LAUSR

We define the notion of strong (r, s, a, b)-stability concerning compact space-like hypersurfaces immersed in the de Sitter space $$\mathbb {S}^{n+1}_1$$S1n+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 $$M^n$$Mn contained in a a chronological future (or past) of $$\mathbb {S}^{n+1}_1$$S1n+1, with positive $$(s+1)$$(s+1)th curvature and such that $$H\le 1$$H≤1, must be a totally umbilical round sphere.