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Pinching of the first eigenvalue of the Laplacian and almost-Einstein hypersurfaces of the Euclidean space

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Abstract

In this article, we prove new pinching theorems for the first eigenvalue λ1(M) of the Laplacian on compact hypersurfaces of the Euclidean space. These pinching results are associated with the upper bound for λ1(M) in terms of higher order mean curvatures H k . We show that under a suitable pinching condition, the hypersurface is diffeomorpic and almost-isometric to a standard sphere. Moreover, as a corollary, we show that a hypersurface of the Euclidean space which is almost-Einstein is diffeomorpic and almost-isometric to a standard sphere.

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Authors and Affiliations

  1. Institut Élie Cartan, UMR 7502, Nancy-Université, CNRS, INRIA, B.P. 239, Vandœuvre lès Nancy Cedex, 54506, France

    Julien Roth

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  1. Julien Roth
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Correspondence to Julien Roth.

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Roth, J. Pinching of the first eigenvalue of the Laplacian and almost-Einstein hypersurfaces of the Euclidean space. Ann Glob Anal Geom 33, 293–306 (2008). https://doi.org/10.1007/s10455-007-9086-4

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  • DOI: https://doi.org/10.1007/s10455-007-9086-4

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