Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds

2013/03/29 by David Borthwick, Colin Guillarmou, Borthwick, David +1
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Spectral Theory in Mathematical Physics #math.DG #math.SP #msc:58J50

paper · pdf · doi:10.48550/arxiv.1303.7471

40 pages. Updated with a minor correction to the main estimate plus a new estimate

arxiv created 2013/04/17 · arxiv updated 2013/04/18

Abstract

On geometrically finite hyperbolic manifolds Γ\backslash H^d, including those with non-maximal rank cusps, we give upper bounds on the number N(R) of resonances of the Laplacian in disks of size R as R→ ∞. In particular, if the parabolic subgroups of Γ satisfy a certain Diophantine condition, the bound is N(R)= O(R^d (log R)^d+1).

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