On the Morse index of free-boundary CMC hypersurfaces in the upper hemisphere

We prove results for free-boundary hypersurfaces in the upper unit hemisphere S^n+1+ of R^n+2. First we show that if the norm squared of the second fundamental form is constant, the Morse index of a free-boundary minimal hypersurface S^n+1+ equals: 1 if is a totally geodesic equator, n+1 if is half of the Clifford torus, or it is at least 2 (n+1) when is not totally geodesic. Next we prove an estimate for the first eigenvalue ₁ of the second variation's Jacobi operator, and show that ₁ -2n if is not totally geodesic, with equality iff is half of the minimal Clifford torus. Furthermore, ₁ = -n iff is totally geodesic. Finally, if is not totally umbilical the Morse index is at least n+1, with equality precisely when is the upper H-torus. For totally umbilical hypersurfaces the Morse index is 1. We also prove an upper bound for the first eigenvalue of free-boundary CMC hypersurfaces, where equality corresponds to totally umbilical hypersurfaces.