Willmore-Chen tubes on homogeneous spaces in warped product spaces

We present a new method to obtain Willmore-Chen submanifolds in spaces endowed with warped product metrics and fibers being a given homogeneous space. The main points are: First the invariance of the variational problem of Willmore-Chen with respect to the conformal changes in the ambient space metric. Second, the principle of symmetric criticality which allows us to relate the problem with that for generalized elastic curves in the conformal structure on the base. We obtain some applications of our method, including one, to get a rational one parameter family of Willmore tori in the standard 3-sphere shaped on an associated family of closed free elastic curves in the standard hyperbolic 2-plane. We also get a 3-dimensional Riemannian manifold which is foliated with leaves being nontrivial Willmore tori.