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.     

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.     

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.     

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.     

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.     

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.     

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.     

Extrinsic curvatures of distributions of arbitrary codimension

In this article, using the generalized Newton transformation, we define higher order mean curvatures of distributions of arbitrary codimension and we show that they agree with the ones from Brito and Naveira [F. Brito, A.M. Naveira, Total extrinsic curvature of certain distributions on closed spaces of constant curvature, Ann. Global Anal. Geom., 18 (2000) 371–383]. We also introduce higher order mean curvature vector fields and we compute their divergence for certain distributions and using this we obtain total extrinsic mean curvatures.     

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.     

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.     

Spacelike surfaces with positive definite second fundamental form in 3D spacetimes

For a spacelike surface with positive definite second fundamental form in any 3-dimensional Lorentzian manifold, a new formula relating its mean and Gauss curvature with the Gauss curvature of the second fundamental form is obtained. As an application, necessary and sufficient conditions are established in order to prove that such a compact spacelike surface is totally umbilical.     

Positivity of quasilocal mass

Motivated by the important work of Brown and York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with a positive intrinsic curvature in a spacetime. We show that the quasilocal energy of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface.     

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.     

The structure of the space of solutions of Einstein’s equations coupled with scalar fields

Following the work of Arms, Fischer, Marsden and Moncrief, it is proved that the space of solutions of Einstein’s equations coupled with self-gravitating mass-less scalar fields has conical singularities at each spacetime possessing a compact Cauchy surface of constant mean curvature and a nontrivial set of simultaneous Killing fields, either all spacelike or including one (independent) timelike.     

Constant Mean Curvature Spacelike Surfaces in Three-Dimensional Generalized Robertson–Walker Spacetimes

Several uniqueness and non-existence results on complete constant mean curvature spacelike surfaces lying between two slices in certain three-dimensional generalized Robertson–Walker spacetimes are given. They are obtained from a local integral estimation of the squared length of the gradient of a distinguished smooth function on a constant mean curvature spacelike surface, under a suitable curvature condition on the ambient spacetime. As a consequence, all the entire bounded solutions to certain family of constant mean curvature spacelike surface differential equations are found.     

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.     

Initial Data Engineering

We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hypersurface.Partially supported by a Polish Research Committee grant 2 P03B 073 24Partially supported by the NSF under Grants PHY-0099373 and PHY-0354659Partially supported by the NSF under Grant DMS-0305048 and the UW Royalty Research Fund     

Existence of Constant Mean Curvature Foliations in Spacetimes with Two-Dimensional Local Symmetry

It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the mean curvature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity. The simplest examples occur when the spatial topology is that of a torus, with the standard global Killing vectors, but more exotic topologies are also covered. In the course of the proof it is shown that in this class of spacetimes a kind of positive mass theorem holds. The symmetry singles out a compact surface passing through any given point of spacetime and the Hawking mass of any such surface is non-negative. If the Hawking mass of any one of these surfaces is zero then the entire spacetime is flat. Received: 15 July 1996 / Accepted: 12 March 1997     

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.