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.
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.
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.
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 …
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.
The paper classifies actions on anti de Sitter spacetimes up to orbit equivalence.
problem Classifying cohomogeneity one actions on anti de Sitter spacetimes.
method Classification of actions by connected closed Lie subgroups of U(1,n) on AdS2n+1. result Classification up to orbit equivalence of actions by Lie subgroups of U(1,n) on AdS2n+1. The positive energy theorem is proven for certain spacetimes with irregular curvature.
problem Proving the positive energy theorem for spacetimes with irregular curvature.
method Weak asymptotically anti-de Sitter initial data sets with distributional curvature under weak dominant energy condition.
result Positive energy theorem established for weakly irregular spacetimes.
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.
We define the total energy-momenta for (4+1)-dimensional asymptotically anti-de Sitter spacetimes, and prove the positive energy theorem for such spacetimes.
Defines quasi-local energy for dS/AdS spacetimes.
problem Conservation of energy in curved spacetimes with cosmological constant.
method Optimal isometric embedding and Killing fields transplantation.
result Generalizes quasi-local energy definition for dS/AdS.
Study of quasilocal mass using isometric embedding in various spacetimes.
problem Understanding quasilocal mass in different spacetimes.
method Application of isometric embedding theory to quasilocal mass.
result Recent progress in quasilocal mass calculations with specific spacetimes.
Paper proves rigidity of certain manifold immersions into specific spacetimes.
problem Rigidity of isometric immersions into light cones.
method Analyzes Riemannian manifolds of dimension n-1 into light cones of n+1 spacetimes.
result Shows rigidity of isometric immersions for n ≥ 3.
We study spacelike hypersurfaces in anti-De Sitter spacetime that evolve by the Lagrangian angle of their Gauß maps.
The paper defines mass for certain spacetimes with positive cosmological constant.
problem Defining mass for specific spacetimes with positive cosmological constant.
method Proposes and discusses a mass definition for compact static metrics with positive cosmological constant.
result Characterizes de Sitter solution as the only static vacuum metric with zero mass.
This paper continues the investigation of constant mean curvature (CMC) time functions in maximal globally hyperbolic spatially compact spacetimes of constant sectional curvature, which was started in math.DG/0604486. In that paper, the case of flat spacetimes was considered, and in the present paper, the remaining cas…
Study reveals how boundary wave operator determines metric properties on anti-de Sitter spacetimes.
problem Determining metric properties on anti-de Sitter spacetimes.
method Analysis of Klein-Gordon equation and Dirichlet-to-Neumann map.
result Determines the Taylor series of the bulk metric at the boundary.
Study of quasi-local energy limit near anti de-Sitter space for spacetimes with negative cosmological constant.
problem Evaluate the quasi-local energy near anti de-Sitter space for spacetimes with negative cosmological constant.
method Introduced a new quasi-local energy for spacetimes with a negative cosmological constant. Studied the small sphere limit using a canonical family of surfaces and solved the optimal embedding equation.
result The limit of the quasi-local energy recovers the stress-energy tensor of the matter field at a point in the spacetime.
Research proves unique continuation for Einstein-vacuum equations on aAdS spacetimes.
problem Establishing rigorous mathematical statements for AdS/CFT correspondence.
method Novel Carleman estimates and unique continuation results for wave equations on aAdS spacetimes.
result Proved a unique continuation result for the Einstein-vacuum equations from aAdS conformal boundaries.
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…
Survey on extending Kleinian group theory to Lorentzian anti-de Sitter space.
problem Applying classical Kleinian group theory to Lorentzian anti-de Sitter space fails due to improper discontinuity and dependence of accumulation points.
method Introducing causality notions to extend limit sets and regularity domains to achronal subgroups.
result The theory of limit sets and regularity domains extends naturally to achronal subgroups in globally hyperbolic spacetimes.
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. …
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 deduce some rigidity results in warped product spaces under normal variations of CMC hypersurfaces. In particular, we prove the existence of one-parameter families locally rigid on the spatial fiber of Anti-de Sitter Schwarzschild spacetime and one-parameter families with bifurcation points on the spa…
The paper extends foliation by constant mean curvature surfaces to spacetimes with particles.
problem Extending foliation to spacetimes with cone singularities.
method Analyzes globally hyperbolic spacetimes with cone singularities.
result Unique foliation by constant mean curvature surfaces in spacetimes with particles.
Proves uniqueness of certain spacetime solutions with extremal horizons.
problem Proving uniqueness of extremal Schwarzschild de Sitter spacetime solutions.
method Analytic proof in four and higher dimensions, spectral problem for hyperbolic surfaces.
result Proves extremal Schwarzschild de Sitter solutions are unique up to identifications.
