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

In this paper, we give pinching theorems for the first nonzero eigenvalue λ1(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 ε > 0, there exists a constant Cε depending on the dimension n of M and the L∞-norm of the mean curvature H, so that if the L2p-norm ||H||2p (p ≥ 2) of H satisfies n||H ||2p2 − Cε < λ1(M), then the Hausdorff-distance between M and a round sphere of radius (n/λ1(M))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 ||2p2 − C < λ1(M) implies that M is diffeomorphic to an n-dimensional sphere.