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

Let χ : 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 Js be the Jacobi operator of M. In 2004, L. J. Alías, A. Brasil and L. A. M. Sousa studied the first eigenvalue of Js 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 Js 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 Js of M and give an optimal upper bound for the second eigenvalue of Js. [ABSTRACT FROM AUTHOR]