Study finds necessary conditions for black hole geometries to asymptotically approach Kerr-de Sitter spacetime.
problem Understanding the asymptotic behavior of black hole geometries.
method Used hidden symmetry and conformal geometry technology to find necessary conditions.
result Necessary conditions for black hole geometries to asymptotically approach Kerr-de Sitter spacetime.
Proves wave equation solutions in Kerr-de Sitter spacetime have specific asymptotic expansions.
problem Analyzing solutions to wave equations in Kerr-de Sitter spacetime.
method Developed a Fredholm setup for quasinormal modes and analyzed trapping of lightlike geodesics.
result Proves asymptotic expansions of wave equation solutions up to a decay order.
Proves analyticity of quasinormal modes in Kerr and Kerr-de Sitter spacetimes.
problem Analyticity of quasinormal modes in extreme Kerr and Kerr-de Sitter spacetimes.
method Observation of stable radial point source/sink structure in bicharacteristic flow; recent microlocal analysis result by Galkowski and Zworski.
result Quasinormal modes are real analytic in subextremal Kerr and Kerr-de Sitter spacetimes.
New proof of stability for expanding Kerr-de Sitter spacetimes with smoothness at the boundary.
problem Stability of expanding region of Kerr-de Sitter spacetimes.
method Modified generalized harmonic gauge, local stability near conformal boundary, smoothness down to future conformal boundary.
result Smoothness of conformally rescaled metric down to future conformal boundary with mild singularity.
Stability of black holes proven in full subextremal range with positive cosmological constant.
problem Stability of Kerr-de Sitter black holes in the full subextremal range.
method Similar to previous proof in slowly rotating case, with implementation of constraint damping and verification of subprincipal symbol condition.
result Stability of Kerr-de Sitter black holes proven in the full subextremal range.
Constructs many black hole spacetimes in de Sitter space.
problem Creating well-controlled many black hole spacetimes in de Sitter space.
method Gluing Schwarzschild-de Sitter or Kerr-de Sitter black hole metrics into neighborhoods of points on the future conformal boundary of de Sitter space, under certain balance conditions.
result Solves the Einstein equation directly for the metric, given scattering data at the future conformal boundary.
New methods show quasinormal modes can be defined using various stationary Killing vectors.
problem Proving asymptotic expansions for wave equations in Kerr-de Sitter spacetimes.
method New definition of quasinormal modes using different stationary Killing vectors.
result Horizon Killing vector fields work for analysis, simplifying the problem.
We establish the full global non-linear stability of the Kerr-de Sitter family of black holes, as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta, and without any symmetry assumptions on the initial data. We achieve this by extendin…
A generalisation of the four-dimensional Kerr-de Sitter metrics to include a NUT charge is well known, and is included within a class of metrics obtained by Plebanski. In this paper, we study a related class of Kerr-Taub-NUT-de Sitter metrics in arbitrary dimensions D \ge 6, which contain three non-trivial continuous p…
We give the general Kerr-de Sitter metric in arbitrary spacetime dimension D\ge 4, with the maximal number [(D-1)/2] of independent rotation parameters. We obtain the metric in Kerr-Schild form, where it is written as the sum of a de Sitter metric plus the square of a null geodesic vector, and in generalised Boyer-Lind…
In this global study of solutions to the linear wave equation on Schwarzschild de Sitter spacetimes we attend to the cosmological region of spacetime which is bounded in the past by cosmological horizons and to the future by a spacelike hypersurface at infinity. We prove an energy estimate capturing the expansion of th…
We prove the global non-linear stability, without symmetry assumptions, of slowly rotating charged black holes in de Sitter spacetimes in the context of the initial value problem for the Einstein-Maxwell equations: If one perturbs the initial data of a slowly rotating Kerr-Newman-de Sitter (KNdS) black hole, then in a …
We obtain infinite classes of new Einstein-Sasaki metrics on complete and non-singular manifolds. They arise, after Euclideanisation, from BPS limits of the rotating Kerr-de Sitter black hole metrics. The new Einstein-Sasaki spaces L^{p,q,r} in five dimensions have cohomogeneity 2, and U(1) x U(1) x U(1) isometry group…
This paper is motivated by the non-linear stability problem for the expanding region of Kerr de Sitter cosmologies in the context of Einstein's equations with positive cosmological constant. We show that under dynamically realistic assumptions the conformal Weyl curvature of the spacetime decays towards future null inf…
In this paper we describe a new method for analyzing the Laplacian on asymptotically hyperbolic spaces, which was introduced recently by the author. This new method in particular constructs the analytic continuation of the resolvent for even metrics (in the sense of Guillarmou), and gives high energy estimates in strip…
Constructs solutions of Einstein equations for black holes gluing along timelike geodesics.
problem Constructing solutions of Einstein equations for black holes gluing along timelike geodesics.
method Constructs solutions gε of the Einstein equations describing a mass ε Kerr black hole traveling along a timelike geodesic C. result Constructs true solutions gε of the Einstein equations for black holes gluing along timelike geodesics. 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.
