Closed trajectories of a particle model on null curves in anti-de Sitter 3-space

Uniqueness of de Sitter space

All inextendible null geodesics in four dimensional de Sitter space dS^4 are complete and globally achronal. This achronality is related to the fact that all observer horizons in dS^4 are eternal, i.e. extend from future infinity scri^+ all the way back to past infinity scri^-. We show that the property of having a null line (inextendible achronal null geodesic) that extends from scri^- to scri^+ characterizes dS^4 among all globally hyperbolic and asymptotically de Sitter spacetimes satisfying the vacuum Einstein equations with positive cosmological constant. This result is then further exten…

The horizontal heat kernel on the quaternionic anti de-Sitter spaces and related twistor spaces

The geometry of the quaternionic anti-de Sitter fibration is studied in details. As a consequence, we obtain formulas for the horizontal Laplacian and subelliptic heat kernel of the fibration. The heat kernel formula is explicit enough to derive small time asymptotics. Related twistor spaces and corresponding heat kernels are also discussed and the connection to the quaternionic magnetic Laplacian is done.

Evolution of spacelike surfaces in anti-De Sitter space by their Lagrangian angle

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

Lightlike hypersurfaces along spacelike submanifolds 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 Legendrian singularities.

A geometrical correspondence between maximal surfaces in anti-De Sitter space-time and minimal surfaces in $\mathbb{H}^2\times \mathbb{R}$

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.

Positive energy theorem for asymptotically anti-de Sitter spacetimes with distributional curvature

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

关于闭反德西特3-流形的Tholozan体积公式

On Tholozan's volume formula for closed anti-de-Sitter 3-manifolds

本文利用Chern-Simons理论为闭反德西特3-流形的Tholozan体积公式提供了一种新的证明。

Pseudo-hyperbolic Gauss maps of Lorentzian surfaces in anti-de Sitter space

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 Gauss map along the parallel family of such oriented Lorentzian surfaces in the $3$-dimensional anti-de Sitter space. Furthermore, we investigate the type number of the pseudo-hyperbolic Gauss map of one of Lorentzian hypersurfaces of B-scroll type in a general dimensional anti-de Si…

Positive energy theorem for (4+1)-dimensional asymptotically 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.

Positive energy-momentum theorem in asymptotically anti de Sitter space-times

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: if the energy-momentum is degenerate (in a certain sense) then our 3-manifold can be isometrically embedded in anti de Sitter.

Families of equivariant differential operators and anti de Sitter spaces

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 asymptotics of eigenfunctions on anti de Sitter spaces.

Timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space

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 so-called Lawson correspondence, between timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space and timelike minimal surfaces in Minkowski 3-space. The hyperbolic Gauss map of timelike surfaces in anti-de Sitter 3-space, which is a close analogue of the classical…

On perturbations of the Schwarzschild anti-de Sitter spaces of positive mass

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.

The positive energy theorem for asymptotically 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 also show that the determinant of energy-momentum endomorphism ${\bf Q}$ is the geometric invariant of asymptotically anti-de Sitter spacetimes.

On Isometric Embeddings into Anti-de Sitter Space-times

We show that any metric on $S^2$ with Gauss curvature $K \geq -κ$ admits a $C^{1,1}$-isometric embedding into the hyperbolic space with sectional curvature $-κ$. We also give a sufficient condition for a metric on $S^2$ to be isometrically embedded into anti-de Sitter spacetime with the prescribed cosmological time function.

The space of Lorentzian flat tori in anti-de Sitter 3-space

We describe the space of isometric immersions from the Lorentz plane $L^2$ 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 immersed in the anti de Sitter space in terms of pairs of closed curves with wave front singularities in the hyperbolic plane satisfying some compatibility conditions.

Inverse curvature flow in anti-de Sitter-Schwarzschild manifold

In this paper, we consider the inverse hessian quotient curvature flow with star-shaped initial hypersurface in anti-de Sitter-Schwarzschild manifold. We prove that the solution exists for all time, and the second fundamental form converges to identity exponentially fast.

On the structure of asymptotically de Sitter and anti-de Sitter spaces

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 of Balasubramanian-deBoer-Minic.

