## Illinois Journal of Mathematics

### On the geometry of constant mean curvature one surfaces in hyperbolic space

#### Abstract

We give a geometric classification of regular ends with constant mean curvature $1$ and finite total curvature, embedded in hyperbolic space. We prove that each such end is either asymptotic to a catenoid cousin or asymptotic to a horosphere. We also study symmetry properties of constant mean curvature $1$ surfaces in hyperbolic space associated to minimal surfaces in Euclidean space. We describe the constant mean curvature $1$ surfaces in $\hi3$ associated to the family of surfaces in $\m3$ that is isometric to the helicoid.

#### Article information

Source
Illinois J. Math., Volume 45, Number 2 (2001), 371-401.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138346

Digital Object Identifier
doi:10.1215/ijm/1258138346

Mathematical Reviews number (MathSciNet)
MR1878610

Zentralblatt MATH identifier
0997.53042

#### Citation

Sa Earp, Ricardo; Toubiana, Eric. On the geometry of constant mean curvature one surfaces in hyperbolic space. Illinois J. Math. 45 (2001), no. 2, 371--401. doi:10.1215/ijm/1258138346. https://projecteuclid.org/euclid.ijm/1258138346