Asymptotic behavior of flat surfaces in hyperbolic 3space
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Abstract
In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3space <I>H</I><SUP>3</SUP>. 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 <I>pitch</I> <I>p</I>) 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<<I>p</I>≤0. If the singular set accumulates at the end, the pitch <I>p</I> is a positive rational number not equal to 1. Choosing appropriate positive integers <I>n</I> and <I>m</I> so that <I>p</I>=<I>n</I>⁄<I>m</I>, suitable slices of the end by horospheres are asymptotic to <I>d</I>coverings (<I>d</I>times wrapped coverings) of epicycloids or <I>d</I>coverings of hypocycloids with 2<I>n</I><SUB>0</SUB> cusps and whose normal directions have winding number <I>m</I><SUB>0</SUB>, where <I>n</I>=<I>n</I><SUB>0</SUB><I>d</I>, <I>m</I>=<I>m</I><SUB>0</SUB><I>d</I> (<I>n</I><SUB>0</SUB>, <I>m</I><SUB>0</SUB> are integers or halfintegers) and <I>d</I> is the greatest common divisor of <I>m</I>−<I>n</I> and <I>m</I>+<I>n</I>. 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.
Journal

 Journal of the Mathematical Society of Japan

Journal of the Mathematical Society of Japan 61(3), 799852, 20090701
The Mathematical Society of Japan