The $L^p$-spectrum of the Laplacian on forms over warped products and Kleinian groups

Abstract

In this article, we generalize the set of manifolds over which the $L^p$-spectrum of the Laplacian on $k$-forms depends on $p$. We will consider the case of manifolds that are warped products at infinity and certain quotients of Hyperbolic space. In the case of warped products at infinity we prove that the $L^p$-spectrum of the Laplacian on $k$-forms contains a parabolic region which depends on $k$, $p$ and the limiting curvature $a_0$ at infinity. For $M=\mathbb{H}^{N+1}/\Gamma $ with $\Gamma$ a geometrically finite group such that $M$ has infinite volume and no cusps, we prove that the $L^p$-spectrum of the Laplacian on $k$-forms is a exactly a parabolic region together with a set of isolated eigenvalues on the real line.