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. This bound is attained if and only if, (varphi : M rightarrow S^{n+1}(1)) is isometric to (S^1(r)times S^{n-1}(sqrt{1-r^2})) when (Hne 0) or (varphi : M rightarrow S^{n+1}(1)) is isometric to a Clifford torus ( S^{n-k}left( sqrt{dfrac{n-k}{n}}right) times S^kleft( sqrt{dfrac{k}{n}}right) ), for (k=1, 2, ldots , n-1) when (H=0).