A characterization of Clifford hypersurfaces among embedded constant mean curvature hypersurfaces in a unit sphere

Let Sigma be an n(>= 3)-dimensional compact embedded hypersurface in a unit sphere with constant mean curvature H >= 0 and with two distinct principal curvatures lambda and mu of multiplicity n - 1 and 1, respectively. It is known that if lambda > mu, there exist many compact embedded constant mean curvature hypersurfaces [ 26]. In this paper, we prove that if lambda > mu, then Sigma is congruent to a Clifford hypersurface. The proof is based on the arguments used by Brendle [10].