An inequality between Willmore functional and Weyl functional for submanifolds in space forms (Q1042463)

The authors use tensor calculus to prove the following inequality between the Willmore and the Weyl functional of an \(n\)-dimensional (\(n\geq4\)) submanifold \(M\) in an \((n+p)\)-dimensional space form: \(W(M)\geq(2{n-2\over n-1})^{-n/4}\nu(g)\), with equality for totally umbilic submanifolds and ``higher dimensional Clifford tori'' \(S^m(a)\times S^m(\sqrt{1-a^2})\subset S^{2m+1}(1)\), \(0<a<1\). Here, the Weyl functional is defined by \(\nu(g)=\int_M\|W_g\|^{n/2}dv_g\), where \(g\) denotes the induced metric on \(M\) and \(W_g\) its \((4,0)\) Weyl curvature tensor.