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These notes summarize the papers [, ] on the analysis of resolvent, Eisenstein series and scattering operator for geometrically finite hyperbolic quotients with rational non-maximal rank cusps. They complete somehow the talk given at the PDE seminar of Ecole Polytechnique in october 2005.
Following joint work with Dyatlov [], we describe the semi-classical measures associated with generalized plane waves for metric perturbation of , under the condition that the geodesic flow has trapped set of Liouville measure .
Let be a complete noncompact manifold of dimension at least 3 and an asymptotically conic metric on , in the sense that compactifies to a manifold with boundary so that becomes a scattering metric on . We study the resolvent kernel and Riesz transform of the operator , where is the positive Laplacian associated to and is a real potential function smooth on and vanishing at the boundary.
In our first paper we assumed that has neither zero modes nor a zero-resonance...
On geometrically finite hyperbolic manifolds , including those with non-maximal rank cusps, we give upper bounds on the number of resonances of the Laplacian in disks of size as . In particular, if the parabolic subgroups of satisfy a certain Diophantine condition, the bound is .
For odd-dimensional Poincaré–Einstein manifolds , we study the set of harmonic -forms (for ) which are (with ) on the conformal compactification of . This set is infinite-dimensional for small but it becomes finite-dimensional if is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology
and the kernel of the Branson–Gover [3] differential operators on the conformal infinity . We also relate the set of forms in the kernel of to the conformal...
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