SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN A SPHERE

Let M be an n-dimensional closed hypersurface with constant mean curvature H satisfying vertical bar H vertical bar <= epsilon(n) in a unit sphere Sn+1, n <= 7, and S the square of the length of the second fundamental form of M. 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 equivalent to S-0 and M is isometric to a Clifford hypersurface, where epsilon(n) is a sufficiently small constant depending on n and S-0 + n(3)/2(n-1) H-2 + n(n-2)/2(n-1) root n(2)H(4) + 4(n - 1)H-2.