Asymptotic behavior of flat surfaces in hyperbolic 3-space

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Abstract

In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space <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 half-integers) 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), 799-852, 2009-07-01

    The Mathematical Society of Japan

References:  13

Cited by:  1

Codes

  • NII Article ID (NAID)
    10026998469
  • NII NACSIS-CAT ID (NCID)
    AA0070177X
  • Text Lang
    ENG
  • Article Type
    Journal Article
  • ISSN
    00255645
  • NDL Article ID
    10286803
  • NDL Source Classification
    ZM31(科学技術--数学)
  • NDL Call No.
    Z53-A209
  • Data Source
    CJP  CJPref  NDL  J-STAGE 
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