A note on spacelike hypersurfaces and timelike conformal vectors

TL;DR

This paper establishes rigidity results for spacelike hypersurfaces in certain warped product spacetimes and spacetimes with conformal timelike vectors, showing they must be slices under specific curvature and completeness conditions.

Contribution

It proves new rigidity theorems for spacelike hypersurfaces in doubly warped products and spacetimes with conformal timelike vectors, extending previous geometric results.

Findings

Compact spacelike hypersurfaces are slices if mean curvature condition is met.

Completeness and noncompactness still yield rigidity under certain assumptions.

Rigidity also applies to hypersurfaces in spacetimes with conformal, expanding timelike vectors.

Abstract

Any compact spacelike hypersurface immersed in a doubly warped product spacetime $I{}_{h} \times_{\rho} \mathbb{P}$ with nondecreasing warping factor $\rho$ must be a spacelike slice, provided that the mean curvature satisfies $H\geq\rho'/h\rho$ everywhere on the hypersurface. The conclusion also holds, under suitable assumptions on the immersion, when the hypersurface is complete and noncompact. A similar rigidity property is shown for compact hypersurfaces in spacetimes carrying a conformal, strictly expanding, timelike vector field.