SECOND EIGENVALUE OF A JACOBI OPERATOR OF HYPERSURFACES WITH CONSTANT SCALAR CURVATURE

Let x : M --> Sn+1(1) be an n-dimensional compact hypersurface with constant scalar curvature n(n - 1)r, r >= 1, in a unit sphere Sn+1(1), n >= 5, and let J(s) be the Jacobi operator of M. In 2004, L. J. Alias, A. Brasil and L. A. M. Sousa studied the first eigenvalue of J(s) of the hypersurface with constant scalar curvature n(n - 1) in Sn+1(1), n >= 3. In 2008, Q.-M. Cheng studied the first eigenvalue of the Jacobi operator J(s) of the hypersurface with constant scalar curvature n(n - 1)r, r > 1, in Sn+1(1). In this paper, we study the second eigenvalue of the Jacobi operator J(s) of M and give an optimal upper bound for the second eigenvalue of J(s).