Spherical volume and spherical Plateau problem
Antoine Song1
1 California Institute of Technology 177 Linde Hall, #1200 E. California Blvd., Pasadena, CA 91125 (USA)
Séminaire de théorie spectrale et géométrie, Tome 37 (2021-2022), pp. 177-234.

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:

  • the intrinsic uniqueness of spherical Plateau solutions for negatively curved, locally symmetric, closed oriented manifolds,
  • the intrinsic uniqueness of spherical Plateau solutions for all 3-dimensional closed oriented manifolds,
  • the construction of higher-dimensional analogues of hyperbolic Dehn fillings.

We also propose some open questions.

Publié le :
DOI : 10.5802/tsg.386

Antoine Song 1

1 California Institute of Technology 177 Linde Hall, #1200 E. California Blvd., Pasadena, CA 91125 (USA) 
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Antoine Song. Spherical volume and spherical Plateau problem. Séminaire de théorie spectrale et géométrie, Tome 37 (2021-2022), pp. 177-234. doi : 10.5802/tsg.386. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.386/

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