Spacelike Hypersurfaces Research Articles

Height estimates and half-space theorems for spacelike hypersurfaces in generalized Robertson–Walker spacetimes

In this paper, we obtain height estimates for spacelike hypersurfaces Σn of constant k-mean curvature, 1⩽k⩽n, in a generalized Robertson–Walker spacetime −I×ρPn and with boundary contained in a slice {s}×Pn. As an application, we obtain some information on the topology at infinity of complete spacelike hypersurfaces of constant k-mean curvature properly immersed in a spatially closed generalized Robertson–Walker spacetime. Finally, using a version of the Omori–Yau maximum principle for the Laplacian and for more general elliptic trace-type differential operators, some non-existence results are also obtained.

Evolutes of spacelike hypersurfaces in anti-de Sitter space

We study spacelike hypersurfaces in anti-de Sitter space from the view point of the Lagrangian/Legendrian singularity theory.

On the geometry of null hypersurfaces in Minkowski space

The present work is divided into three parts. First we study the null hypersurfaces of the Minkowski space R1n+2, classifying all rotation null hypersurfaces in R1n+2. In the second part we start our analysis of the submanifold geometry of the null hypersurfaces. In the particular case of the (n+1)-dimensional light cone, we characterize its totally umbilical spacelike hypersurfaces, show the existence of non-totally umbilical ones and give a uniqueness result for the minimal spacelike rotation surfaces in the 3-dimensional light cone. In the third and final part we consider an isolated umbilical point on a spacelike surface immersed in the 3-dimensional light cone of R14 and obtain the differential equation of the principal configuration associated to this point, showing that every classical generic Darbouxian principal configuration appears in this context.

Partial generalizations of some conjectures in Lorentzian manifolds

In this paper, we mainly investigate complete or compact spacelike hypersurfaces with constant mean curvature or constant scalar curvature in Lorentzian manifolds L1n+1. We give a new estimate of the Laplacian ΔS of the squared length S of the second fundamental form of such spacelike hypersurfaces. Finally, we give partial generalizations of some Conjectures in Lorentzian manifolds L1n+1.

Conformal Geometry of Hypersurfaces in Lorentz Space Forms

Let be a space-like hypersurface without umbilical points in the Lorentz space form . We define the conformal metric and the conformal second fundamental form on the hypersurface, which determines the hypersurface up to conformal transformation of . We calculate the Euler-Lagrange equations of the volume functional of the hypersurface with respect to the conformal metric, whose critical point is called a Willmore hypersurface, and we give a conformal characteristic of the hypersurfaces with constant mean curvature and constant scalar curvature. Finally, we prove that if the hypersurface with constant mean curvature and constant scalar curvature is Willmore, then is a hypersurface in .

Complete spacelike hypersurfaces in the de Sitter space

In this paper, by modifying Cheng--Yau's technique to complete spacelike hypersurfaces in the de Sitter ($n+1$)-space $S_{1}^{n+1}(1)$, we prove a rigidity under the hypothesis of the mean curvature and the normalized scalar curvature being linearly related. As a corollary, we have the Theorem 1.1 of [3].

PARABOLICITY OF SPACELIKE HYPERSURFACES IN GENERALIZED ROBERTSON–WALKER SPACETIMES: APPLICATIONS TO UNIQUENESS RESULTS

We study non-compact complete spacelike hypersurfaces in generalized Robertson–Walker spacetimes of arbitrary dimension whose fiber is parabolic. Under boundedness assumptions on the warping function restricted on a spacelike hypersurface and on the hyperbolic angle of the hypersurface, we prove that a complete spacelike hypersurface is parabolic if the Riemannian universal covering of the fiber is so. As an application of this new technique, several uniqueness results on complete maximal spacelike hypersurfaces are obtained. Also, the corresponding Calabi–Bernstein problems are solved.

Uniqueness of charged static asymptotically flat black holes in dynamical Chern-Simons gravity

Making use of the conformal positive energy theorem, we prove the uniqueness of four-dimensional static electrically charged black holes being the solution of Chern-Simons modified gravity equations of motion. We assume that black hole spacetime contains an asymptotically flat spacelike hypersurface with compact interior and nondegenerate components of the event horizon.

Curvature estimates of hypersurfaces in the Minkowski space

A class of curvature estimates of spacelike admissible hypersurfaces related to translating solitons of the higher order mean curvature flow in the Minkowski space is obtained, which may offer an idea to study an open question of the existence of hypersurfaces with the prescribed higher mean curvature in the Minkowski space.

