Spacelike Hypersurfaces Research Articles

An application of maximum principle to space-like hypersurfaces with constant mean curvature in anti-de Sitter space

In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space \({H^{n+1}_1(-1)}\). we prove that if a complete space-like hypersurface with constant mean curvature \({x:\mathbf M\rightarrow H^{n+1}_1(-1) }\) has two distinct principal curvatures λ, μ, and inf|λ − μ| > 0, then x is the standard embedding \({ H^{m} (-\frac{1}{r^2})\times H^{n-m} ( -\frac{1}{1 - r^2} )}\) in anti-de Sitter space \({ H^{n+1}_1 (-1) }\).

Quantizing Hořava-Lifshitz gravity via causal dynamical triangulations

We extend the discrete Regge action of causal dynamical triangulations to include discrete versions of the curvature squared terms appearing in the continuum action of $(2+1)$-dimensional projectable Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz gravity. Focusing on an ensemble of spacetimes whose spacelike hypersurfaces are two-spheres, we employ Markov chain Monte Carlo simulations to study the path integral defined by this extended discrete action. We demonstrate the existence of known and novel macroscopic phases of spacetime geometry, and we present preliminary evidence for the consistency of these phases with solutions to the equations of motion of classical Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz gravity. Apparently, the phase diagram contains a phase transition between a time-dependent de Sitter-like phase and a time-independent phase. We speculate that this phase transition may be understood in terms of deconfinement of the global gravitational Hamiltonian integrated over a spatial two-sphere.

Spacelike hypersurfaces with negative total energy in de Sitter spacetime

De Sitter spacetime can be separated into two parts along two kinds of hypersurfaces and the half-de Sitter spacetimes are covered by the planar and hyperbolic coordinates, respectively. Two positive energy theorems were proved previously for certain \documentclass[12pt]{minimal}\begin{document}$\mathcal P$\end{document}P-asymptotically de Sitter and \documentclass[12pt]{minimal}\begin{document}$\mathcal H$\end{document}H-asymptotically de Sitter initial data sets by the second author and collaborators. These initial data sets are asymptotic to time slices of the two kinds of half-de Sitter spacetimes, respectively, and their mean curvatures are bounded from above by certain constants. While the mean curvatures violate these conditions, the spacelike hypersurfaces with negative total energy in the two kinds of half-de Sitter spacetimes are constructed in this short paper.

Space-like surfaces with free boundary in the Lorentz–Minkowski space

We investigate a variational problem in the Lorentz–Minkowski space whose critical points are space-like surfaces with a constant mean curvature and making a constant contact angle with a given support surface along its common boundary. We show that if the support surface is a pseudosphere, then the surface is a planar disc or a hyperbolic cap. We also study the problem of space-like hypersurfaces with free boundary in the higher dimensional Lorentz–Minkowski space .

Compact space-like hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

A new class of $(n+1)$-dimensional Lorentz spaces of index $1$ is introduced which satisfies some geometric conditions and can be regarded as a generalization of Lorentz space form. Then, the compact space-like hypersurface with constant scalar curvature of this spaces is investigated and a gap theorem for the hypersurface is obtained.

On the geometry of maximal spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime

In this paper, we establish new characterizations of totally geodesic spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime, which is supposed to obey the null convergence condition. As applications, we get nonparametric results concerning to entire maximal vertical graphs in a such ambient spacetime. Proceeding, we obtain a lower estimate of the index of relative nullity of complete r-maximal spacelike hypersurfaces immersed in Robertson–Walker spacetimes of constant sectional curvature. In particular, we prove a sort of weak extension of the classical Calabi–Bernstein theorem.

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.

