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 ${\lambda }_1(M)$ 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 $\epsilon >0$, there exists a constant $C_{\epsilon }$ depending on the dimension $n$ of $M$ and the $L_{\infty }$-norm of the mean curvature $H$, so that if the $L_{2p}$-norm $\Vert H{\Vert }_{2p}$ ($p\ge 2$) of $H$ satisfies $n\Vert H{\Vert }_{2p}^2-C_{\epsilon }<{\lambda }_1(M)$, then the Hausdorff-distance between $M$ and a round sphere of radius $(n/{\lambda }_1(M){)}^{1/2}$ is smaller than $\epsilon$. Furthermore, we prove that if $C$ is a small enough constant depending on $n$ and the $L_{\infty }$-norm of the second fundamental form, then the pinching condition $n\Vert H{\Vert }_{2p}^2-C<{\lambda }_1(M)$ implies that $M$ is diffeomorphic to an $n$-dimensional sphere.