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A half-space type property in the Euclidean sphere

Marco Antonio Lázaro Velásquez (2022)

Archivum Mathematicum

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.

A new characterization of r -stable hypersurfaces in space forms

H. F. de Lima, M. A. Velásquez (2011)

Archivum Mathematicum

In this paper we study the r -stability of closed hypersurfaces with constant r -th mean curvature in Riemannian manifolds of constant sectional curvature. In this setting, we obtain a characterization of the r -stable ones through of the analysis of the first eigenvalue of an operator naturally attached to the r -th mean curvature.

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