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…
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48 results for “anti-de Sitter 3-space”
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.
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.
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.
Lorentz harmonic maps into the hyperbolic plane and spacelike surfaces in anti-de Sitter 3-spacemath.DG
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.
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.
We construct minimal surfaces in hyperbolic and anti-de Sitter 3-space with the topology of a -punctured sphere by loop group factorization methods. The end behavior of the surfaces is based on the asymptotics of Delaunay-type surfaces, i.e., rotational symmetric minimal cylinders. The minimal surfaces in $\mathrm{H…
We describe the space of isometric immersions from the Lorentz plane 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…
Crooked planes are piecewise linear surfaces that were introduced by Drumm in the early 1990s to construct fundamental domains for properly discontinuous actions of free groups on Minkowski 3-space. In a previous paper, we introduced analogues of these surfaces, called AdS crooked planes, in the 3-dimensional anti-de S…
The paper defines analogs of volume and action for curves in flag manifolds.
problem Investigating invariants for curves in flag manifolds.
method Using the correspondence between anti-de Sitter 3-space and (1,1)-conformal metrics, defining analogs of $\cW$-volume, Epstein surfaces, and Liouville action.
result Obtained finite invariants for positive curves in flag manifolds.
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.
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.
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.
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.
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.
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.
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.
The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space and anti-de Sitter 3-space is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula wh…
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.
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.
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
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 -slice in anti-de Sitter spacetime. In particular, when , it generalizes Chruściel-Maerten-Tod's inequality in the center of AdS mass coordinates. We …
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.
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…
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.
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…
Study - symbols linking anti-de Sitter tetrahedra to hyperbolic geometry.
problem Analyzing - symbols for quantum invariants.
method Examining asymptotics and analytic extensions of -symbols.
result Connection between anti-de Sitter tetrahedra and hyperbolic geometry.
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.
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 -2-type in De Sitter and anti De Sitter spaces are either standard products, scrolls, or have non-constant curvature properties.
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 -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 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 -eigenfunctions and stable -eigenvalues.
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…
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.
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.
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.
Anti-de Sitter spacetimes embed cone-metrics as bent Cauchy surfaces.
problem Embedding cone-metrics in anti-de Sitter spacetimes.
method Proving embeddings using Fuchsian representations and GHMC spacetimes.
result Unique embeddings of cone-metrics in GHMC anti-de Sitter spacetimes.
Positive energy theorem for asymptotically anti-de Sitter spacetimes with distributional curvaturegr-qc
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.
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: …
Paper constructs geometric transitions between hyperbolic and Anti-de Sitter geometries.
problem Transitioning between hyperbolic and Anti-de Sitter geometries.
method Construction via Half-pipe geometry on with cone singularities.
result Deformation of convex core structure as bending laminations collapse.
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.
Degeneration of globally hyperbolic maximal anti-de Sitter structures along pinching sequencesmath.DG
Let be a closed oriented surface of genus at least . Using the parameterisation of the deformation space of globally hyperbolic maximal anti-de Sitter structures on by the cotangent bundle over the Teichmüller space of , we study the behaviour of these geometric structures along pinching…
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.
We study geodesics along a noncompact Kerr-Newman instanton, where the asymptotic geometry is either de Sitter or anti-de Sitter. We use first integrals for the Hamilton-Jacobi equation to characterize trajectories both near and away from horizons. We study the interaction of geodesics with special features of the metr…