Geodesic
A half-space type property in the Euclidean sphere
Archivum mathematicum, Tome 58 (2022) no. 1, pp. 49-63 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.5817/AM2022-1-49
Classification : 53C21, 53C42
Keywords: Euclidean sphere; closed hypersurfaces; $(r+1)$-th mean curvature; strong $r$-stability; geodesic spheres; upper (lower) domain enclosed by a geodesic sphere 
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Velásquez, Marco Antonio Lázaro. A half-space type property in the Euclidean sphere. Archivum mathematicum, Tome 58 (2022) no. 1, pp. 49-63. doi: 10.5817/AM2022-1-49

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