Spacelike Hypersurfaces Research Articles

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.

Precise finite speed and uniqueness in the Cauchy problem for symmetrizable hyperbolic systems

Precise finite speed, in the sense of the domain of influence being a subset of the union of influence curves through the support of the initial data, is proved for hyperbolic systems symmetrized by pseudodifferential operators in the spatial variables. From this, uniqueness in the Cauchy problem at spacelike hypersurfaces is derived by a Hölmgren style duality argument. Sharp finite speed is derived from an estimate for propagation in each direction. Propagation in a fixed direction is proved by regularizing the problem in the orthogonal directions. Uniform estimates for the regularized equations are proved using pseudodifferential techniques of Beals-Fefferman type.

A note on time-symmetric hypersurfaces in the Schwarzschild geometry

In this note, we show that any inextensible time-symmetric space-like hypersurface of differentiability class Ck, k ⩾ 2, isometrically embedded in the maximal Schwarzschild geometry must intersect the bifurcation sphere.

Foliation of the Kottler–Schwarzschild–de Sitter spacetime by flat spacelike hypersurfaces

There exist Kruskal like coordinates for the Reissner–Nordstrom (RN) black hole spacetime which are regular at coordinate singularities. Non-existence of such coordinates for the extreme RN black hole spacetime has already been shown. Also the Carter coordinates available for the extreme case are not manifestly regular at the coordinate singularity, therefore, a numerical procedure was developed to obtain free fall geodesics and flat foliation for the extreme RN black hole spacetime. The Kottler–Schwarzschild–de Sitter (KSSdS) spacetime geometry is similar to the RN geometry in the sense that, like the RN case, there exist non-singular coordinates when there are two distinct coordinate singularities. There are no manifestly regular coordinates for the extreme KSSdS case. In this paper foliation of all the cases of the KSSdS spacetime by flat spacelike hypersurfaces is obtained by introducing a non-singular time coordinate.

On spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms

We investigate the spacelike hypersurfaces in Lorentzian space forms N 1 n + 1 ( c ) ( n ⩾ 4 ) with two distinct non-simple principal curvatures without the assumption that the (high order) mean curvature is constant. We prove that any spacelike hypersurface in Lorentzian space forms with two distinct non-simple principal curvatures is locally conformal to the Riemannian product of two constant curved manifolds. We also obtain some characterizations for hyperbolic cylinders in Lorentzian space forms in terms of the trace free part of the second fundamental form.

Maximal surfaces and the universal Teichmüller space

We show that any element of the universal Teichmuller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in AdSn+1, any subset E of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface. In AdS3, if E is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature. As a by-product, we find a simple characterization of quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional projective geometry.

Geometric analysis of Lorentzian distance function on spacelike hypersurfaces

Some analysis on the Lorentzian distance in a spacetime with controlled sectional (or Ricci) curvatures is done. In particular, we focus on the study of the restriction of such distance to a spacelike hypersurface satisfying the Omori-Yau maximum principle. As a consequence, and under appropriate hypotheses on the (sectional or Ricci) curvatures of the ambient spacetime, we obtain sharp estimates for the mean curvature of those hypersurfaces. Moreover, we also give a sufficient condition for its hyperbolicity.

On the [formula omitted]-stability of spacelike hypersurfaces

In this paper we study the strong stability of spacelike hypersurfaces with constant r -th mean curvature in Robertson–Walker spacetimes. In particular, we treat the case in which the ambient spacetime is the de Sitter space.

New characterizations of totally geodesic hypersurfaces in anti-de Sitter space [formula omitted

By extending Yau’s technique, we obtain rigidity results to complete maximal spacelike hypersurfaces in anti-de Sitter space, imposing suitable conditions on both the norm of the second fundamental form and a certain height function naturally attached to the hypersurface. We also characterize complete maximal spacelike graphs satisfying a certain assumption on the gradient of the function which determines the graph.

