First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature

Let $M$ be an $n$-dimensional compact hypersurface with constant scalar curvature $n(n-1)r$, $r> 1$, in a unit sphere $S^{n+1}(1)$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral $\int_MHdM$ of the mean curvature $H$. In this paper, we first study the eigenvalue of the Jacobi operator $J_s$ of $M$. We derive an optimal upper bound for the first eigenvalue of $J_s$, and this bound is attained if and only if $M$ is a totally umbilical and non-totally geodesic hypersurface or $M$ is a Riemannian product $S^m(c)\times S^{n-m}(\sqrt{1-c^2})$, $1\leq m\leq n-1$.