A characterization of quadric constant mean curvature hypersurfaces of spheres

Let $\phi:M\to\mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be an immersion of a complete $n$-dimensional oriented manifold. For any $v\in\mathbb{R}^{n+2}$, let us denote by $\ell_v:M\to\mathbb{R}$ the function given by $\ell_v(x)=\phi(x),v$ and by $f_v:M\to\mathbb{R}$, the function given by $f_v(x)=\nu(x),v$, where $\nu:M\to\mathbb{S}^{n}$ is a Gauss map. We will prove that if $M$ has constant mean curvature, and, for some $v\ne{\bf 0}$ and some real number $\lambda$, we have that $\ell_v=\lambda f_v$, then, $\phi(M)$ 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 $M^n$ in $\mathbb{S}^{n+1}$ 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

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Seja 'M POT.N', n'>OU='3 uma hipersuperficie de esfera euclidiana unitaria 'S POT.N+1' (1) e denotemos, respectivamente, por r, h e s a sua curvatura escalar, curvatura media e o quadrado da norma da segunda forma fundamental. No apendice respondemos afirmativamente a conjectura de chern para alguns tipos de hipersuperficies compactas, minimas e de dupin, ie, hipersuperficies 'M POT.N' de 'S POT.N+1' (1) cujas curvaturas principais sao constantes ao longo de suas linhas de curvaturanot availabl