On the Dirichlet problem at infinity on three-manifolds of negative curvature

In this paper we prove that for a three-manifold with finite expansive ends and curvature bounded above by a negative constant, the Dirichlet problem at infinity can be solved, and hence that such manifolds posses a wealth of bounded non constant harmonic functions. In the case of infinitely many expansive ends, we show that the Dirichlet problem at infinity is solvable for continuous boundary data at infinity which is bounded from below.