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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Discrete flat surfaces and linear Weingarten surfaces in hyperbolic 3-space


Authors: T. Hoffmann, W. Rossman, T. Sasaki and M. Yoshida
Journal: Trans. Amer. Math. Soc. 364 (2012), 5605-5644
MSC (2010): Primary 53A10; Secondary 53A30, 53A35, 52C99
Published electronically: June 7, 2012
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05698-4
PII: S 0002-9947(2012)05698-4
Received by editor(s): January 6, 2010
Published electronically: June 7, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.