ON COMPLETE SPACELIKE HYPERSURFACES WITH CONSTANT m-TH MEAN CURVATURE IN AN ANTI-DE SITTER SPACE

We investigate complete spacelike hypersurfaces in an Anti-de Sitter space with constant m-th mean curvature and two distinct principal curvatures. By using Otsuki's idea, we obtain some global classification results. For their application, we obtain some characterizations for hyperbolic cylinders. We prove that the only complete spacelike hypersurfaces in Anti-de Sitter (n + 1)-spaces (n ≥ 3) of constant mean curvature or constant scalar curvature with two distinct principal curvatures λ and μ satisfying inf (λ - μ)2 > 0 are the hyperbolic cylinders. It is a little surprising that the corresponding result does not hold for m-th mean curvature when m > 2. We also obtain some global rigidity results for hyperbolic cylinders in terms of square length of the second fundamental form.

On the scalar curvature of hypersurfaces in spaces with a Killing field

Abstract We consider compact hypersurfaces in an (n + 1)-dimensional either Riemannian or Lorentzian space endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the simpler case when is a product space, allows us to derive some interesting consequences in terms of the scalar curvature of the hypersurface. For instance, when n = 2 and is either the sphere or the real projective plane , we characterize the slices of the trivial totally geodesic foliation as the only compact two-sided surfaces with constant Gaussian curvature in the Riemannian product such that its angle function does not change sign. When n ≥ 3 and is a compact Einstein Riemannian manifold with positive scalar curvature, we also characterize the slices as the only compact two-sided hypersurfaces with constant scalar curvature in the Riemannian product whose angle function does not change sign. Similar results are also established for spacelike hypersurfaces in a Lorentzian product 𝕄 × ℝ1.

On Geometric Problems Related to Brown-York and Liu-Yau Quasilocal Mass

We discuss some geometric problems related to the definitions of quasilocal mass proposed by Brown-York \cite{BYmass1} \cite{BYmass2} and Liu-Yau \cite{LY1} \cite{LY2}. Our discussion consists of three parts. In the first part, we propose a new variational problem on compact manifolds with boundary, which is motivated by the study of Brown-York mass. We prove that critical points of this variation problem are exactly static metrics. In the second part, we derive a derivative formula for the Brown-York mass of a smooth family of closed 2 dimensional surfaces evolving in an ambient three dimensional manifold. As an interesting by-product, we are able to write the ADM mass \cite{ADM61} of an asymptotically flat 3-manifold as the sum of the Brown-York mass of a coordinate sphere $S_r$ and an integral of the scalar curvature plus a geometrically constructed function $\Phi(x)$ in the asymptotic region outside $S_r $. In the third part, we prove that for any closed, spacelike, 2-surface $\Sigma$ in the Minkowski space $\R^{3,1}$ for which the Liu-Yau mass is defined, if $\Sigma$ bounds a compact spacelike hypersurface in $\R^{3,1}$, then the Liu-Yau mass of $\Sigma$ is strictly positive unless $\Sigma$ lies on a hyperplane. We also show that the examples given by \'{O} Murchadha, Szabados and Tod \cite{MST} are special cases of this result.

Spacelike Hypersurfaces with Constant Mean Curvature in the Steady State Space

In this paper we obtain height estimates concerning to a compact spacelike hypersurface $\Sigma^n$ immersed with constant mean curvature $H$ in the Steady State space $\mathcal H^{n+1}$, when its boundary is contained into some hyperplane of this spacetime. As a first application of these results, when $\Sigma^n$ has spherical boundary, we establish relations between its height and the radius of its boundary. Moreover, under a certain restriction on the Gauss map of $\Sigma^n$, we obtain a sharp estimate for $H$. Finally, we also apply our estimates to describe the end of a complete spacelike hypersurface and to get theorems of characterization concerning to spacelike hyperplanes in $\mathcal H^{n+1}$.

