topological invariant called the spherical volume. In principle, its solutions should be realized by minimal surfaces in quotients of spheres. We explain that in many geometrically interesting cases, those solutions are essentially unique. We start with a review of the Ambrosio-Kirchheim theory of metric currents, and the barycenter map method developed by Besson-Courtois-Gallot. Then, we outline the following applications: (1) the intrinsic uniqueness of spherical Plateau solutions for negatively curved, locally symmetric, closed oriented manifolds, (2) the intrinsic uniqueness of spherical Plateau solutions for all 3-dimensional closed oriented manifolds, (3) the construction of higher-dimensional analogues of hyperbolic Dehn fillings. We also propose some open questions.

Spherical volume and spherical Plateau problem

TL;DR

This work introduces the spherical Plateau problem as an infinite‑dimensional variational model for a topological invariant called the spherical volume, defined via integral currents on the spherical quotient . Central technical tools include Ambrosio–Kirchheim metric currents, Wenger’s intrinsic flat compactness, and a barycenter map with a sharp Jacobian bound that underpins intrinsic uniqueness results. The authors prove intrinsic uniqueness of spherical Plateau solutions for closed hyperbolic manifolds (and extend the paradigm to 3‑manifolds), establishing an intrinsic rigidity that ties the spherical Plateau solution to canonical geometric structures scaled by or in dimension 3. They also develop an analogue of Dehn fillings via Plateau‑level constructions, showing asymptotic rigidity: as the filling tori injectivity radii blow up, the spherical Plateau solutions converge to the underlying hyperbolic core, thereby providing a higher‑dimensional counterpart to Thurston’s Dehn filling phenomenon. Together, these results connect a variational, current‑theoretic framework with classical rigidity, entropy, and geometric decomposition phenomena, and open several questions about well‑ordering and extrinsic rigidity in broader settings.

Abstract

Given a closed oriented manifold or more generally a group homology class, we introduce the spherical Plateau problem, which is a variational problem corresponding to a topological invariant called the spherical volume. In principle, its solutions should be realized by minimal surfaces in quotients of spheres. We explain that in many geometrically interesting cases, those solutions are essentially unique. We start with a review of the Ambrosio-Kirchheim theory of metric currents, and the barycenter map method developed by Besson-Courtois-Gallot. Then, we outline the following applications: (1) the intrinsic uniqueness of spherical Plateau solutions for negatively curved, locally symmetric, closed oriented manifolds, (2) the intrinsic uniqueness of spherical Plateau solutions for all 3-dimensional closed oriented manifolds, (3) the construction of higher-dimensional analogues of hyperbolic Dehn fillings. We also propose some open questions.
Paper Structure (34 sections, 16 theorems, 207 equations)

This paper contains 34 sections, 16 theorems, 207 equations.

Table of Contents

  1. Preliminaries on metric integral currents
  2. Basic definitions
  3. Rectifiable sets and integral currents
  4. Weak and flat topology
  5. The area and coarea formulas
  6. The slicing theorem for integral currents
  7. Integral current spaces and intrinsic flat topology
  8. Approximation of integral currents by polyhedral chains
  9. Preliminaries on the barycenter map
  10. Definition of the barycenter map
  11. Notations and a priori Lipschitz bounds
  12. The Jacobian bound
  13. The spherical Plateau problem
  14. Definitions for the spherical volume and spherical Plateau solutions
  15. More details on the set of cycles $\mathscr{C}(h)$.
  16. ...and 19 more sections

Key Result

Theorem 1

If $(M,g_0)$ is a closed, oriented, locally symmetric manifold of dimension $n\geq 3$, with negative curvature between $-4$ and $-1$, then any spherical Plateau solution for $h_M$ is intrinsically isomorphic to $(M,\frac{h(g_0)^2}{4n} g_0)$.

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5: Wenger11SW11
  • Lemma 1.6
  • Definition 2.1
  • ...and 26 more