Compact spacelike surfaces whose mean curvature function satisfies a nonlinear inequality in a 3-dimensional Generalized Robertson-Walker spacetime

Spacelike surfaces in Generalized Robertson-Walker spacetimes whose mean curvature function satisfies a natural nonlinear inequality are analyzed. Several uniqueness and nonexistence results for such compact spacelike surfaces are proved. In the nonparametric case, new Calabi-Bernstein type problems are solved as a consequence

On the First eigenvalue of the Laplace operator for Compact Spacelike submanifolds in Lorentz-Minkowski Spacetime Lm\mathbb{L}^{m}

By means of a family of counter-examples, it is shown that the Reilly upper bound for the first eigenvalue of the Laplace operator for a compact submanifold in Euclidean space does not work for $n$-dimensional compact spacelike submanifolds of Lorentz-Minkowski spacetime $\mathbb{L}^m$, $m\geq n+2$. We develop a new suitable technique, based on an integral formula on compact spacelike sections of the light cone in $\mathbb{L}^m$. Then, a family of extrinsic upper bounds for the first eigenvalue of the Laplace operator for a compact spacelike submanifold in $\mathbb{L}^m$ is proved. For each one of these inequalities, becoming an equality can be characterized in geometric terms. In particular, the eigenvalue achieves one of these upper bounds if and only if the submanifold lies minimally in certain hypersphere of a spacelike hyperplane

A new method to construct spacetimes with a spacelike circle action

A new general procedure to construct realistic spacetimes is introduced. It is based on the null congruence on a time-oriented Lorentzian manifold associated to a certain timelike vector field. As an application, new examples of stably causal Petrov type D spacetimes which obey the timelike convergence condition and which admit an isometric spacelike circle action are obtained.Comment: 18 page

Completeness of trajectories of relativistic particles under stationary magnetic fields

The second order differential equation $\frac{D\dot\gamma}{dt}(t) = F_{\gamma(t)}(\dot\gamma(t)) - \nabla V(\gamma(t))$ on a Lorentzian manifold describes, in particular, the dynamics of particles under the action of a electromagnetic field $F$ and a conservative force $-\nabla V$. We provide a first study on the extendability of its solutions, by imposing some natural assumptions

A probabilistic methodology for multilabel classification

Multilabel classification is a relatively recent subfield of machine learning. Unlike to the classical approach, where instances are labeled with only one category, in multilabel classification, an arbitrary number of categories is chosen to label an instance. Due to the problem complexity (the solution is one among an exponential number of alternatives), a very common solution (the binary method) is frequently used, learning a binary classifier for every category, and combining them all afterwards. The assumption taken in this solution is not realistic, and in this work we give examples where the decisions for all the labels are not taken independently, and thus, a supervised approach should learn those existing relationships among categories to make a better classification. Therefore, we show here a generic methodology that can improve the results obtained by a set of independent probabilistic binary classifiers, by using a combination procedure with a classifier trained on the co-occurrences of the labels. We show an exhaustive experimentation in three different standard corpora of labeled documents (Reuters-21578, Ohsumed-23 and RCV1), which present noticeable improvements in all of them, when using our methodology, in three probabilistic base classifiers.Comment: 14 pages, 1 figure, under revie

Remarks on the completeness of trajectories of accelerated particles in Riemannian manifolds and plane waves

Recently, classical results on completeness of trajectories of Hamiltonian systems obtained at the beginning of the seventies, have been revisited, improved and applied to Lorentzian Geometry. Our aim here is threefold: to give explicit proofs of some technicalities in the background of the specialists, to show that the introduced tools allow to obtain more results for the completeness of the trajectories, and to apply these results to the completeness of spacetimes that generalize classical plane and pp-waves

Compact spacelike surfaces in four-dimensional Lorentz-Minkowski spacetime with a non-degenerate lightlike normal direction

A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field $\eta$. Several sufficient assumptions on such a surface with non-degenerate $\eta$-second fundamental form are established to prove that it must be a totally umbilical round sphere. With this aim, a new formula which relates the Gauss curvatures of the induced metric and of the $\eta$-second fundamental form is developed. Then, totally umbilical round spheres are characterized as the only compact spacelike surfaces through the lightcone such that its $\eta$-second fundamental form is non-degenerate and has constant Gauss curvature two. Another characterizations of totally umbilical round spheres in terms of the Gauss-Kronecker curvature of $\eta$ and the area of the $\eta$-second fundamental form are also given

The Gauss-Landau-Hall problem on Riemannian surfaces

We introduce the notion of Gauss-Landau-Hall magnetic field on a Riemannian surface. The corresponding Landau-Hall problem is shown to be equivalent to the dynamics of a massive boson. This allows one to view that problem as a globally stated, variational one. In this framework, flowlines appear as critical points of an action with density depending on the proper acceleration. Moreover, we can study global stability of flowlines. In this equivalence, the massless particle model correspond with a limit case obtained when the force of the Gauss-Landau-Hall increases arbitrarily. We also obtain new properties related with the completeness of flowlines for a general magnetic fields. The paper also contains new results relative to the Landau-Hall problem associated with a uniform magnetic field. For example, we characterize those revolution surfaces whose parallels are all normal flowlines of a uniform magnetic field.Comment: 20 pages, LaTe

On maximal hypersurfaces in Lorentz manifolds admitting a parallel lightlike vector field

We study constant mean curvature spacelike hypersurfaces and in particular maximal hypersurfaces immersed in pp-wave spacetimes satisfying the timelike convergence condition. We prove the non-existence of compact spacelike hypersurfaces whose constant mean curvature is non-zero and also that every compact maximal hypersurface is totally geodesic. Moreover, we give an extension of the classical Calabi-Bernstein theorem to this class of pp-wave spacetimes

Uniqueness of complete maximal hypersurfaces in spatially open (n+1)(n+1)-dimensional Robertson-Walker spacetimes with flat fiber

In this paper, under natural geometric and physical assumptions we provide new uniqueness and non-existence results for complete maximal hypersurfaces in spatially open Robertson-Walker spacetimes whose fiber is flat. Moreover, our results are applied to relevant spacetimes as the steady state spacetime, Einstein-de Sitter spacetime and radiation models.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1511.01422, arXiv:1401.768