Abstract:We prove a local minimizing property for strictly stable free-boundary minimal hypersurfaces in the relative current setting. Let $\Sigma^n$ be a compact, two-sided, properly embedded free-boundary minimal hypersurface in a compact Riemannian manifold $(N^{n+1},\partial N)$. If $\Sigma$ is strictly stable, then, in a sufficiently small free-boundary adapted tubular neighborhood $K_r$, the relative cycle $\llbracket\Sigma\rrbracket$ 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 $\Sigma$ 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 $\Sigma$.
| Subjects: | Differential Geometry (math.DG) |
| Cite as: | arXiv:2605.25585 [math.DG] |
| (or arXiv:2605.25585v1 [math.DG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25585 arXiv-issued DOI via DataCite (pending registration) |
From: Hangyue Zhu [view email]
[v1]
Mon, 25 May 2026 08:34:58 UTC (38 KB)
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