Exotic open $4$-manifolds which are non-leaves

We study the possibility of realizing exotic smooth structures on punctured simply connected $4$-manifolds as leaves of a codimension one foliation on a compact manifold. In particular, we show the existence of uncountably many smooth open $4$-manifolds which are not diffeomorphic to any leaf of a codimension one transversely $C^2$ foliation on a compact manifold. These examples include some exotic ${\mathbb{R}}^4$’s and exotic cylinders $S^3\times \mathbb{R}$.