A note on spacelike hypersurfaces and timelike conformal 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 $H\ge {\rho }^{\prime }/h\rho$ 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.

This is a preprint of the article published in Recent Advances in Pure and Applied Mathematics, RSME Springer Series, 2020, (135-147). ISBN: 978-3-030-41320-0