Discrete flat surfaces and linear Weingarten surfaces in hyperbolic 3-space
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- by T. Hoffmann, W. Rossman, T. Sasaki and M. Yoshida PDF
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Abstract:
We define discrete flat surfaces in hyperbolic $3$-space $\mathbb {H}^3$ from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature $1$ surfaces in $\mathbb {H}^3$, and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfaces. Along the way, we also examine discrete linear Weingarten surfaces of Bryant type in $\mathbb {H}^3$, and consider an example of a discrete flat surface related to the Airy equation that exhibits swallowtail singularities and a Stokes phenomenon.References
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Additional Information
- T. Hoffmann
- Affiliation: Department of Mathematics, Munich Technical University, 85748 Garching, Germany
- Email: tim.hoffmann@ma.tum.de
- W. Rossman
- Affiliation: Department of Mathematics, Kobe University, Kobe 657-8501, Japan
- Email: wayne@math.kobe-u.ac.jp
- T. Sasaki
- Affiliation: Department of Mathematics, Kobe University, Kobe 657-8501, Japan
- Email: sasaki@math.kobe-u.ac.jp
- M. Yoshida
- Affiliation: Department of Mathematics, Kyushu University, Fukuoka 819-0395, Japan
- Email: myoshida@math.kyushu-u.ac.jp
- Received by editor(s): January 6, 2010
- Published electronically: June 7, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 5605-5644
- MSC (2010): Primary 53A10; Secondary 53A30, 53A35, 52C99
- DOI: https://doi.org/10.1090/S0002-9947-2012-05698-4
- MathSciNet review: 2946924