Impossible Metric Conditions on Exotic R4's

Abstract

There are many theorems in the differential geometry literature of the following sort.

Let M be a complete Riemannian manifold with some conditions on various curvatures, diameters, volumes, etc. Then M is homotopy equivalent to a finite CW complex, or M is the interior of a compact, topological manifold with boundary.

At first glance it seems unlikely that such theorems have anything to say about smooth manifolds homeomorphic to $\mathbb{R}^4$. However, there is a common theme to all the proofs which forbids the existence of such metrics on most (and possibly all) exotic $\mathbb{R}^4$’s.