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Thierry Barbot

Publications and source records attributed to Thierry Barbot.

30 records · Page 2Linked to original sources
arXiv2008

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

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 the 3-dimensional Minkowski space for datas that are invariant under the action of a co-compact Fuchsian group.

gr-qc↗
arXiv2007

Quasi-Fuchsian AdS representations are Anosov

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. It also includes: -- A construction of Dirichlet domains in the context of anti-de Sitter geometry, -- A proof that spatially compact globally hyperbolic anti-de Sitter spacetimes with acausal limit set admit locally CAT(-1) Cauchy hypersurfaces.

math.DG↗
arXiv2007

A primer on the (2+1) Einstein universe

The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on dimension $2 + 1$, in which there is a rich interplay with symplectic geometry.

math.DG↗
arXiv2007

Notes on a paper of Mess

These notes are a companion to the article "Lorentz spacetimes of constant curvature" by Geoffrey Mess, which was first written in 1990 but never published. Mess' paper will appear together with these notes in a forthcoming issue of Geometriae Dedicata.

math.DG↗
arXiv2007

Cosmological time versus CMC time II: the de Sitter and anti-de Sitter cases

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 the work consists of proving the required curvature estimates for these level sets. The nonzero curvature case presents significant new difficulties, in part due to the fact that the topological structure of nonzero constant curvature spacetimes is much richer than that of the flat spacetimes. Further, the timelike convergence condition fails for de Sitter spacetimes, and hence uniqueness for CMC hypersurfaces fails in general. We characterize those de Sitter spacetimes which admit CMC time functions (which are automatically unique), as well as those which admit CMC foliations but no CMC time function.

math.DG↗
arXiv2006

Cosmological time versus CMC time I: Flat spacetimes

This paper gives a new proof that maximal, globally hyperbolic, flat spacetimes of dimension $n\geq 3$ with compact Cauchy hypersurfaces are globally foliated by Cauchy hypersurfaces of constant mean curvature, and that such spacetimes admit a globally defined constant mean curvature time function precisely when they are causally incomplete. The proof, which is based on using the level sets of the cosmological time function as barriers, is conceptually simple and will provide the basis for future work on constant mean curvature time functions in general constant curvature spacetimes, as well for an analysis of the asymptotics of constant mean foliations.

math.DG↗
arXiv2006

Causal properties of AdS-isometry groups I: Causal actions and limit sets

We study the causality relation in the 3-dimensional anti-de Sitter space AdS and its conformal boundary Ein. To any closed achronal subset $Λ$ in ${Ein}\_2$ we associate the invisible domain $E(Λ)$ from $Λ$ in AdS. We show that if $Γ$ is a torsion-free discrete group of isometries of AdS preserving $Λ$ and is non-elementary (for example, not abelian) then the action of $Γ$ on $E(Λ)$ is free, properly discontinuous and strongly causal. If $Λ$ is a topological circle then the quotient space $M\_Λ(Γ) = Γ\backslash{E}(Λ)$ is a maximal globally hyperbolic AdS-spacetime admitting a Cauchy surface $S$ such that the induced metric on $S$ is complete. In a forthcoming paper we study the case where $Γ$ is elementary and use the results of the present paper to define a large family of AdS-spacetimes including all the previously known examples of BTZ multi-black holes.

math.GT↗
arXiv2006

Causal properties of AdS-isometry groups II: BTZ multi black-holes

This paper is the continuation of [8]. We essentially prove that the familly of strongly causal spacetimes defined in [8] associated to generic achronal subsets in Ein contains all the examples of BTZ multi black-holes. It provides new elements for the global description of these multi black-holes. We also prove that any strongly causal spacetime locally modelled on the anti-de Sitter space admits a well-defined maximal strongly causal conformal boundary locally modelled on Ein.

math.GT↗
arXiv2005

Three-dimensional Anosov flag manifolds

Let $Γ$ be a surface group of higher genus. Let $ρ\_0: Γ\to {PGL}(V)$ be a discrete faithful representation with image contained in the natural embedding of ${SL}(2, {\mathbb R})$ in ${PGL}(3, {\mathbb R})$ as a group preserving a point and a disjoint projective line in the projective plane. We prove that such a representation is $(G,Y)$-Anosov (following the terminology of \cite{labourieanosov}), where $Y$ is the frame bundle. More generally, we prove that all the deformations $ρ: Γ\to {PGL}(3, {\mathbb R})$ studied in \cite{barflag} are $(G,Y)$-Anosov. As a corollary, we obtain all the main results of \cite{barflag}, and extend them to any small deformation of $ρ\_0$, not necessarily preserving a point or a projective line in the projective space: in particular, there is a $ρ(Γ)$-invariant solid torus $Ω$ in the flag variety. The quotient space $ρ(Γ)\backslashΩ$ is a flag manifold, naturally equipped with two 1-dimensional transversely projective foliations arising from the projections of the flag variety on the projective plane and its dual; if $ρ$ is strongly irreducible, these foliations are not minimal. More precisely, if one of these foliations is minimal, then it is topologically conjugate to the strong stable foliation of a double covering of a geodesic flow, and $ρ$ preserves a point or a projective line in the projective plane. All these results hold for any $(G,Y)$-Anosov representation which is not quasi-Fuchsian, i.e., does not preserve a strictly convex domain in the projective plane.

math.RT↗
arXiv2004

Globally Hyperbolic Flat Spacetimes

We consider (flat) Cauchy-complete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes - model spacetimes - including four subfamilies: translation spacetimes, Misner spacetimes, unipotent spacetimes, and Cauchy-hyperbolic spacetimes (the last family - undoubtfull the most interesting one - is a generalization of standart spacetimes defined by G. Mess). We prove that, up to finite coverings and (twisted) products by euclidean linear spaces, any Cauchy-complete GH spacetime can be isometrically embedded in a model spacetime, or in a twisted product of a Cauchy-hyperbolic spacetime by flat euclidean torus. We obtain as a corollary the classification of maximal GH spacetimes admitting closed Cauchy hypersurfaces. We also establish the existence of CMC foliations on every model spacetime.

math.GT↗
arXiv2001

Flag Structures on Seifert Manifolds

We consider faithful projective actions of a cocompact lattice of SL(2,R) on the projective plane, with the following property: there is a common fixed point, which is a saddle fixed point for every element of infinite order of the the group. Typical examples of such an action are linear actions, ie, when the action arises from a morphism of the group into GL(2,R), viewed as the group of linear transformations of a copy of the affine plane in RP^{2}. We prove that in the general situation, such an action is always topologically linearisable, and that the linearisation is Lipschitz if and only if it is projective. This result is obtained through the study of a certain family of flag structures on Seifert manifolds. As a corollary, we deduce some dynamical properties of the transversely affine flows obtained by deformations of horocyclic flows. In particular, these flows are not minimal.

math.DS↗