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 Sn+1( 1). We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral integral(M) HdM 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 Js, 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) x Sn-m(root 1-c(2)), 1 <= m <= n-1.