Willmore tori and Willmore-Chen submanifolds in pseudo-Riemannian spaces

Abstract

We exhibit a new method to find Willmore tori and Willmore-Chen submanifolds in spaces endowed with pseudo-Riemannian warped product metrics, whose fibres are homogeneous spaces. The chief points are the invariance of the involved variational problems with respect to the conformal changes of the metrics on the ambient spaces and the principle of symmetric criticality. They allow us to relate the variational problems with that of generalized elastic curves in the conformal structure of the base space. Among other applications we get a rational one-parameter family of Willmore tori in the standard anti De Sitter 3-space shaped on an associated family of closed free elastic curves in the once punctured standard 2-sphere. We also obtain rational one-parameter families of Willmore-Chen submanifolds in standard pseudo-hyperbolic spaces. As an application of a general approach to our method, we give nice examples of pseudo-Riemannian 3-spaces which are foliated with leaves being non-trivial Willmore tori. More precisely, the leaves of this foliation are Willmore tori which are conformal to non-zero constant mean curvature flat tori.