  In this paper we give pinching theorems for the first nonzero eigenvalue of
the Laplacian on the compact hypersurfaces of ambient spaces with bounded
sectional curvature. As application we deduce rigidity results for stable
constant mean curvature hypersurfaces $M$ of these spaces $N$. Indeed, we prove
that if $M$ is included in a ball of radius small enough then the
Hausdorff-distance between $M$ and a geodesic sphere $S$ of $N$ is small.
Moreover $M$ is diffeomorphic and quasi-isometric to $S$. As other application,
we give rigidity results for almost umbilic hypersurfaces.
