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 Gauss map is considered. It is discussed that the hyperbolic Gauss map plays an important role in the study of timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space. In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gauss map and timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space is studied.
This paper studies timelike minimal surfaces in the De Sitter space $\mathbb S^3_1(1) \subset \mathbb R^4_1$ via a complex variable. Using complex analysis and stereographic projection of lightlike vectors we obtain a representation formula. Real and complex special quadrics in $\mathbb CP^3$ are identified with the grassmannians of spacelike and timelike oriented 2-planes of $\mathbb R^4_1$, and the normal frame is written in terms of certain complex valued functions $x$ and $y$, which may be considered holomorphic functions as a special case. Then several results describing the analytic rest…
In this article, we study spacelike and timelike rotational surfaces in a 3--dimensional de Sitter space $\mathbb{S}^3_1$ which are the orbit of a regular curve under the action of the orthogonal transformation of 4--dimensional Minkowski space $\mathbb{E}^4_1$ leaving a spacelike, a timelike or a degenerate plane pointwise fixed. We determine the profile curve of such Weingarten rotational surfaces parameterized by the principal curvature. Then, we classify spacelike and timelike Weingarten rotational surface in $\mathbb{S}^3_1$ with the principal curvatures $κ$ and $λ$ satisfying $κ=aλ+b$ or…
We prove that an isometric immersion of a timelike surface in four-dimensional Minkowski space is equivalent to a normalized spinor field which is a solution of a Dirac equation on the surface. Using the quaternions and the complex numbers, we obtain a spinor representation formula that relates the spinor field and the isometric immersion. Applying the representation formula, we deduce a new spinor representation of a timelike surface in three-dimensional De Sitter space; we give a formula for the Laplacian of the Gauss map of a minimal timelike surface in four-dimensional Minkowski space in t…
Timelike surfaces in the three-dimensional Heisenberg group with left invariant semi-Riemannian metric are studied. In particular, non-vertical timelike minimal surfaces are characterized by the non-conformal Lorentz harmonic maps into the de Sitter two phere. On the basis of the characterization, the generalized Weierstrass type representation will be established through the loop group decompositions.
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 sufficient conditions are obtained for a hypersurface to be timelike, null, or spacelike in the generalized model; in the non-null case, the geometry is given by a warped product. Several examples of timelike, null, and spacelike hypersurfaces are presented. Lastly, we calculate t…
A world sheet in anti-de Sitter space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds. We consider the family of lightlike hypersurfaces along spacelike submanifolds in the world sheet. The locus of the singularities of lightlike hypersurfaces along spacelike submanifolds forms the caustic of the world sheet. This notion is originally introduced by Bousso and Randall in theoretical physics. In this paper we give a mathematical framework for the caustics of world sheets as an application of the theory of graph-like Legendrian unfoldings.
Using techniques of integrable systems, we study a Weierstrass representation formula for timelike surfaces with prescribed mean curvature in Minkowski 3-space. It is shown that timelike minimal surfaces are obtained by integrating a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in Minkowski 3-space. The relationship between timelike minimal surfaces and bosonic Nambu-Goto string worldsheets in spacetime is also discussed in the appendix.
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 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.
We prove uniqueness, existence, and regularity results for maximal hypersurfaces in spacetimes with a conformal completion at timelike infinity and asymptotically constant scalar curvature, as relevant for asymptotically AdS spacetimes. This work is dedicated to Yvonne Choquet-Bruhat on the occasion of her upcoming 99th birthday.
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 cases of negative curvature (i.e. anti-de Sitter) spacetimes and postitive curvature (i.e. de Sitter) spacetimes is dealt with. As in the flat case, the existence of CMC time functions is obtained by using the level sets of the cosmological time function as barriers. A major part of…
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…
The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space, de Sitter 3-space, and Minkowski motion group 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 which is the unificaton of representation formulas for minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds is obtained. The normal Gauß map of minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds and its harmonicity are discuss…
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.
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.
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.
In the first part of this work we show a uniqueness result for globally hyperbolic spacetimes with a spacelike conformal boundary satisfying the vacuum Einstein equations with positive cosmological constant. Then we present applications of this result in the contexts of cosmic censorship and the initial value problem in general relativity. Further extensions of this result to non-vacuum solutions of the Einstein equations are also studied. On the other hand, the second half of this thesis deals with spacetimes with timelike boundary. We first develop the causal theory of such spacetimes and th…
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.
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.
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.
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.
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.
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.
A timelike space is a Hausdorff topological space equipped with a partial order relation $<$ and a distance function $ρ$ satisfying a collection of axioms including a set of compatibility conditions between the partial order relation and the distance function. The distance function is defined only on a subset of the product of the space with itself that contains the diagonal, namely, $ρ(x,y)$ is defined if and only if $x<y$ or $x=y$. Distances between pairs of distinct points in a triple $x,y,z$, whenever these distances are defined, satisfy the so-called \emph{time inequality}, which is a rev…
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.
