UC Irvine

In this text, we survey some basic results related to the New Weyl criterion for the essential spectrum of the Laplace-Beltrami operator on Riemannian manifolds. We then use the language of Gromov-Hausdorff convergence to prove a spectral gap theorem. Finally, we discuss the Hodge-Laplacian operator on differential k-forms, and generalize our main theorem to that setting.