We employ min-max techniques to show that the unit ball in $\mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected and arbitrary genus.

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 this work that of a hypersurface n-dimensional Riemannnian Schwarzschild space respect to variations are tangential along horizon is zero, for \(n\ge 4\). Moreover, show there exist non-compact hypersurface...