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.
Citation
Ronaldo F. de Lima. "On surfaces with constant mean curvature in hyperbolic space." Illinois J. Math. 47 (4) 1079 - 1098, Winter 2003. https://doi.org/10.1215/ijm/1258138092
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