The Laplace spectrum on conformally compact manifolds

Abstract

We consider the spectrum of the Laplace operator acting on L p \mathcal {L}^p over a conformally compact manifold for 1 ≤ p ≤ ∞ 1 \leq p \leq \infty . We prove that for p ≠ 2 p \neq 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 p , the dimension of the manifold, and the values of the sectional curvatures approaching the boundary.