Smooth Euclidean 4-spaces with few symmetries

We say that a topologically embedded 3-sphere in a smoothing of Euclidean 4-space is a barrier provided, roughly, no diffeomorphism of the 4-manifold moves the 3-sphere off itself. In this paper we construct infinitely many one parameter families of distinct smoothings of 4-space with barrier 3-spheres. \par The existence of barriers implies, amongst other things, that the isometry group of these manifolds, in any smooth metric, is finite. In particular, S^1 can not act smoothly and effectively on any smoothing of 4-space with barrier 3-spheres.