arXiv research

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

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

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…

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…

Classification of specific pseudo-Riemannian manifolds.

Paper constructs geometric transitions between hyperbolic and Anti-de Sitter geometries.

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

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

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

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

We provide the first examples of geometric transition from hyperbolic to anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic 4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a 3-dimensional…

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

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

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

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

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

Study para-hyperKähler geometry of anti-de Sitter structures.

Solves Dirichlet problem for prescribed scalar curvature 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 …

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

Using the parameterisation of the deformation space of GHMC anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle of the Teichmüller space of $S$, we study how some geometric quantities, such as the Lorentzian Hausdorff dimension of the limit set, the width of the convex core and the Hölder exponen…

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

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…

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…

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…

As is well-known for compact Riemann surfaces, eigenvalues of the Laplacianbare distributed discretely and most of eigenvalues vary viewed as functions on the Teichmuller space. We discuss a new feature in the Lorentzian geometry, or more generally, in pseudo-Riemannian geometry. One of the distinguished features is th…

Study $b$-$6j$ symbols linking anti-de Sitter tetrahedra to hyperbolic geometry.

Introduces Anti-de Sitter geometry and its connection to Teichmüller theory.

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

Study of longest arcs and cut loci in deformed 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.

Let $Σ$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti-de Sitter structures on $Σ\times \mathbb{R}$ and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of $Σ$ and as the b…

Let $Σ$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce wild globally hyperbolic anti-de Sitter structures on $Σ\times \mathbb{R}$ and provide two parameterisations of their deformation space: as a quotient of the product of two copies of the Teichmüller space of crowned …

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

Study character varieties of a Coxeter group in hyperbolic and Anti-de Sitter spaces.

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…

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

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

Anti-de Sitter spacetimes embed cone-metrics as bent Cauchy surfaces.

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…

We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.

We define the total energy-momenta for (4+1)-dimensional asymptotically anti-de Sitter spacetimes, and prove the positive energy theorem for such spacetimes.

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

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: …

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

Geometric transformations on null curves in AdS induce KdV solutions.