  In this paper, we give pinching Theorems for the first nonzero eigenvalue
$\lambda$ 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 $\|H\|\_{2p}$ ($p\geq 2$) of $H$ satisfies
$n\|H\|\_{2p}-C\_{\epsilon}<\lambda$, then the Hausdorff-distance between $M$
and a round sphere of radius $(n/\lambda)^{1/2}$ 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 $n\|H\|\_{2p}-C<\la$ implies that $M$ is diffeomorphic to an
$n$-dimensional sphere.
