Spacelike metric foliations

In this paper, we investigate spacelike metric foliations in lightlike complete spacetimes. When such a foliation satisfies the strong energy condition Ric^V (e) ≥ 0 for timelike vectors e, it must be totally geodesic, and the metric is of higher rank, in the sense that each point of the spacetime is contained inside a flat, totally geodesic, timelike rectangle. If in addition Ric^V(e) = 0, then the metric is (at least locally) a product metric, with the leaves of the foliation tangent to one of the factors.     

Closed spaces in cosmology

This paper deals with two aspects of relativistic cosmologies with closed spatial sections. These spacetimes are based on the theory of general relativity, and admit a foliation into space sectionsS(t), which are spacelike hypersurfaces satisfying the postulate of the closure of space: eachS(t) is a three-dimensional closed Riemannian manifold. The topics discussed are: (i) a comparison, previously obtained, between Thurston geometries and Bianchi-Kantowski-Sachs metrics for such three-manifolds is here clarified and developed; and (ii) the implications of global inhomogeneity for locally homogeneous three-spaces of constant curvature are analyzed from an observational viewpoint.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Lichnerowicz-York equation and conformal deformations on maximal slicings in asymptotically flat space-times

The solvability of the Lichnerowicz-York equation is discussed on each sliceS _t=IR³ of a spacelike, asymptotically Euclidean maximal foliation {S _τ}. Following Cantor, the problem is reduced to a discussion of the properties of a smooth, time-dependent, family of conformal transformations,ø _t, relating the physical metrich _tofS _t to a metric ? _t =ø ⁴h_t, with vanishing scalar curvature. An estimate is provided for infø _t. This allows us to examine the properties of the scale geometry on eachS _twhen strong field regions are probed. It is shown that in such regions ? _t tends to become degenerate exponentially as a suitable average of the scalar curvature of (S _t, h _t ) increases. This is interpreted as representing the approach to a singular regime for (S _t, h _t ). An estimate is also provided for the lapse function-N _t defining {S _t}. This is found to be in agreement with a similar estimate suggested, on heuristic grounds, by Smarr and York. This latter result indicates that asymptotically flat maximal slicings in general (but not always) avoid reaching regions where the above singular regime is approached.     

On gravity’s role in the genesis of rest masses of classical fields

It is shown that in the Einstein-conformally coupled Higgs–Maxwell system with Friedman–Robertson–Walker symmetries the energy density of the Higgs field has stable local minimum only if the mean curvature of the \(t=\mathrm{const}\) hypersurfaces is less than a finite critical value \(\chi _c\), while for greater mean curvature the energy density is not bounded from below. Therefore, there are extreme gravitational situations in which even quasi-locally defined instantaneous vacuum states of the Higgs sector cannot exist, and hence one cannot at all define the rest mass of all the classical fields. On hypersurfaces with mean curvature less than \(\chi _c\) the energy density has the ‘wine bottle’ (rather than the familiar ‘Mexican hat’) shape, and the gauge field can get rest mass via the Brout–Englert–Higgs mechanism. The spacelike hypersurface with the critical mean curvature represents the moment of ‘genesis’ of rest masses.     

A slice theorem for the space of solutions of Einstein's equations

A slice for the action of a group G on a manifold X at a point x ? X is, roughly speaking, a submanifold S_x which is transverse to the orbits of G near x. Ebin and Palais proved the existence of a slice for the diffeomorphism group of a compact manifold acting on the space of all Riemannian metrics. We prove a slice theorem for the group $\text{D}$ of diffeomorphisms of spacetime acting on the space $\text{E}$ of spatially compact, globally hyperbolic solutions of Einstein's equations. New difficulties beyond those encountered by Ebin and Palais arise because of the Lorentz signature of the spacetime metrics in $\text{E}$ and because $\text{E}$ is not a smooth manifold- it is known to have conical singularities at each spacetime metric with symmetries. These difficulties are overcome through the use of the dynamic formulation of general relativity as an infinite dimensional Hamiltonian system (ADM formalism) and through the use of constant mean curvature foliations of the spacetimes in $\text{E}$. (We devote considerable space to a review and extension of some special properties of constant mean curvature surfaces and foliations that we need.) The conical singularity structure of $\text{E}$, the sympletic aspects of the ADM formalism, and the uniqueness of constant mean curvature foliations play key roles in the proof of the slice theorem for the action of $\text{D}$ on $\text{E}$. As a consequence of this slice theorem, we find that the space $\text{D}$ = $\text{E}$/$\text{D}$ of gravitational degrees of freedom is a stratified manifold with each stratum being a sympletic manifold. The spaces for homogeneous cosmologies of particular Bianchi types give rise to special finite dimensional symplectic strata in this space $\text{G}$. Our results should extend to such coupled field theories as the Einstein-Yang-Mills equations, since the Yang-Mills system in a given background spacetime admits a slice theorem for the action of the gauge transformation group on the space of Yang-Mills solutions, since there is a satisfactory Hamiltonian treatment of the Einstein-Yang-Mills system, and since the singularity structure of the solution set is known.     

