Weyl Law and convergence in the classical limit for min-max nonlocal minimal surfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912643116171264 |
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| author | Florit-Simon, Enric |
| author_facet | Florit-Simon, Enric |
| contents | We study nonlocal minimal surfaces as a new approximation theory for the area functional, and more specifically in the context of Yau's conjecture on the existence of minimal surfaces in closed three-dimensional manifolds. This programme offers an alternative to the Almgren--Pitts and Allen--Cahn approaches, with advantageous features both from the existence and regularity viewpoints.
We build on recent work in which the author and collaborators constructed infinitely many nonlocal $s$-minimal hypersurfaces (via min-max methods) on any closed $n$-dimensional Riemannian manifold $M$, establishing a full analogue of Yau's conjecture for $s\in(0,1)$.
The present article first proves a Weyl-type Law for the fractional perimeters of these hypersurfaces. The rest -- and main part -- of the article is devoted to obtaining uniform estimates (in the classical limit $s\to 1$) for min-max $s$-minimal surfaces in closed three-manifolds, eventually establishing their convergence to smooth, classical minimal surfaces. We recover in particular recent results on existence, generic density and equidistribution of minimal surfaces, which are a strong form of Yau's conjecture in this setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_12162 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weyl Law and convergence in the classical limit for min-max nonlocal minimal surfaces Florit-Simon, Enric Differential Geometry Analysis of PDEs We study nonlocal minimal surfaces as a new approximation theory for the area functional, and more specifically in the context of Yau's conjecture on the existence of minimal surfaces in closed three-dimensional manifolds. This programme offers an alternative to the Almgren--Pitts and Allen--Cahn approaches, with advantageous features both from the existence and regularity viewpoints. We build on recent work in which the author and collaborators constructed infinitely many nonlocal $s$-minimal hypersurfaces (via min-max methods) on any closed $n$-dimensional Riemannian manifold $M$, establishing a full analogue of Yau's conjecture for $s\in(0,1)$. The present article first proves a Weyl-type Law for the fractional perimeters of these hypersurfaces. The rest -- and main part -- of the article is devoted to obtaining uniform estimates (in the classical limit $s\to 1$) for min-max $s$-minimal surfaces in closed three-manifolds, eventually establishing their convergence to smooth, classical minimal surfaces. We recover in particular recent results on existence, generic density and equidistribution of minimal surfaces, which are a strong form of Yau's conjecture in this setting. |
| title | Weyl Law and convergence in the classical limit for min-max nonlocal minimal surfaces |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2406.12162 |