# Stable maximal hypersurfaces in Lorentzian spacetimes

## TL;DR

This paper investigates the stability properties of maximal hypersurfaces in Lorentzian spacetimes, providing characterizations, conditions for stability, and rigidity results relevant to general relativity.

## Contribution

It offers new criteria for stability of maximal hypersurfaces under curvature assumptions and extends the analysis to higher-order mean curvature stability.

## Key findings

- Characterization of stability in constant sectional curvature spacetimes
- Sufficient conditions for stability under the Timelike Convergence Condition
- Rigidity results and height estimates in GRW spacetimes

## Abstract

We study the geometry of stable maximal hypersurfaces in a variety of spacetimes satisfying various physically relevant curvature assumptions, for instance the Timelike Convergence Condition (TCC). We characterize stability when the target space has constant sectional curvature as well as give sufficient conditions on the geometry of the ambient spacetime (e.g., the validity of TCC) to ensure stability. Some rigidity results and height estimates are also proven in GRW spacetimes. In the last part of the paper we consider $k$-stability of spacelike hypersurfaces, a concept related to mean curvatures of higher orders.