A Local Minimizing Property of Strictly Stable Free Boundary Minimal Hypersurfaces

Abstract

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 $Σ$.