SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN UNIT SPHERES

Let M be an n-dimensional closed hypersurface with constant mean curvature H in a unit sphere S(n+1), n <= 8, and S the squared length of the second fundamental form of M. If vertical bar H vertical bar <= epsilon(n), then there exists a positive constant alpha(n, H), which depends only on n and H, such that if S(0) <= S <= S(0) + alpha(n, H), then S equivalent to S(0) and M is isometric to a Clifford hypersurface, where epsilon(n) is a positive constant depending only on n and S(0) = n + n(3)/2(n-1)H(2) + n(n-2)/2(n-1)root n(2)H(4) + 4(n-1)H(2).