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Stable constant mean curvature hypersurfaces in the real projective space

Authors:	Luis J Alías Aldir Brasil Jr Oscar Perdomo

Institution:	(1) Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Espinardo, Spain;(2) Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, 60455-760 Fortaleza-Ce, Brazil;(3) Departamento de Matemáticas, Universidad del Valle, Cali, Colombia

Abstract:	In this paper, we prove that the only compact two-sided hypersurfaces with constant mean curvature H which are weakly stable in $\mathbb{RP}^{n+1}$ and have constant scalar curvature are (i) the twofold covering of a totally geodesic projective space; (ii) the geodesic spheres in $\mathbb{RP}^{n+1}$ ; and (iii) the quotient to $\mathbb{RP}^{n+1}$ of the hypersurface $\mathbb{S}^{k}(r)\times\mathbb{S}^{n-k}(\sqrt{1-r^2})\hookrightarrow\mathbb{S}^{n+1}$ obtained as the product of two spheres of dimensions k and n − k, with k = 1,..., n − 1, and radii r and $\sqrt{1-r^2}$ , respectively, with $\sqrt{k/(n+2)}\leqslant r\leqslant\sqrt{(k+2)/(n+2)}$ .

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