Improved fractal Weyl bounds for hyperbolic manifolds. With an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich

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.