# Spacelike hypersurfaces in standard static spacetimes

## TL;DR

This paper investigates spacelike hypersurfaces in static spacetimes, establishing conditions under which they are maximal or are slices, based on curvature bounds and mean curvature estimates.

## Contribution

It provides new curvature-based criteria for maximality and slicing of spacelike hypersurfaces in static spacetimes, extending previous geometric results.

## Key findings

- Complete CMC hypersurfaces with bounded hyperbolic angle or height are maximal.
- Complete maximal hypersurfaces not intersecting a slice are themselves slices in certain spacetimes.
- General mean curvature and gradient estimates are developed for spacelike hypersurfaces.

## 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.