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

TL;DR

This paper establishes upper bounds on the number of resonances for the Laplacian on geometrically finite hyperbolic manifolds, including those with non-maximal rank cusps, under certain Diophantine conditions.

Contribution

It provides the first known upper bounds for resonance counts on these manifolds, extending previous results to include non-maximal rank cusps and specific Diophantine conditions.

Findings

Resonance count grows at most like R^d (log R)^{d+1} under certain conditions

Includes manifolds with non-maximal rank cusps in the analysis

Provides explicit upper bounds for resonance distribution

Abstract

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