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…
Proves optimal isoperimetric inequality in de Sitter space.
problem Optimal isoperimetric inequality for specific hypersurfaces in de Sitter space.
method Analyzes spacelike, compact, star-shaped, and 2-convex hypersurfaces in de Sitter space.
result Proves an optimal isoperimetric inequality for the specified hypersurfaces.
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.
Estimates for spacelike hypersurfaces in de Sitter space.
problem Isometric embedding problem in de Sitter space.
method Obtained Weyl-type estimates and closedness in the space of embeddings.
result Weyl-type estimates and closedness in the space of embeddings.
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.
Combines non-Euclidean and de Sitter geometries on the plane.
problem Exploring Penrose's Conformal Cyclic Cosmology.
method Geometric model combining Beltrami-Klein and de Sitter spaces.
result Discovery of hidden ${f G}_2$ symmetry in de Sitter spaces.
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. 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 flow on de Sitter space for convex hypersurfaces.
problem Behavior of locally constrained inverse curvature flow in de Sitter space.
method Analyze flow on de Sitter space with specific initial conditions and inequalities.
result Derive Alexandrov-Fenchel type inequalities.
The paper characterizes timelike rectifying curves in De Sitter 3-space.
problem Characterizing timelike rectifying curves in De Sitter 3-space.
method Defining timelike rectifying curves and conical surfaces, providing characterizations and results.
result Characterizations and results of timelike rectifying curves in De Sitter 3-space.
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 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.
Paper proves rigidity for certain spacelike hypersurfaces in de Sitter space.
problem Proving rigidity for hypersurfaces with curvature restrictions.
method Analogue to Guan and Shen's theorem for Riemannian space forms.
result Rigidity theorem for locally isometric hypersurfaces in de Sitter space.
Study on compact biconservative hypersurfaces in de Sitter space.
problem Understanding compact biconservative hypersurfaces in de Sitter space.
method Investigation of hypersurfaces with constant scalar curvature under geometric constraints.
result Extension of rigidity properties of biconservative hypersurfaces.
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.
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.
The paper proves uniqueness of closed timelike geodesics on de-Sitter tori with one singularity.
problem Uniqueness of closed timelike geodesics on de-Sitter tori with one singularity.
method Introduced the notion of timelike marked length spectrum and constructed length-twist coordinates.
result Uniqueness of closed timelike geodesics in their free homotopy class.
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 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…
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 higher-dimensional Kerr-NUT-de Sitter spacetime describes the general rotating asymptotically de Sitter black hole with NUT parameters. It is known that such a spacetime possesses a rank-2 closed conformal Killing-Yano (CKY) tensor as a ``hidden'' symmetry which provides the separation of variables for the geodesic…
The paper proves geometric inequalities for pinched convex hypersurfaces in de Sitter space.
problem Geometric inequalities for convex hypersurfaces in de Sitter space.
method Locally constrained flows with initial compact spacelike hypersurfaces pinched in de Sitter space.
result Established geometric inequalities related to quermassintegrals and weighted curvature integrals.
In this paper, we generalize the defining equation for de Sitter space by replacing the de Sitter radius with a function f satisfying certain conditions; each resulting hypersurface is diffeomorphic to de Sitter space, and has a geometry (and causal character) which is controlled by the choice of f. Necessary and s…
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.
Adds charged black holes to de Sitter space.
problem Gluing charged black holes into de Sitter space.
method Extends Hintz's result to Einstein-Maxwell system with positive cosmological constant, using Reissner-Nordström or Kerr--Newman--de Sitter black holes.
result Determines conditions for gluing black holes into de Sitter space.
Study characterizes conformal boundaries of de Sitter spacetimes.
problem Characterize conformal infinity of asymptotically de Sitter spacetimes.
method Derive constraints relating stress-energy tensor to conformal geometric data using higher conformal fundamental forms.
result Constraints on stress-energy tensor relate to conformal geometric data.
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.