Yau's conjecture for nonlocal minimal surfaces
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866915386827472896 |
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| author | Caselli, Michele Florit-Simon, Enric Serra, Joaquim |
| author_facet | Caselli, Michele Florit-Simon, Enric Serra, Joaquim |
| contents | We introduce nonlocal minimal surfaces on closed manifolds and establish a far-reaching Yau-type result: in every closed, $n$-dimensional Riemannian manifold we construct infinitely many nonlocal $s$-minimal surfaces. We prove that, when $s\in (0,1)$ is sufficiently close to $1$, the constructed surfaces are smooth for $n=3$ and $n=4$, while for $n\ge 5$ they are smooth outside of a closed set of dimension $n-5$. Moreover, we prove surprisingly strong regularity and rigidity properties of finite Morse index $s$-minimal surfaces such as a "finite Morse index Bernstein-type result". These properties make nonlocal minimal surfaces ideal objects on which to apply min-max variational methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07100 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Yau's conjecture for nonlocal minimal surfaces Caselli, Michele Florit-Simon, Enric Serra, Joaquim Differential Geometry Analysis of PDEs We introduce nonlocal minimal surfaces on closed manifolds and establish a far-reaching Yau-type result: in every closed, $n$-dimensional Riemannian manifold we construct infinitely many nonlocal $s$-minimal surfaces. We prove that, when $s\in (0,1)$ is sufficiently close to $1$, the constructed surfaces are smooth for $n=3$ and $n=4$, while for $n\ge 5$ they are smooth outside of a closed set of dimension $n-5$. Moreover, we prove surprisingly strong regularity and rigidity properties of finite Morse index $s$-minimal surfaces such as a "finite Morse index Bernstein-type result". These properties make nonlocal minimal surfaces ideal objects on which to apply min-max variational methods. |
| title | Yau's conjecture for nonlocal minimal surfaces |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2306.07100 |