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Let M be a compact n + 1-dimensional manifold with boundary S. A Riemannian metric g on the interior M is conformally compact if for any defining function r of the boundary g = rg extends to a C metric on M. The conformal class of the metric g|s is invariantly defined and is called the conformal infinity of g. We also assume that |dr|| = 1 on E. This condition is invariantly defined and is satisfied if g is conformally compact Einstein, i.e., Ric(flf) = -ng. It was proved by Mazzeo [7] that the continuous spectrum of the Laplacian on such a Riemannian manifold consists of the ray [n/4,oo) with no embedded eigenvalues; however, in general there may be finitely many eigenvalues in the interval (0,n/4). If the metric is conformally compact Einstein, one might expect a relationship between its spectrum and global conformal invariants of its conformal infinity. This is justified by the following beautiful theorem proved by Lee [6].

A new proof of Lee’s theorem on the spectrum of conformally compact Einstein manifolds