Year: 2024
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Song, Antoine
Institution: California Institute of Technology
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4391949109