Miguel A. Meroño

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.

A criterion for reduction of variables in the Willmore-Chen variational problem and its applications

We exhibit a criterion for a reduction of variables for WillmoreChen submanifolds in conformal classes associated with generalized KaluzaKlein metrics on flat principal fibre bundles. Our method relates the variational problem of Willmore-Chen with an elasticity functional defined for closed curves in the base space. The main ideas involve the extrinsic conformal invariance of the Willmore-Chen functional, the large symmetry group of generalized Kaluza-Klein metrics and the principle of symmetric criticality. We also obtain interesting families of elasticae in both lens spaces and surfaces of revolution (Riemannian and Lorentzian). We give applications to the construction of explicit examples of isolated Willmore-Chen submanifolds, discrete families of Willmore-Chen submanifolds and foliations whose leaves are Willmore-Chen submanifolds.

Solutions of the Betchov-Da Rios soliton equation: a Lorentzian approach

Abstract The purpose of this paper is to find out explicit solutions of the Betchov-Da Rios soliton equation in three-dimensional Lorentzian space forms. We start with non-null curves and obtain solutions living in certain flat ruled surfaces in l3 and h13, as well as in r3 and s3. Next we take a null curve and have got solutions lying in the associated B-scrolls in l3, s13 and h13. It should be pointed out that we extend previous results already obtained, and as far as we know, this is the first time that solutions in the De Sitter 3-space appear in the literature. Soliton solutions are characterized as null geodesics in B-scrolls.