A characterization of quadric constant mean curvature hypersurfaces of spheres

TL;DR

This paper characterizes certain constant mean curvature hypersurfaces in spheres, showing they are either totally umbilical or Clifford hypersurfaces, and establishes a lower bound on their stability index.

Contribution

It proves a geometric characterization of CMC hypersurfaces with specific linear relations involving position and Gauss map functions, and derives stability index bounds.

Findings

Hypersurfaces satisfying the linear relation are either totally umbilical or Clifford.

The stability index of certain CMC hypersurfaces is at least 2n+4.

The result applies to hypersurfaces with constant scalar curvature.

Abstract

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 $2n+4$.