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

### Bruno Colbois

Université de Neuchâtel, Switzerland### Jean-Francois Grosjean

Université Henri Poincaré, Vandoeuvre Les Nancy, France

## Abstract

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.

## Cite this article

Bruno Colbois, Jean-Francois Grosjean, A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space. Comment. Math. Helv. 82 (2007), no. 1, pp. 175–195

DOI 10.4171/CMH/88