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

Abstract

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 $R\to \infty$, the number of resonances in the box $[R,R+1]+i[-\beta ,0]$ is $\mathcal{O}(R^{m(\beta ,\delta )+})$, where the exponent $m(\beta ,\delta )=\min (2\delta +2\beta +1-n,\delta )$ changes its behavior at $\beta =\frac{n-1}{2}-\frac{\delta }{2}$. In the case $\delta <\frac{n-1}{2}$, we also give an improved resolvent upper bound in the standard resonance free strip $\{\Im \lambda >\delta -\frac{n-1}{2}\}$. Both results use the fractal uncertainty principle point of view recently introduced in [DyZa]. The appendix presents numerical evidence for the Weyl upper bound.