Marginally trapped submanifolds in space forms with arbitrary signature

We give explicit representation formulas for marginally trapped submanifolds of co-dimension two in pseudo-Riemannian spaces with arbitrary signature and constant sectional curvature. This paper is dedicated to the memory of Franki Dillen, 1963-2013.Comment: 13 Pages. Third version: few typos corrected and one short comment section added. arXiv admin note: text overlap with arXiv:1209.511

Construction of Hamiltonian-minimal Lagrangian submanifolds in complex Euclidean space

We describe several families of Lagrangian submanifolds in the complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of the odd-dimensional unit sphere.Comment: 23 pages, 5 figures, Second version. Changes in statement and proof of Corollary

Data degradation to enhance privacy for the Ambient Intelligence

Increasing research in ubiquitous computing techniques towards the development of an Ambient Intelligence raises issues regarding privacy. To gain the required data needed to enable application in this Ambient Intelligence to offer smart services to users, sensors will monitor users' behavior to fill personal context histories. Those context histories will be stored on database/information systems which we consider as honest: they can be trusted now, but might be subject to attacks in the future. Making this assumption implies that protecting context histories by means of access control might be not enough. To reduce the impact of possible attacks, we propose to use limited retention techniques. In our approach, we present applications a degraded set of data with a retention delay attached to it which matches both application requirements and users privacy wishes. Data degradation can be twofold: the accuracy of context data can be lowered such that the less privacy sensitive parts are retained, and context data can be transformed such that only particular abilities for application remain available. Retention periods can be specified to trigger irreversible removal of the context data from the system

Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Given an oriented Riemannian surface $(\Sigma, g)$, its tangent bundle $T\Sigma$ enjoys a natural pseudo-K\"{a}hler structure, that is the combination of a complex structure \J, a pseudo-metric \G with neutral signature and a symplectic structure \Om. We give a local classification of those surfaces of $T\Sigma$ which are both Lagrangian with respect to \Om and minimal with respect to \G. We first show that if $g$ is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly. As an application, we show that motions of surfaces in $\R^3$ or $\R^3_1$ induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in $T\S^2$ or T \H^2 respectively. We relate the area of the congruence to a second-order functional $\mathcal{F}=\int \sqrt{H^2-K} dA$ on the original surface.Comment: 22 pages, typos corrected, results streamline

Cyclic and ruled Lagrangian surfaces in complex Euclidean space

We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve of the 3-sphere or a Legendrian curve of the anti de Sitter 3-space. We also describe ruled Lagrangian surfaces. Finally we characterize those cyclic and ruled Lagrangian surfaces which are solutions to the self-similar equation of the Mean Curvature Flow. Finally, we give a partial result in the case of Hamiltonian stationary cyclic surfaces