arXiv research

In this work we study spacelike hypersurfaces immersed in spatially open standard static spacetimes with complete spacelike slices. Under appropriate lower bounds on the Ricci curvature of the spacetime in directions tangent to the slices, we prove that every complete CMC hypersurface having either bounded hyperbolic a…

New rigidity results for submanifolds in GRW spacetimes.

We show that a given $x:M^n\to \bar{M}^{n+1}=I\times_φF^{n}$ closed spacelike hypersurface with constant mean curvature $H$ and warping function $φ$ satisfying φ''\geq \ma\{Hφ'',0\}$is either minimal or a spacelike slice$\{t_{o}\}\times F$for some$t_{o}\in I$.

In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker (GRW) spacetimes. In particular, we consider the following question: Under what conditions must a compact spacelike hypersurface with constant higher order mean curvatur…

We continue work initiated in a 1990 preprint of Mess giving a geometric parameterization of the moduli space of classical solutions to Einstein's equations in 2+1 dimensions with cosmological constant 0 or -1 (the case +1 has been worked out in the interim by the present author). In this paper we make a first step tow…

Study on spacelike submanifolds in generalized Schwarzschild spacetimes with lightlike foliations.

A characterization of the foliation by spacelike slices of an $(n+1)$-dimensional spatially closed Generalized Robertson-Walker spacetime is given by means of studying a natural mean curvature type equation on spacelike graphs. Under some natural assumptions, of physical or geometric nature, all the entire solutions of…

Rigidity results for hypersurfaces in warped spacetimes.

We solve the spacelike, spherically symmetric, constant mean curvature hypersurfaces in the maximally extended Reissner-Nordstrom spacetime with the charge smaller than the mass. Based on these results, we construct constant mean curvature foliations with fixed or varied mean curvature in each slice in this spacetime.

Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weig…

Study of mean curvature flow in de Sitter space, showing convergence to flat slicing.

We construct a $T$-axisymmetric, spacelike, spherically symmetric, constant mean curvature hypersurfaces foliation in the Kruskal extension with properties that the mean curvature varies in each slice and ranges from minus infinity to plus infinity. This family of hypersurfaces extends the CMC foliation discussions pos…

We study the limit of quasilocal mass defined in [4] and [5] for a family of spacelike 2-surfaces in spacetime. In particular, we show the limit coincides with the ADM mass at spatial infinity. The limit for coordinate spheres of a boosted slice of the Schwarzchild solution is computed explicitly and shown to give the …

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 $¶$-asymptotically de Sitter and $\H$-asymptotically de Sitter initial…

In the Friedmann Model of the universe, cosmologists assume that spacelike slices of the universe are Riemannian manifolds of constant sectional curvature. This assumption is justified via Schur's Theorem by stating that the spacelike universe is locally isotropic. Here we define a Riemannian manifold as almost locally…

Study shows nonextendibility of warped spacelike singularities in specific spacetimes.

A time-flat condition on spacelike 2-surfaces in spacetime is considered here. This condition is analogous to constant torsion condition for curves in three dimensional space and has been studied in [2, 4, 5, 12, 13]. In particular, any 2-surface in a static slice of a static spacetime is time-flat. In this article, we…

We prove the mean curvature flow of a spacelike graph in $(Σ_1\times Σ_2, g_1-g_2)$ of a map $f:Σ_1\to Σ_2$ from a closed Riemannian manifold $(Σ_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(Σ_2,g_2)$ with bounded curvature tensor and derivatives, and with sectional curvatures satisfying $K_2\leq K_1$,…

We give general sufficient conditions for the existence of trapped surfaces due to concentration of matter in spherically symmetric initial data sets satisfying the dominant energy condition. These results are novel in that they apply and are meaningful for arbitrary spacelike slices, that is they do not require any au…

The folk questions in Lorentzian Geometry, which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime $(M,g)$ admits a smooth time function $τ$ whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth…

Unified framework for photon and massive particle hypersurfaces in stationary spacetimes.

