Spacelike Hypersurfaces Research Articles

A Jang equation approach to the Penrose inequality

We introduce a generalized version of the Jang equation, designedfor the general case of the Penrose Inequality in the setting ofan asymptotically flat space-like hypersurface of a spacetimesatisfying the dominant energy condition. The appropriateexistence and regularity results are established in the specialcase of spherically symmetric Cauchy data, and are applied to givea new proof of the general Penrose Inequality for these data sets.When appropriately coupled with an inverse mean curvature flow,analogous existence and regularity results for the associatedsystem of equations in the nonspherical setting would yield aproof of the full Penrose Conjecture. Thus it remains as animportant and challenging open problem to determine whether thissystem does indeed admit the desired solutions.

On spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

The purpose of this paper is to study compact or complete spacelike hypersurfaces with constant normalized scalar curvature in a locally symmetric Lorentz space satisfying some curvature conditions. We give an optimal estimate of the squared norm of the second fundamental form of such hypersurfaces. Furthermore, the totally umbilical hypersurfaces are characterized.

Positive mass theorems for asymptotically de Sitter spacetimes

We use planar coordinates as well as hyperbolic coordinates to separate the de Sitter spacetime into two parts. These two ways of cutting the de Sitter give rise to two different spatial infinities. For spacetimes which are asymptotic to either half of the de Sitter spacetime, we are able to provide definitions of the total energy, the total linear momentum, the total angular momentum, respectively. And we prove two positive mass theorems, corresponding to these two sorts of spatial infinities, for spacelike hypersurfaces whose mean curvatures are bounded by certain constant from above.

Lower bounds for eigenvalues of the Dirac-Witten operator

We get optimal lower bounds for the eigenvalues of the Dirac-Witten operator of compact (with or without boundary) spacelike hypersurfaces of Lorentian manifold satisfying certain conditions, just in terms of the mean curvature and the scalar curvature and the spinor energy-momentum tensor. In the limiting case, the spacelike hypersurface is either maximal and Einstein manifold with positive scalar curvature or Ricci-flat manifold with nonzero constant mean curvature.

On complete spacelike hypersurfaces with two distinct principal curvatures in Lorentz–Minkowski space

We investigate complete spacelike hypersurfaces in Lorentz–Minkowski space with two distinct principal curvatures and constant m th mean curvature. By using Otsuki’s idea, we obtain the global classification result. As their applications, we obtain some characterizations for hyperbolic cylinders. We prove that the only complete spacelike hypersurfaces in Lorentz–Minkowski ( n + 1 ) -spaces ( n ≥ 3 ) of nonzero constant m th mean curvature ( m ≤ n − 1 ) with two distinct principal curvatures λ and μ satisfying inf ( λ − μ ) 2 > 0 are the hyperbolic cylinders. We also obtain a global characterization for hyperbolic cylinder H n − 1 ( c ) × R in terms of square length of the second fundamental form.

Stable homology as an indicator of manifoldlikeness in causal set theory

We present a computational tool that can be used to obtain the ‘spatial’ homology groups of a causal set. Localization in the causal set is seeded by an inextendible antichain, which is the analogue of a spacelike hypersurface, and a one-parameter family of nerve simplicial complexes is constructed by ‘thickening’ this antichain. The associated homology groups can then be calculated using existing homology software, and their behaviour studied as a function of the thickening parameter. Earlier analytical work showed that for an inextendible antichain in a causal set which can be approximated by a globally hyperbolic spacetime region, there is a one-parameter sub-family of these simplicial complexes which are homological to the continuum, provided the antichain satisfies certain conditions. Using causal sets that are approximated by a set of 2D spacetimes, our numerical analysis suggests that these conditions are generically satisfied by inextendible antichains. In both 2D and 3D simulations, as the thickening parameter is increased, the continuum homology groups tend to appear as the first region in which the homology is constant, or ‘stable’, above the discreteness scale. Below this scale, the homology groups fluctuate rapidly as a function of the thickening parameter. This provides a necessary though not sufficient criterion to test for manifoldlikeness of a causal set.

