Formulas derived for operators on forms in anti-de Sitter spaces.
problem Operators on forms in anti-de Sitter spaces.
method Explicit formulas for codifferential and Laplace-de Rham operators.
result Formulas for restriction and continuation between spaces.
Study subelliptic heat kernel on octonionic anti-de Sitter space.
problem Heat kernel of octonionic anti-de Sitter space.
method Lift Laplacian of octonionic hyperbolic space and use sub-Laplacian.
result Two integral representations for subelliptic heat kernel.
Link between Teichmüller and anti de Sitter geometry via length functions.
problem Understanding the geometry of Teichmüller space and anti de Sitter manifolds.
method Establishing a connection between Teichmüller space and anti de Sitter geometry through length functions.
result New purely anti de Sitter proofs of Teichmüller theory results.
Classification of specific pseudo-Riemannian manifolds.
problem Classifying conformally flat generalized Ricci recurrent pseudo-Riemannian manifolds.
method Complete classification through mathematical analysis.
result Conformally flat generalized Ricci recurrent pseudo-Riemannian manifolds are either de Sitter or anti-de Sitter spacetimes.
Solves Dirichlet problem for prescribed scalar curvature in Anti-de Sitter space
problem Dirichlet problem for prescribed scalar curvature in Anti-de Sitter space
method Fully non-linear elliptic equation
result Solves if datas are strictly convex
The paper classifies surfaces with constant curvature in 3D De Sitter and anti De Sitter spaces.
problem Classifying surfaces with constant curvature in 3D De Sitter and anti De Sitter spaces.
method Analyzing conditions equivalent to constant principal curvature, mean curvature, and second mean curvature.
result Surfaces of L1-2-type in De Sitter and anti De Sitter spaces are either standard products, scrolls, or have non-constant curvature properties. 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.
problem Characterizing and classifying hypersurfaces in Anti-de Sitter space.
method Analytic foliation and classification of hypersurfaces based on their asymptotic boundary.
result Every admissible sphere is the boundary of a unique hypersurface for any given mean curvature.
Study null surfaces of pseudo-spherical curves in anti-de Sitter space.
problem Characterize null surfaces of pseudo-spherical spacelike framed curves in anti-de Sitter 3-space.
method Introduced nullcone fronts, classified singularities, defined Anti-de Sitter distance-squared functions.
result Relate singularities of nullcone fronts to those of framed curves.
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…
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…
Study examines solutions to Jang equation on anti-de Sitter spacetimes.
problem Existence and properties of solutions to the generalized Jang equation.
method Rigorous analysis in asymptotically anti-de Sitter setting.
result Provides solutions for a broad class of asymptotic conditions.
Proof of Tholozan's volume formula for anti-de-Sitter 3-manifolds.
problem Volume calculation for closed anti-de-Sitter 3-manifolds.
method Chern-Simons theory approach.
result Proof of Tholozan's volume formula.
The paper adapts metrics to anti-de Sitter structures, characterizing their degeneracies.
problem Characterizing degeneracies of metrics on anti-de Sitter structures.
method Adapting Hitchin component metrics to anti-de Sitter structures.
result Characterized degeneracies of the pressure metric and showed the Loftin metric is nowhere degenerate.
Integrable flows on null curves in anti-de Sitter 3-space studied.
problem Analyzing integrable flows on null curves in anti-de Sitter 3-space.
method Formulated integrable flows related to the KdV hierarchy on null curves exploiting the geometry of anti-de Sitter 3-space.
result Explicitly found closed stationary solutions in terms of periodic solutions of a Lamé equation.
We study spacelike hypersurfaces in anti-De Sitter spacetime that evolve by the Lagrangian angle of their Gauß maps.
Study on linear independence of Poincaré series for anti-de Sitter 3-manifolds.
problem Linear independence of generalized Poincaré series for anti-de Sitter 3-manifolds.
method Analysis of eigenfunctions and Laplacian on anti-de Sitter 3-manifolds.
result Unbounded multiplicities of eigenvalues for L2-eigenfunctions and stable L2-eigenvalues. Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.
problem Analyzing the relationship between Lorentz harmonic maps and spacelike surfaces.
method Using loop group techniques, develop DPW-type representations and solve Cauchy problems.
result Establish a correspondence between Lorentz harmonic maps and spacelike immersions, leading to families of surfaces of constant Gauss curvature.
We describe the space of isometric immersions from the Lorentz plane L2 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…
Geometric transformations on null curves in AdS induce KdV solutions.
problem Transforming null curves in AdS to solve KdV equations.
method Geometric transformations that induce Bäcklund transformations for KdV.
result Satisfies permutability theorem for null curves with constant bending.
Helix surfaces in Anti-de Sitter space maintain constant Gaussian curvature.
problem Understanding helix surfaces in Anti-de Sitter space with Berger-like metrics.
method Proved helix surfaces have constant Gaussian curvature and described them explicitly.
result Explicit local description of helix surfaces in terms of isometries and curves.
