Resonances for manifolds hyperbolic at infinity: optimal lower bounds on order of growth

TL;DR

This paper proves that for a broad class of hyperbolic-at-infinity manifolds, the number of resolvent resonances grows at the maximal possible rate, establishing optimal lower bounds on their growth order.

Contribution

It establishes the generic maximal order of growth for the resonance counting function on conformally compact hyperbolic manifolds.

Findings

Resonance counting function has maximal order of growth $(n+1)$.

Maximal growth is generic among such manifolds.

Provides optimal lower bounds on resonance growth.

Abstract

Suppose that $(X, g)$ is a conformally compact $(n+1)$-dimensional manifold that is hyperbolic at infinity in the sense that outside of a compact set $K \subset X$ the sectional curvatures of $g$ are identically equal to minus one. We prove that the counting function for the resolvent resonances has maximal order of growth $(n+1)$ generically for such manifolds.