## Illinois Journal of Mathematics

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

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

Ricardo Sa Earp and Eric Toubiana

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/ijm/1258138346

**Mathematical Reviews number (MathSciNet)**

MR1878610

**Zentralblatt MATH identifier**

0997.53042

**Subjects**

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

#### 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