A half-space type property in the Euclidean sphere

Velásquez, Marco Antonio Lázaro

Archivum mathematicum, Tome 58 (2022) no. 1, pp. 49-63.

Résumé

We study the notion of strong $r$-stability for the context of closed hypersurfaces ${\mathrm{\Sigma }}^n$ ($n\ge 3$) with constant $(r+1)$-th mean curvature $H_{r+1}$ immersed into the Euclidean sphere ${\mathbb{S}}^{n+1}$, where $r\in \{1,\dots ,n-2\}$. 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.

Détail

DOI : 10.5817/AM2022-1-49

Classification : 53C21, 53C42
Mots-clés : Euclidean sphere; closed hypersurfaces; $(r+1)$-th mean curvature; strong $r$-stability; geodesic spheres; upper (lower) domain enclosed by a geodesic sphere