Abstract
We obtain a reduction of variables criterion for 4-dimensional Willmore-Chen submanifolds associated with the generalized Kaluza-Klein conformal structures on the 7-sphere. This argument connects the variational problem of Willmore-Chen with a variational problem for closed curves into 4-spheres. It involves an elastic energy functional with potential. The method is based on the extrinsic conformal invariance of the Willmore-Chen variational problem, and the principle of symmetric criticality. It also uses several techniques from the theory of pseudo-Riemannian submersions. Furthermore, we give some applications, in particular, a result of existence for constant mean curvature Willmore-Chen submanifolds which is essentially supported on the nice geometry of closed helices in the standard 3-sphere.
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