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].     

A generalization of the concept of constant mean curvature and canonical time

Some compact spaces of achronal hypersurfaces are constructed in various types of space-time. A variational principle is introduced on these spaces, smooth extremals of which are spacelike hypersurfaces of constant mean curvature. The integrand of the variational principle is shown to be upper semicontinuous and the direct methods of the calculus of variations are applied to obtain aC ⁰ extremal, which is defined to be a spacelike hypersurface of generalized constant mean curvature. The family of such hypersurfaces generated by altering the value of the mean curvature is discussed and the mean curvature itself is shown to have many of the properties of a canonical time coordinate.     

Comparison theorems in Lorentzian geometry and applications to spacelike hypersurfaces

In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some estimates on the higher order mean curvatures of spacelike hypersurfaces satisfying a Omori-Yau maximum principle for certain elliptic operators.     

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.     

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.     

The upperbound of the volume expansion rate in a Lorentzian manifold

Using the differential equation obtained from spacelike level hypersurfaces in a Lorentzian manifold, the volume expansion rate of an achronal spacelike hypersurface orthogonal to a timelike geodesic is investigated in terms of the integral Ricci and scalar curvature bound.     

Relative Lorentzian volume comparison with integral Ricci and scalar curvature bound

A relative Lorentzian volume comparison estimate between spacelike hypersurfaces is studied with the integral curvature bound in terms of Ricci and Scalar curvature which generalize the Bishop–Gromov volume comparison theorem.     

Spacelike parallels and evolutes in Minkowski pseudo-spheres

We consider extrinsic differential geometry on spacelike hypersurfaces in Minkowski pseudo-spheres (hyperbolic space, de Sitter space and the lightcone). In the previous paper [S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, Preprint] we have shown a basic Legendrian duality theorem between pseudo-spheres. We define the spacelike parallels by using the basic Legendrian duality theorem. This definition unifies the notions of parallels of spacelike hypersurfaces in pseudo-spheres. We also define the evolute as the locus of singularities of the spacelike parallels. These notions are investigated as applications of Lagrangian or Legendrian singularity theory. We consider geometric properties of non-singular spacelike hypersurfaces corresponding to singularities of spacelike parallels or evolutes.     

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.     

Spacelike Hypersurfaces with Free Boundary in the Minkowski Space under the Effect of a Timelike Potential

In this paper we consider a variational problem for spacelike hypersurfaces in the (n + 1)-dimensional Lorentz-Minkowski space , whose critical points are hypersurfaces supported in a spacelike hyperplane Π determined by two facts: the mean curvature is a linear function of the distance to Π and the hypersurface makes a constant angle with Π along its boundary. We prove that the hypersurface is rotational symmetric with respect to a straight-line orthogonal to Π and that each (non-empty) intersection with a parallel hyperplane to Π is a round (n − 1)-sphere. A similar result is proved for hypersurfaces trapped between two parallel hyperplanes.     

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.     

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.     

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.     

Semi-symmetric Lorentzian hypersurfaces in Lorentzian space forms

In the present paper, we shall investigate and classify Lorentzian hypersurfaces in Lorentzian space forms satisfying some curvature conditions. We shall focus especially on semi-symmetric Lorentzian hypersurfaces. Those of constant curvature and those so-called “good” are explicitly described and classified. We shall also classify Einstein–Lorentzian hypersurfaces.     

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.     

A method of reduction of Einstein's equations of evolution and a natural symplectic structure on the space of gravitational degrees of freedom

A program is outlined which addresses the problem of thereduction of Einstein's equations, namely, that of writing Einstein's vacuum equations in (3+1)-dimensions as anunconstrained dynamical system where the variables are thetrue degrees of freedom of the gravitational field. Our analysis is applicable for globally hyperbolic Ricci-flat spacetimes that admit constant mean curvature compact orientable spacelike Cauchy hypersurfaces M with degM=0 andM not diffeomorphic toF ₆, the underlying manifold of a certain compact orientable flat affine 3-manifold. We find that for these spacetimes, modulo the extended Poincaré conjecture and the use of local cross-sections rather than a global cross-section, (3+1)-reduction can be completed much as in the (2+1)-dimensional case. In both cases, one gets as the reduced phase space the cotangent bundleT ^* T _M of theTeichmüller space T _M of conformal structures onM, whereM is a given initial constant mean curvature compact orientable spacelike Cauchy hypersurface in a spacetime (V, g _V ), and one gets reduction of the full classical non-reduced Hamiltonian system with constraints to a reduced Hamiltonian system without constraints onT ^* T _M . For these reduced systems, the time parameter is the parameter of a family of monotonically increasing constant mean curvature compact orientable spacelike Cauchy hypersurfaces in a neighborhood of a given initial one. In the (2+1)-dimensional case, the Hamiltonian is the area functional of these hypersurfaces, and in the (3+1)-dimensional case, the Hamiltonian is the volume functional of these hypersurfaces.     

Local foliations and optimal regularity of Einstein spacetimes

We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the existence of a local coordinate chart defined in a ball with definite size in which the metric coefficients have optimal regularity. The proof is based on quantitative estimates for, on one hand, a constant mean curvature (CMC) foliation by spacelike hypersurfaces defined locally near the observer and, on the other hand, the metric in local coordinates that are spatially harmonic in each CMC slice. The results and techniques in this paper should be useful in the context of general relativity for investigating the long-time behavior of solutions to the Einstein equations.     

Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hypersurfaces: Topologies and boundary conditions

Spacetimes with closed spacelike hypersurfaces and spacelike two-parameter isometry groups are investigated to determine their possible global structures. It is shown that the two spacelike Killing vectors always commute with each other. Connected group-invariant spacelike hypersurfaces must be homeomorphic to S¹ ? S¹ ? S¹ (three-torus), S¹ ? S² (three-handle), S³ (three-sphere), or to a manifold which is covered by one of these. The spacetime metric and Einstein equations are simplified in the absence of nongravitational sources and used to establish the impossibility of topology change as well as other limitations on global structure. Regularity conditions for spacetimes of this type are derived and shown to be compatible with Einstein's equations.     

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.     

The search for new representations of the Wheeler-DeWitt equation using the first order formalism

We obtain all possible representations of the Wheeler-DeWitt equation that may arise from the development of the first-order formalism for the quantization of gravity. They are the usual one, the modified extrinsic curvature representation and the intrisic and extrinsic time representations, which arise naturally in this formalism. We show that in the intrinsic time representation the factor-ordering problem is restricted to the pair consisting of the determinant of the metric of the spacelike hypersurfaces and its canonical momentum. Contrary to what was suggested in an earlier paper, no new representations can be obtained in this formalism or by considering actions differing by a total divergence.