Let Ï:MâSn+1âRn+2 be an immersion of a
complete n-dimensional oriented manifold. For any vâRn+2, let
us denote by âvâ:MâR the function given by
âvâ(x)=Ï(x),v and by fvâ:MâR, the function given by
fvâ(x)=Îœ(x),v, where Îœ:MâSn is a Gauss map. We will prove
that if M has constant mean curvature, and, for some vî =0 and some
real number λ, we have that âvâ=λfvâ, then, Ï(M) is
either a totally umbilical sphere or a Clifford hypersurface. As an
application, we will use this result to prove that the weak stability index of
any compact constant mean curvature hypersurface Mn in Sn+1
which is neither totally umbilical nor a Clifford hypersurface and has constant
scalar curvature is greater than or equal to 2n+4.Comment: Final version (February 2008). To appear in the Journal of Geometric
Analysi