Existence of minimal hypersurfaces with non-empty free boundary for generic metrics

TL;DR

For most Riemannian metrics on a compact manifold with boundary, there exist free boundary minimal hypersurfaces intersecting any given open subset of the boundary, in dimensions 3 to 7.

Contribution

This paper proves the generic existence of free boundary minimal hypersurfaces intersecting arbitrary boundary subsets for a broad class of metrics.

Findings

Existence of free boundary minimal hypersurfaces for generic metrics.

Hypersurfaces intersect any open boundary subset.

Results hold in dimensions 3 to 7.

Abstract

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a compact manifold with boundary $(M^{n+1},\partial M)$, $3\leq (n + 1)\leq 7$, we prove that, for any open subset $V$ of $\partial M$, there exists a compact, properly embedded free boundary minimal hypersurface intersecting $V$.