A Local Minimizing Property of Strictly Stable Free Boundary Minimal Hypersurfaces
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866916044693569536 |
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| author | Jiao, Xiaoxiang Zhu, Hangyue |
| author_facet | Jiao, Xiaoxiang Zhu, Hangyue |
| contents | We prove a local minimizing property for strictly stable free-boundary minimal hypersurfaces in the relative current setting. Let $Σ^n$ be a compact, two-sided, properly embedded free-boundary minimal hypersurface in a compact Riemannian manifold $(N^{n+1},\partial N)$. If $Σ$ is strictly stable, then, in a sufficiently small free-boundary adapted tubular neighborhood $K_r$, the relative cycle $\llbracketΣ\rrbracket$ is the unique mass minimizer in its relative $\mathbb Z_2$-homology class in $(K_r,K_r\cap\partial N)$.
We further prove a relative flat-neighborhood version, and apply this to obtain an index-one conclusion for a multiplicity-one realization of the first free-boundary width under the standard generic hypotheses.
The main point is to bridge the gap between strict stability, which is a smooth graphical condition, and local minimality among relative cycles. We prove that any relative mass minimizer in the same class converges to $Σ$ with multiplicity one as a varifold, satisfies a uniform first variation bound, and hence becomes a small free-boundary graph by Allard--Gr{ü}ter--Jost regularity. The conclusion then follows from the strict stability of $Σ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_25585 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Local Minimizing Property of Strictly Stable Free Boundary Minimal Hypersurfaces Jiao, Xiaoxiang Zhu, Hangyue Differential Geometry We prove a local minimizing property for strictly stable free-boundary minimal hypersurfaces in the relative current setting. Let $Σ^n$ be a compact, two-sided, properly embedded free-boundary minimal hypersurface in a compact Riemannian manifold $(N^{n+1},\partial N)$. If $Σ$ is strictly stable, then, in a sufficiently small free-boundary adapted tubular neighborhood $K_r$, the relative cycle $\llbracketΣ\rrbracket$ is the unique mass minimizer in its relative $\mathbb Z_2$-homology class in $(K_r,K_r\cap\partial N)$. We further prove a relative flat-neighborhood version, and apply this to obtain an index-one conclusion for a multiplicity-one realization of the first free-boundary width under the standard generic hypotheses. The main point is to bridge the gap between strict stability, which is a smooth graphical condition, and local minimality among relative cycles. We prove that any relative mass minimizer in the same class converges to $Σ$ with multiplicity one as a varifold, satisfies a uniform first variation bound, and hence becomes a small free-boundary graph by Allard--Gr{ü}ter--Jost regularity. The conclusion then follows from the strict stability of $Σ$. |
| title | A Local Minimizing Property of Strictly Stable Free Boundary Minimal Hypersurfaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.25585 |