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Alfonso Romero

Publications and source records attributed to Alfonso Romero.

At least 19 recordsLinked to original sources
arXiv2025

Spacelike Submanifolds of Codimension Two with Parallel Mean Curvature Vector Field in Lorentz-Minkowski Spacetime Contained in the Light Cone

A general integral inequality is established for compact spacelike submanifolds of codimension two in the Lorentz-Minkowski spacetime under the assumption that the mean curvature vector field is parallel. This inequality is then used to derive a rigidity result. Specifically, we obtain a complete characterization of all compact spacelike submanifolds with parallel mean curvature vector field that lie in the light cone of the Lorentz-Minkowski spacetime: they must be totally umbilical spheres contained in a spacelike hyperplane in Lorentz-Minkowski spacetime.

math.DG↗
arXiv2024

A Gauss-Bonnet-Chern type obstruction for Killing vector fields on Lorentzian manifolds

A new curvature obstruction to the existence of a timelike (resp. causal) Killing or homothetic vector field $X$ on an even-dimensional (odd-dimensional) Lorentzian manifold, in terms of its timelike (resp. null) sectional curvature is given. As a consequence for the compact case, the well-known Gauss-Bonnet-Chern obstruction to the existence of semi-Riemannian metrics is extended from non-zero constant sectional curvature to non-zero timelike sectional curvature on $X$.

math.DG↗
arXiv2024

Exploring new extrinsic upper bounds on the first eigenvalue of the Laplace operator for compact submanifolds in Euclidean spaces

Upper bounds of the first non-trivial eigenvalue $λ_1$ of the Laplace operator of a compact submanifold $M^n$ of Euclidean space $\R^{m+1}$, by means of a new technique, are obtained. Each of the upper bounds of $λ_1$ depends on the length of mean curvature vector field, the dimension $n$, the volume of $M^n$, and of a vector of $\R^{m+1}$. When $M^n$ does not lie minimally in a hypersphere of $\R^{m+1}$, classical Reilly's inequality \cite{Re} is improved and new upper bounds are explicitly computed. For instance, considering a torus of revolution whose generating circle has a radius of $1$ and is centered at distance $\sqrt{2}$ from the axis of revolution, we find $λ_1 < \frac{4}{3}(\sqrt{2}-1)\approx 0.552284$, whereas Reilly's upper bound gives $λ_1 < 1/\sqrt{2}\approx 0.707106$.

math.DG↗
arXiv2022

Rigidity results for complete spacelike submanifolds in plane fronted waves

New rigidity results for complete non-compact spacelike submanifolds of arbitrary codimension in plane fronted waves are obtained. Under appropriate assumptions, we prove that a complete spacelike submanifold in these spacetimes is contained in a characteristic lightlike hypersurface. Moreover, for a complete codimension two extremal submanifold in a plane fronted wave we show sufficient conditions to guarantee that it is a (totally geodesic) wavefront.

math.DG↗
arXiv2021

Complete stationary spacelike surfaces in an $n$-dimensional Generalized Robertson-Walker spacetime

Several uniqueness results for non-compact complete stationary spacelike surfaces in an $n(\geq 3)$-dimensional Generalized Robertson Walker spacetime are obtained. In order to do that, we assume a natural inequality involving the Gauss curvature of the surface, the restrictions of the warping function and the sectional curvature of the fiber to the surface. This inequality gives the parabolicity of the surface. Using this property, a distinguished non-negative superharmonic function on the surface is shown to be constant, which implies that the stationary spacelike surface must be totally geodesic. Moreover, non-trivial examples of stationary spacelike surfaces in the four dimensional Lorentz-Minkowski spacetime are exposed to show that each of our assumptions is needed.

math.DG↗
arXiv2020

New examples of Moser-Bernstein type problems for some nonlinear elliptic partial differential equations arising in geometry

A family of nonlinear partial differential equations of divergence form is considered. Each one is the Euler-Lagrange equation of a natural Riemaniann variational problem of geometric interest. New uniqueness results for the entire solutions of these equations on a parabolic Riemaniann manifold of arbitrary dimension are given. In particular, several Moser-Bernstein type theorems are proved.

math.DG↗
arXiv2019

On uniqueness of the foliation by comoving observers restspaces of a Generalized Robertson Walker spacetime

A characterization of the foliation by spacelike slices of an $(n+1)$-dimensional spatially closed Generalized Robertson-Walker spacetime is given by means of studying a natural mean curvature type equation on spacelike graphs. Under some natural assumptions, of physical or geometric nature, all the entire solutions of such an equation are obtained. In particular, the case of entire spacelike graphs in de Sitter spacetime is faced and completely solved by means of a new application of a known integral formula.

math.DG↗
arXiv2019

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

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.

math.DG↗
arXiv2016

Uniqueness of complete maximal hypersurfaces in spatially open $(n+1)$-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.

math.DG↗
arXiv2016

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.

math.DG↗
arXiv2016

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 $η$. Several sufficient assumptions on such a surface with non-degenerate $η$-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 $η$-second fundamental form is developed. Then, totally umbilical round spheres are characterized as the only compact spacelike surfaces through the lightcone such that its $η$-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 $η$ and the area of the $η$-second fundamental form are also given.

math.DG↗
arXiv2014

Constant mean curvature spacelike hypersurfaces in Lorentzian warped products and Calabi-Bernstein type problems

In this paper we provide several uniqueness and non-existence results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian warped products under appropriate geometric assumptions. As a consequence of this parametric study, we obtain very general uniqueness and non-existence results for a large family of uniformly elliptic EDP's, so solving the Calabi-Bernstein problem in a wide family of spacetimes.

math.DG↗
arXiv2013

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.

math.DG↗
arXiv2012

Completeness of trajectories of relativistic particles under stationary magnetic fields

The second order differential equation $\frac{D\dotγ}{dt}(t) = F_{γ(t)}(\dotγ(t)) - \nabla V(γ(t))$ 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.

math.DG↗
arXiv2012

Completeness of the Trajectories of Particles Coupled to a General Force Field

We analyze the extendability of the solutions to a certain second order differential equation on a Riemannian manifold $(M,g)$, which is defined by a general class of forces (both prescribed on $M$ or depending on the velocity). The results include the general time-dependent anholonomic case, and further refinements for autonomous systems or forces derived from a potential are obtained. These extend classical results for Lagrangian and Hamiltonian systems. Several examples show the optimality of the assumptions as well as the applicability of the results, including an application to relativistic pp-waves.

math.DS↗
arXiv2012

On spacelike surfaces in 4-dimensional Lorentz-Minkowski spacetime through a lightcone

On any spacelike surface in a lightcone of four dimensional Lorentz-Minkowski space a distinguished smooth function is considered. It is shown how both extrinsic and intrinsic geometry of such a surface is codified by this function. The existence of a local maximum is assumed to decide when the spacelike surface must be totally umbilical, deriving a Liebmann type result. Two remarkable families of examples of spacelike surfaces in a lightcone are explicitly constructed. Finally, several results which involve the first eigenvalue of the Laplace operator of a compact spacelike surface in a lightcone are obtained.

math.DG↗
arXiv2007

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.

gr-qc↗