Eigenfunction Expansions for Hyperbolic Laplacians
The geometry near infinity of a non-compact complete Riemannian manifold is reflected in the essential spectrum its Laplace operator. The nature of this correspondence, however, is subtle, and the manifolds whose Laplace operators have a well understood spectral theory are ones whose structure near infinity is simple. In this lecture I would like to explain some recent work of P. Hislop, P. Perry and myself on the Laplacian for three-dimensional non-compact hyperbolic manifolds. These manifolds are are simple near infinty, in the sense of being geometrically finite, but somewhat less simple than the ones that had been considered previously, in that they have cusps of non-maximal rank.
KeywordsLaplace Operator Eisenstein Series Hyperbolic Manifold Kleinian Group Eigenfunction Expansion
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