Wave Decay on Convex Co-Compact Hyperbolic Manifolds

摘要

For convex co-compact hyperbolic quotients , 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 where and . We explain, in terms of conformal theory of the conformal infinity of X, the special cases , where the leading asymptotic term vanishes. In a second part, we show for all 0}$$" align="middle" border="0"> the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip src="/content/Q752K318970X078W/220_2008_706_Article_IEq6.gif" alt="$${{-ndelta-epsilon < rm Re(lambda) . As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f. Communicated by P. Sarnak