We discuss several aspects of the relation between asymptotically AdS and asymptotically dS spacetimes including: the continuation between these types of spaces, the global stability of asymptotically dS spaces and the structure of limits within this class, holographic renormalization, and the maximal mass conjecture o…
Unified description of tetrahedra in various spacetimes.
problem Characterizing tetrahedra with lightlike faces in different spacetimes.
method Using a generalized cross-ratio and edge lengths/dihedral angles to describe tetrahedra and their duals.
result Generalized ideal tetrahedra are the duals of tetrahedra with lightlike faces.
The paper proves a rigidity theorem for null geodesic travel times in asymptotically AdS spacetimes.
problem Determining if a spacetime is conformally AdS based on null geodesic travel times.
method Analyzing all null geodesics from a point to its antipodal point, considering various spacetime conditions.
result The spacetime is conformally AdS if and only if all null geodesics from a point refocus at its antipodal point.
Study lightlike hypersurfaces in anti-de Sitter space.
problem Characterize singularities of lightlike hypersurfaces.
method Investigate lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space.
result Characterize singularities of lightlike hypersurfaces.
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.
Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.
problem Investigating singularities of evolutes and focal surfaces of pseudo-spherical framed immersions.
method Introduced pseudo-spherical non-null framed curves, defined moving frames, and analyzed evolutes and focal surfaces.
result Evolutes of pseudo-spherical framed immersions are the sets of singular points of their focal surfaces.
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 geodesics on Kerr-Newman instantons with de Sitter or anti-de Sitter asymptotics.
problem Characterize geodesics on Kerr-Newman instantons with specific asymptotic geometries.
method Use first integrals for the Hamilton-Jacobi equation to characterize trajectories.
result Characterized geodesics, including stable equilibrium orbits, in Kerr-Newman instantons with de Sitter or anti-de Sitter asymptotics.
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.
Proves properties of maximal hypersurfaces in specific spacetimes.
problem Maximal hypersurfaces in asymptotically AdS spacetimes.
method Uniqueness, existence, and regularity results via mathematical proofs.
result Proves uniqueness, existence, and regularity of maximal hypersurfaces.
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.
Let Gamma be a cocompact lattice in SO(1,n). A representation rho: Gamma \to SO(2,n) is quasi-Fuchsian if it is faithfull, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space - a particular case is the case of Fuchsian representations, ie. composition of the inclusions of Gamma in SO(1,n) …
The study explores unique spacetimes with boundary conditions in relativity.
problem Cosmic censorship and initial value problem in general relativity.
method Global hyperbolic spacetimes with conformal boundaries were analyzed using the vacuum Einstein equations.
result Uniqueness results for globally hyperbolic spacetimes with spacelike conformal boundaries were established.
The paper defines a new coordinate system for anti-de Sitter structures.
problem Understanding the geometry of anti-de Sitter spaces.
method Combining recent work on anti-de Sitter structures with classical theory.
result Introduced a coordinate system resembling Fenchel-Nielsen coordinates.
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.
Inverse curvature flow studied in anti-de Sitter-Schwarzschild manifold.
problem Analyzing curvature flow in a specific spacetime geometry.
method Inverse hessian quotient curvature flow with star-shaped initial hypersurface.
result The solution exists for all time and converges exponentially fast to the identity.
The paper analyzes Gauss maps of Lorentzian surfaces in anti-de Sitter space.
problem Determining the type numbers of pseudo-hyperbolic Gauss maps of Lorentzian surfaces.
method Investigates the type numbers of pseudo-hyperbolic Gauss maps of oriented Lorentzian surfaces with constant curvatures in anti-de Sitter space.
result Investigates the behavior of type numbers of pseudo-hyperbolic Gauss maps along parallel families of surfaces.
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.
Procedure maps quantum systems to curved spacetimes with resonant frequencies.
problem Mapping quantum mechanics to curved spacetimes with resonant frequencies.
method Klein-Gordonization procedure, reducing to nonlinear elliptic equation.
result Large family of spacetimes with resonant spectra for massless wave equations.
The paper generalizes metrics on hyperbolic and anti-de Sitter spaces with curved boundaries.
problem Determining metrics on the boundary of convex sets in hyperbolic and anti-de Sitter spaces.
method Analyzing metrics on the boundary of convex subsets with specific curvature and boundary conditions.
result Every quasisymmetric map can be realized as the gluing map at infinity for a quasicircle boundary.
De Sitter spacetime can be separated into two parts along two kinds of hypersurfaces and the half-de Sitter spacetimes are covered by the planar and hyperbolic coordinates respectively. Two positive energy theorems were proved previously for certain ¶-asymptotically de Sitter and $\H$-asymptotically de Sitter initial…
Study pinching sequences to understand degeneration of anti-de Sitter structures.
problem Understanding the degeneration of anti-de Sitter structures along pinching sequences.
method Parameterization of deformation space and analysis of pinching sequences.
result Regular anti-de Sitter structures appear as limiting points.