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July, 2009 Asymptotic behavior of flat surfaces in hyperbolic 3-space
Masatoshi KOKUBU, Wayne ROSSMAN, Masaaki UMEHARA, Kotaro YAMADA
J. Math. Soc. Japan 61(3): 799-852 (July, 2009). DOI: 10.2969/jmsj/06130799

Abstract

In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3 -space H 3 . Gálvez, Martínez and Milán showed that when the singular set does not accumulate at an end, the end is asymptotic to a rotationally symmetric flat surface. As a refinement of their result, we show that the asymptotic order (called pitch p ) of the end determines the limiting shape, even when the singular set does accumulate at the end. If the singular set is bounded away from the end, we have - 1 < p 0 . If the singular set accumulates at the end, the pitch p is a positive rational number not equal to 1 . Choosing appropriate positive integers n and m so that p = n / m , suitable slices of the end by horospheres are asymptotic to d -coverings ( d -times wrapped coverings) of epicycloids or d -coverings of hypocycloids with 2 n 0 cusps and whose normal directions have winding number m 0 , where n = n 0 d , m = m 0 d ( n 0 , m 0 are integers or half-integers) and d is the greatest common divisor of m - n and m + n . Furthermore, it is known that the caustics of flat surfaces are also flat. So, as an application, we give a useful explicit formula for the pitch of ends of caustics of complete flat fronts.

Citation

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Masatoshi KOKUBU. Wayne ROSSMAN. Masaaki UMEHARA. Kotaro YAMADA. "Asymptotic behavior of flat surfaces in hyperbolic 3-space." J. Math. Soc. Japan 61 (3) 799 - 852, July, 2009. https://doi.org/10.2969/jmsj/06130799

Information

Published: July, 2009
First available in Project Euclid: 30 July 2009

zbMATH: 1177.53059
MathSciNet: MR2552916
Digital Object Identifier: 10.2969/jmsj/06130799

Subjects:
Primary: 53C42
Secondary: 53A35

Rights: Copyright © 2009 Mathematical Society of Japan

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Vol.61 • No. 3 • July, 2009
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