JournalsjemsVol. 18, No. 5pp. 997–1041

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

  • David Borthwick

    Emory University, Atlanta, USA
  • Colin Guillarmou

    Ecole Normale Superieure, Paris, France
Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds cover
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Abstract

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

Cite this article

David Borthwick, Colin Guillarmou, Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds. J. Eur. Math. Soc. 18 (2016), no. 5, pp. 997–1041

DOI 10.4171/JEMS/607