## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 47, Number 3 (2003), 667-698.

### Flux for Bryant surfaces and applications to embedded ends of finite total curvature

#### Abstract

We compute the flux of Killing fields through ends of constant mean curvature $1$ in hyperbolic space, and we prove a result conjectured by Rossman, Umehara and Yamada: the flux matrix defined by these authors is equivalent to the flux of Killing fields. We next give a geometric description of embedded ends of finite total curvature. In particular, we show that if such an end is asymptotic to a catenoid cousin, then we can associate an axis to it. We also compute the flux of Killing fields through these ends, and we deduce some geometric properties and analogies to minimal surfaces in Euclidean space.

#### Article information

**Source**

Illinois J. Math., Volume 47, Number 3 (2003), 667-698.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258138187

**Digital Object Identifier**

doi:10.1215/ijm/1258138187

**Mathematical Reviews number (MathSciNet)**

MR2007230

**Zentralblatt MATH identifier**

1043.53010

**Subjects**

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

Secondary: 53A35: Non-Euclidean differential geometry 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

#### Citation

Daniel, Benoît. Flux for Bryant surfaces and applications to embedded ends of finite total curvature. Illinois J. Math. 47 (2003), no. 3, 667--698. doi:10.1215/ijm/1258138187. https://projecteuclid.org/euclid.ijm/1258138187