Stability of marginally outer trapped surfaces and symmetries

We study the properties of stable, strictly stable and locally outermost marginally outer trapped surfaces in spacelike hypersurfaces of spacetimes possessing certain symmetries such as isometries, homotheties and conformal Killings. We first obtain results for general diffeomorphisms in terms of the so-called metric deformation tensor and then particularize to different types of symmetries. In particular, we find restrictions at the surfaces on the vector field generating the symmetry. Some consequences are discussed. As an application, we present a result on non-existence of stable marginally outer trapped surfaces in slices of FLRW.

Evolution of Spacelike Hypersurfaces by Mean Curvature Minus External Force Field in Minkowski Space

Abstract We study the evolution of spacelike hypersurface moving by mean curvature minus an external force field in Minkowski space. It is shown that the flow will exist for all time if the initial spacelike surface has two compact spacelike barrier surfaces and the external force field satisfies a weak convexity condition.

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.

Singularities of lightcone Gauss images of spacelike hypersurfaces in de Sitter space

We define the notions of lightcone Gauss images of spacelike hypersurfaces in de Sitter space. We investigate the relationships between singularities of these maps and geometric properties of spacelike hypersurfaces as an application of the theory of Legendrian singularities. We classify the singularities and give some examples in the generic case in de Sitter 3-space.

Integral formulas for closed spacelike hypersurfaces in anti-de Sitter space H 1 n+1 (−1)

In this paper, we study closed k-maximal spacelike hypersurfaces Mn in anti-de Sitter space H1n+1 (−1) with two distinct principal curvatures and give some integral formulas about these hypersurfaces.

Constructing spherical traversable wormholes: an initial value approach

The construction of spherical wormholes is formulated as an initial value problem with the throat serving as an initial value surface. We model exotic matter with a stress tensor that relative to the orthonormal frame of Killing observers reduces to a diagonal form. This assumption combined with staticity, spherical symmetry and a (2 + 1 + 1) decomposition of Einstein's equations yields a set of first-order, coupled ordinary differential equations and an algebraic constraint. The dynamical equations involve the gradient Λ of the red-shift factor, the tension τ, the area A and the extrinsic curvature K of the SO(3) orbits as embedded on the hypersurfaces orthogonal to the timelike Killing field. The throat dictates a set of initial conditions which in combination with the dynamical equations yield a well-defined initial value problem. We study this initial value problem for the special case where the energy density ρc2 and Λ are a priori specified. Provided that ρc2 does not become large and positive over extended domains, we demonstrate the global existence of asymptotically flat, wormholes have a throat of prescribed area A(0). For any density profile decaying according to ρc2 = O(r−(2+ϵ)), where r is a radial coordinate tied to an asymptotic cartesian system, the resulting wormhole has finite Komar mass MK, but diverging ADM mass MADM. Finiteness of MADM requires . Besides this discrepancy, the non-trivial topology of the spacelike hypersurfaces orthogonal to the Killing fields, inserts an element of ambiguity in the definition of the Komar mass originating in the inability to normalize simultaneously the timelike Killing vector at both asymptotically flat ends.

Complete spacelike hypersurfaces with constant mean curvature in anti-de Sitter space

In this paper, we investigate the complete spacelike hypersurfaces with constant mean curvature and two distinct principal curvatures in an anti-de Sitter space. We give a characterization of hyperbolic cylinder and prove the conjecture in a paper by L. F. Cao and G. X. Wei [J. Math. Anal. Appl., 2007, 329(1): 408–414].

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.

Space-like hypersurfaces in locally symmetric Lorentz space

Let M be an n-dimensional space-like hypersurface in a locally symmetric Lorentz space, with n(n−1)R=κH(κ>0) and satisfying certain additional conditions on the sectional curvature. Denote by S and H the squared norm of the second fundamental form and the mean curvature of M, respectively. We show that if the mean curvature is nonnegative and attains its maximum on M, then: (1) if H2<4(n−1)c/n2, M is totally umbilical; (2) if H2=4(n−1)c/n2, M is totally umbilical or is an isoparametric hypersurface; (3) if H2>4(n−1)c/n2 and S satisfies some pinching conditions, M is totally umbilical or is an isoparametric hypersurface. doi:10.1017/S1446181108000254

Spacelike hypersurfaces of prescribed Gauss-Kronecker curvature in exterior domains

In this paper, we prove an existence theorem for spacelike convex hypersurfaces of prescribed Gauss curvature in Minkowski space.

