Estimates for the first eigenvalue of Jacobi operator on hypersurfaces with constant mean curvature in spheres

In this paper, we study the first eigenvalue of Jacobi operator on an n-dimensional non-totally umbilical compact hypersurface with constant mean curvature H in the unit sphere Sn+1(1). We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature H and the dimension n. This bound is attained if and only if, phi : M -> Sn+1(1) is isometric to S-1(r) x Sn-1(root 1 - r(2)) when H not equal 0 or phi : M -> Sn+1(1) is isometric to a Clifford torus Sn-k (root n - k/n) x S-k (root k/n), for k = 1, 2, ... , n - 1 when H = 0.