Stable compact spacelike hypersurfaces in the de Sitter space as maxima of a linear combination of area and volume

We define the notion of strong (r, s, a, b)-stability concerning compact space-like hypersurfaces immersed in the de Sitter space S-1(n+1). We study the variational problem of maximizing a certain Jacobi functional given by a linear combination of area and volume. Under a suitable constraint on a constant that appears in the computation of the second variation of this functional, we prove that a compact space-like hypersurface M-n contained in a a chronological future (or past) of S-1(n+1), with positive (s + 1)th curvature and such that H <= 1, must be a totally umbilical round sphere.