Foliations of space-times by spacelike hypersurfaces of constant mean curvature

The foliations under discussion are of two different types, although in each case the leaves areC ² spacelike hypersurfaces of constant mean curvature. For manifolds, such as that of the Friedmann universe with closed spatial sections, which are topologicallyI×S ³,I an open interval, the leaves will be spacelike hypersurfaces without boundary and the foliation will fill the manifold. In the case of the domain of dependence of a spacelike hypersurface,S, with boundaryB, the leaves will be spacelike hypersurfaces with boundary,B, and the foliation will fillD(S). It is shown that a local energy condition ensures that the constant mean curvature increases monotonically with time through such foliations and that, in the case of a foliation whose leaves are spacelike hypersurfaces without boundary in a manifold where this energy condition is satisfied globally, the foliation is unique.     

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 the mean curvature of spacelike submanifolds in semi-Riemannian manifolds

In this paper we establish some estimates for the higher-order mean curvature of a complete spacelike hypersurface in spacetimes with sectional curvature satisfying certain condition. We also obtain the estimate for the mean curvature of a complete spacelike submanifold in semi-Riemannian space forms.     

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.     

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].     

Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

In this paper, the complete spacelike hypersurface with constant normal scalar curvature in a locally symmetric Lorentz space is discussed. Several classified theorems are obtained using the operator _L1$L_1$ introduced by S.Y. Cheng and S.T. Yau (1977) [4].     

Algebraic integrability conditions for Killing tensors on constant sectional curvature manifolds

We use an isomorphism between the space of valence two Killing tensors on an n$n$-dimensional constant sectional curvature manifold and the irreducible GL(n+1)$GL(n+1)$-representation space of algebraic curvature tensors in order to translate the Nijenhuis integrability conditions for a Killing tensor into purely algebraic integrability conditions for the corresponding algebraic curvature tensor, resulting in two simple algebraic equations of degree two and three. As a first application of this we construct a new family of integrable Killing tensors.     

Volume comparison between spacelike hypersurfaces in a Lorentzian manifold with integral Ricci curvature bounds

We obtain the volume comparison between spacelike hypersurfaces in a Lorentzian manifold with integral Ricci and mean curvature bounds. Also we extend volume comparisons to weighted volume comparisons with integral norms of the generalized Ricci tensor.     

Maximal hypersurfaces and foliations of constant mean curvature in general relativity

We prove theorems on existence, uniqueness and smoothness of maximal and constant mean curvature compact spacelike hypersurfaces in globally hyperbolic spacetimes. The uniqueness theorem for maximal hypersurfaces of Brill and Flaherty, which assumed matter everywhere, is extended to spacetimes that are vacuum and non-flat or that satisfy a generic-type condition. In this connection we show that under general hypotheses, a spatially closed universe with a maximal hypersurface must be Wheeler universe; i.e. be closed in time as well. The existence of Lipschitz achronal maximal volume hypersurfaces under the hypothesis that candidate hypersurfaces are bounded away from the singularity is proved. This hypothesis is shown to be valid in two cases of interest: when the singularities are of strong curvature type, and when the singularity is a single ideal point. Some properties of these maximal volume hypersurfaces and difficulties with Avez' original arguments are discussed. The difficulties involve the possibility that the maximal volume hypersurface can be null on certain portions; we present an incomplete argument which suggests that these hypersurfaces are always smooth, but prove that an a priori bound on the second fundamental form does imply smoothness. An extension of the perturbation theorem of Choquet-Bruhat, Fischer and Marsden is given and conditions under which local foliations by constant mean curvature hypersurfaces can be extended to global ones is obtained.     

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.     

Keller magnetic fields in space of constant holomorphic sectional curvature

Keller magnetic fields on Keller manifolds of constant holomorphic curvature with an arbitrary signature are considered. The metric of the Keller space of constant holomorphic curvature is obtained in general form. Simulation of the Keller magnetic fields of by means of H-projective curves of a flat space is used to reduce the study of the trajectory equations to a study of one ordinary second-order differential equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 13–19, January, 1996.     

Complete spacelike hypersurfaces in orthogonally splitted spacetimes

We provide some “half-space theorems” for spacelike complete non-compact hypersurfaces into orthogonally splitted spacetimes. In particular we generalize some recent work of Rubio and Salamanca on maximal spacelike compact hypersurfaces. Beside compactness, we also relax some of their curvature assumptions and even consider the case of nonconstant mean curvature bounded from above. The analytic tools used in various arguments are based on some forms of the weak maximum principle.     

On the topology of spacelike hypersurfaces,singularities, and black holes

Sufficient topological conditions and some reasonable geometric properties near infinity are given on a partial Cauchy surface to study the occurrence of singularities and the topological structure of smooth event horizons. Conditions are given on electrovac space-times to show the horizon is a sphere without assuming the solution is stationary.     

On curvature homogeneous three-dimensional Lorentzian manifolds

The aim of this paper is the study of three-dimensional Lorentzian manifolds whose Ricci tensor has three equal constant eigenvalues, whose associated eigenspace is two-dimensional. A complete local classification of this class of curvature homogeneous manifolds is presented. It turns out that, if the eigenvalue is zero, these are exactly the curvature homogeneous manifolds modelled on an indecomposable, non-irreducible Lorentzian symmetric space, which were first studied in Cahen etaal. (1990), and the techniques presented in this paper can therefore be applied to obtain a complete (local) classification of these manifolds, and to construct a number of new examples of such manifolds.     

Evolutes of spacelike hypersurfaces in anti-de Sitter space

We study spacelike hypersurfaces in anti-de Sitter space from the view point of the Lagrangian/Legendrian singularity theory.     

Spacelike hypersurfaces with constant higher order mean curvature in Minkowski space–time

In this paper, we develop a series of general integral formulae for compact spacelike hypersurfaces with hyperplanar boundary in the (n+1)-dimensional Minkowski space–time . As an application of them, we prove that the only compact spacelike hypersurfaces in having constant higher order mean curvature and spherical boundary are the hyperplanar balls (with zero higher order mean curvature) and the hyperbolic caps (with nonzero constant higher order mean curvature). This extends previous results obtained by the first author, jointly with Pastor, for the case of constant mean curvature [J. Geom. Phys. 28 (1998) 85] and the case of constant scalar curvature [Ann. Global Anal. Geom. 18 (2000) 75].     

Geometric inequalities for spacelike hypersurfaces in the Minkowski spacetime

We derive a linear isoperimetric inequality and some geometric inequalities for properly located compact achronal spacelike hypersurfaces via a Minkowski-type integral formula in the Minkowski spacetime.