$r$-Stable Hypersurfaces in Conformally Stationary Spacetimes

TL;DR

This paper investigates the stability of certain spacelike hypersurfaces with constant r-th mean curvature in specific spacetimes, providing a characterization via eigenvalues and applying results to de Sitter space.

Contribution

It offers a new characterization of r-stability for hypersurfaces in conformally stationary spacetimes with constant sectional curvature, including de Sitter space.

Findings

Characterization of r-stability through eigenvalues of an associated operator.

Application of stability results to the de Sitter space.

Analysis of hypersurfaces with constant r-th mean curvature in curved spacetimes.

Abstract

In this paper we study the r-stability of closed spacelike hypersurfaces with constant $r$-th mean curvature in conformally stationary spacetimes of constant sectional curvature. In this setting, we obtain a characterization of $r-$stability through the analysis of the first eigenvalue of an operator naturally attached to the $r$-th mean curvature. As an application, we treat the case in which the spacetime is the de Sitter space.