A Characterization of Quadric Constant Mean Curvature Hypersurfaces of Spheres

摘要

Let $phi:Mto mathbb{S}^{n+1}subsetmathbb{R}^{n+2}$ be an immersion of a complete n-dimensional oriented manifold. For any v∈? n+2, let us denote by ? v :M→? the function given by ? v (x)=〈φ(x),v〉 and by f v :M→?, the function given by f v (x)=〈ν(x),v〉, where $nu:Mto mathbb{S}^{n+1}subsetmathbb{R}^{n+2}$ is a Gauss map. We will prove that if M has constant mean curvature, and, for some v≠0 and some real number λ, we have that ? v =λ f v , then, φ(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.