arXiv research

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

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

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

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

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.

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

Classification of specific pseudo-Riemannian manifolds.

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…

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…

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…

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

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.

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…

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

Integrable flows on null curves in anti-de Sitter 3-space studied.

We study spacelike hypersurfaces in anti-De Sitter spacetime that evolve by the Lagrangian angle of their Gauß maps.

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…

Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.

Study examines solutions to Jang equation on anti-de Sitter spacetimes.

Geometric transformations on null curves in AdS induce KdV solutions.

Study on linear independence of Poincaré series for anti-de Sitter 3-manifolds.

Proof of Tholozan's volume formula for anti-de-Sitter 3-manifolds.

Helix surfaces in Anti-de Sitter space maintain constant Gaussian curvature.

This paper proves a positive energy-momentum theorem for oriented Riemannian 3-manifolds that are asymptotic to a standard hyperbolic slice in anti de Sitter space-time. Analogously to the original Witten's proof in the asymptotically flat case, this result relies on spinorial methods. We also give a rigidity theorem: …

Let $S$ be a closed oriented surface of genus at least $2$. Using the parameterisation of the deformation space of globally hyperbolic maximal anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle over the Teichmüller space of $S$, we study the behaviour of these geometric structures along pinching…

The study calculates the index distribution of Brownian loops in various geometrical settings.

In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space $H^{n+1}_1(-1)$. we prove that if a complete space-like hypersurface with constant mean curvature $x:\mathbf M\rightarrow H^{n+1}_1(-1)$ has two distinct principal curvatures $λ,μ$, and inf$|λ-μ|>0$, then $x$ is the sta…

Study domination between non-Fuchsian surface group representations and anti-de Sitter geometry.

Paper extends positive energy theorem to anti-de Sitter spacetimes.

Proof of Thurston's earthquake theorem using Anti-de Sitter geometry.

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…

Paper finds minimum volume for specific anti-de Sitter 3-manifolds.

Study classifies metrics on anti-de Sitter spacetime with specific symmetries.

Anti-de Sitter spacetimes with convex boundaries are uniquely determined by their boundary metrics.

Study on curvature bounds for specific hypersurfaces in Anti-de Sitter space.

Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.

We establish the inequality for Henneaux-Teitelboim's total energy-momentum for asymptotically anti-de Sitter initial data sets which are asymptotic to arbitrary $t$-slice in anti-de Sitter spacetime. In particular, when $t=0$, it generalizes Chruściel-Maerten-Tod's inequality in the center of AdS mass coordinates. We …

The paper constructs convex subsets in anti-de Sitter space with specific metrics on boundaries.

In a recent paper, Q. Mérigot proved that representations in SO(2,n) of uniform lattices of SO(1,n) which are Anosov in the sense of Labourie are quasi-Fuchsian, i.e. are faithfull, discrete, and preserve an acausal subset in the boundary of anti-de Sitter space. In the present paper, we prove the reverse implication. …

Fixed points found in Teichmüller space via anti-de Sitter geometry.

Study of longest arcs and cut loci in deformed anti de-Sitter spaces.

We show that any metric on $S^2$ with Gauss curvature $K \geq -κ$ admits a $C^{1,1}$-isometric embedding into the hyperbolic space with sectional curvature $-κ$. We also give a sufficient condition for a metric on $S^2$ to be isometrically embedded into anti-de Sitter spacetime with the prescribed cosmological time fun…

We study the existence of closed trajectories of a particle model on null curves in anti-de Sitter 3-space defined by a functional which is linear in the curvature of the particle path. Explicit expressions for the trajectories are found and the existence of infinitely many closed trajectories is proved.

In this paper we prove the Penrose inequality for metrics that are small perturbations of the Schwarzschild anti-de Sitter metrics of positive mass. We use the existence of a global foliation by weakly stable constant mean curvature spheres and the monotonicity of the Hawking mass.

The paper connects Weil-Petersson homeomorphisms to maximal surfaces in anti-de Sitter space.

Using symmetric space techniques, we show that closed orbits of the Iwasawa subgroups of $SO(2,l-1)$ naturally define singularities of a black hole causal structure in anti-de Sitter spaces in $l \geq 3$ dimensions. In particular, we recover for $l=3$ the non-rotating massive BTZ black hole. The method presented here i…