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arXiv · 1909.06622

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

Ken'ichi Yoshida
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Abstract

For 3-dimensional hyperbolic cone structures with cone angles $θ$, local rigidity is known for $0 \leq θ\leq 2π$, but global rigidity is known only for $0 \leq θ\leq π$. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most $π$ 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 $2π$. 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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Ken'ichi Yoshida. 2022-10-13. Degeneration of 3-dimensional hyperbolic cone structures with decreasing cone angles. https://doi.org/10.1090/ecgd%2F375

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