## Illinois Journal of Mathematics

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

Ronaldo F. de Lima

#### 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

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

Digital Object Identifier
doi:10.1215/ijm/1258138092

Mathematical Reviews number (MathSciNet)
MR2036991

Zentralblatt MATH identifier
1055.53048

#### 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