Existence and uniqueness of CMC parabolic graphs in \(\mathbb H^3\) with boundary data satisfying the bounded slope condition (Q1038521)

The authors consider a bounded simply connected domain \(\Omega\), with \(C^{2,\alpha}\) boundary, contained in a totally geodesic plane in \(3\)-dimensional hyperbolic space. For each \(\phi\in C^{2,\alpha}(\partial \Omega)\), there exists a non-negative constant \(K\) such that \(\phi\) satisfies an appropriate form of the bounded slope condition for the constant \(K\). The authors' main result is that, if \(H\) is such that \(H\geq K/\sqrt{1+K^2}\) and if the curvature of \(\partial\Omega\) in \({\mathbb H}^3\) with respect to the inner orientation exceeds \( \exp[ \text{diam}\,(\Omega)] \, (H+1)\, (1+K^2)^2 + 1\), where \(\text{diam}\,(\cdot)\) denotes the hyperbolic diameter, then there exists a unique \(u\in C^{2,\alpha}\), with \(u|_{\partial \Omega} = \phi\), the parabolic graph of which has constant mean curvature \(H\). The authors state that their work is based on ideas of \textit{J. Ripoll} [J. Differ. Equations 181, No.~1, 230--241 (2002; Zbl 1257.35093)].