Rigidity of polyhedral surfaces, III

Feng Luo

Differential Geometry mathscidoc:1912.43780

Geometry & Topology, 15, (4), 2299-2319, 2011.12
This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P Bowers and K Stephenson in [Mem. Amer. Math. Soc. 170, no. 805, Amer. Math. Soc.(2004)] as a generalization of Andreev and Thurstons circle packing. They conjectured that inversive distance circle packings are rigid. We prove this conjecture using recent work of R Guo [Trans. Amer. Math. Soc. 363 (2011) 47574776] on the variational principle associated to the inversive distance circle packing. We also show that each polyhedral metric on a triangulated surface is determined by various discrete curvatures that we introduced in [arXiv 0612.5714], verifying a conjecture in [arXiv 0612.5714]. As a consequence, we show that the discrete Laplacian operator determines a spherical polyhedral metric.
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@inproceedings{feng2011rigidity,
  title={Rigidity of polyhedral surfaces, III},
  author={Feng Luo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205752116240344},
  booktitle={Geometry & Topology},
  volume={15},
  number={4},
  pages={2299-2319},
  year={2011},
}
Feng Luo. Rigidity of polyhedral surfaces, III. 2011. Vol. 15. In Geometry & Topology. pp.2299-2319. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205752116240344.
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