The Bochner-Hartogs dichotomy for bounded geometry hyperbolic Kähler manifolds
arXiv:1506.04328
Abstract
The main result is that for a connected hyperbolic complete Kähler 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.