A characterization of quadric constant mean curvature hypersurfaces of spheres

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dc.creatorAlias, Luis J.
dc.creatorBrasil Jr., Aldir
dc.creatorPerdomo, Oscar
dc.date2008-02-22
dc.date.accessioned2026-07-07T12:41:55Z
dc.date.available2026-07-07T12:41:55Z
dc.descriptionLet $ϕ:M\to\mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be an immersion of a complete $n$-dimensional oriented manifold. For any $v\in\mathbb{R}^{n+2}$, let us denote by $\ell_v:M\to\mathbb{R}$ the function given by $\ell_v(x)=ϕ(x),v$ and by $f_v:M\to\mathbb{R}$, the function given by $f_v(x)=ν(x),v$, where $ν:M\to\mathbb{S}^{n}$ is a Gauss map. We will prove that if $M$ has constant mean curvature, and, for some $v\ne{\bf 0}$ and some real number $λ$, we have that $\ell_v=λf_v$, then, $ϕ(M)$ is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface $M^n$ in $\mathbb{S}^{n+1}$ which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to $2n+4$.
dc.descriptionFinal version (February 2008). To appear in the Journal of Geometric Analysis
dc.identifierhttps://arxiv.org/abs/0802.3310
dc.identifierhttp://arxiv.org/abs/0802.3310
dc.identifierJournal of Geometric Analysis 18 (2008), 687--703
dc.identifierdoi:10.1007/s12220-008-9029-8
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/219904
dc.subjectDifferential Geometry
dc.subject53C42; 53A10
dc.titleA characterization of quadric constant mean curvature hypersurfaces of spheres
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