Generic Scarring for Minimal Hypersurfaces in Manifolds Thick at Infinity with a Thin Foliation at Infinity
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914631540277248 |
|---|---|
| author | Li, Xingzhe |
| author_facet | Li, Xingzhe |
| contents | We show generic scarring phenomenon for minimal hypersurfaces in a class of complete non-compact manifolds. In particular, we prove that for any metric $g$ in a $C^{\infty}$-generic subset of the family of complete metrics which are thick at infinity with a thin foliation at infinity on a fixed $M^{n+1}$ of dimension $3 \leq (n + 1) \leq 7$, to any connected, closed, embedded, $2$-sided, stable minimal hypersurface $S \subset (M, g)$, there exists a sequence of closed, embedded, minimal hypersurfaces $\{Σ_{k}\}$ scarring along $S$, in the sense that the area of $Σ_{k}$ diverges to infinity, and when properly renormalized, $Σ_{k}$ converges to $S$ as varifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_03591 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generic Scarring for Minimal Hypersurfaces in Manifolds Thick at Infinity with a Thin Foliation at Infinity Li, Xingzhe Differential Geometry Geometric Topology We show generic scarring phenomenon for minimal hypersurfaces in a class of complete non-compact manifolds. In particular, we prove that for any metric $g$ in a $C^{\infty}$-generic subset of the family of complete metrics which are thick at infinity with a thin foliation at infinity on a fixed $M^{n+1}$ of dimension $3 \leq (n + 1) \leq 7$, to any connected, closed, embedded, $2$-sided, stable minimal hypersurface $S \subset (M, g)$, there exists a sequence of closed, embedded, minimal hypersurfaces $\{Σ_{k}\}$ scarring along $S$, in the sense that the area of $Σ_{k}$ diverges to infinity, and when properly renormalized, $Σ_{k}$ converges to $S$ as varifolds. |
| title | Generic Scarring for Minimal Hypersurfaces in Manifolds Thick at Infinity with a Thin Foliation at Infinity |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2312.03591 |