Cosmological singularity theorems such as that of Hawking and Penrose assume local curvature conditions as well as global ones like the existence of a compact (achronal) slice. Here, we prove a new singularity theorem for chronological spacetimes that satisfy what we call a `past null focusing' condition. Such a condit…

We establish a black hole uniqueness theorem for Schwarzschild-de Sitter spacetime, also called Kottler spacetime, which satisfies Einstein's field equations of general relativity with positive cosmological constant. Our result concerns the class of static vacuum spacetimes with compact spacelike slices and regular max…

Develops weak formulation for spacelike flows in pseudo-Euclidean space.

We consider compact hypersurfaces in an $(n+1)$-dimensional either Riemannian or Lorentzian space $N^{n+1}$ endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the simpler case when $N=M^n\times R$ is a product space, allows us to derive some inte…

A normal field on a spacelike surface in $R^4_1$ is called bi-normal if $K^ν$, the determinant of Weingarten map associated with $ν$, is zero. In this paper we give a relationship between the spacelike pseudo-planar surfaces and spacelike pseudo-umbilical surfaces, then study the bi-normal fields on spacelike ruled sur…

In this paper, we define dual geodesic trihedron(dual Darboux frame) of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim …

Study spacelike loxodromes on helicoidal surfaces in Lorentzian n-space.

New gravitational solitons and infinite topological manifolds found.

The paper studies mean curvature flow of spacelike-convex submanifolds in pseudo-Euclidean space.

In this paper, we give the characterizations of spacelike B2-slant helix by means of curvatures of the spacelike curve in Minkowski 4-space. Furthermore, we give the integral characterization of the spacelike B2-slant helix.

Three explicit families of spacelike Zoll surface admitting a Killing field are provided. It allows to prove the existence of spacelike Zoll surface not smoothly conformal to a cover of de Sitter space as well as the existence of Lorentzian Möbius strips of non constant curvature all of whose spacelike geodesics are cl…

In this paper, we investigate the parametric version and non-parametric version of rigidity theorem of spacelike translating solitons in pseudo-Euclidean space $\mathbb{R}^{m+n}_{n}$. Firstly, we classify $m$-dimensional complete spacelike translating solitons in $\mathbb{R}^{m+n}_{n}$ by affine technique and classical…

In this paper, we study entire spacelike translating solitons in Minkowski space. By constructing convex spacelike solutions to (1.3) in bounded convex domains, we obtain many entire smooth convex strictly spacelike translating solitons by prescribing boundary data at infinity.

In this paper, using the classifications of timelike and spacelike ruled surfaces, we define and study the Mannheim offsets of spacelike ruled surfaces in Minkowski 3-space. We give the conditions for spacelike offset surfaces to be developable.

Given a spacelike 2-surface $Σ$ in a spacetime $N$ and a constant future timelike unit vector $T_0$ in $\R^{3,1}$, we derive upper and lower estimates of Wang-Yau quasilocal energy $E(Σ, X, T_0)$ for a given isometric embedding $X$ of $Σ$ into a flat 3-slice in $\R^{3,1}$. The quantity $E(Σ, X, T_0)$ itself depends …

Estimates for spacelike hypersurfaces in de Sitter space.

In this paper, we investigate the relations between the pitch, the angle of pitch and drall of parallel ruled surface of a closed spacelike curve with a spacelike binormal in dual Lorentzian space.

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 ex…

Proves inequality for maximal spacelike submanifolds in Minkowski space.

The paper proves the existence of specific spacelike hypersurfaces in Minkowski space.

For a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz-Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilic point, which was developed by Choe \cite{Choe}. Using the concept of the rotation index at the interior and bou…

Study of spacelike submanifolds in spherical RW spacetime, proving a Lorentzian Takahashi theorem.

Study proves spacelike self-shrinkers are hyperplanes under certain conditions.

Defines horocyclic evolutes, parallels, and involutes of spacelike frontals in hyperbolic 2-space.

Study finds rigid submanifolds in spacelike waves under specific conditions.

We generalize a Bernstein-type result due to Albujer and Alías, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $Σ_1\times \mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete spacelike graphic submanifold with parallel mean curvature, defined by a map $f: Σ_1\to Σ_2$ …

Proves stability of Minkowski space for specific initial data.