Riemannian domains and complete spacelike hypersurfaces with one end in steady state type spacetimes

We deal with generalized linear Weingarten spacelike hypersurfaces immersed into a steady state type spacetime $-\mathbb R\times_{e^t}M^n$, namely, spacelike hypersurfaces of $-\mathbb R\times_{e^t}M^n$ that admit a linear combination of their higher order mean curvatures. First, supposing that the fiber $M^n$ has nonnegative constant sectional curvature, we establish characterization results concerning Riemannian domains of $-\mathbb R\times_{e^t}M^n$, namely, domains of the spacelike slices $\{t\}\times M^n$, with $t\in \mathbb R^n$. Then we apply such characterizations to study the uniqueness of complete generalized linear Weingarten spacelike hypersurfaces with one end.