The Laplace spectrum on conformally compact manifolds

We consider the spectrum of the Laplace operator acting on ${\mathcal{L}}^p$ over a conformally compact manifold for $1\le p\le \mathrm{\infty }$. We prove that for $p\mathrm{\ne }2$ this spectrum always contains an open region of the complex plane. We further show that the spectrum is contained within a certain parabolic region of the complex plane. These regions depend on the value of $p$, the dimension of the manifold, and the values of the sectional curvatures approaching the boundary.