Observer-dependent Gauss–Codazzi formalism for null hypersurfaces in the space–time

We introduce a new Gauss–Codazzi framework for null hypersurfaces in the space–time. First, with the use of space–time splitting techniques, and working within the framework of general coordinates of the ambient space–time, we generalize the second fundamental form and the Ricci and Gauss–Codazzi formulae of a non-null hypersurface Σ to a neighbourhood of it. Then in a similar way we introduce a second fundamental form analogue for the null hypersurface case, and deduce the corresponding Ricci and Gauss–Codazzi formulae.     

Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces

In this paper we obtain a height estimate concerning compact space-like hypersurfaces Σ ⁿ immersed with some positive constant r-mean curvature into an (n + 1)-dimensional Lorentzian product space , and whose boundary is contained into a slice {t} × M ⁿ . By considering the hyperbolic caps of the Lorentz–Minkowski space , we show that our estimate is sharp. Furthermore, we apply this estimate to study the complete space-like hypersurfaces immersed with some positive constant r-mean curvature into a Lorentzian product space. For instance, when the ambient space–time is spatially closed, we show that such hypersurfaces must satisfy the topological property of having more than one end which constitutes a necessary condition for their existence.     

Closed time-like curves in flat Lorentz space–times

We consider the region of closed time-like curves (CTCs) in three-dimensional flat Lorentz space–times. The interest in this global geometrical feature goes beyond the purely mathematical one. Such space–times are lower-dimensional toy models of sourceless Einstein gravity or cosmology. In three dimensions all such space–times are known: they are quotients of Minkowski space by a suitable group of Poincaré isometries. The presence of CTCs would indicate the possibility of “time machines”, a region of space–time where an object can travel along in time and revisit the same event. Such space–times also provide a testbed for the chronology protection conjecture, which suggests that quantum back reaction would eliminate CTCs. In particular, our interest in this note will be to find the set free of CTCs for , where is modeled on Minkowski space and γ is a Poincaré transformation. We describe the set free of CTCs where γ is hyperbolic, parabolic, and elliptic.     

Remarks on compact spacelike hypersurfaces in de Sitter space with constant higher order mean curvature

It is shown that a compact spacelike hypersurface which is contained in the chronological future (or past) of an equator of de Sitter space is a totally umbilical round sphere if one of the mean curvatures H_l does not vanish and the ratio H_k/H_l is constant for some k, l, 1≤l<k≤n. This extends the previous result in [J. Geom. Phys. 31 (1999) 195, Theorem 7].     

Flat lightlike hypersurfaces in Lorentz–Minkowski 4-space

The lightlike hypersurfaces in Lorentz–Minkowski space are of special interest in Relativity Theory. In particular, the singularities of these hypersurfaces provide good models for the study of different horizon types. We introduce the notion of flatness for these hypersurfaces and study their singularities. The classification result asserts that a generic classification of flat lightlike hypersurfaces is quite different from that of generic lightlike hypersurfaces.     

Complete spacelike hypersurfaces in conformally stationary Lorentz manifolds

We derive, for the square operator of Yau, an analogue of the Omori–Yau maximum principle for the Laplacian. We then apply it to obtain nonexistence results concerning complete noncompact spacelike hypersurfaces immersed with constant higher order mean curvature in a conformally stationary Lorentz manifold.     

Some estimates for the curvatures of complete spacelike hypersurfaces in generalized Robertson–Walker spacetimes

In this paper we obtain some estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in a generalized Robertson–Walker spacetime, under certain assumptions on the warped function of the ambient space. Our results will be an application of a generalized maximum principle due to Omori.