Hopf submanifolds in \(S^7\) which are Willmore-Chen submanifolds (Q1127655)

The Willmore-Chen variational problem is associated with the conformal total mean curvature functional. In this paper, the authors obtain a criterion to reduce variables for Willmore-Chen submanifolds (critical points of the above mentioned functional), in the standard conformal structure \([g]\) on \(\mathbb{S}^7\). By using the principle of symmetric criticality [\textit{R. S. Palais}, Commun. Math. Phys. 69, 19-30 (1979; Zbl 0417.58007)], the problem is reduced to a variational one associated with an elastic energy functional acting on closed curves on \(\mathbb{S}^4\). The paper also shows the first known examples of non-trivial Willmore-Chen submanifolds. They constitute a rational one-parameter family of 4-dimensional Willmore-Chen submanifolds in \((\mathbb{S}^7, [g])\) which have constant mean curvature in \((\mathbb{S}^7,g)\).