Scattering theory without injectivity radius assumptions and spectral stability for the Ricci flow
by Batu Güneysu and Anton Thalmaier

We prove a completely new integral criterion for the existence and completeness of the wave operators W±(Δh,Δg,Ig,h) corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators Δj, j = g, h, that are induced by two quasi-isometric complete Riemannian metrics g and h on an open manifold M. In particular, this result provides a criterion for the absolutely continuous spectra of Δg and Δh to coincide. Our proof relies on estimates that are obtained using a probabilistic Bismut type formula for the gradient of a heat semigroup. Unlike all previous results, our integral criterion only requires some lower control on the Ricci curvatures and some upper control on the heat kernels, but no control at all on the injectivity radii. As a consequence, we obtain a stability result for the absolutely continuous spectrum under a Ricci flow.

The paper is available here:

Batu Güneysu
Anton Thalmaier

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