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首页> 外文期刊>Journal of the Mathematical Society of Japan >Asymptotic behavior of flat surfaces in hyperbolic 3-space
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Asymptotic behavior of flat surfaces in hyperbolic 3-space

机译：双曲3空间中平面的渐近行为

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摘要

In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space H~3. Galvez. Martinez and Milan 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 2n_n cusps and whose normal directions have winding number m_0, where n = n_od. m = rn_od (n_o, 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.
机译：在本文中，我们研究了双曲3空间H〜3中平面正则端的渐近行为。加尔维斯马丁内斯和米兰证明了当奇数集没有结束时。末端渐近于旋转对称的平面。作为其结果的改进，我们表明，即使奇异集合确实在末端累积，末端的渐近顺序（称为间距p）也决定了极限形状。如果奇数集的末尾有界，则我们有-1 ≤0。如果奇数集在末尾累积，则音调p是不等于1的正有理数。选择适当的正整数n和m p = n / m，合适的球形圈末端切片是渐近线的d覆盖层（d倍包裹的覆盖层）或具有2n_n尖点且法线方向为缠绕数m_0的下摆线的d覆盖层渐近。点头。 m = rn_od（n_o，m_0是整数或半整数），d是m_n和m + n的最大公约数。此外，已知平坦表面的焦散也是平坦的。所以。作为一种应用，我们给出了一个有用的显式公式，用于表示完全平直的苛性碱的末端间距。

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