Journal of the Mathematical Society of Japan

Asymptotic behavior of flat surfaces in hyperbolic 3-space

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. 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 $2n_{0}$ 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.

Article information

Source
J. Math. Soc. Japan Volume 61, Number 3 (2009), 799-852.

Dates
First available in Project Euclid: 30 July 2009

https://projecteuclid.org/euclid.jmsj/1248961479

Digital Object Identifier
doi:10.2969/jmsj/06130799

Mathematical Reviews number (MathSciNet)
MR2552916

Zentralblatt MATH identifier
1177.53059

Citation

KOKUBU, Masatoshi; ROSSMAN, Wayne; UMEHARA, Masaaki; YAMADA, Kotaro. Asymptotic behavior of flat surfaces in hyperbolic 3-space. J. Math. Soc. Japan 61 (2009), no. 3, 799--852. doi:10.2969/jmsj/06130799. https://projecteuclid.org/euclid.jmsj/1248961479

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