On the first Hodge eigenvalue of isometric immersions

We give an extrinsic upper bound for the first positive eigenvalue of the Hodge Laplacian acting on $p$-forms on a compact manifold without boundary isometrically immersed in $\mathbf R^n$or $\mathbf S^n$. The upper bound generalizes an estimate of Reilly for functions; it depends on the mean value of the squared norm of the mean curvature vector of the immersion and on the mean value of the scalar curvature. In particular, for minimal immersions into a sphere the upper bound depends only on the degree, the dimension and the mean value of the scalar curvature.