Erratum to “Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes”. Math. Proc. Camb. Phil. Soc. (2012), 152, 365–383.

The proof of Corollary 4⋅3 in our paper [1] is not correct because there is a mistake in the expression given for ∥X* ∧ Y*∥2 on page 374. In fact, the correct expression for this term is \begin{eqnarray*} \norm{X^*\wedge Y^*}^2 & = & \norm{X^*}^2\norm{Y^*}^2-\pair{X^*,Y^*}^2\\ {} & = & 1+\pair{X,T}^2+\pair{Y,T}^2\geq 1, \end{eqnarray*} and then the inequality (4⋅9) is no longer true. Observe that all the previous reasoning before the wrong expression for ∥X* ∧ Y*∥2 is correct.

On the rigidity of complete spacelike hypersurfaces immersed in a generalized Robertson-Walker spacetime

We study the problem of uniqueness of complete noncompact spacelike hypersurfaces immersed in a generalized Robertson-Walker spacetime, which is supposed to satisfy the so-called null convergence condition. By extending a technique of S.T. Yau and imposing a suitable restriction on the norm of the gradient of the height function of the hypersurface, we obtain rigidity theorems in such ambient spacetimes. Furthermore, we also establish nonparametric results concerning entire vertical graphs.

Ricci curvature of spacelike hypersurfaces in de Sitter space

In this paper, we study the spacelike hypersurfaces in de Sitter space S1n+1(c), and give some estimates on the Ricci curvature tensor and the square of its length. By these estimates, many properties of spacelike hypersurfaces are derived, that is, the compactness and convexity of spacelike hypersurfaces, the finiteness of fundamental group. We also get an integral inequality on eigenvalue of the Laplacian operator and a sufficient and necessary condition for such hypersurfaces to be totally geodesic.

Scalar field perturbations with arbitrary potentials in quantum backgrounds

In this paper it is shown how to obtain, without ever using the background classical equations of motion, a simple second order Hamiltonian involving the Mukhanov-Sasaki variable describing quantum linear scalar perturbations for the case of scalar fields with arbitrary potentials and arbitrary spacelike hyper-surfaces. It is a generalization of previous works, where the scalar field potential was absent and the spacelike hypersurfaces were flat. This was possible due to the implementation of a new method, together with the Faddeev-Jackiw procedure for the constraint reduction. The resulting Hamiltonian can then be used to study the evolution of quantum cosmological perturbations in quantum backgrounds.

Simple, explicitly time-dependent, and regular solutions of the linearized vacuum Einstein equations in Bondi-Sachs coordinates

Perturbations of the linearized vacuum Einstein equations on a null cone in the Bondi-Sachs formulation of General Relativity can be derived from a single master function with spin weight two, which is related to the Weyl scalar \Psi_0, and which is determined by a simple wave equation. Utilizing a standard spin representation of the tensors on a sphere and two different approaches to solve the master equation, we are able to determine two simple and explicitly time-dependent solutions. Both solutions, of which one is asymptotically flat, comply with the regularity conditions at the vertex of the null cone. For the asymptotically flat solution we calculate the corresponding linearized perturbations, describing all multipoles of spin-2 waves that propagate on a Minkowskian background spacetime. We also analyze the asymptotic behavior of this solution at null infinity using a Penrose compactification, and calculate the Weyl scalar, \Psi_4. Because of its simplicity, the asymptotically flat solution presented here is ideally suited for testbed calculations in the Bondi-Sachs formulation of numerical relativity. It may be considered as a sibling of the well-known Teukolsky-Rinne solutions, on spacelike hypersurfaces, for a metric adapted to null hypersurfaces.

Conditions for low-redshift positive apparent acceleration in smooth inhomogeneous models

It is known that a smooth LTB model cannot have a positive apparent central acceleration. Using a local Taylor expansion method we study the low-redshift conditions to obtain an apparent negative deceleration parameter $q^{app}(z)$ derived from the luminosity distance $D_L(z)$ for a central observer in a LTB space, confirming that central smoothness implies a positive central deceleration. Since observational data is only available at redshift greater than zero we find the critical values of the parameters defining a centrally smooth LTB model which give a positive apparent acceleration at $z>0$, providing a graphical representation of the conditions in the $q_0^{app},q_1^{app}$ plane, which are respectively the zero and first order terms of the central Taylor expansion of $q^{app}(z)$. We finally derive a coordinate independent expression for the apparent deceleration parameter based on the expansion of the relevant functions in red-shift rather than in the radial coordinate. We calculate $q^{app}(z)$ with two different methods to solve the null geodesic equations, one based on a local central expansion of the solution in terms of cosmic time and the other one using the exact analytical solution in terms of generalized conformal time. %The expansion of the solution in terms of cosmic time is quite useful also for other applications requiring foliation %of space-time in space-like hyper-surfaces, such as spatial averaging, which is much more difficult to study using the %analytical solution in terms of the generalized conformal time coordinate.

