New Calabi–Bernstein type results in Lorentzian product spaces with density

Abstract

We investigate the rigidity of complete spacelike hypersurfaces immersed in a weighted Lorentzian product space of the type R1×Pfn, endowed with a weight function f which does not depend on the parameter t∈R and whose Riemannian fiber Pn has nonnegative Bakry–Émery–Ricci tensor. In this direction, supposing that the f-mean curvature is constant and assuming appropriate constraints on the norm of the gradient of f, we prove that such a spacelike hypersurface must be a slice of the ambient space. As application of our investigation, we obtain new Calabi–Bernstein type results concerning entire spacelike graphs constructed over Pn. Our approach is based on an extension for the drift Laplacian of a generalized maximum principle of Akutagawa (1987) and a Liouville-type result due to Pigola et al. (2005).