The gap phenomenon for extremal submanifolds in a sphere (Q531713)

Let \(M\) be an \(n\)-dimensional submanifold of an \((n+p)\)-dimensional unit sphere \(S\). By using the mean curvature and the squared length of the second fundamental form of \(M\), the authors define a functional \(\int _M(S-nH^2)\,dv\), which for \(n = 2\) becomes the Willmore functional. A critical point with respect to this functional is called an extremal submanifold. The authors prove a global pinching theorem and a pointwise pinching theorem for extremal submanifolds in \(S\).