Global rigidity of 3-dimensional cone-manifolds

Hartmut Weiss Universit¨at M¨unchen

Differential Geometry mathscidoc:1609.10029

Journal of Differential Geometry, 76, (3), 495-523, 2007
We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles ≤ π (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles ≤ π, possibly with boundary consisting of totally geodesic hyperbolic turnovers.To that end we first generalize the local rigidity result contained in [Wei] to the setting of hyperbolic cone-3-manifolds of finite volume as above. We then use the techniques developed in [BLP] to deform the cone-manifold structure to a complete non-singular or a geometric orbifold structure, where global rigidity holds due to Mostow-Prasad rigidity, cf. [Mos], [Pra], in the hyperbolic case,resp. [deR], cf. also [Rot], in the spherical case. This strategy has already been implemented successfully by [Koj] in the compact hyperbolic case if the singular locus is a link using HodgsonKerckhoff local rigidity, cf. [HK].
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@inproceedings{hartmut2007global,
  title={GLOBAL RIGIDITY OF 3-DIMENSIONAL CONE-MANIFOLDS},
  author={Hartmut Weiss},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908203158863803686},
  booktitle={Journal of Differential Geometry},
  volume={76},
  number={3},
  pages={495-523},
  year={2007},
}
Hartmut Weiss. GLOBAL RIGIDITY OF 3-DIMENSIONAL CONE-MANIFOLDS. 2007. Vol. 76. In Journal of Differential Geometry. pp.495-523. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908203158863803686.
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