We give necessary conditions, for a closed smooth simply connected 4-manifold X, to contain a collection of surfaces L such that X-L admits a complete finite volume hyperbolic structure. We then show that examples of non-compact hyperbolic 4-manifolds constructed by Ivanšić, and Ivanšić, Ratcliffe and Tschantz, give rise to examples of such link complements in #_2kS² x S².

The author thanks Andras Juhasz, Marc Lackenby, and John Parker for their valuable comments on an earlier version of this work. A thanks must also be given to Igor Belegradek for providing a useful reference on eta invariants. Finally, the author wishes to thank the referees for their comments and corrections to the paper.