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For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on acompact 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 $V$.

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