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: …
Study domination between non-Fuchsian surface group representations and anti-de Sitter geometry.
problem Domination problem between non-Fuchsian representations of closed surface groups.
method Analysis of branched harmonic immersions and construction of anti-de Sitter 3-manifolds.
result Found that representations admitting branched harmonic immersions dominate other representations, and constructed large families of branched anti-de Sitter 3-manifolds.
Paper extends positive energy theorem to anti-de Sitter spacetimes.
problem Proving positive energy theorem for weighted anti-de Sitter spacetimes.
method Generalized positive energy theorem for 3D anti-de Sitter initial data sets.
result Positive energy theorem proved for weighted anti-de Sitter spacetimes.
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×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.
problem Calculating the distribution of the index of Brownian loops in specific geometrical settings.
method Analysis based on the geometry of Hopf and anti-de Sitter fibrations, and the relationship between winding and area forms.
result Explicit formulas and asymptotics for the distribution of the index of the Brownian loop.
Paper finds minimum volume for specific anti-de Sitter 3-manifolds.
problem Finding the minimum volume of globally hyperbolic anti-de Sitter 3-manifolds.
method Analyzes maximal globally hyperbolic Cauchy-compact anti-de Sitter 3-manifolds.
result Minimum volume is π²|χ(M)|, attained for Fuchsian manifolds.
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…
Study classifies metrics on anti-de Sitter spacetime with specific symmetries.
problem Classifying metrics with specific symmetries on anti-de Sitter spacetime.
method Used classification techniques for pseudo-Riemannian and almost contact metric structures.
result Obtained classifications of homogeneous structures on anti-de Sitter spacetime.
Proof of Thurston's earthquake theorem using Anti-de Sitter geometry.
problem Proving Thurston's earthquake theorem for orientation-preserving homeomorphisms.
method Using the bi-invariant geometry of Anti-de Sitter three-space.
result Provided a proof of Thurston's earthquake theorem.
Anti-de Sitter spacetimes with convex boundaries are uniquely determined by their boundary metrics.
problem Identifying anti-de Sitter spacetimes based on boundary properties.
method Proving rigidity for spacetimes with holonomy close to Fuchsian and convex boundaries.
result Globally hyperbolic compact anti-de Sitter spacetimes are determined by their boundary metrics.
In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space H1n+1(−1). we prove that if a complete space-like hypersurface with constant mean curvature x:M→H1n+1(−1) has two distinct principal curvatures λ,μ, and inf∣λ−μ∣>0, then x is the sta…
Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.
problem Understanding extremal trajectories and reachable set on anti-de Sitter plane.
method Geometric control theory and differential geometry.
result Construction of optimal synthesis and description of Lorentzian distance.
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 …
Proves metrics of curvature ≤ -1 on surfaces are isometric to space-like convex surfaces in anti-de Sitter space.
problem Proving metrics of curvature ≤ -1 on closed surfaces.
method Approximation using isometric immersion of smooth metrics by Labourie--Schlenker.
result Any metric with curvature ≤ -1 on a closed surface is isometric to a space-like convex surface in anti-de Sitter space.
Study on curvature bounds for specific hypersurfaces in Anti-de Sitter space.
problem Bounding principal curvatures of constant mean curvature hypersurfaces.
method Generalized convex hull concept and quantitative estimates based on width.
result Explicit bounds on sectional curvature and quasiconformal dilatation.
The paper constructs convex subsets in anti-de Sitter space with specific metrics on boundaries.
problem Creating convex subsets with prescribed metrics on boundaries in anti-de Sitter space.
method Using quasi-symmetric maps and properties of hyperbolic metrics, the paper constructs convex subsets with specific metrics on boundaries.
result Existence of globally hyperbolic convex subsets with prescribed metrics on boundaries.
Study b-6j symbols linking anti-de Sitter tetrahedra to hyperbolic geometry.
problem Analyzing b-6j symbols for quantum invariants. method Examining asymptotics and analytic extensions of 6j-symbols. result Connection between anti-de Sitter tetrahedra and hyperbolic geometry.
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.
problem Finding fixed points in Teichmüller space using earthquakes.
method Left earthquakes along measured laminations, using anti-de Sitter geometry.
result Composition of left earthquakes has a fixed point.
Study of longest arcs and cut loci in deformed anti de-Sitter spaces.
problem Existence and properties of time-like cycles in deformed Lorentzian manifolds.
method Analysis of universal covering, admissible curves, and Lorentzian geodesics.
result Identification of cut time and cut locus in deformed anti de-Sitter spaces.
Introduces Anti-de Sitter geometry and its connection to Teichmüller theory.
problem None explicitly stated, focuses on introducing concepts.
method Broad introduction to Anti-de Sitter geometry, focusing on dimension three and Teichmüller theory.
result Classification of maximal globally hyperbolic Cauchy compact manifolds and construction of the Gauss map.
We show that any metric on S2 with Gauss curvature K≥−κ admits a C1,1-isometric embedding into the hyperbolic space with sectional curvature −κ. We also give a sufficient condition for a metric on S2 to be isometrically embedded into anti-de Sitter spacetime with the prescribed cosmological time fun…
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.