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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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
Trans. Amer. Math. Soc. 364 (2012), 5605-5644 Request permission

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
  • 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