## Journal of the Mathematical Society of Japan

### 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.

#### Article information

Source
J. Math. Soc. Japan Volume 61, Number 3 (2009), 799-852.

Dates
First available in Project Euclid: 30 July 2009

http://projecteuclid.org/euclid.jmsj/1248961479

Digital Object Identifier
doi:10.2969/jmsj/06130799

Mathematical Reviews number (MathSciNet)
MR2552916

Zentralblatt MATH identifier
05603963

#### Citation

KOKUBU, Masatoshi; ROSSMAN, Wayne; UMEHARA, Masaaki; YAMADA, Kotaro. Asymptotic behavior of flat surfaces in hyperbolic 3-space. J. Math. Soc. Japan 61 (2009), no. 3, 799--852. doi:10.2969/jmsj/06130799. http://projecteuclid.org/euclid.jmsj/1248961479.

#### References

• B. Daniel, Flux for Bryant surfaces and applications to embedded ends of finite total curvature, Illinois J. Math., 47 (2003), 667–698.
• R. Sa Earp and E. Toubiana, On the geometry of constant mean curvature one surfaces in hyperbolic space, Illinois J. Math., 45 (2001), 371–401.
• J. A. Gálvez, A. Martínez and F. Milán, Flat surfaces in the hyperbolic 3-space, Math. Ann., 316 (2000), 419–435.
• N. Korevaar, R. Kusner and B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom., 30 (1989), 465–503.
• M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat fronts in hyperbolic space, Pacific J. Math., 221 (2005), 303–351.
• M. Kokubu, W. Rossman, M. Umehara and K. Yamada, Flat fronts in hyperbolic 3-space and their caustics, J. Math. Soc. Japan, 59 (2007), 265–299.
• M. Kokubu, M. Umehara and K. Yamada, An elementary proof of Small's formula for null curves in $\mathit{PSL}(2,\mbi{C})$ and an analogue for Legendrian curves in $\mathit{PSL}(2,\mbi{C})$, Osaka J. Math., 40 (2003), 697–715.
• M. Kokubu, M. Umehara and K. Yamada, Flat fronts in hyperbolic 3-space, Pacific J. Math., 216 (2004), 149–175.
• P. Roitman, Flat surfaces in hyperbolic 3-space as normal surfaces to a congruence of geodesics, Tôhoku Math. J., 59 (2007), 21–37.
• W. Rossman, M. Umehara and K. Yamada, Flux for mean curvature 1 surfaces in hyperbolic 3-space, and applications, Proc. Amer. Math. Soc., 127 (1999), 2147–2154.
• K. Saji, M. Umehara and K. Yamada, The geometry of fronts, Ann. of Math., 169 (2009), 491–529.
• M. Umehara and K. Yamada, Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space, Ann. of Math., 137 (1993), 611–638.
• M. Umehara and K. Yamada, Surfaces of constant mean curvature $c$ in $H^{3}(-c^{2})$ with prescribed hyperbolic Gauss map, Math. Ann., 304 (1996), 203–224.