12 research outputs found

    Non-rigidity of spherical inversive distance circle packings

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    We give a counterexample of Bowers-Stephenson's conjecture in the spherical case: spherical inversive distance circle packings are not determined by their inversive distances.Comment: 6 pages, one pictur

    On the infinitesimal rigidity of polyhedra with vertices in convex position

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    Let P⊂R3P \subset \R^3 be a polyhedron. It was conjectured that if PP is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. PP can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assumption of codecomposability. The proof relies on a result of independent interest concerning the Hilbert-Einstein function of a triangulated convex polyhedron. We determine the signature of the Hessian of that function with respect to deformations of the interior edges. In particular, if there are no interior vertices, then the Hessian is negative definite.Comment: 12 page

    Fuchsian polyhedra in Lorentzian space-forms

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    Let S be a compact surface of genus >1, and g be a metric on S of constant curvature K\in\{-1,0,1\} with conical singularities of negative singular curvature. When K=1 we add the condition that the lengths of the contractible geodesics are >2\pi. We prove that there exists a convex polyhedral surface P in the Lorentzian space-form of curvature K and a group G of isometries of this space such that the induced metric on the quotient P/G is isometric to (S,g). Moreover, the pair (P,G) is unique (up to global isometries) among a particular class of convex polyhedra, namely Fuchsian polyhedra. This extends theorems of A.D. Alexandrov and Rivin--Hodgson concerning the sphere to the higher genus cases, and it is also the polyhedral version of a theorem of Labourie--Schlenker

    Hyperbolic manifolds with convex boundary

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    Let (M,∂M)(M, \partial M) be a compact 3-manifold with boundary, which admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on MM such that the boundary is smooth and strictly convex. We show that the induced metrics on the boundary are exactly the metrics with curvature K>−1K>-1, and that the third fundamental forms of \dr M are exactly the metrics with curvature K2πK2\pi. Each is obtained exactly once. Other related results describe existence and uniqueness properties for other boundary conditions, when the metric which is achieved on \dr M is a linear combination of the first, second and third fundamental forms.Comment: Check the updated version(s) on http://picard.ups-tlse.fr/~schlenker/ Version 2: an error corrected. Version 3: simpler main statement, small corrections, more details on one technical statement. Version 5: one error correcte

    Réalisation de métriques sur les surfaces compactes

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    Un polyèdre fuchsien de l'espace hyperbolique est une surface polyèdrale invariante sous l'action d'un groupe fuchsien d'isométries (c.a.d. un groupe d'isométries qui laissent globalement invariante une surface totalement géodésique et sur laquelle il agit de manière cocompacte). La métrique induite sur un polyèdre fuchsien convexe est isométrique à une métrique hyperbolique avec des singularités coniques de courbure singulière positive sur une surface compacte de genre >1. On démontre que ces métriques sont en fait réalisées par un unique polyèdre fuchsien convexe (modulo les isométries globales). Ce résultat étend un théorème célèbre de A.D. Alexandrov. On montre aussi que chaque métrique à courbure constante avec des courbures singulières négatives sur une surface compacte de genre >1 peut-être réalisée par un unique polyèdre ``fuchsien'' convexe dans un espace modèle lorentzien. Finalement on présente des extensions possibles de ces résultats, ce qui amène à des énoncés généraux sur la réalisation de métriques sur les surfaces.A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic metric with conical singularities of positive singular curvature on a compact surface of genus >1. We prove that these metrics are actually realised by exactly one convex Fuchsian polyhedron (up to global isometries). This extends a famous theorem of A.D. Alexandrov. We also prove that any constant curvature metric with conical singularities of negative singular curvature on a compact surface of genus >1 can be realised by a unique convex ``Fuchsian'' polyhedron in a Lorentzian space-form. Finally we present some possible expansion of these results, and this leads to general statements about realisation of metrics on surfaces

    A rigidity criterion for non-convex polyhedra

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    11 pages, 1 image. Revised versions will be posted on http://picard.ups-tlse.fr/~schlenker v2: one statement corrected, ref. addedLet PP be a (non necessarily convex) embedded polyhedron in R3\R^3, with its vertices on an ellipsoid. Suppose that the interior of PP can be decomposed into convex polytopes without adding any vertex. Then PP is infinitesimally rigid. More generally, let PP be a polyhedron bounding a domain which is the union of polytopes C1,...,CnC_1, ..., C_n with disjoint interiors, whose vertices are the vertices of PP. Suppose that there exists an ellipsoid which contains no vertex of PP but intersects all the edges of the CiC_i. Then PP is infinitesimally rigid. The proof is based on some geometric properties of hyperideal hyperbolic polyhedra
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