Lp-spectral theory for the Laplacian on forms

In this article, we find sufficient conditions on an open Riemannian manifold so that a Weyl criterion holds for the $L^p$-spectrum of the Laplacian on k-forms, and also prove the decomposition of the $L^p$-spectrum depending on the order of the forms. We then show that the resolvent set of an operator such as the Laplacian on $L^p$ lies outside a parabola whenever the volume of the manifold has an exponential volume growth rate, removing the requirement on the manifold to be of bounded geometry. We conclude by providing a detailed description of the $L^p$ spectrum of the Laplacian on k-forms over hyperbolic space.