A half-space type property in the Euclidean sphere

Marco Antonio Lázaro Velásquez

Address: Departamento de Matemática, Universidade Federal de Campina Grande, 58.409-970 Campina Grande, Paraíba, Brazil

E-mail: marco.velasquez@mat.ufcg.edu.br

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.

AMSclassification: primary 53C42; secondary 53C21.

Keywords: Euclidean sphere, closed hypersurfaces, (r+1)-th mean curvature, strong r-stability, geodesic spheres, upper (lower) domain enclosed by a geodesic sphere.

DOI: 10.5817/AM2022-1-49