arXiv research

The paper revisits Markowitz's pseudodistance on pseudo-Riemannian manifolds.

In this paper, under natural geometric and physical assumptions we provide new uniqueness and non-existence results for complete maximal hypersurfaces in spatially open Robertson-Walker spacetimes whose fiber is flat. Moreover, our results are applied to relevant spacetimes as the steady state spacetime, Einstein-de Si…

Study finds unique and non-existent constant mean curvature hypersurfaces in specific spacetimes.

Study maximal hypersurfaces in open spacetimes using a maximum principle.

Formula for spacelike submanifolds in warped products.

Estimates for spacelike hypersurfaces in de Sitter space.

Proves optimal isoperimetric inequality in de Sitter space.

The paper characterizes timelike rectifying curves in De Sitter 3-space.

Classification of specific pseudo-Riemannian manifolds.

Study flow on de Sitter space for convex hypersurfaces.

The paper classifies surfaces with constant curvature in 3D De Sitter and anti De Sitter spaces.

Paper proves rigidity for certain spacelike hypersurfaces in de Sitter space.

Formulas derived for operators on forms in anti-de Sitter spaces.

Constructs many black hole spacetimes in de Sitter space.

Combines non-Euclidean and de Sitter geometries on the plane.

Study subelliptic heat kernel on octonionic anti-de Sitter space.

Study on compact biconservative hypersurfaces in de Sitter space.

Solves Dirichlet problem for prescribed scalar curvature in Anti-de Sitter space

The paper proves geometric inequalities for pinched convex hypersurfaces in de Sitter space.

The paper proves inequalities for convex hypersurfaces in spheres and hyperbolic spaces.

In this paper, we generalize the defining equation for de Sitter space by replacing the de Sitter radius with a function $f$ satisfying certain conditions; each resulting hypersurface is diffeomorphic to de Sitter space, and has a geometry (and causal character) which is controlled by the choice of $f$. Necessary and s…

Anti-de Sitter space is the Lorentzian space form with negative curvature. In this paper we consider lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space with general codimension. In particular, we investigate the singularities of lightlike hypersurfaces as an application of the theory of Legend…

Study on constant mean curvature hypersurfaces in Anti-de Sitter space.

It is shown that timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in PSL(2,R) via Bryant type representation formulae. These formulae are used to investigate an explicit one-to-one correspondence, the s…

Characterizes curves for minimal surfaces in de Sitter space.

A new proof of Friedrich's theorem on the existence and stability of asymptotically de Sitter spaces in 3+1 dimensions is given, which extends to all even dimensions. In addition, we characterize the possible limits of spaces which are globally asymptotically de Sitter, to the past and future.

Crooked planes were defined by Drumm to bound fundamental polyhedra in Minkowski space for Margulis spacetimes. They were extended by Frances to closed polyhedral surfaces in the conformal compactification of Minkowski space (Einstein space) which we call crooked surfaces. The conformal model of anti-de Sitter space is…

Adds charged black holes to de Sitter space.

We show existence of constant mean curvature 1 surfaces in both hyperbolic 3-space and de Sitter 3-space with two complete embedded ends and any positive genus up to genus twenty. We also find another such family of surfaces in de Sitter 3-space, but with a different non-embedded end behavior.

We prove existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymp…

The paper solves a conjecture about spacelike hypersurfaces in de Sitter space.

Study null surfaces of pseudo-spherical curves in anti-de Sitter space.

We demonstrate a family of Strichartz estimates for the conformally invariant Klein-Gordon equation on a class of asymptotically de Sitter spaces with C^2 metrics by using well-known local Strichartz estimates and a rescaling argument. This class of metrics includes de Sitter space. We also give an application of the e…

In this paper, we determine the type numbers of the pseudo-hyperbolic Gauss maps of all oriented Lorentzian surfaces of constant mean and Gaussian curvatures and non-diagonalizable shape operator in the $3$-dimensional anti-de Sitter space. Also, we investigate the behavior of type numbers of the pseudo-hyperbolic Gaus…

A geometrical correspondence between maximal surfaces in anti-De Sitter space-time and minimal surfaces in the Riemannian product of the hyperbolic plane and the real line is established. New examples of maximal surfaces in anti-De Sitter space-time are obtained in order to illustrate this correspondence.

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

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 s…

We prove existence of compact spacelike hypersurfaces with prescribed k - curvature in de Sitter space, where the prescription function depends on both space and the tilt function.

Link between Teichmüller and anti de Sitter geometry via length functions.

In this paper we combine our recent work on regular globally hyperbolic maximal anti-de Sitter structures with the classical theory of globally hyperbolic maximal Cauchy-compact anti-de Sitter manifolds in order to define an augmented moduli space. Moreover, we introduce a coordinate system in this space that resembles…

We state several sufficient conditions for compact spacelike surface in the3-dimensional de Sitter space to be totally geodesic or spherical.

We prove pinching estimates for dual flows provided the curvature function used in the inverse flow in de Sitter space is convex.

Study singularities of CMC 1 surfaces in de Sitter space.

Defining Lorentzian Sabban frame of the unit speed time-like curves on de Sitter 2-space $\mathbb{S}^{2}_{1}$ and introducing space-like height function on the unit speed time-like curves on $\mathbb{S}^{2}_{1}$, the invariants of the unit speed time-like curves on $\mathbb{S}^{2}_{1}$ and geometric properties of de Si…

The paper adapts metrics to anti-de Sitter structures, characterizing their degeneracies.

We describe the space of isometric immersions from the Lorentz plane $L^2$ into the 3-dimensional anti-de Sitter space, and solve several open problems of this context raised by M. Dajczer and K. Nomizu in 1981. We also obtain from the above result a description of the space of Lorentzian flat tori isometrically immers…

A correct description of the past and the future event horizons for timelike geodesics in de Sitter space-time of the first kind is given.

In this paper, we study the Gauss map of the surfaces in the de Sitter space-time $\mathbb S^4_1(1)$. First, we prove that a space-like surface lying in the de Sitter space-time has pointwise 1-type Gauss map if and only if it has parallel mean curvature vector. Then, we obtain the complete classification of the quasi-…