Abstract
We define the notion of strong (r, s, a, b)-stability concerning compact space-like hypersurfaces immersed in the de Sitter space \(\mathbb {S}^{n+1}_1\). 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 \(\mathbb {S}^{n+1}_1\), with positive \((s+1)\)th curvature and such that \(H\le 1\), must be a totally umbilical round sphere.
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Velásquez, M.A.L., de Lima, H.F., da Silva, J.F. et al. Stable compact spacelike hypersurfaces in the de Sitter space as maxima of a linear combination of area and volume. manuscripta math. 159, 229–245 (2019). https://doi.org/10.1007/s00229-018-1047-7
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DOI: https://doi.org/10.1007/s00229-018-1047-7