# 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 $λ_{1}(M)$ 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 $C_{ε}$ 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 $n∥H∥_{2p}−C_{ε}<λ_{1}(M)$, then the Hausdorff-distance between $M$ and a round sphere of radius $(n/λ_{1}(M))_{1/2}$ 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 $n∥H∥_{2p}−C<λ_{1}(M)$ 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