The main result is that for a connected hyperbolic complete Kahler manifold with bounded geometry of order two and exactly one end, either the first compactly supported cohomology with values in the structure sheaf vanishes or the manifold admits a proper holomorphic mapping onto a Riemann surface.
Simple approaches to the proofs of the L^2 Castelnuovo-de Franchis theorem and the cup product lemma which give n
