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 Sⁿ⁺¹₊ of Rⁿ⁺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 Σ⊂ Sⁿ⁺¹₊ equals: $1$ if Σ is a totally geodesic equator, $n+1$ if Σ is half of the Clifford torus, or it is at least $2(n+1)$ when Σ is not totally geodesic. Next we prove an estimate for the first eigenvalue λ₁ of the second variation's Jacobi operator, and show that λ₁ ≤ -2n if Σ is not totally geodesic, with equality iff Σ is half of the minimal Clifford torus. Furthermore, λ₁ = -n iff Σ is totally geodesic. Finally, if Σ is not totally umbilical the Morse index is at least $n+1$, with equality precisely when Σ 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.