A half-space type property in the Euclidean sphere

Abstract

We study the notion of strong $r$-stability for the context of closed hypersurfaces $\Sigma ^n$ ($n\ge 3$) with constant $(r+1)$-th mean curvature $H_{r+1}$ immersed into the Euclidean sphere $\mathbb{S}^{n+1}$, where $r\in \lbrace 1,\ldots ,n-2\rbrace $. In this setting, under a suitable restriction on the $r$-th mean curvature $H_r$, we establish that there are no $r$-strongly stable closed hypersurfaces immersed in a certain region of $\mathbb{S}^{n+1}$, a region that is determined by a totally umbilical sphere of $\mathbb{S}^{n+1}$. We also provide a rigidity result for such hypersurfaces.