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

摘要

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 delta of its limit set. More precisely, we show that as R ->infinity, the number of resonances in the box [R, R+1]+i[-beta, 0] is O(R-m(beta,R- delta)+), where the exponent m(beta, delta) = min(2 delta+ 2 beta +1- n, delta) changes its behavior at beta = (n - 1)/2 - delta/2. In the case S < (n - 1)/2, we also give an improved resolvent upper bound in the standard resonance free strip {Im lambda > delta - (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.