SPACELIKE SUBMANIFOLDS IN DE SITTER SPACE

We investigate the differential geometry of spacelike submanifolds of codimension at least two in de Sitter space as an application of the theory of Legendrian singularities. We also discuss related geometric property of spacelike hypersurfaces in de Sitter space. Mathematics Subject Classification (2000): 53A35, 53B30, 58C25.

Complete spacelike hypersurfaces with constant mean curvature in [formula omitted

In this paper, by applying the Omori–Yau generalized maximum principle for complete Riemannian manifolds, we obtain Bernstein-type results concerning complete spacelike hypersurfaces with constant mean curvature immersed in the Lorentzian product space − R × H n .

A gap theorem for complete space-like hypersurface with constant scalar curvature in locally symmetric Lorentz spaces

Let M^n be a complete space-like hypersurface with constant scalar curvature in locally symmetric Lorentz space N^{n+1}_1, S be the squared norm of the second fundamental form of M^n in N^{n+1}_1. In this paper, we obtain a gap property of S: if nP\leq \sup S\leq D(n,P) for some constants P and D(n, P), then either \sup S=nP and M^n is totally umbilical, or \sup S=D(n, P) and M^n has two distinct principal curvatures.

Spacelike submanifolds in de Sitter space

AbstractWe investigate the differential geometry of spacelike submanifolds of codimension at least two in de Sitter space as an application of the theory of Legendrian singularities. We also discuss related geometric property of spacelike hypersurfaces in de Sitter space.

Nonconformally flat initial data for binary compact objects

A new method is described for constructing initial data for a binary neutron-star (BNS) system in quasi-equilibrium circular orbit. Two formulations for non-conformally flat data, waveless (WL) and near-zone helically symmetric (NHS), are introduced; in each formulation, the Einstein-Euler system, written in 3+1 form on an asymptotically flat spacelike hypersurface, is exactly solved for all metric components, including the spatially non-conformally flat potentials, and for irrotational flow. A numerical method applicable to both formulations is explained with an emphasis on the imposition of a spatial gauge condition. Results are shown for solution sequences of irrotational BNS with matter approximated by parametrized equations of state that use a few segments of polytropic equations of state. The binding energy and total angular momentum of solution sequences computed within the conformally flat -- Isenberg-Wilson-Mathews (IWM) -- formulation are closer to those of the third post-Newtonian (3PN) two point particles up to the closest orbits, for the more compact stars, whereas sequences resulting from the WL/NHS formulations deviate from the 3PN curve even more for the sequences with larger compactness. We think it likely that this correction reflects an overestimation in the IWM formulation as well as in the 3PN formula, by $\sim 1$ cycle in the gravitational wave phase during the last several orbits. The work suggests that imposing spatial conformal flatness results in an underestimate of the quadrupole deformation of the components of binary neutron-star systems in the last few orbits prior to merger.

Codimension-2 surfaces and their Hilbert spaces: low-energy clues for holography from general covariance

We argue that the holographic principle may be hinted at already from low-energy considerations, assuming diffeomorphism invariance, quantum mechanics and Minkowski-like causality. We consider the states of finite spacelike hypersurfaces in a diffeomorphism-invariant QFT. A low-energy regularization is assumed. We note a natural dependence of the Hilbert space on a codimension-2 boundary surface. The Hilbert product is defined dynamically, in terms of transition amplitudes which are described by a path integral. We show that a canonical basis is incompatible with these assumptions, which opens the possibility for a smaller Hilbert-space dimension than canonically expected. We argue further that this dimension may decrease with surface area at constant volume, hinting at holographic area proportionality. We draw comparisons with other approaches and setups, and propose an interpretation for the non-holographic space of graviton states at asymptotically Minkowski null infinity.

Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

In this paper, the complete spacelike hypersurface with constant normal scalar curvature in a locally symmetric Lorentz space is discussed. Several classified theorems are obtained using the operator L 1 introduced by S.Y. Cheng and S.T. Yau (1977) [4].