Scattering resonances in the vanishing curvature limit

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On a closed Riemannian manifold, the eigenvalues and eigenfunctions of the Laplacian encode information about the geometry of the manifold. On an open manifold the spectrum of the Laplacian is not discrete in general, however in certain circumstances there may exist a class of objects called "scattering resonances" which play a somewhat analogous role to the eigenvalues in the closed setting. In particular, Mazzeo-Melrose showed that scattering resonances can be defined for manifolds whose open end is asymptotic to a space of constant negative curvature by means of an analytic continuation argument. More recently Vasy has developed a robust Fredholm theory for the scattering resonances, making use of the tools of microlocal analysis. In the limit where the curvature at infinity tends to zero these methods break down, but nevertheless it is known that at least some of the scattering resonances are well behaved in the limit. The goal of this project is to see whether methods in the spirit of Vasy's work can be developed that permit the limit where the curvature at infinity vanishes to be studied.