Compactness and generic finiteness for free boundary minimal hypersurfaces, I
Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical convergence away from finitely many points. For a generic ambient metric, we prove that there are only finitely many such hypersurfaces. The key to proving the last statement is to show that the limit of a sequence of such hypersurfaces always inherits a non-trivial Jacobi field.