Optimal choices of reference for a quasilocal energy: Spherically symmetric spacetimes

For a given timelike displacement vector, the covariant Hamiltonian quasilocal energy expression requires a proper choice of reference spacetime. We propose a program for determining the reference by embedding a neighborhood of the two-sphere boundary in the dynamic spacetime into a Minkowski reference, so that the two-sphere is embedded isometrically, and then extremizing the energy to determine the embedding variables. Applying this idea to Schwarzschild spacetime, we found that for each given future timelike displacement vector our program gives a unique energy value. The static observer measures the maximal energy. Applied to the Friedmann-Lema\^{\i}tre-Robertson-Walker spacetime, we find that the maximum energy value is non-negative; the associated displacement vector is the unit dual mean curvature vector; and the expansion of the two-sphere boundary matches that of its reference image. For these spherically symmetric cases the reference determined by our program is equivalent to isometrically matching the geometry at the two-sphere boundary and taking the displacement vector to be orthogonal to the spacelike constant coordinate time hypersurface, like the timelike Killing vector of the Minkowski reference.

Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes

AbstractIn this paper we analyse the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing some previous results given in [2]. We next extend these results to the complete noncompact case. In that case, our approach is based on the use of a generalized version of the Omori–Yau maximum principle for trace type differential operators, recently given by the authors in [3].

R-stable spacelike hypersurfaces in conformally stationary spacetimes

r-stable spacelike hypersurfaces in conformally stationary spacetimes

Expanding Lorentzian Wormholes in R2 Gravity

We present traversable, Lorentzian wormholes in the framework of R 2 gravity. These wormholes are obtained by assuming constant Ricci scalar and trace-less equation of state. The metric is asymptotically RW, and locally that of a Lorentzian wormhole. Dynamics of the cosmological background, as well as the expanding wormhole are discussed. Weak energy condition near the wormhole throat is also examined. It is shown that the positive energy density condition can be satisfied all-over a spacelike hypersurface at large enough cosmological times and small enough, positive cosmological constant. Subject Index: 450, 453

Extensions of the mandala of Legendrian dualities for pseudo-spheres in Lorentz–Minkowski space

In this paper we define one-parameter families of Legendrian double fibrations in the products of pseudo-spheres in Lorentz–Minkowski space which are the extensions of four Legendrian double fibrations in the previous research (Izumiya, 2009 [9]). We show that these are contact diffeomorphic to each other. Moreover, we construct one-parameter families of new extrinsic differential geometries on spacelike hypersurfaces in these pseudo-spheres as applications of such extensions of the Legendrian double fibrations.

Rigidity theorem for complete spacelike hypersurfaces with constant scalar curvature in Lorentz spaces

In this paper, a complete space-like hypersurface with constant normal scalar curvature in a locally symmetric Lorentz space is discussed. The rigidity theorem is proved by using the operator □ introduced by S. Y. Cheng and S. T. Yau, and the result is a partially affirmative answer to the question posed by Haizhong Li in 1997.

Witten spinors on maximal, conformally flat hypersurfaces

The boundary conditions that exclude zeros of the solutions of the Witten equation (and hence guarantee the existence of a 3-frame satisfying the so-called special orthonormal frame gauge conditions) are investigated. We determine the general form of the conformally invariant boundary conditions for the Witten equation, and find the boundary conditions that characterize the constant and the conformally constant spinor fields among the solutions of the Witten equations on compact domains in extrinsically and intrinsically flat, and on maximal, intrinsically globally conformally flat spacelike hypersurfaces, respectively. We also provide a number of exact solutions of the Witten equation with various boundary conditions (both at infinity and on inner or outer boundaries) that single out nowhere vanishing spinor fields on the flat, non-extreme Reissner–Nordström and Brill–Lindquist data sets. Our examples show that there is an interplay between the boundary conditions, the global topology of the hypersurface and the existence/non-existence of zeros of the solutions of the Witten equation.

On spacelike hypersurfaces with constant scalar curvature in the anti-de Sitter space

In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space H 1 n + 1 ( − 1 ) ( n ⩾ 3 ) with constant scalar curvature and with two principal curvatures. Moreover, we prove that if M n is a complete spacelike hypersurface with constant scalar curvature n ( n − 1 ) R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n − 1 , then R < ( n − 2 ) c / n . Additionally, we also obtain several rigidity theorems for such hypersurfaces.

