Appears in collection : Not Only Scalar Curvature Seminar

In this talk, we will show that the resolvent set of the Laplacian on $L^p$ integrable $k$-forms 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. Moreover, 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 we provide a detailed description of the $L^p$ spectrum of the Laplacian on $k$-forms over hyperbolic space. The above results are joint work with Zhiqin Lu. We will also see a recent result with Julie Rowlett for the $L^p$ spectrum of conformally compact manifolds.