Improved fractal Weyl bounds for hyperbolic manifolds. With an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich
Name
1512.00836.pdf
Description
Submitted version
Size
5.12 MB
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Adobe PDF
Checksum (MD5)
56ae83cb5b5c51b08c1d3fc0be2c8a81
Author(s)
Dyatlov, Semyon
Borthwick, David
Weich, Tobias
Date Issued
2019
Journal
Journal of the European Mathematical Society
Publisher
European Mathematical Publishing House
Version
Original manuscript
Abstract
© European Mathematical Society 2019. We give a new fractal Weyl upper bound for resonances of convex co-compact hyperbolic manifolds in terms of the dimension n of the manifold and the dimension δ of its limit set. More precisely, we show that as R → ∞, the number of resonances in the box [R, R+1]+i[−β, 0] is O(R m(β,δ)+ ), where the exponent m(β, δ) = min(2δ + 2β + 1 − n, δ) changes its behavior at β = (n − 1)/2 − δ/2. In the case δ < (n − 1)/2, we also give an improved resolvent upper bound in the standard resonance free strip {Im λ > δ − (n − 1)/2}. Both results use the fractal uncertainty principle point of view recently introduced in [DyZa]. The appendix presents numerical evidence for the Weyl upper bound.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.4171/JEMS/867
