Rigidity of spacelike hypersurfaces in spatially weighted generalized Robertson-Walker spacetimes

TL;DR

This paper investigates the rigidity of complete spacelike hypersurfaces in weighted generalized Robertson-Walker spacetimes using maximum principles, establishing conditions under which these hypersurfaces are slices and deriving new Calabi-Bernstein results.

Contribution

It introduces new rigidity criteria for hypersurfaces in weighted GRW spacetimes and extends Calabi-Bernstein type theorems to this setting.

Findings

Hypersurfaces are slices under certain curvature and weight conditions.

Established new Calabi-Bernstein type theorems for entire graphs.

Applied maximum principles to derive geometric rigidity results.

Abstract

Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weight function and on the f-mean curvature, we establish sufficient conditions to guarantee that such a hypersurface must be a slice of the ambient space. In this setting, we also obtain new Calabi-Bernstein type results concerning entire graphs in a spatially weighted GRW spacetime.