Asymptotic behavior of flat surfaces in hyperbolic 3-space

TL;DR

This paper studies the asymptotic shapes of flat surfaces in hyperbolic 3-space, revealing how the 'pitch' determines their limiting form, especially when singularities accumulate at the ends, with applications to caustics.

Contribution

It refines previous results by characterizing the asymptotic behavior of flat surface ends based on the pitch, including cases with accumulating singularities, and provides explicit formulas for caustics.

Findings

Ends with bounded singular sets have -1<p<=0.

Ends with accumulating singular sets have positive rational pitch p.

Slices of ends are asymptotic to coverings of epicycloids or hypocycloids.

Abstract

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.