Local Variational Properties of Non-degenerate Free Boundary Minimal Hypersurfaces

Abstract

We establish local variational characterizations of non-degenerate free-boundary minimal hypersurfaces in compact Riemannian manifolds with boundary. For a two-sided strictly stable hypersurface, we show that it is the unique mass minimizer in its relative homology class, both in a small tubular and flat-neighborhood. As an application, for generic metrics in dimensions $3\le n+1\le 7$, we show that the first free-boundary width is realized by a multiplicity-one hypersurface of Morse index one. For an orientable non-degenerate free-boundary hypersurface of Morse index $k>0$, we construct a canonical local $k$-parameter family and obtain a local min--max characterization.