On the global evolution problem in 2 + 1 gravity

Anderson, Lars; Moncrief, Vincent; Tromba, Anthony J.

doi:10.1016/s0393-0440(97)87804-7

Cited by 57 publications

(98 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In case n ≥ 3, this metric is the unique hyperbolic metric on M , while in case n = 2, this metric corresponds to a point in the Teichmuller space Teich(M ) of M . This is a partial generalization of the results for the case n = 2 proved in [4]. In that paper it was also proved that in the direction τ ց −∞, towards the singularity, the Teichmuller class of the induced metric on M τ diverges, in the sense that it leaves every compact subset of Teich(M ), as τ ց −∞.…”

Section: Introductionsupporting

confidence: 65%

“…This is proved by showing that the Dirichlet energy E, which is a proper function on Teich(M ) [16, §3], see also [18], diverges as τ ց −∞. However, the work in [4] does not give a detailed picture of the geometry of the CMC hypersurfaces M τ for τ ց −∞. It is the purpose of this paper to study the detailed asymptotic behavior of the geometry of M τ in the case when V is simplicial.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Constant mean curvature foliations of simplicial flat spacetimes

Andersson¹

2005

Communications in Analysis and Geometry

View full text Add to dashboard Cite

Abstract. Benedetti and Guadagnini [5] have conjectured that the constant mean curvature foliation Mτ in a 2 + 1 dimensional flat spacetime V with compact hyperbolic Cauchy surfaces satisfies limτ→−∞ ℓM τ = sT , where ℓM τ and sT denote the marked length spectrum of Mτ and the marked measure spectrum of the R-tree T , dual to the measured foliation corresponding to the translational part of the holonomy of V , respectively. We prove that this is the case for n + 1 dimensional, n ≥ 2, simplicial flat spacetimes with compact hyperbolic Cauchy surface. A simplicial spacetime is obtained from the Lorentz cone over a hyperbolic manifold by deformations corresponding to a simple measured foliation.

show abstract

Section: Introductionsupporting

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Constant mean curvature foliations of simplicial flat spacetimes

Andersson¹

2005

Communications in Analysis and Geometry

View full text Add to dashboard Cite

show abstract

“…The projection π : AdS 3 → ADS 3 is a two-fold covering. The spacetime ADS 3 is time-orientable (7) , since the antipodal map of R 4 preserves the time-orientation of AdS 3 . The isometric action of (SL(2, R) × SL(2, R))/I on AdS 3 induces an action of PSL(2, R) × PSL(2, R) on ADS 3 .…”

Section: Geometry Of Mghc Adsmentioning

confidence: 99%

Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space

Barbot¹,

Béguin²,

Zeghib³

2011

Ann. inst. Fourier

100

View full text Add to dashboard Cite

We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with constant Gauss curvature. In the constant negative curvature case, such a foliation exists outside the convex core. The existence of these foliations, together with a theorem of C. Gerhardt, yield several corollaries. For example, they allow to solve the Minkowski problem in Min3 for datas that are invariant under the action of a co-compact Fuchsian group.-the Minkowski problem in the 3-dimensional Minkowski space.The results concerning the last two items are essentially application of the first one. STATEMENTS OF RESULTS K-slicings of MGHC spacetimes with constant curvature.The following is our main result: Theorem 2.1. Let M be a 3-dimensional non-elementary MGHC spacetime with constant curvature Λ. If Λ ≥ 0, reversing the time orientation if necessary, we assume that M is future complete.• If Λ ≥ 0 (flat case or locally de Sitter case), then M admits a unique K-slicing. The leaves of this slicing are the level sets of a K-time ranging over (−∞, −Λ). • If Λ < 0 (locally anti-de Sitter case), then M does not admit any global K-slicing, but each of the two connected component of the complement of the convex core 2 of M admits a unique K-slicing. The leaves of the K-slicing of the past of the convex core are the level sets of a K-time ranging over (−∞, 0). The leaves of the K-slicing of the future of the convex core are the level sets of a reverse K-time 3 ranging over (−∞, 0).Let us make a few comments on this result.Gaussian curvature. Since the Gaussian curvature of a surface is R = Λ + κ, the Gaussian curvature of the leaves of the K-slicings provided by Theorem 2.1 varies in (−∞, 0) when Λ ≥ 0, and in (−∞, Λ) when Λ < 0.CMC times. It is our interest on CMC-times that led us to extend our attention to more general geometric times. Existence of CMC times on MGHC spacetimes of constant non-positive curvature and any dimension was proved in [3,4]. For spacetimes locally modelled on the de Sitter space, there are some restrictions but not in dimension 3.Regularity. The slicings provided by Theorem 2.1 are continuous. It follows from their uniqueness (i.e. they are canonical). Extra smoothness is not automatic (e.g. the cosmological time is C 1,1 , but not C 2 ).Here, we can hope that our K-slicings are (real) analytic, and even more, they depend analytically on the spacetime (within the space of MGHC spacetimes of curvature Λ and fixed topology). All this depends on consideration of Einstein equations in a K-gauge.Non-standard isometric immersions of H 2 in Min 3 . It was observed by Hano and Nomizu [38], that the hyperbolic plane H 2 admits non standard isometric immersions in the Minkowski space Min 3 (i.e. different from the hyperbola, up to a Lorentz motion).

