## Degeneration of 3-dimensional hyperbolic cone structures with decreasing cone angles

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- 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.

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## 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