Wave decay on convex co-compact hyperbolic manifolds

TL;DR

This paper studies wave decay on convex co-compact hyperbolic manifolds, revealing how the decay rate depends on the Hausdorff dimension of the limit set and establishing the existence of infinitely many resonances.

Contribution

It provides a detailed asymptotic analysis of wave solutions, links decay behavior to conformal infinity, and proves the existence of infinitely many resonances in specific strips.

Findings

Wave decay rate depends on the Hausdorff dimension of the limit set.

Special cases where the leading term vanishes are explained via conformal theory.

Infinitely many resonances exist in certain complex strips, affecting wave decay.

Abstract

For convex co-compact hyperbolic quotients $X=\Gamma\backslash\hh^{n+1}$, we analyze the long-time asymptotic of the solution of the wave equation $u(t)$ with smooth compactly supported initial data $f=(f_0,f_1)$. We show that, if the Hausdorff dimension $\delta$ of the limit set is less than $n/2$, then $u(t) = C_\delta(f) e^{(\delta-\ndemi)t} / \Gamma(\delta-n/2+1) + e^{(\delta-\ndemi)t} R(t)$ where $C_{\delta}(f)\in C^\infty(X)$ and $||R(t)||=\mc{O}(t^{-\infty})$. We explain, in terms of conformal theory of the conformal infinity of $X$, the special cases $\delta\in n/2-\nn$ where the leading asymptotic term vanishes. In a second part, we show for all $\eps>0$ the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip $\{-n\delta-\eps<\Re(\la)<\delta\}$. As a byproduct we obtain a lower bound on the remainder $R(t)$ for generic initial data $f$.