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 ${\mathbb{S}}_{+}^{n+1}$ of ${\mathbb{R}}^{n+2}$. First we show that if the norm squared of the second fundamental form is constant, the Morse index of a free-boundary minimal hypersurface $\mathrm{\Sigma }\subset {\mathbb{S}}_{+}^{n+1}$ equals: $1$ if $\mathrm{\Sigma }$ is a totally geodesic equator, $n+1$ if $\mathrm{\Sigma }$ is half of the Clifford torus, or it is at least $2(n+1)$ when $\mathrm{\Sigma }$ is not totally geodesic. Next we prove an estimate for the first eigenvalue ${\lambda }_1$ of the second variation’s Jacobi operator, and show that ${\lambda }_1\le -2n$ if $\mathrm{\Sigma }$ is not totally geodesic, with equality iff $\mathrm{\Sigma }$ is half of the minimal Clifford torus. Furthermore, ${\lambda }_1=-n$ iff $\mathrm{\Sigma }$ is totally geodesic. Finally, if $\mathrm{\Sigma }$ is not totally umbilical the Morse index is at least $n+1$, with equality precisely when $\mathrm{\Sigma }$ is the upper $\mathrm{H}$-torus. For totally umbilical hypersurfaces the Morse index is $1$. We also prove an upper bound for the first eigenvalue of free-boundary $\mathrm{C}\mathrm{M}\mathrm{C}$ hypersurfaces, where equality corresponds to totally umbilical hypersurfaces.