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

Full-text: Open access


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.

Article information

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

First available in Project Euclid: 30 July 2009

Permanent link to this document

Digital Object Identifier

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.

Export citation