In this paper, we prove new pinching theorems for the first eigenvalue of the
Laplacian on compact hypersurfaces of the Euclidean space. These pinching
results are associated with the upper bound for the first eigenvalue in terms
of higher order mean curvatures. We show that under a suitable pinching
condition, the hypersurface is diffeomorpic and almost isometric to a standard
sphere. Moreover, as a corollary, we show that a hypersurface of the Euclidean
space which is almost Einstein is diffeomorpic and almost isometric to a
standard sphere.Comment: 18 page