Anti-de Sitter空间中预定标量曲率的Dirichlet问题

The Dirichlet problem for the prescribed scalar curvature in Anti-de Sitter space

研究Anti-de Sitter空间中类空超曲面标量曲率预定的完全非线性二阶椭圆方程Dirichlet问题，在数据严格凸时得到解。

An Application of Maximum Principle to space-like Hypersurfaces with Constant Mean Curvature in Anti-de Sitter Space

In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space $H^{n+1}_1(-1)$. we prove that if a complete space-like hypersurface with constant mean curvature $x:\mathbf M\rightarrow H^{n+1}_1(-1) $ has two distinct principal curvatures $λ,μ$, and inf$|λ-μ|>0$, then $x$ is the standard embedding $ H^{m} (-\frac{1}{r^2})\times H^{n-m} (-\frac{1}{1 - r^2})$in anti-de Sitter space $ H^{n+1}_1 (-1)$.

Degeneration of globally hyperbolic maximal anti-de Sitter structures along pinching sequences

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 sequences. We show, in particular, that the regular globally hyperbolic anti-de Sitter structures introduced in arXiv:1806.08176 naturally appear as limiting points.

Cohomogeneity one actions on anti de Sitter spacetimes

In this paper we classify, up to orbit equivalence, cohomogeneity one actions of connected closed Lie subgroups of $U(1,n)$ on the $(2n+1)$-dimensional anti de Sitter spacetime $AdS^{2n+1}$. We also give some new examples of nonproper cohomogeneity one actions on $AdS^{n+1}$ and determine parabolic Lie subgroups of $SO(2,n)$ and their orbits in $AdS^{n+1}$

关于渐近反德西特初始数据的广义Jang方程解的存在性与性质

On the existence and properties of solutions of the generalized Jang equation with respect to asymptotically anti-de Sitter initial data

在$3\leq n\leq 7$维反德西特时空的恒定时间切片上，针对一类非常一般的渐近行为，严格分析了广义Jang方程，并讨论了其在渐近反德西特初始数据集时空正质量定理中的潜在应用。

Geodesics on a Kerr-Newman-(anti-)de Sitter instanton

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 metric, particularly regions of angular degeneracy or "theta horizons" in the de Sitter case. Finally, we characterize a number of stable equilibrium orbits.

Fenchel-Nielsen coordinates on the augmented moduli space of anti-de Sitter structures

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 the complex Fenchel-Nielsen coordinates for hyperbolic quasi-Fuchsian manifolds.

Teichmüller 与反德西特几何中的长度函数

Length functions in Teichmüller and anti de Sitter geometry

本文建立了 Teichmüller 空间长度函数与反德西特 3-流形几何的联系，并以此给出长度函数凸性与二阶变分界的新证明，同时引入剪切-弯曲坐标。

非Fuchs表示之间的支配关系与反de Sitter几何

Domination between non-Fuchsian representations and anti-de Sitter geometry

本文受曲面群表示相关研究启发，研究闭曲面群非Fuchs表示间的新支配问题，对容许分支调和浸入的表示求解该问题，证明其余情形不可普遍求解，还关联到带奇点反de Sitter 3流形并构造多族相关分支流形。

加权渐近反德西特时空的正能量定理

The positive energy theorem for weighted asymptotically anti-de Sitter spacetimes

本文针对三维渐近反德西特初始数据集，在加权主能量条件下证明其正能量定理可推广至加权情形。

Crooked surfaces and anti-de Sitter geometry

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 the interior of the quotient of Einstein space by an involution fixing an Einstein plane. The purpose of this note is to show that the crooked planes defined in anti-de Sitter space recently by Danciger-Guéritaud-Kassel lift to restrictions of crooked surfaces in Einstein space …

洛伦兹反德西特平面

Lorentzian anti-de Sitter plane

针对反德西特平面上的二维洛伦兹问题，采用几何控制理论与微分几何方法，完成标架构建、轨迹计算等多项工作，推进了相关几何问题的研究。

On non-degeneracy of Riemannian Schwarzschild-anti de Sitter metrics

We prove that the $TT$-gauge-fixed linearised Einstein operator is non-degenerate for Riemannian Kottler ("Schwarzschild-anti de Sitter") metrics with dimension- and topology-dependent ranges of mass parameter. We provide evidence that this remains true for all such metrics except the spherical ones with a critical mass.

具有凸边界的反德西特(2+1)时空在Fuchsian轨迹附近的刚性

Rigidity of anti-de Sitter (2+1)-spacetimes with convex boundary near the Fuchsian locus

本文证明了在Fuchsian轨迹附近，具有严格凸类空边界（光滑或多面体）的整体双曲紧致反德西特(2+1)时空由边界诱导度量唯一确定。