  Let (X,g) be a metrically complete, simply connected Riemannian manifold with
bounded geometry and pinched negative curvature, i.e. there are constants a>b>0
such that -a^2<K<-b^2 for all sectional curvatures K. Here bounded geometry is
used in the sense that all covariant derivatives of the Riemannian curvature
tensor are bounded and the injectivity radius is uniformly bounded below by a
positive constant. We show that there are no superexponentially decaying
eigenfunctions of the Laplacian of g. We also show the analogous conclusion for
other geometric operators, and prove a theorem with the assumptions and
conclusions localized near infinity.

Analysis of PDEs;Differential Geometry