Journal of the
Korean Mathematical Society JKMS

In this paper, we study complete Riemannian immersions into a semi-Riemannian warped product obeying suitable curvature constraints. Under appropriate differential inequalities involving higher order mean curvatures, we establish rigidity and nonexistence results concerning these immersions. Applications to the cases that the ambient space is either an Einstein manifold, a steady state type spacetime or a pseudo-hyperbolic space are given, and a particular investigation of entire graphs constructed over the fiber of the ambient space is also made. Our approach is based on a parabolicity criterion related to a linearized differential operator which is a divergence-type operator and can be regarded as a natural extension of the standard Laplacian.

Keywords: Semi-Riemannian warped products, Einstein manifolds, Riemannian immersions, entire graphs, higher order mean curvatures, parabolicity criterion

MSC numbers: Primary 53C42; Secondary 53C50

Supported by: The first author was partially supported by CAPES, Brazil. The second and third authors are partially supported by CNPq, Brazil, grants 301970/2019-0 and 306524/2022-8, respectively. The third author is also partially supported by FAPESQ-PB, Brazil, grant 3025/2021.