## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 47, Number 4 (2003), 1079-1098.

### On surfaces with constant mean curvature in hyperbolic space

#### Abstract

It is shown that for a complete surface with constant mean curvature $H>1$ in $\mathbb{H}\kern0.5pt ^3$ with boundary and finite index the distance function to the boundary is bounded. We apply this result to establish a sharp height estimate for certain geodesic graphs with noncompact boundary. We also show that a geodesically complete, embedded surface in $\mathbb{H}\kern0.5pt ^3$ with constant mean curvature $H>1$ and bounded Gaussian curvature is proper and has an $\epsilon -$tubular neighborhood on its mean convex side that is embedded. Finally, we use this last result to obtain a monotonicity formula for such a surface.

#### Article information

**Source**

Illinois J. Math., Volume 47, Number 4 (2003), 1079-1098.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258138092

**Digital Object Identifier**

doi:10.1215/ijm/1258138092

**Mathematical Reviews number (MathSciNet)**

MR2036991

**Zentralblatt MATH identifier**

1055.53048

**Subjects**

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

Secondary: 53A35: Non-Euclidean differential geometry

#### Citation

de Lima, Ronaldo F. On surfaces with constant mean curvature in hyperbolic space. Illinois J. Math. 47 (2003), no. 4, 1079--1098. doi:10.1215/ijm/1258138092. https://projecteuclid.org/euclid.ijm/1258138092