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