A half-space type property in the Euclidean sphere

Marco Antonio Lázaro Velásquez

Archivum Mathematicum (2022)

  • Volume: 058, Issue: 1, page 49-63
  • ISSN: 0044-8753

Abstract

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We study the notion of strong r -stability for the context of closed hypersurfaces Σ n ( n ≥ 3 ) with constant ( r + 1 ) -th mean curvature H r + 1 immersed into the Euclidean sphere 𝕊 n + 1 , where r ∈ { 1 , ... , 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 𝕊 n + 1 , a region that is determined by a totally umbilical sphere of 𝕊 n + 1 . We also provide a rigidity result for such hypersurfaces.

How to cite

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Velásquez, Marco Antonio Lázaro. "A half-space type property in the Euclidean sphere." Archivum Mathematicum 058.1 (2022): 49-63. <http://eudml.org/doc/298246>.

@article{Velásquez2022,
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.},
author = {Velásquez, Marco Antonio Lázaro},
journal = {Archivum Mathematicum},
keywords = {Euclidean sphere; closed hypersurfaces; $(r+1)$-th mean curvature; strong $r$-stability; geodesic spheres; upper (lower) domain enclosed by a geodesic sphere},
language = {eng},
number = {1},
pages = {49-63},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A half-space type property in the Euclidean sphere},
url = {http://eudml.org/doc/298246},
volume = {058},
year = {2022},
}

TY - JOUR
AU - Velásquez, Marco Antonio Lázaro
TI - A half-space type property in the Euclidean sphere
JO - Archivum Mathematicum
PY - 2022
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 058
IS - 1
SP - 49
EP - 63
AB - 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.
LA - eng
KW - Euclidean sphere; closed hypersurfaces; $(r+1)$-th mean curvature; strong $r$-stability; geodesic spheres; upper (lower) domain enclosed by a geodesic sphere
UR - http://eudml.org/doc/298246
ER -

References

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