A half-space theorem for graphs of constant mean curvature \(0<H<\frac{1}{2}\) in \(\mathbb{H}^{2}\times\mathbb{R}\) (Q5965357)

The authors study the case of a graph with constant mean curvature. More precisely they consider graphs over unbounded domains of \(\mathbb{H}^2\)-hyperbolic plane with constant mean curvature \(0< H<{1\over 2}\) (the domains are some ``ideal polygons'' with edges of constant curvature). In this case they prove a result similar to one of the other authors. Their main result is the following:NEWLINENEWLINE Let \(D\subset\mathbb{H}^2\) be a Scherk-type domain and \(u\) be Scherk-type solution over \(D\) (for some value \(0<H<{1\over 2}\)). Denote by \(\Sigma= \text{Graph}(u)\). If \(S\) is a properly immersed constant mean curvature surface contained in \(D\times\mathbb{R}\) and above \(\Sigma\), then \(S\) is a vertical translate of \(\Sigma\).