On surfaces with constant mean curvature in hyperbolic space (Q1419637)

In this paper, the author discusses surfaces with constant mean curvature \(>1\) in the hyperbolic space \({\mathbb H}^3.\) As the first main result, he shows that for a complete surface with constant mean curvature \(>1\) in \({\mathbb H}^3\) with boundary and finite index, the distance function to the boundary is bounded. Here, completeness means that all divergent paths have infinite length. He applies this result to establish a sharp height estimate \(\tanh^{-1}(1/H)\) for a geodesic graph defined on an \textit{unbounded} domain with boundary with constant mean curvature \(H>1\) and finite index. The same height estimate is known to hold for compact geodesic graphs with constant mean curvature \(H>1\) without finite index condition [see \textit{N. J. Korevaar}, \textit{R. Kusner}, \textit{W. H. Meeks} and \textit{B. Solomon}, Am. J. Math. 114, 1--43 (1992; Zbl 0757.53032)]. He also shows that a geodesically complete embedded surface with constant mean curvature \(>1\) and bounded Gaussian curvature in \({\mathbb H}^3\) is proper and has an \(\varepsilon\)-tubular neighborhood on its mean convex side that is embedded. He uses this result to obtain a monotonicity formula for such a surface.