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

Abstract: 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.