Spectral Gaps on Riemannian Manifolds

We survey some basic results related to the new Weyl criterion for the essential spectrum of the Laplacian on Riemannian manifolds. We then use the language of Gromov-Hausdorff convergence to prove a spectral gap theorem.  Finally, we make the necessary adjustments to extend our main results and construct a class of complete noncompact manifolds with an arbitrarily large number of gaps in the spectrum of the Hodge Laplacian acting on differential forms.