Entire unbounded constant mean curvature Killing graphs (Q2399629)

Let \(N^{n+1}=M^n \times \mathbb R\) denote a complete warped product manifold and \(u \in C^2(M)\). The \textit{Killing graph} associated to \(u\) is defined by \(\Sigma (u)=\left\{ \Psi(u(x),x): x \in M^n \right\}\), where \(s\) is the parameter on \(\mathbb R\), \(Y =\partial /\partial s\) and (identifying \(M\) with the slice \(M\times \{0\}\)) \(\Psi: \mathbb R\times M^n \to N^{n+1}\) denotes the flux generated by \(Y\). In a previous work [J. Math. Pures Appl. (9) 103, No. 1, 219--227 (2015; Zbl 1333.53087)], the same authors gave conditions under which an entire bounded Killing graph of constant mean curvature is a slice. In the present paper, they improve this result obtaining some sufficient conditions for an entire constant mean curvature Killing graph, lying inside a possible unbounded region, to be necessarily a slice.