Intrinsic and extrinsic geometries of a tidally deformed black hole

A description of the event horizon of a perturbed Schwarzschild black hole is provided in terms of the intrinsic and extrinsic geometries of the null hypersurface. This description relies on a Gauss–Codazzi theory of null hypersurfaces embedded in spacetime, which extends the standard theory of spacelike and timelike hypersurfaces involving the first and second fundamental forms. We show that the intrinsic geometry of the event horizon is invariant under a reparameterization of the null generators, and that the extrinsic geometry depends on the parameterization. Stated differently, we show that while the extrinsic geometry depends on the choice of gauge, the intrinsic geometry is gauge invariant. We apply the formalism to solutions to the vacuum field equations that describe a tidally deformed black hole. In a first instance, we consider a slowly varying, quadrupolar tidal field imposed on the black hole, and in a second instance, we examine the tide raised during a close parabolic encounter between the black hole and a small orbiting body.

Complete spacelike hypersurfaces in a Robertson–Walker spacetime

AbstractIn this paper, as a suitable application of the well-known generalized maximum principle of Omori–Yau, we obtain uniqueness results concerning to complete spacelike hypersurfaces with constant mean curvature immersed in a Robertson–Walker (RW) spacetime. As an application of such uniqueness results for the case of vertical graphs in a RW spacetime, we also get non-parametric rigidity results.

Manifestly gauge invariant theory of the nonlinear cosmological perturbations in the leading order of the gradient expansion

In the full nonlinear cosmological perturbation theory in the leading order of the gradient expansion, all the types of the gauge invariant perturbation variables are defined. The metric junction conditions across the spacelike transition hypersurface are formulated in a manifestly gauge invariant manner. It is manifestly shown that all the physical laws such as the evolution equations, the constraint equations, and the junction conditions can be written using the gauge invariant variables which we defined only. Based on the existence of the universal adiabatic growing mode in the nonlinear perturbation theory and the $\ensuremath{\rho}$ philosophy where the physical evolution are described using the energy density $\ensuremath{\rho}$ as the evolution parameter, we give the definitions of the adiabatic perturbation variable and the entropic perturbation variables in the full nonlinear perturbation theory. In order to give the analytic order estimate of the nonlinear parameter ${f}_{\mathrm{NL}}$, we present the exponent evaluation method. As the models where ${f}_{\mathrm{NL}}$ changes continuously and becomes large, using the $\ensuremath{\rho}$ philosophy, we investigate the non-Gaussianity induced by the entropic perturbation of the component which does not govern the cosmic energy density, and we show that in order to obtain the significant non-Gaussianity it is necessary that the scalar field which supports the entropic perturbation is extremely small compared with the scalar field which supports the adiabatic perturbation.

Slant geometry of spacelike hypersurfaces in the lightcone

In this paper, we consider one-parameter families of new extrinsic differential geometries on spacelike hypersurfaces in the lightcone. These geometries are constructed by applying two of one-parameter families of Legendrian dualities between pseudo-spheres in Lorentz-Minkowski space. These Legendrian dualities have been recently given as a part of extensions of the mandala of Legendrian dualities in the previous research of the authors.

On the geometry of conformally stationary Lorentz spaces

We study several aspects of the geometry of conformally stationary Lorentz manifolds, and particularly of GRW spaces, due to the presence of a closed conformal vector field. More precisely, we begin by extending a result of J. Simons on the minimality of cones in Euclidean space to these spaces, and apply it to the construction of complete, noncompact minimal Lorentz submanifolds of both de Sitter and anti-de Sitter spaces. Then we state and prove very general Bernstein-type theorems for spacelike hypersurfaces in conformally stationary Lorentz manifolds, one of which not assuming the hypersurface to be of constant mean curvature. Finally, we study the strong r-stability of spacelike hypersurfaces of constant r-th mean curvature in a conformally stationary Lorentz manifold of constant sectional curvature, extending previous results in the current literature.