The spectrum of an asymptotically hyperbolic Einstein manifold

Abstract

This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its ``ideal boundary'' at infinity. It follows from work of R. Mazzeo that the essential spectrum of such a metric on an $(n+1)$-dimensional manifold is the ray $[n^2/4,\infty)$, with no embedded eigenvalues; however, in general there may be discrete eigenvalues below the continuous spectrum. The main result of this paper is that, if the Yamabe invariant of the conformal structure on the boundary is non-negative, then there are no such eigenvalues. This generalizes results of R. Schoen, S.-T. Yau, and D. Sullivan for the case of hyperbolic manifolds.