  In this paper we prove that in a three-manifold with finitely many expansive
ends, such that each end has a neighborhood where the curvature is bounded
above by a negative constant, the Dirichlet problem at infinity is solvable,
and hence that such manifolds posses a wealth of bounded non constant harmonic
functions (and thus, Liouville's theorem does not hold). In the case of
infinitely many expansive ends, we show that if each end has a neighborhood
where the curvature is bounded above by a negative constant, then the Dirichlet
problem at infinity is solvable for continuous boundary data at infinity which
is uniformly bounded. Our method is based on a result that does not require
explicit curvature assumptions, and hence it can be applied to other
situations: we present an example of a metric on an end with curvature of
indefinite sign (no matter how long we go along the end) for which the
Dirichlet Problem at Infinity is solvable with respect to that end. We also
present a related result in the case of surfaces with a pole which generalises
a celebrated criteria of Milnor.
