Spherical volume and spherical Plateau problem
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.
Cite
@article{arxiv.2202.10636,
title = {Spherical volume and spherical Plateau problem},
author = {Antoine Song},
journal= {arXiv preprint arXiv:2202.10636},
year = {2025}
}
Comments
v2: Content restructured. v3: Some corrections added, many of them due to Cosmin Manea. v4: Added Subsection 4.2 which explains how Besson-Courtois-Gallot use the spherical volume to prove the volume entropy inequality. v5: Added a new rigidity result, Corollary 4.3, and polished writing. Title changed. v4: Minor edits. To appear in S\'eminaire de th\'eorie spectrale et g\'eom\'etrie