Degeneration of 3-dimensional hyperbolic cone structures with decreasing cone angles
HTML articles powered by AMS MathViewer
- by Ken’ichi Yoshida PDF
- Conform. Geom. Dyn. 26 (2022), 182-193 Request permission
Abstract:
For 3-dimensional hyperbolic cone structures with cone angles $\theta$, local rigidity is known for $0 \leq \theta \leq 2\pi$, but global rigidity is known only for $0 \leq \theta \leq \pi$. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most $\pi$ do not degenerate in deformations decreasing cone angles to zero.
In this paper, we give an example of a degeneration of hyperbolic cone structures with decreasing cone angles less than $2\pi$. These cone structures are constructed on a certain alternating link in the thickened torus by gluing four copies of a certain polyhedron. For this construction, we explicitly describe the isometry types on such a hyperbolic polyhedron.
References
- Colin Adams, Aaron Calderon, and Nathaniel Mayer, Generalized bipyramids and hyperbolic volumes of alternating $k$-uniform tiling links, Topology Appl. 271 (2020), 107045, 28. MR 4046919, DOI 10.1016/j.topol.2019.107045
- E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (125) (1970), 256–260 (Russian). MR 0273510
- Abhijit Champanerkar, Ilya Kofman, and Jessica S. Purcell, Geometrically and diagrammatically maximal knots, J. Lond. Math. Soc. (2) 94 (2016), no. 3, 883–908. MR 3614933, DOI 10.1112/jlms/jdw062
- Abhijit Champanerkar, Ilya Kofman, and Jessica S. Purcell, Geometry of biperiodic alternating links, J. Lond. Math. Soc. (2) 99 (2019), no. 3, 807–830. MR 3977891, DOI 10.1112/jlms.12195
- Raquel Díaz, Non-convexity of the space of dihedral angles of hyperbolic polyhedra, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 9, 993–998 (English, with English and French summaries). MR 1485617, DOI 10.1016/S0764-4442(97)89092-1
- Craig D. Hodgson and Steven P. Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998), no. 1, 1–59. MR 1622600
- Ivan Izmestiev, Examples of infinitesimally flexible 3-dimensional hyperbolic cone-manifolds, J. Math. Soc. Japan 63 (2011), no. 2, 581–598. MR 2793111
- Sadayoshi Kojima, Deformations of hyperbolic $3$-cone-manifolds, J. Differential Geom. 49 (1998), no. 3, 469–516. MR 1669649
- Alexander Kolpakov and Bruno Martelli, Hyperbolic four-manifolds with one cusp, Geom. Funct. Anal. 23 (2013), no. 6, 1903–1933. MR 3132905, DOI 10.1007/s00039-013-0247-2
- Rafe Mazzeo and Grégoire Montcouquiol, Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra, J. Differential Geom. 87 (2011), no. 3, 525–576. MR 2819548
- G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004
- Gopal Prasad, Strong rigidity of $\textbf {Q}$-rank $1$ lattices, Invent. Math. 21 (1973), 255–286. MR 385005, DOI 10.1007/BF01418789
- Roland K. W. Roeder, John H. Hubbard, and William D. Dunbar, Andreev’s theorem on hyperbolic polyhedra, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 3, 825–882 (English, with English and French summaries). MR 2336832, DOI 10.5802/aif.2279
- Hartmut Weiss, Global rigidity of 3-dimensional cone-manifolds, J. Differential Geom. 76 (2007), no. 3, 495–523. MR 2331529
- Hartmut Weiss, The deformation theory of hyperbolic cone-3-manifolds with cone-angles less than $2\pi$, Geom. Topol. 17 (2013), no. 1, 329–367. MR 3035330, DOI 10.2140/gt.2013.17.329
Additional Information
- Ken’ichi Yoshida
- Affiliation: Center for Soft Matter Physics, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan
- MR Author ID: 1044051
- ORCID: 0000-0002-7478-603X
- Email: yoshida.kenichi@ocha.ac.jp
- Received by editor(s): January 23, 2021
- Received by editor(s) in revised form: November 22, 2021, and August 2, 2022
- Published electronically: October 12, 2022
- Additional Notes: This work is supported by JSPS KAKENHI Grant Numbers 15H05739 and 19K14530, and JST CREST Grant Number JPMJCR17J4.
- © Copyright 2022 American Mathematical Society
- Journal: Conform. Geom. Dyn. 26 (2022), 182-193
- MSC (2020): Primary 57M50, 52B10
- DOI: https://doi.org/10.1090/ecgd/375
- MathSciNet review: 4495836