Wave Decay on Convex Co-Compact Hyperbolic Manifolds

Abstract

For convex co-compact hyperbolic quotients ${X=\Gamma\backslash\mathbb {H}^{n+1}}$ , 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 ${u(t)=C_\delta(f)e^{(\delta-\frac{n}{2})t}/\Gamma(\delta-n/2+1)+e^{(\delta-\frac{n}{2})t}R(t),}$ where ${C_{\delta}(f)\in C^\infty(X)}$ and ${||R(t)||=\mathcal{O}(t^{-\infty})}$ . We explain, in terms of conformal theory of the conformal infinity of X, the special cases ${\delta\in n/2-\mathbb {N}}$ , where the leading asymptotic term vanishes. In a second part, we show for all ${\epsilon > 0}$ the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip ${\{-n\delta-\epsilon < \rm Re(\lambda) < \delta\}}$ . As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f.