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.