The geometry at infinity of LW-spacelike submanifolds in semi-Riemannian space forms (Q6614517)

Let \(\mathbb{Q}^{n+p}_{p}(c)\) be the \((n + p)\)-dimensional connected semi-Riemannian manifold with index \(p\) and constant sectional curvature \(c\in \mathbb{R}\). An \(n\)-dimensional submanifold \(M^{n}\) immersed in \(\mathbb{Q}^{n+p}_{p}(c)\) is said to be space-like if the induced metric on \(M^{n}\) is positive definite. A space-like submanifold of \(\mathbb{Q}^{n+p}_{p}(c)\) is said to be linear Weingarten (for short, an LW-space-like submanifold) when its mean and normalized scalar curvatures are linearly related.\N\NThe paper under review studies complete noncompact LW-space-like submanifolds in \(\mathbb{Q}^{n+p}_{p}(c)\) with parallel normalized mean curvature vector. Under suitable restrictions on the behavior of the mean curvature at infinity and values of the norm of the traceless part of the second fundamental form, the authors show that these surfaces must be isometric to one of the following hyperbolic cylinders:\N\begin{itemize}\N\item \(\mathbb{R}^{n-1} \times \mathbb{H}^{1}(c_2)\), with \(c_2 < 0\), when \(c = 0\);\N\item \(\mathbb{S}^{n-1}(c_1) \times \mathbb{H}^{1}(c_2)\), with \(c_1 > 0\), \(c_2 < 0\) and \(1/c_1 + 1/c_2 = 1/c\), when \(c > 0\); \N\item \(\mathbb{H}^{n-1}(c_1) \times \mathbb{H}^{1}(c_2)\), with \(c_1 < 0\), \(c_2 < 0\) and \(1/c_1 + 1/c_2 = 1/c\), when \(c < 0\).\N\end{itemize}