## Rigidity of circle polyhedra in the $2$-sphere and of hyperideal polyhedra in hyperbolic $3$-space

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- by John C. Bowers, Philip L. Bowers and Kevin Pratt PDF
- Trans. Amer. Math. Soc.
**371**(2019), 4215-4249 Request permission

## Abstract:

We generalize Cauchy’s celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb {E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb {S}^{2}$. We prove that any two convex and proper nonunitary*c*-polyhedra with Möbius-congruent faces that are consistently oriented are Möbius congruent. Our result implies the global rigidity of convex inversive distance circle packings in the Riemann sphere, as well as that of certain hyperideal hyperbolic polyhedra in $\mathbb {H}^{3}$.

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## Additional Information

**John C. Bowers**- Affiliation: Department of Computer Science, James Madison University, Harrisonburg, Virginia 22807
- MR Author ID: 1056730
- Email: bowersjc@jmu.edu
**Philip L. Bowers**- Affiliation: Department of Mathematics, The Florida State University, Tallahassee, Florida 32306
- MR Author ID: 40455
- Email: bowers@math.fsu.edu
**Kevin Pratt**- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
- MR Author ID: 1207765
- Email: kevin.pratt@uconn.edu
- Received by editor(s): June 1, 2017
- Received by editor(s) in revised form: December 6, 2017
- Published electronically: September 25, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**371**(2019), 4215-4249 - MSC (2010): Primary 52C26
- DOI: https://doi.org/10.1090/tran/7483
- MathSciNet review: 3917221