r-Stablity Index of r-Maximal Closed Hypersurfaces in de Sitter Spaces

It is well-known that some of minimal (or maximal) hypersurfaces are stable. However, there is growing recognition on unstable hypersurfaces by introducing the concept of index of stability for minimal ones. For instance, the index of stability for minimal hypersurefces in Euclidean n-sphere has been defined by J. Simons  and followed by many people. Also, Barros and Sousa have studied a high order extention of index as the concept of r-index (i.e. index of r-stability) on r-minimal hypersurfaces of n-sphere. They gave low bonds for r-stability index of r-minimal hypersurfaces in Euclidean sphere. In this paper, we low bounds for the r-stability index of r-maximal closed spacelike hypersurfaces in the de Sitter space.