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.
In this note we relate the geometric notion of fill radius with the fundamental group of the manifold. We prove: Suppose that a closed Riemannian manifold M satisfies the property that its universal cover has bounded fill radius. Then the fundamental
group of M is virtually free. We explain the relevance of this theorem to some conjectures on positive isotropic curvature and 2-positive Ricci curvature.
Simple approaches to the proofs of the L^2 Castelnuovo-de Franchis theorem and the cup product lemma which give n
