Entire constant mean curvature graphs in \(\mathbb{H}^2\times\mathbb{R}\) (Q2123066)

Graphs having constant mean curvature \(H\) are called \(H\)-graphs. Previously, the only known examples of entire \(H\)-graphs with \(0<H<\frac{1}{2}\) were conformally hyperbolic invariant surfaces. When \(H=0\), the examples are minimal graphs constructed by P. Collin and the second author. In this paper, for each \(0\leq H<\frac{1}{2}\), the authors construct entire \(H\)-graphs in \(\mathbb H^2\times\mathbb R\) that are parabolic and not invariant by one-parameter groups of isometries of \(\mathbb H^2\times\mathbb R\). Moreover, their asymptotic boundaries are \((\partial_{\infty}\mathbb H^2)\times\mathbb R\).