On Vanishing Theorems for Vector Bundle Valued <Emphasis Type="Italic">p</Emphasis>-Forms and their Applications

Let F : [0, ∞) → [0, ∞) be a strictly increasing C ² function with F(0) = 0. We unify the concepts of F-harmonic maps, minimal hypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, and introduce F-Yang-Mills fields, F-degree, F-lower degree, and generalized Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on manifolds. When \({F(t)=t, \frac 1p(2t)^{\frac p2}, \sqrt{1+2t} -1,}\) and \({1-\sqrt{1-2t},}\) the F-Yang-Mills field becomes an ordinary Yang-Mills field, p-Yang-Mills field, a generalized Yang-Mills-Born-Infeld field with the plus sign, and a generalized Yang-Mills-Born-Infeld field with the minus sign on a manifold respectively. We also introduce the E _F,g ?energy functional (resp. F-Yang-Mills functional) and derive the first variational formula of the E _F,g ?energy functional (resp. F-Yang-Mills functional) with applications. In a more general frame, we use a unified method to study the stress-energy tensors that arise from calculating the rate of change of various functionals when the metric of the domain or base manifold is changed. These stress-energy tensors are naturally linked to F-conservation laws and yield monotonicity formulae, via the coarea formula and comparison theorems in Riemannian geometry. Whereas a “microscopic” approach to some of these monotonicity formulae leads to celebrated blow-up techniques and regularity theory in geometric measure theory, a “macroscopic” version of these monotonicity inequalities enables us to derive some Liouville type results and vanishing theorems for p?forms with values in vector bundles, and to investigate constant Dirichlet boundary value problems for 1-forms. In particular, we obtain Liouville theorems for F?harmonic maps (which include harmonic maps, p-harmonic maps, exponentially harmonic maps, minimal graphs and maximal space-like hypersurfaces, etc.), F?Yang-Mills fields, extended Born-Infeld fields, and generalized Yang-Mills-Born-Infeld fields (with the plus sign and with the minus sign) on manifolds, etc. As another consequence, we obtain the unique constant solution of the constant Dirichlet boundary value problems on starlike domains for vector bundle-valued 1-forms satisfying an F-conservation law, generalizing and refining the work of Karcher and Wood on harmonic maps. We also obtain generalized Chern type results for constant mean curvature type equations for p?forms on \({\mathbb{R}^m}\) and on manifolds M with the global doubling property by a different approach. The case p = 0 and \({M=\mathbb{R}^m}\) is due to Chern.     

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.     

Systematic behavior ofB(E2) values in the yrast bands of doubly even nuclei

The experimental information onB(E2) transition rates in the yrast bands of doubly even nuclei (126≦A≦184) is systematized. The strength functionS _exp≡B(E2,I→I?2)×E(I→I?2) is found to reveal characteristic behavior significant for structure studies of yrast bands. The energy-weightedB(E2,I→I?2) values (S _exp) and 2?/?²(?: moment of inertia) are plotted versus the rotational frequency squared ?²ω² for each nucleus. In strongly deformed nuclei (N≧90), theS _exp curves smoothly increase for low rotational frequencies suggesting that up to spin valuesI≈8 the ratioQ ₀ ² ? is nearly constant (Q ₀: quadrupole moment). This is not the case in nuclei with a soft core (N≦88). In the relevant discussion, the hydrodynamical model as well as the CAP effect are considered. The results in the backbending region are qualitatively discussed in terms of the two-band crossing model. Evidence is found supporting the prediction of an oscillating behavior of the yrast-yrare interaction.     

The space of (generalized) Taub-Nut spacetimes

In this paper we show how to construct all analytic solutions of the vacuum Einstein equations having a compact Cauchy horizon diffeomorphic to S³ and ruled by closed null generators which fiber the horizon in the sense of Hopf. The set of (inequivalent) solutions is infinite dimensional, contains the two parameter Taub-NUT family as a special case, and may be uniquely parameterized by a pair of arbitrary, real analytic functions on S² (modulo an action of the conformal group of S²). The horizon of each such solution is necessarily a Killing horizon (as proven recently by Isenberg and the author) and is shown here always to be a «crushingå horizon in the sense of Eardley and Smarr. Some recent results of Gerhardt are used to show that a neighborhood of the horizon (in the globally hyperbolic region) is always foliated by constant mean curvature hypersurfaces.The possible isometry groups of the solutions considered are characterized in terms of isometries of the determining «Cauchy dataå which is specified on the horizons themselves.     

Foliations by Stable Spheres with Constant Mean Curvature for Isolated Systems with General Asymptotics

We prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the Regge–Teitelboim condition near infinity. It is known that the (Hamiltonian) center of mass is well-defined for manifolds satisfying this condition. We also show that the foliation is asymptotically concentric, and its geometric center is the center of mass. The construction of the foliation generalizes the results of Huisken–Yau, Ye, and Metzger, where strongly asymptotically flat manifolds and their small perturbations were studied.