First eigenvalue pinching for Euclidean hypersurfaces via k-th mean curvatures

In this article, we prove pinching theorems for the first eigenvalue $$\lambda _1(M)$$λ1(M) of the Laplacian on compact Euclidean hypersurfaces involving the integrals of $$k$$k-th mean curvature. Particularly, we show that under a suitable pinching condition, the hypersurface is starshaped and almost-isometric to a standard sphere. Based on our theorems, we prove some pinching results for the almost-Einstein and almost-umbilical hypersurfaces in Euclidean space.