  We prove a curvature pinching result for the Ricci flow on asymptotically
flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has
scale-invariant integral norm of curvature sufficiently pinched relative to the
inverse of its Sobolev constant, then the Ricci flow starting from this
manifold exists for all positive times and converges to flat Euclidean space.
In particular our result implies that the initial manifold must have been
diffeomorphic to $\mathbb{R}^n$.
