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 $H^{3}$. 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 $-1<p\le 0$. 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 $n=n_{0}d$, $m=m_{0}d$ ($n_{0}$, $m_{0}$ 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.