A pinching theorem for the first eigenvalue of the laplacian on hypersurface of the euclidean space

2006/09/18 by Bruno Colbois, Colbois, Bruno, Jean-Francois Grosjean +1
Mathematics · #53A07 #53C21 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53A07 #msc:53C21

paper · pdf · doi:10.48550/arxiv.math/0609494

arxiv created 2006/09/18 · arxiv updated 2009/12/01

Abstract

In this paper, we give pinching Theorems for the first nonzero eigenvalue λ 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 ε>0, there exists a constant C_ε depending on the dimension n of M and the L_∞-norm of the mean curvature H, so that if the L_2p-norm ‖H‖_2p (p≥ 2) of H satisfies n‖H‖_2p-C_ε<λ, then the Hausdorff-distance between M and a round sphere of radius (n/λ)^1/2 is smaller than ε. Furthermore, we prove that if C is a small enough constant depending on n and the L_∞-norm of the second fundamental form, then the pinching condition n‖H‖_2p-C<\la implies that M is diffeomorphic to an n-dimensional sphere.

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