We will show that, in contrast with the \(3\)-dimensional case, the Morse index of a free boundary rotationally symmetric totally geodesic hypersurface of the \(n\)-dimensional Riemannnian Schwarzschild space with respect to variations that are tangential along the horizon is zero, for \(n\geq 4\). Moreover, we will show that there exist non-compact free boundary minimal hypersurfaces which are not totally geodesic, \(n\geq 8\), with Morse index equal to \(0\). Also, for \(n\geq 4\), there exist infinitely many non-compact free boundary minimal hypersurfaces, which are not congruent to each other, with infinite Morse index. Finally, we will discuss the density at infinity of a free boundary minimal hypersurface with respect to a minimal cone constructed over a minimal hypersurface of the unit Euclidean sphere. We obtain a lower bound for the density in terms of the area of the boundary of the hypersurface and the area of the minimal hypersurface in the unit sphere. This lower bound is optimal in the sense that only minimal cones achieve it.

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