A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space

• Autores: Bruno Colbois, Jean-Francois Grosjean
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 82, Nº 1, 2007, págs. 175-195
• Idioma: inglés
• DOI: 10.4171/cmh/88
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• Resumen

  • In this paper, we give pinching theorems for the first nonzero eigenvalue ?1(M) of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of M is 1 then, for any e > 0, there exists a constant Ce depending on the dimension n of M and the L8-norm of the mean curvature H, so that if the L2p-norm ||H||2p (p = 2) of H satisfies n||H ||2p2 - Ce < ?1(M), then the Hausdorff-distance between M and a round sphere of radius (n/?1(M))1/2 is smaller than e. Furthermore, we prove that if C is a small enough constant depending on n and the L8-norm of the second fundamental form, then the pinching condition n||H ||2p2 - C < ?1(M) implies that M is diffeomorphic to an n-dimensional sphere.