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 $S^{n+1}(1)$. We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature $H$ and the dimension $n$. 
Daguang Chen and Qing-Ming Cheng
math.DG