Asymptotic behavior of flat surfaces in hyperbolic 3-space
Asymptotic behavior of flat surfaces in hyperbolic 3-space
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DOI：
10.2969/jmsj/06130799
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发表时间：
2007-08
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影响因子：
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通讯作者：
M. Kokubu;W. Rossman;M. Umehara;Kotaro Yamada
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作者：
M. Kokubu;W. Rossman;M. Umehara;Kotaro Yamada
本文研究了双曲三维空间H^3中平面正则端的渐近性质。Galvez， Martinez和Milan证明了当奇异集在某一端不积累时，该端点渐近于旋转对称平面。作为他们的结果的改进，我们证明了端点的渐近阶数（称为“节距”p）决定了极限形状，即使奇异集确实在端点累积。如果奇异集与端点有界，则有-1<p<=0。如果奇异集在最后累积，则音高p是一个不等于1的正有理数。选取适当的正整数n和m，使p=n/m，使圆球端部的适当切片渐近于表摆线的d-覆盖（d次包裹覆盖）或有2n_0个顶点的次摆线的d-覆盖，其法线方向圈数为m_0，其中n=n_0d, m=m_0d （n_0， m_0为整数或半整数），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。因此，作为一个应用，我们给出了一个有用的显式公式来计算完全平坦锋面的焦散端点的节距。
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_0 cusps and whose normal directions have winding number m_0, where n=n_0d, m=m_0d (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.