A characterization of quadric constant mean curvature hypersurfaces of spheres

Abstract

Let $\varphi :M\to {\mathbb{S}}^{n+1}\subset {\mathbb{R}}^{n+2}$ 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 ${\ell }_v(x)=\varphi (x),v$ 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 $v\ne 0$ and some real number $\lambda$, we have that ${\ell }_v=\lambda f_v$, then, $\varphi (M)$ 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 $2n+4$.

Final version (February 2008). To appear in the Journal of Geometric Analysis