Alfonso Romero

Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes

A new technique is introduced in order to solve the following question:When is a complete spacelike hypersurface of constant mean curvature in a generalized Robertson-Walker spacetime totally umbilical and a slice? (Generalized Robertson-Walker spacetimes extend classical Robertson-Walker ones to include the cases in which the fiber has not constant sectional curvature.) First, we determine when this hypersurface must be compact. Then, all these compact hypersurfaces in (necessarily spatially closed) spacetimes are shown to be totally umbilical and, except in very exceptional cases, slices. This leads to proof of a new Bernstein-type result. The power of the introduced tools is also shown by reproving and extending several known results.

On some real hypersurfaces of a complex hyperbolic space

Given a real hypersurface of a complex hyperbolic space #x2102;?Hn,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H12n+1.Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of ℂHn.

Integral formulas for compact space-like hypersurfaces in de Sitter space: Applications to the case of constant higher order mean curvature

Abstract In this paper we develop some integral formulas for compact space-like hypersurface in de Sitter space S 1 n+1 and apply them in order to characterized thetotally umbilical round spheres of S 1 n+1 as the only compact space-like hypersurfaces with constant higher order mean curvature under some appropriate hypothesis. In particular, for hypersurfaces contained in the chronological future (or past) of an equator of S 1 n+1 we prove that the only compact space-like hypersurface with a constant higher order mean curvature are the totally umbilical round spheres.

Completeness of compact Lorentz manifolds admitting a timelike conformal Killing vector field

It is proved that every compact Lorentz manifold admitting a timelike conformai Killing vector field is geodesically complete. So, a recent result by Kamishima in J. Differential Geometry [37 (1993), 569-601] is widely extended. Recently, it has been proved in [2] that a compact Lorentz manifold of constant curvature admitting a timelike Killing vector field is (geodesically) complete. It is natural to think about the importance of the assumption on the curvature in this result. Moreover, we could ask ourselves if there is a more general condition than the existence of a timelike Killing vector field. The answers to these questions are given in the following Theorem. Let (M, g) be a compact Lorentz manifold which admits a timelike conformai Killing vector field K. Then (M, g) is geodesically complete. Proof. We are going to see that any geodesic y : [0, e[-> M, 0 0. Thus, we have only to check that g(K, y) is bounded. Taking into account the fact that Lng = o • g for some function o , we obtain that j-(g(K, y) = ^C-ooy, so (d/dt)g(K, y) and, as a consequence g(K, y) is bounded on [0, b[. Received by the editors October 20, 1993 and, in revised form, January 25, 1994. 1991 Mathematics Subject Classification. Primary 53C50, 53C22.

Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker spacetimes

A new technique for the study of noncompact complete spacelike hypersurfaces in generalized Robertson–Walker (GRW) spacetimes whose fiber is a parabolic Riemannian manifold is introduced. This class of spacetimes allows us to model open universes which extend to spacelike closed GRW spacetimes from the viewpoint of the geometric analysis of the fiber, and which, unlike those spacetimes, could be compatible with the holographic principle. First, under reasonable assumptions on the restriction of the warping function to the spacelike hypersurface and on the hyperbolic angle between the unit normal vector field and a certain timelike vector field, a complete spacelike hypersurface in a spatially parabolic GRW spacetime is shown to be parabolic, and the existence of a simply connected parabolic spacelike hypersurface in a GRW spacetime also leads to the parabolicity of its fiber. Then, all the complete maximal hypersurfaces in spatially parabolic GRW spacetimes are determined in several cases, extending, in particular, to this family of open cosmological models several well-known uniqueness results for the case of spatially closed GRW spacetimes. Moreover, new Calabi–Bernstein problems are solved.

Magnetic vortex filament flows

We exhibit a variational approach to study the magnetic flow associated with a Killing magnetic field in dimension 3. In this context, the solutions of the Lorentz force equation are viewed as Kirchhoff elastic rods and conversely. This provides an amazing connection between two apparently unrelated physical models and, in particular, it ties the classical elastic theory with the Hall effect. Then, these magnetic flows can be regarded as vortex filament flows within the localized induction approximation. The Hasimoto transformation can be used to see the magnetic trajectories as solutions of the cubic nonlinear Schrodinger equation showing the solitonic nature of those.

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 corresponds with a limit case obtained when the force of the Gauss-Landau-Hall magnetic field increases arbitrarily. We also obtain properties related with the completeness of flowlines for general magnetic fields. The paper also contains 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.

A Berger-Green type inequality for compact Lorentzian manifolds

We give a Lorentzian metric on the null congruence associated with a timelike conformal vector field. A Liouville type theorem is proved and a boundedness for the volume of the null congruence, analogous to a well-known Berger-Green theorem in the Riemannian case, will be derived by studying conjugate points along null geodesics. As a consequence, several classification results on certain compact Lorentzian manifolds without conjugate points on its null geodesics are obtained. Finally, several properties of null geodesics of a natural Lorentzian metric on each odd-dimensional sphere have been found.

New examples of Calabi-Bernstein problems for some nonlinear equations

Abstract All the entire solutions to the maximal surface equation in certain 3-dimensional Lorentzian manifolds, obeying the null energy condition, are obtained. Thus, we solve new Calabi–Bernstein problems. As a consequence, the corresponding parametric versions are also given. The behaviour of this Calabi–Bernstein property with respect to an special family of C 2 -perturbations of Lorentz–Minkowski space L 3 is investigated

Simple proof of Calabi-Bernstein’s Theorem on maximal surfaces

In the references cited above, this result appears as either a particular case of some much more general theorems or stated in terms of local complex representation of the surface. However, a direct simple proof would be desirable to be easily understood for beginning researchers. The proof we present here uses only Liouville’s Theorem on harmonic functions on R. Thus, it is simple and complex function theory is not needed. This proof is inspired by [Ch]. Roughly, the key steps of our proof are: (1) On any maximal surface there exists a positive harmonic function, which is constant if and only if the surface is totally geodesic. (2) The metric of any spacelike graph is globally conformally related to a metric g∗, which is complete when the graph is entire. (3) On any maximal graph the metric g∗ is flat.