Algebraic zero mean curvature hypersurfaces in de Sitter and anti de Sitter spaces

In this paper we provide a family of algebraic space-like surfaces in the three dimensional anti de Sitter space that shows that this Lorentzian manifold admits algebraic maximal examples of any order. Then, we classify all the space-like order two algebraic maximal hypersurfaces in the anti de Sitter N-dimensional space. Finally, we provide two families of examples of Lorentzian order two algebraic zero mean curvature hypersurfaces in the de Sitter space.     

On closed anti de Sitter spacetimes

This article deals with the structure of the fundamental group of compact anti de Sitter spacetimes, i.e. Lorentz manifolds with constant negative curvature. Algebraically such a manifold is the quotient of the universal cover of the homogeneous space by a discrete group acting properly and co-compactly on it. This exists if and only if is even. Indeed, as this was observed by Kulkarni, is contained in , and acts properly transitively on . It then suffices to take as a co-compact lattice in . The results of the present article give evidence to the question: in dimension , are all compact anti de Sitter spacetimes constructed in this way? Received: 18 May 1996 / Revised version: 3 January 1997     

Duality between hyperbolic and de Sitter geometry

We study the trigonometry on the de Sitter surface. Since this surface carries a metric of Lorentzian signature, care has to be taken when defining lengths and angles. We provide trigonometric formulae for triangles of all causality types. This is basically achieved by transferring the concept of polar triangles from spherical geometry into the Minkowski space. As a byproduct, we obtain a new simple proof of the hyperbolic law of cosines for angles.     

Anti de Sitter horospherical flat timelike surfaces

We investigate a special timelike surfaces in Anti de Sitter 3-space.We call such a timelike surface an Anti de Sitter horospherical flat surface which belongs to a class of surfaces given by one parameter families of Anti de Sitter horocycle.We give a generic classification of singularities and study the geometric properties of such surfaces from the viewpoint of Legendrian singularity theory.     

Strichartz Estimates on Asymptotically de Sitter Spaces

In this article we prove a family of local (in time) weighted Strichartz estimates with derivative losses for the Klein–Gordon equation on asymptotically de Sitter spaces and provide a heuristic argument for the non-existence of a global dispersive estimate on these spaces. The weights in the estimates depend on the mass parameter and disappear in the “large mass” regime. We also provide an application of these estimates to establish small-data global existence for a class of semilinear equations on these spaces.     

Approximate representation of the de Sitter group

The purpose of this paper is to develop a theory of approximate representations of the de Sitter group considered as a perturbation of the Poincaré group. This approach simplifies investigation of relativistic effects pertaining to the mechanics in the de Sitter universe. Utility of the approximate approach is manifest if one compares the transformations of the de Sitter group with their approximate representations.     

广义de Sitter空间中的类时超曲面

利用奇点理论研究广义de Sitter空间中的类时超曲面.介绍类时超曲面的局部微分几何,定义了广义de Sitter高斯像及广义de Sitter高度函数,研究广义deSitter高度函数族的性质及广义de Sitter高斯像的几何意义,介绍了一种证明高度函数为Morse族的新方法.最后研究了类时超曲面的通有性质.     

Linear Weingarten spacelike submanifolds in de Sitter space

Let M ⁿ be a spacelike linear Weingarten submanifold in a de Sitter space ${S^{n+p}_{p}(1)}$ with R?=?aH?+?b, where R and H are the normalized scalar curvature and the length of the mean curvature vector respectively. In this paper, we give intrinsic and extrinsic conditions for M ⁿ to be totally umbilical, respectively.     

关于de Sitter空间中伪脐类空子流形

本文研究de Sitter空间中的紧致伪脐类空子流形,得到了这类空子流形的一个积分不等式及其一些刚性定理．     

On space-like hypersurfaces in the de Sitter space

This paper gives the intrinsic conditions for a compact space-like hypersurface in a de Sitter space to be isometric to a sphere.     

Cohomogeneity one de Sitter space S n 1

In this paper we study the cohomogeneity one de Sitter space S1 n. We consider the actions in both proper and non-proper cases. In the first case we characterize the acting groups and orbits and we prove that the orbit space is homeomorphic to R. In the latter case we determine the groups and consequently the orbits in some different cases and prove that the orbit space is not Hausdorff.     

Singularities of Anti de Sitter torus Gauss maps

We study timelike surfaces in Anti de Sitter 3-space as an application of singularity theory. We define two mappings associated to a timelike surface which are called Anti de Sitter nullcone Gauss image and Anti de Sitter torus Gauss map. We also define a family of functions named Anti de Sitter null height function on the timelike surface. We use this family of functions as a basic tool to investigate the geometric meanings of singularities of the Anti de Sitter nullcone Gauss image and the Anti de Sitter torus Gauss map.     

关于de Sitter空间中类空子流形的一些刚性定理

研究了deSitter空间中具有常数量曲率的类空子流形,利用活动标架的方法,证明了这类子流形的某些刚性定理,推广了已有的一些结果．     

Particle Decays and Stability on the de Sitter Universe

We study particle decay in de Sitter space–time as given by first-order perturbation theory in a Lagrangian interacting quantum field theory. We study in detail the adiabatic limit of the perturbative amplitude and compute the “phase space” coefficient exactly in the case of two equal particles produced in the disintegration. We show that for fields with masses above a critical mass m _c there is no such thing as particle stability, so that decays forbidden in flat space–time do occur here. The lifetime of such a particle also turns out to be independent of its velocity when that lifetime is comparable with de Sitter radius. Particles with mass lower than critical have a completely different behavior: the masses of their decay products must obey quantification rules, and their lifetime is zero.     

Weingarten rotation surfaces in 3-dimensional de Sitter space

In the 3-dimensional de Sitter Space , a surface is said to be a spherical (resp. hyperbolic or parabolic) rotation surface, if it is a orbit of a regular curve under the action of the orthogonal transformations of the 4-dimensional Minkowski space which leave a timelike (resp. spacelike or degenerate) plane pointwise fixed. In this paper, we give all spacelike and timelike Weingarten rotation surfaces in .     

Semiclassical analysis of pair creation in de Sitter space

We study a massive scalar field theory in de Sitter space. Using a worldline instanton approach, we calculate the pair creation probability in the weak-field limit. In addition to the exponential factor, we derive the pre-exponential factor. In this method, the pair creation probability for an odd-dimensional space vanishes. We find the leading contribution to the two-point correlator in the φ³ theory.     

Time-like hypersurfaces in de Sitter space and Legendrian singularities

We construct a basic framework for studying the extrinsic differential geometry on time-like hypersurfaces in the de Sitter space from the viewpoint of the theory of Legendrian singularities. As an application, we study the contact of time-like hypersurfaces with flat totally umbilic time-like hypersurfaces in the de Sitter space. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.