Complete Vertical Graphs with Constant Mean Curvature in Semi-Riemannian Warped Products

In this paper we study complete vertical graphs of constant mean curvature in the Hyperbolic and Steady State spaces. We first derive suitable formulas for the Laplacians of the height function and of a support-like function naturally attached to the graph; then, under appropriate restrictions on the values of the mean curvature and the growth of the height function, we obtain necessary conditions for the existence of such a graph. In the two-dimensional case we apply this analytical framework to state and prove Bernstein-type results in each of these ambient spaces. This paper deals with complete non-compact constant mean curvature graphs over a horosphere of the Hyperbolic space, as well as over horizontal hyperplanes (slices) in the Steady State space. In connection with our work, L. Al´oas and M. Dajczer (cf. (2)) studied properly immersed complete surfaces of the 3−dimensional Hyperbolic space contained between two horospheres, obtaining a Bernstein-type result for the case of constant mean curvature between −1 and 1. In de Sitter space, K. Akutagawa (cf. (5)) proved that complete spacelike hypersurfaces having constant mean curvature in a specific interval of the real line are totally umbilical. Also for de Sitter space, among other interesting results S. Montiel (cf. (14)) proves that, under an appropriate restriction on their Hyperbolic Gauss map, complete spacelike Hypersurfaces of constant mean curvature greater than or equal to 1 must actually have mean curvature 1. For the Lorentz case, our motivation to restrict attention to the Steady State space comes from the fact that there exists a natural duality between the Gauss maps of Riemannian hypersurfaces of this space and those of the Hyperbolic space, provided we model these as hyperquadrics of the Lorentz-Minkowski space (cf. section 5). Besides, in physical context the Steady State space appears naturally as an exact solution for the Einstein equations, being a cosmological model where matter is supposed to travel along geodesics normal to horizontal hyperplanes; these, in turn, serve as the initial data for the Cauchy problem associated to those equations (cf. (8), chapter 5).

Scalar perturbations in scalar field quantum cosmology

In this paper it is shown how to obtain the simplest equations for the Mukhanov-Sasaki variables describing quantum linear scalar perturbations in the case of scalar fields without potential term. This was done through the implementation of canonical transformations at the classical level, and unitary transformations at the quantum level, without ever using any classical background equation, and it completes the simplification initiated in investigations by Langlois [D. Langlois, Classical Quantum Gravity 11, 389 (1994).], and Pinho and Pinto-Neto [E. J. C. Pinho and N. Pinto-Neto, Phys. Rev. D 76, 023506 (2007).] for this case. These equations were then used to calculate the spectrum index ${n}_{s}$ of quantum scalar perturbations of a nonsingular inflationary quantum background model, which starts at infinity past from flat space-time with Planckian size spacelike hypersurfaces, and inflates due to a quantum cosmological effect, until it makes an analytical graceful exit from this inflationary epoch to a decelerated classical stiff matter expansion phase. The result is ${n}_{s}=3$, incompatible with observations.

Legendrian Dualities and Spacelike Hypersurfaces in the Lightcone

We show four Legendrian dualities between pseudo-spheres in Minkowski space as a basic theorem. We can apply such dualities for constructing extrinsic differential geometry of spacelike hypersurfaces in pseudo-spheres. In this paper we stick to spacelike hypersurfaces in the lightcone and establish an extrinsic differential geometry which we call the lightcone differential geometry.

Compact Spacelike Hypersurfaces with Constant Mean Curvature in the Antide Sitter Space

We obtain a height estimate concerning to a compact spacelike hypersurfaceΣnimmersed with constant mean curvatureHin the anti-de Sitter spaceℍ1n+1, when its boundary∂Σis contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic spaceℍn. Our estimate depends only on the value ofHand on the geometry of∂Σ.As applications of our estimate, we obtain a characterization of hyperbolic domains ofℍ1n+1and nonexistence results in connection with such types of hypersurfaces.