show abstract

“…[6] wherein a non-zero cosmological constant was also allowed for. The main technique for this argument involved the use of the Dirichlet energy on Teichmüller space, exploiting its known properties as a proper function, to bound the motion to the interior of Teichmüller space for all values of mean curvature τ in the range (−∞, 0) and then to show that this motion captures the maximal Cauchy development of every solution.…”

Section: The Reduced Hamiltonianmentioning

confidence: 99%

“…The opposite limit τ 0 corresponds to that of infinite cosmological expansion for which the geometric area of Σ blows up but for which the induced conformal geometry always asymptotes to an interior point of Teichmüller space (which together with an associated asymptotic "velocity" is determined by the chosen point of T * T (Σ)). It is known from earlier work that the range (−∞, 0) always exhausts the maximal Cauchy development for each vacuum solution [6].…”

Section: Introductionmentioning

confidence: 99%

Relativistic Teichmüller theory: a Hamilton-Jacobi approach to 2+1–dimensional Einstein gravity

Moncrief

2007

Surveys in Differential Geometry

View full text Add to dashboard Cite

Abstract. We consider vacuum spacetimes in 2 + 1 dimensions defined on manifolds of the form M = Σ× R where Σ is a compact, orientable surface of genus > 1. By exploiting the harmonicity properties of the Gauss map for an arbitrary constant mean curvature (CMC) slice in such a spacetime we relate the Hamiltonian dynamics of the corresponding reduced Einstein equations to some fundamental results in the Teichmüller theory of harmonic maps. In particular we show, expanding upon an argument sketched by Puzio, that a global complete solution to the Hamilton-Jacobi equation for the reduced Einstein equations can be expressed in terms of the Dirichlet energy for harmonic maps defined over the surface Σ. While in principle this complete solution to the HamiltonJacobi equation determines all the solution curves to the reduced Einstein equations, one can derive a more explicit characterization of these curves through the solution of an associated (parametrized) Monge-Ampêre equation. Using the latter we define a corresponding family of "ray structures" on the Teichmüller space of the chosen 2-manifold Σ. These ray structures are similar but complementary to a different family of such ray structures defined by M. Wolf and we herein derive a "relativistic interpretation" of both sets. We also use our Hamilton-Jacobi results to define complementary families of Lagrangian foliations of the cotangent bundle of the Teichmüller space of Σ and to provide the corresponding "relativistic interpretation" of the leaves of these foliations.

show abstract

On the global evolution problem in 2 + 1 gravity

Cited by 57 publications

References 3 publications

Constant mean curvature foliations of simplicial flat spacetimes

Constant mean curvature foliations of simplicial flat spacetimes

Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space

Relativistic Teichmüller theory: a Hamilton-Jacobi approach to 2+1–dimensional Einstein gravity

Contact Info

Product

Resources

About