Wave Decay on Convex Co-Compact Hyperbolic Manifolds

For convex co-compact hyperbolic quotients X = Gamma\Hn+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-(n/2)t)/Gamma(delta - n/2 + 1) + e((delta- n/2)t) R(t), where C-delta(f) is an element of C-infinity(X) and ||R(t)|| = O(t(-infinity)). We explain, in terms of conformal theory of the conformal infinity of X, the special cases delta is an element of n/2-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 < Re(lambda) < delta}. As a byproduct we obtain a lower bound on the remainder R( t) for generic initial data