Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces |
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Authors: | J-F Grosjean J Roth |
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Institution: | 1. Institut élie Cartan (Mathématiques), Université Henri Poincaré Nancy I, B.P. 239, 54506, Vand?uvre-les-Nancy Cedex, France 2. LAMA, Université Paris-Est - Marne-la-Vallée, 5 bd Descartes, Cité Descartes, Champs-sur-Marne, 77454, Marne-la-Vallée, France
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Abstract: | In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on compact hypersurfaces of ambient spaces with bounded sectional curvature. As an application we deduce a rigidity result for stable constant mean curvature hypersurfaces M of these spaces N. Indeed, we prove that if M is included in a ball of radius small enough then the Hausdorff-distance between M and a geodesic sphere S of N is small. Moreover M is diffeomorphic and quasi-isometric to S. As other application, we obtain rigidity results for almost umbilic hypersurfaces. |
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