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

Abstract

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

7 pages; comments are welcome!