Hyperbolic groups and spherical minimal surfaces
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916128667729920 |
|---|---|
| author | Song, Antoine |
| author_facet | Song, Antoine |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10869 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyperbolic groups and spherical minimal surfaces Song, Antoine Differential Geometry Group Theory Representation Theory 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. |
| title | Hyperbolic groups and spherical minimal surfaces |
| topic | Differential Geometry Group Theory Representation Theory |
| url | https://arxiv.org/abs/2402.10869 |