Nonlocal approximation of minimal surfaces: optimal estimates from stability
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911217878040576 |
|---|---|
| author | Chan, Hardy Dipierro, Serena Serra, Joaquim Valdinoci, Enrico |
| author_facet | Chan, Hardy Dipierro, Serena Serra, Joaquim Valdinoci, Enrico |
| contents | Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren-Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new proof of Yau's conjecture for generic metrics and establish the multiplicity one conjecture. The primary goal of this paper is to set the ground for a new approximation based on nonlocal minimal surfaces. More precisely, we prove that if $\partial E$ is a stable $s$-minimal surface in $B_1\subset \mathbb R^3$ then:
- $\partial E\cap B_{1/2}$ enjoys a $C^{2,α}$ estimate that is robust as $s\uparrow 1$ (i.e. uniform in $s$);
- the distance between different connected components of~$\partial E\cap B_{1/2}$ must be at least of order~$(1-s)^{\frac 1 2}$ (optimal sheet separation estimate);
- interactions between multiple sheets at distances of order $(1-s)^{\frac 1 2}$ are described by the Dávila--del Pino--Wei system. A second important goal of the paper is to establish that hyperplanes are the only stable $s$-minimal hypersurfaces in $\mathbb R^4$, for $s\in(0,1)$ sufficiently close to $1$. This is done by exploiting suitable modifications of the results described above. In this application, it is crucially used that our curvature and separations estimates hold without any assumption on area bounds (in contrast to the analogous estimates for Allen-Cahn). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06328 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonlocal approximation of minimal surfaces: optimal estimates from stability Chan, Hardy Dipierro, Serena Serra, Joaquim Valdinoci, Enrico Differential Geometry Analysis of PDEs Minimal surfaces in closed 3-manifolds are classically constructed via the Almgren-Pitts approach. The Allen-Cahn approximation has proved to be a powerful alternative, and Chodosh and Mantoulidis (in Ann. Math. 2020) used it to give a new proof of Yau's conjecture for generic metrics and establish the multiplicity one conjecture. The primary goal of this paper is to set the ground for a new approximation based on nonlocal minimal surfaces. More precisely, we prove that if $\partial E$ is a stable $s$-minimal surface in $B_1\subset \mathbb R^3$ then: - $\partial E\cap B_{1/2}$ enjoys a $C^{2,α}$ estimate that is robust as $s\uparrow 1$ (i.e. uniform in $s$); - the distance between different connected components of~$\partial E\cap B_{1/2}$ must be at least of order~$(1-s)^{\frac 1 2}$ (optimal sheet separation estimate); - interactions between multiple sheets at distances of order $(1-s)^{\frac 1 2}$ are described by the Dávila--del Pino--Wei system. A second important goal of the paper is to establish that hyperplanes are the only stable $s$-minimal hypersurfaces in $\mathbb R^4$, for $s\in(0,1)$ sufficiently close to $1$. This is done by exploiting suitable modifications of the results described above. In this application, it is crucially used that our curvature and separations estimates hold without any assumption on area bounds (in contrast to the analogous estimates for Allen-Cahn). |
| title | Nonlocal approximation of minimal surfaces: optimal estimates from stability |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2308.06328 |