Spacelike hypersurfaces in standard static spacetimes

Abstract

In this work we study spacelike hypersurfaces immersed in spatially open standard static spacetimes with complete spacelike slices. Under appropriate lower bounds on the Ricci curvature of the spacetime in directions tangent to the slices, we prove that every complete CMC hypersurface having either bounded hyperbolic angle or bounded height is maximal. Our conclusions follow from general mean curvature estimates for spacelike hypersurfaces. In case where the spacetime is a Lorentzian product with spatial factor of nonnegative Ricci curvature and sectional curvatures bounded below, we also show that a complete maximal hypersurface not intersecting a spacelike slice is itself a slice. This result is obtained from a gradient estimate for parametric maximal hypersurfaces.

Extremal and weakly trapped submanifolds in generalized Robertson-Walker spacetimes

“…The aforementioned reasons have recently aroused the interest in obtaining new rigidity results for spacelike submanifolds in a wide variety of spacetimes including Lorentzian space forms [11,26], pp-wave spacetimes [27,29], Robertson-Walker spacetimes [7] and standard static spacetimes [15,33] among others.…”

A Note on Spacelike Hypersurfaces and Timelike Conformal Vectors

“…This infinitesimal symmetry is usually assumed when searching for exact solutions of Einstein's field equations [13]. In fact, there is a vast literature concerning the study of spacelike hypersurfaces in spacetimes that admit different causal symmetries (see [8], [9], [10], for instance). We will focus on the study of spacelike hypersurfaces in doubly warped product spacetimes, where natural conformal vector field pertains to this structure.…”