CONSTANT MEAN CURVATURE HYPERSURFACES IN SPHERES

In this paper, we first summarise the progress for the famous Chern conjecture, and then we consider n-dimensional closed hypersurfaces with constant mean curvature H in the unit sphere Sn+1 with n <= 8 and generalise the result of Cheng et al. (Q. M. Cheng, Y. J. He and H. Z. Li, Scalar curvature of hypersurfaces with constant mean curvature in a sphere, Glasg. Math. J. 51(2) (2009), 413-423). In order to be precise, we prove that if vertical bar H vertical bar <= epsilon(n), then there exists a constant delta(n, H) > 0, which depends only on n and H, such that if S-0 <= S <= S-0 + delta(n, H), then S = S-0 and M is isometric to the Clifford hypersurface, where epsilon(n) is a sufficiently small constant depending on n.