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 $\varepsilon>0$, there exists a constant $C_{\varepsilon}$ 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 $n\|H\|{2p}-C_{\varepsilon}<\lambda$, then the Hausdorff-distance between $M$ and a round sphere of radius $(n/\lambda)^{1/2}$ is smaller than $\varepsilon$. 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.
