Rigidity of complete spacelike hypersurfaces with constant weighted mean curvature (Q304549)

A weighted Lorentzian product space with weighted function \(f:P\longrightarrow \mathbb R\) is the triple \((\mathbb R\times P,\left \langle\;,\;\right\rangle,d\mu=e^{-f} d\sigma)\) where \( P\) is a complete \(n\)-dimensional Riemannian manifold, \(d\sigma\) is its standard volume element and \(f\) is a differentiable function. A slice of such a manifold is the hypersurface \(\{t_0\}\times P\) for some \(t_0\in \mathbb R\). The Bakry-Émery-Ricci tensor of this Lorentzian product space is defined as \(\text{Ric}_f=\text{Ric}+\text{Hess}\,f\). The authors study hypersurfaces of the Lorentzian product space having special properties with respect to the Bakry-Émery-Ricci tensor. The main result of the paper is to show that the only complete space-like hypersurfaces in a weighted Lorentzian product space whose Bakry-Émery-Ricci tensor on the fiber \(P\) is nonnegative, having \(\text{Hess}\,f\) bounded below and with constant weighted mean curvature are \textit{slices} of the manifold, under some assumptions on the gradient of the height function.