Umbilicity of Complete LW Spacelike Hypersurfaces Immersed in Certain Einstein Spacetimes

We provide sufficient geometric conditions to guarantee the umbilicity of complete linear Weingarten spacelike hypersurfaces immersed in an Einstein spacetime obeying certain curvature constraints, that were originally considered by Nishikawa (Nagoya Math J 95:117–124, 1984). Our approach is based on suitable maximum principles recently deduced by Alías et al. (J Math Anal Appl 474:242–247, 2019; Ann Mat Pura Appl 200:1637–1650, 2021), which deal with the notions of convergence to zero at infinity and polynomial volume growth of a complete noncompact Riemannian manifold.