1,721,003 research outputs found
Para-hyperkahler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds.
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Area-preserving diffeomorphisms of the hyperbolic plane and K-surfaces in anti-de Sitter space
No full textOn Codazzi Tensors on a Hyperbolic Surface and Flat Lorentzian Geometry
No full textInternational audienceUsing global considerations, Mess proved that the moduli space of globally hyperbolic flat Lorentzian structures on S × R is the tangent bundle of the Teichmüller space of S, if S is a closed surface. One of the goals of this paper is to deepen this surprising occurrence and to make explicit the relation between the Mess parameters and the embedding data of any Cauchy surface. This relation is pointed out by using some specific properties of Codazzi tensors on hyperbolic surfaces. As a by-product we get a new Lorentzian proof of Goldman's celebrated result about the coincidence of the Weil-Petersson symplectic form and the Goldman pairing. In the second part of the paper we use this machinery to get a classification of globally hyperbolic flat space-times with particles of angles in (0, 2π) containing a uniformly convex Cauchy surface. The analogue of Mess' result is achieved showing that the corresponding moduli space is the tangent bundle of the Teichmüller space of a punctured surface. To generalize the theory in the case of particles, we deepen the study of Codazzi tensors on hyperbolic surfaces with cone singularities, proving that the well-known decomposition of a Codazzi tensor in a harmonic part and a trivial part can be generalized in the context of hyperbolic metrics with cone singularities
Fibered spherical 3-orbifolds
Full text linkInternational audienceIn early 1930s Seifert and Threlfall classified up to conjugacy the finite subgroups of SO(4), this gives an algebraic classification of orientable spherical 3-orbifolds. For the most part, spherical 3-orbifolds are Seifert fibered. The underlying topological space and singular set of non-fibered spherical 3-orbifolds were described by Dunbar. In this paper we deal with the fibered case and in particular we give explicit formulae relating the finite subgroups of SO(4) with the invariants of the corresponding fibered 3-orbifolds. This allows to deduce directly from the algebraic classification topological properties of spherical 3-orbifolds
Character varieties of a transitioning Coxeter 4-orbifold
Full text linkInternational audienceIn 2010, Kerckhoff and Storm discovered a path of hyperbolic 4-polytopes eventually collapsing to an ideal right-angled cuboctahedron. This is expressed by a deformation of the inclusion of a discrete reflection group (a right-angled Coxeter group) in the isometry group of hyperbolic 4-space. More recently, we have shown that the path of polytopes can be extended to Anti-de Sitter geometry so as to have geometric transition on a naturally associated 4-orbifold, via a transitional half-pipe structure. In this paper, we study the hyperbolic, Anti-de Sitter, and half-pipe character varieties of Kerckhoff and Storm's right-angled Coxeter group near each of the found holonomy representations, including a description of the singularity that appears at the collapse. An essential tool is the study of some rigidity properties of right-angled cusp groups in dimension four
On the volume of anti-de Sitter maximal globally hyperbolic three-manifolds
No full textpeer reviewedWe study the volume of maximal globally hyperbolic Anti-de Sitter manifolds containing a closed orientable Cauchy surface S, in relation to some geometric invariants depending only on the two points in Teichmüller space of S provided by Mess’ parameterization - namely on two isotopy classes of hyperbolic metrics h and h' on S. The main result of the paper is that the volume coarsely behaves like the minima of the L1-energy of maps from (S, h) to (S, h'). The study of Lp-type energies had been suggested by Thurston, in contrast with the well-studied Lipschitz distance. A corollary of our result shows that the volume of maximal globally hyperbolic Anti-de Sitter manifolds is bounded from above by the exponential of (any of the two) Thurston’s Lipschitz asymmetric distances, up to some explicit constants. Although there is no such bound from below, we provide examples in which this behaviour is actually realized. We prove instead that the volume is bounded from below by the exponential of the Weil-Petersson distance. The proof of the main result uses more precise estimates on the behavior of the volume, which is proved to be coarsely equivalent to the length of the (left or right) measured geodesic lamination of earthquake from (S, h) to (S, h'), and to the minima of the holomorphic 1-energy
Isometry group and mapping class group of spherical 3-orbifolds
Full text linkInternational audienceWe study the isometry group of compact spherical orientable 3-orbifolds S^3/G, where G is a finite subgroup of SO(4), by determining its isomorphism type and, when S^3/G is a Seifert fibrered orbifold, by describing the action on the Seifert fibrations induced by isometric copies of the Hopf fibration of S^3. Moreover, we prove that the inclusion of Isom(S^3/G) into Diff(S^3/G) induces an isomorphism of the π_0 groups, thus proving the π_0-part of the natural generalization of the Smale Conjecture to spherical 3-orbifolds
Geometric transition from hyperbolic to Anti-de Sitter structures in dimension four
Full text linkInternational audienceWe provide the first examples of geometric transition from hyperbolic to Anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic 4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a 3-dimensional ideal cuboc-tahedron. We show the existence of a similar family of collapsing Anti-de Sitter polytopes, and join the two deformations by means of an opportune half-pipe orbifold structure. The desired examples of geometric transition are then obtained by gluing copies of the polytope
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