In contrast with the three-dimensional case (cf. Montezuma in Bull Braz Math Soc), where rotationally symmetric totally geodesic free boundary minimal surfaces have Morse index one; we prove in this work that 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 \(n\ge 4\). Moreover, we show that there exist non-compact free boundary minimal hypersurfaces which are not totally geodesic, \(n\ge 8\), with Morse index equal to 0. In addition, it is shown that, for \(n\ge 4\), there exist infinitely many non-compact free boundary minimal hypersurfaces, which are not congruent to each other, with infinite Morse index. We also study 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.

与三维情况相反（参见 Bull Braz Math Soc 中的 Montezuma），其中旋转对称的完全测地自由边界最小表面的莫尔斯指数为 1；我们在这项工作中证明了n维黎曼施瓦西空间的自由边界旋转对称完全测地超曲面的莫尔斯指数相对于沿地平线切线的变化为零，对于\(n\ge 4\)。此外，我们证明存在不完全测地线的非紧自由边界最小超曲面\(n\ge 8\)，莫尔斯指数等于 0。此外，还表明，对于\(n\ge 4\)，存在无穷多个非紧自由边界极小超曲面，它们彼此不全等，具有无穷大莫尔斯指数。我们还研究了自由边界最小超曲面相对于在单位欧几里得球的最小超曲面上构造的最小锥体的无穷远密度。我们根据超曲面边界的面积和单位球体中最小超曲面的面积获得密度的下限。这个下界是最佳的，因为只有最小的锥才能达到它。