Deformations of hyperbolic convex polyhedra and cone-3-manifolds

被引：0
作者：
Grégoire Montcouquiol
机构：
[1] UMR 8628,Univ Paris
[2] CNRS,Sud, Laboratoire de Mathématiques
来源：
Geometriae Dedicata | 2013年 / 166卷
关键词：
Convex polyhedra; Hyperbolic geometry; Cone-manifolds; Character variety; Surface group representations; 52B10; 53C24;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
The Stoker problem, first formulated in Stoker (Commun. Pure Appl. Math. 21:119–168, 1968), consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means of the double construction, this problem is intimately related to rigidity issues for 3-dimensional cone-manifolds. In Mazzeo and Montcouquiol (J. Differ. Geom. 87(3):525–576, 2011), two such rigidity results were proven, implying that the infinitesimal version of the Stoker conjecture is true in the hyperbolic and Euclidean cases. In this second article, we show that local rigidity holds and prove that the space of convex hyperbolic polyhedra with given combinatorial type is locally parametrized by the set of dihedral angles, together with a similar statement for hyperbolic cone-3-manifolds.
引用
收藏
页码：163 / 183
页数：20
相关论文
共 20 条
  • [1] Andreev E.M.(1970)Convex polyhedra in Lobachevsky spaces Mat. Sb. (N.S.) 81 445-478
  • [2] Culler M.(1986)Lifting representations to covering groups Adv. Math. 59 64-70
  • [3] Culler M.(1983)Varieties of group representations and splittings of 3-manifolds Ann. Math. (2) 117 109-146
  • [4] Shalen P.B.(1984)The symplectic nature of fundamental groups of surfaces Adv. Math. 54 200-225
  • [5] Goldman W.M.(1998)Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery J. Differ. Geom. 48 1-59
  • [6] Hodgson C.(2008)The shape of hyperbolic Dehn surgery space Geom. Topol. 12 1033-1090
  • [7] Kerckhoff S.(2011)Examples of infinitesimally flexible 3-dimensional hyperbolic cone-manifolds. J. Math Soc. Jpn. 63 581-598
  • [8] Hodgson C.(1998)Deformations of hyperbolic 3-cone-manifolds J. Differ. Geom. 49 469-516
  • [9] Kerckhoff S.(2011)Rigidity of cone-3-manifolds and the Stoker problem J. Differ. Geom. 87 525-576
  • [10] Izmestiev I.(2007)Andreev’s theorem on hyperbolic polyhedra Ann. Inst. Fourier (Grenoble) 57 825-882
12