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

Abstract

In this paper, we give pinching theorems for the first nonzero eigenvalue λ 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 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) implies that M is diffeomorphic to an n-dimensional sphere. © Swiss Mathematical Society.