Existence and non-existence for a mean curvature equation in hyperbolic space (Q2583072)

The authors study the Dirichlet problem for hypersurfaces of prescribed mean curvature in the \((n+1)\)-dimensional hyperbolic space \(\mathbb{H}^{n+1}\). In the upper half space model of \(\mathbb{H}^{n+1}\) with coordinates \((x_0,x_1,\dots,x_n)\), \(x_n>0\), they consider graphs \(x_0 =u(x_1,\dots,x_n)\) defined over some bounded open set \(\Omega\) of class \(C^3\) contained in the hyperplane \(x_0=0\). As their main result, the authors claim an existence theorem which, for arbitrary boundary data of class \(C^3\) and any given mean curvature function \(H\in C^2 (\overline\Omega)\) states the existence of a unique solution to the Dirichlet problem provided only that \(H\) satisfies the inequality (*) \(|H(y) |\leq H_C'(y)\) for all points \(y\in\partial\Omega\), where \(H_C'\) denotes the mean curvature of the cylinder \(C=\{(x_0,x)\mid x_0\in\mathbb{R}, x\in \partial\Omega\}\). A theorem in this form cannot be true, since the condition (*) would allow to make \(H\) arbitrarily large in the interior of \(\Omega\). What is missing in the case of nonconstant \(H\) is a condition which would ensure a \(C^0\)-bound for solutions. The reviewer contacted the authors in this matter and they agree to make the following correction: The mean curvature \(H\) should either be constant or satisfy the inequality \(|H(x)|\leq 1\).