Abstract
It is known that the totally umbilical hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S n+1, different from an Euclidean sphere, must have stability index greater than or equal to 1. In this paper we prove that the weak stability index of any non-totally umbilical compact hypersurface \({M \subset S^{{n+1}}}\) with cmc cannot take the values 1, 2, 3 . . . , n.
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Perdomo, O., Brasil, A. Stability index jump for constant mean curvature hypersurfaces of spheres. Arch. Math. 99, 493–500 (2012). https://doi.org/10.1007/s00013-012-0437-4
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DOI: https://doi.org/10.1007/s00013-012-0437-4