Journal of the Mathematical Society of Japan

Asymptotic behavior of flat surfaces in hyperbolic 3-space

Masatoshi KOKUBU, Wayne ROSSMAN, Masaaki UMEHARA, and Kotaro YAMADA

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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 $-1<p\le 0$. 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 $2n_{0}$ 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

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

First available in Project Euclid: 30 July 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53A35: Non-Euclidean differential geometry

flat surface flat front end asymptotic behavior hyperbolic 3-space


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.

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