Rigidity for Closed Totally Umbilical Hypersurfaces in Space Forms |
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Authors: | Xu Cheng Detang Zhou |
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Institution: | 1. Instituto de Matemática, Universidade Federal Fluminense—UFF, Centro, Niterói, RJ, 24020-140, Brazil
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Abstract: | In Perez (Thesis, 2011), Perez proved some L 2 inequalities for closed convex hypersurfaces immersed in the Euclidean space ? n+1, and more generally for closed hypersurfaces with non-negative Ricci curvature, immersed in an Einstein manifold. In this paper, we discuss the rigidity of these inequalities when the ambient manifold is ? n+1, the hyperbolic space ? n+1, or the closed hemisphere \(\mathbb{S}_{+}^{n+1}\) . We also obtain a generalization of Perez’s theorem to the hypersurfaces without the hypothesis of non-negative Ricci curvature. |
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