12 research outputs found
Non-rigidity of spherical inversive distance circle packings
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
Let be a polyhedron. It was conjectured that if is
weakly convex (i. e. its vertices lie on the boundary of a strictly convex
domain) and decomposable (i. e. 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
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
Let be a compact 3-manifold with boundary, which admits a
convex co-compact hyperbolic metric. We consider the hyperbolic metrics on
such that the boundary is smooth and strictly convex. We show that the induced
metrics on the boundary are exactly the metrics with curvature , and that
the third fundamental forms of \dr M are exactly the metrics with curvature
. 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
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
11 pages, 1 image. Revised versions will be posted on http://picard.ups-tlse.fr/~schlenker v2: one statement corrected, ref. addedLet be a (non necessarily convex) embedded polyhedron in , with its vertices on an ellipsoid. Suppose that the interior of can be decomposed into convex polytopes without adding any vertex. Then is infinitesimally rigid. More generally, let be a polyhedron bounding a domain which is the union of polytopes with disjoint interiors, whose vertices are the vertices of . Suppose that there exists an ellipsoid which contains no vertex of but intersects all the edges of the . Then is infinitesimally rigid. The proof is based on some geometric properties of hyperideal hyperbolic polyhedra