Henrique F. de Lima

Complete spacelike hypersurfaces in a Robertson-Walker spacetime

In this paper, as a suitable application of the well-known generalized maximum principle of Omori–Yau, we obtain uniqueness results concerning to complete spacelike hypersurfaces with constant mean curvature immersed in a Robertson–Walker (RW) spacetime. As an application of such uniqueness results for the case of vertical graphs in a RW spacetime, we also get non-parametric rigidity results.

On the rigidity of spacelike hypersurfaces immersed in the steady state space H elevado a n+1

In this paper, as a suitable application of the well known generalized Maximum Principle of Omori{Yau, we obtain rigidity results concerning to complete spacelike hypersurfaces immersed in the half Hn+1 of the de Sitter space Sn+1 1 , which models the so-called steady state space. Moreover, by using an isometrically equivalent model for Hn+1, we extend our results to a wider family of spacetimes. Finally, we also study the uniqueness of entire vertical graphs in such ambient spacetimes.

Complete CMC hypersurfaces in the hyperbolic space with prescribed Gauss mapping

Our aim in this paper is to show that a complete hypersurface x : Mn → Hn+1 immersed with constant mean curvature into the hyperbolic space Hn+1 is totally umbilical provided that its Gauss mapping ν has some suitable behavior. In this setting, our first result requires that the image ν(M) lies in a totally umbilical spacelike hypersurface of the de Sitter space S 1 , while in our second one we suppose that Mn has scalar curvature bounded from below and that ν(M) is contained in the closure of a domain enclosed by a totally umbilical spacelike hypersurface of S 1 determined by some vector a of the Minkowski space Ln+2, with the tangential component of a with respect to Mn having Lebesgue integrable norm.

On the geometry of horospheres

The aim of this paper is to investigate Bernstein-type properties of horospheres of the hyperbolic space H. Our approach is based on the use of appropriate generalized maximum principles in order to obtain new characterization results of such horospheres. Furthermore, by supposing a linear dependence between support functions naturally attached to a hypersurface, we also establish a classification theorem concerning horospheres and hyperbolic cylinders of H. Mathematics Subject Classification (2010). Primary 53C42; Secondary 53B30.

Entire spacelike

In this work we establish suffcient conditions to ensure that an entire spacelike graph immersed with constant mean curvature in a Lorentzian product space, whose Riemannian fiber has sectional curvature bounded from below, must be a trivial slice of the ambient space.

H

In this paper, our aim is to study the geometry of n-dimensional trapped and marginally trapped submanifolds immersed in a Lorentzian space form L1n+p(c) of constant sectional curvature c. In this setting, we establish sufficient conditions to guarantee that a complete trapped submanifold with parallel mean curvature vector in L1n+p(c) must be pseudo-umbilical. Afterwards, we obtain a nonexistence result concerning complete trapped submanifolds in the Lorentz–Minkowski space. Furthermore, under suitable constraints on the Ricci curvature and the second fundamental form, we show that an n-dimensional complete pseudo-umbilical marginally trapped submanifold of L1n+p(c) with parallel mean curvature vector is, in fact, totally umbilical.

-graphs in Lorentzian product spaces

We study complete noncompact spacelike hypersurfaces immersed into conformally stationary spacetimes, equipped with either one or two conformal vector fields. In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of totally geodesic hypersurfaces in terms of their r -th mean curvatures. For instance, for a timelike geodesically complete conformally stationary spacetime endowed with a closed conformal timelike vector field V, under appropriate restrictions on the flow and the norm of the tangential component of V, we are able to prove that totally geodesic spacelike hypersurfaces must be, in fact, leaves of the distribution determined by V. Applications to the so-called generalized Robertson–Walker spacetimes are also given. Furthermore, we extend our approach in order to obtain a lower estimate of the relative nullity index.

On the geometry of trapped and marginally trapped submanifolds in Lorentzian space forms

Our purpose is to study the geometry of linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Einstein spacetime, whose sectional curvature is supposed to obey some standard restrictions. In this setting, by using as main analytical tool a generalized maximum principle for complete non-compact Riemannian manifolds, we establish sufficient conditions to guarantee that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. Applications to the de Sitter space are given.

ON THE TOTALLY GEODESIC SPACELIKE HYPERSURFACES IN CONFORMALLY STATIONARY SPACETIMES

In this paper, we extend a technique due to Romero et al. (Class Quantum Gravity 30:1–13, 2013; Int J Geom Methods Mod Phys 10:1360014, 2013; J Math Anal Appl 419:355–372, 2014) establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed in a weighted generalized Robertson–Walker spacetime whose fiber has