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

Yoshida, Ken’ichi

doi:10.1090/ecgd/375

Degeneration of 3-dimensional hyperbolic cone structures with decreasing cone angles
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by Ken’ichi Yoshida

Conform. Geom. Dyn. 26 (2022), 182-193

DOI: https://doi.org/10.1090/ecgd/375

Published electronically: October 12, 2022

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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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