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We construct examples of codimension two hyperbolic link complements inclosed smooth 4-manifolds with homeomorphism type $\#_{2k}S^2 \times S^2$. Allour examples are based on a construction of J. Ratcliffe and S. Tschantz, whoconstructed 1171 non-compact finite volume hyperbolic 4-manifolds of minimalvolume. We then give necessary conditions for a closed smooth simply connected4-manifold to contain a codimension two link complement that admits ahyperbolic structure.

Complements of tori in $\#_{2k}S^2 \times S^2$ that admit a hyperbolic structure