Wave decay on convex co-compact hyperbolic manifolds

Abstract

For convex co-compact hyperbolic quotients $X=Γ\backslash\hh^{n+1}$, 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 $δ$ of the limit set is less than $n/2$, then $u(t) = C_δ(f) e^{(δ-\ndemi)t} / Γ(δ-n/2+1) + e^{(δ-\ndemi)t} R(t)$ where $C_δ(f)\in C^\infty(X)$ and $||R(t)||=\mc{O}(t^{-\infty})$. We explain, in terms of conformal theory of the conformal infinity of $X$, the special cases $δ\in n/2-\nn$ 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 $\{-nδ-\eps<\Re(\la)<δ\}$. As a byproduct we obtain a lower bound on the remainder $R(t)$ for generic initial data $f$.