Abstract
In (2) Barbosa, do Carmo and Eschenburg characterized the to- tally umbilical spheres as the only weakly stable compact constant mean cur- vature hypersurfaces in the Euclidean sphere Sn+1. In this paper we prove that the weak index of any other compact constant mean curvature hyper- surface Mn in Sn+1 which is not totally umbilical and has constant scalar curvature is greater or equal to n + 2, with equality if and only if M is a constant mean curvature Cliord torus
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