Asymptotic behavior of flat surfaces in hyperbolic 3-space

Abstract

In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space H³. 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 2n₀ cusps and whose normal directions have winding number m₀, where n=n₀d, m=m₀d (n₀, m₀ 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.