The second pinching theorem for hypersurfaces with constant mean curvature in a sphere

We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng-Terng, Wei-Xu, Zhang, and Ding-Xin to the case of hypersurfaces with small constant mean curvature. Let be a compact hypersurface with constant mean curvature in . Denote by the squared norm of the second fundamental form of . We prove that there exist two positive constants and depending only on such that if and , then and is one of the following cases: (i) , ; (ii) . Here and .