Quantum cosmological perturbations of generic fluids in quantum universes

In previous works, it was shown that the Lagrangians and Hamiltonians of cosmological linear scalar, vector and tensor perturbations of homogeneous and isotropic space-times with flat spatial sections containing a perfect fluid can be put in a simple form through the implementation of canonical transformations and redefinitions of the lapse function, without ever using the background classical equations of motion. In this paper, we generalize this result to general fluids, which includes entropy perturbations, and to arbitrary spacelike hyper-surfaces through a new method together with the Faddeev-Jackiw procedure for the constraint reduction. A simple second order Hamiltonian involving the Mukhanov-Sasaki variable is obtained, again without ever using the background equations of motion.

Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker spacetimes

A new technique for the study of noncompact complete spacelike hypersurfaces in generalized Robertson–Walker (GRW) spacetimes whose fiber is a parabolic Riemannian manifold is introduced. This class of spacetimes allows us to model open universes which extend to spacelike closed GRW spacetimes from the viewpoint of the geometric analysis of the fiber, and which, unlike those spacetimes, could be compatible with the holographic principle. First, under reasonable assumptions on the restriction of the warping function to the spacelike hypersurface and on the hyperbolic angle between the unit normal vector field and a certain timelike vector field, a complete spacelike hypersurface in a spatially parabolic GRW spacetime is shown to be parabolic, and the existence of a simply connected parabolic spacelike hypersurface in a GRW spacetime also leads to the parabolicity of its fiber. Then, all the complete maximal hypersurfaces in spatially parabolic GRW spacetimes are determined in several cases, extending, in particular, to this family of open cosmological models several well-known uniqueness results for the case of spatially closed GRW spacetimes. Moreover, new Calabi–Bernstein problems are solved.

Photon counting by inertial and accelerated detectors

Bases of exactly localized Minkowski and Rindler states on spacelike hypersurfaces are used to describe inertial and accelerated photon counting devices. It is found that the spacetime coordinates of photons absorbed by a pair of counteraccelerating detectors in causally disconnected Rindler wedges are correlated. If a photon is absorbed by a single accelerated detector the Minkowski vacuum collapses to a state containing at least one photon and that photon can be absorbed by an inertial detector.

Lower eigenvalue estimates of a Dirac-type operator involving electromagnetic field

We establish the lower bounds for the eigenvalues of the Dirac-type operator associated to the Einstein–Maxwell spin connection of 3-dimensional compact (with or without boundary) spacelike hypersurfaces of a (3+1)-dimensional Lorentzian manifold. We also study the limiting cases.

ON NULL CUT POINTS IN TWO-DIMENSIONAL SPACETIMES

Abstract. It is shown that any two-dimensional globally hyperbolicspacetime with noncompact Cauchy surfaces has no null cut points. 1. IntroductionConjugate point and cut point, defined in terms of Jacobi field along ageodesic and Riemannian distance, play important roles in global analysis ofRiemannian manifold. Likewise, in Lorentzian geometry, they play essentialroles in singularity theory of spacetimes. In connection with singularity theory,causality theory developed by Penrose [6], is an essential tool for global analysisof spacetimes. Lorentzian cut point, defined in terms of Lorentzian distancefunction, has a close relation to conjugate points and it is well-known that thereis no null conjugate points in two-dimensional spacetimes ([1], [4], [5], [6]).In contrast to this, there exists a two-dimensional spacetime in which everynull geodesic has cut points which can be easily seen in two-dimensional Ein-stein static universe. In two-dimensional Einstein static universe, we can seethat the spacetime has compact Cauchy surfaces.In this paper, we show that if a two-dimensional Lorentzian manifold hasnoncompact Cauchysurfaces, then there exist no cut points alongnull geodesicsby use of topological property of its Cauchy surface.2. PreliminariesIn this section, we assume that M is an arbitrary dimensional globally hy-perbolic Lorentzian manifold with a noncompact Cauchy surface Σ. By thework of Bernal and Sa´nchez ([2] and [3]), we can assume that Σ is a smooth,spacelike hypersurface. If there exists a future-directed timelike curve from xto y, then we write x ≪ y and we say that y lies in the chronological future