Eigenvalue and gap estimates of isometric immersions for the Dirichlet-to-Neumann operator acting on p-forms

In this paper, we study the first eigenvalue of the Dirichlet-to-Neumann operator acting on differential forms of a Riemannian manifold with boundary isometrically immersed in some Euclidean space. We give a lower bound of the integral energy of p-forms in terms of its first eigenvalue associated with $(p?1)$-forms. We also find a lower bound for the gap between two consecutive first eigenvalues in terms of the curvature of the boundary.