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Rigidity of minimal hypersurfaces of spheres with constant Ricci curvature
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Minimal hypersurfaces, Spheres, Shape operator, Clifford tori, 2000 Mathematics Subject Classification, Primary: 53C42, Secondary: 53A10 (en)Descargas
Abstract . Let M be a compact oriented minimal hypersurface of the unit n-dimensional sphere Sn. In this paper we will point out that if the Ricci curvature of M is constant, then, we have that either Ric ≡ 1 and M is isometric to an equator or, n is odd, Ric ≡ n3/n2 = and M is isometric to S (n-1)/2 ((√2)⁄2) x S (n-1)/2 ((√2)⁄2). Next, we will prove that there exists a positive number ϵ (n) such that if the Ricci curvature of a minimal hypersurface immersed by first eigenfunctions M satisfies that n-3/n-2 − ϵ (n) Ric ≤ n-3/n-2 + ϵ (n) and the average of the scalar curvature is n-3/n-2 then, the ricci curvature of M must be constant and therefore M must be isometric to S (n-1)/2 ((√2)⁄2) x S (n-1)/2 ((√2)⁄2).
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