The gap phenomenon for extremal submanifolds in a sphere

AbstractLet x:M→Sn+p be an n-dimensional submanifold in the unit sphere Sn+p and denote by H and S the mean curvature and squared length of the second fundamental form of M, respectively. M is called an extremal submanifold if it is a critical point with respect to the functional ∫M(S−nH2)dv. In this paper, we investigate gap phenomenon and prove a global pinching theorem and a pointwise pinching theorem for extremal submanifolds in a unit sphere