Spherical volume and spherical Plateau problem

TL;DR

This paper introduces the spherical Plateau problem, a variational problem linked to spherical volume, with solutions often being minimal surfaces, and explores their uniqueness and applications in geometry.

Contribution

It defines the spherical Plateau problem, connects it to minimal surfaces, and demonstrates its applications in understanding geometric structures and invariants.

Findings

Unique solutions for negatively curved, locally symmetric manifolds

Unique solutions for all 3-dimensional closed manifolds

Construction of higher-dimensional hyperbolic Dehn fillings

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.