  We prove long-time existence of the Ricci flow starting from complete
manifolds with bounded curvature and scale-invariant integral curvature
sufficiently pinched with respect to the inverse of its Sobolev constant.
Moreover, if the curvature is sub-critical $L^p$ integrable, this flow
converges locally smoothly to a limiting metric $g(\infty)$ on $M$ with
$(M,g(\infty))$ isometric to the standard flat $\mathbb{R}^n$, which implies
topological rigidity of $M$. This generalizes work of Chen, who proved
analogous results for asymptotically flat manifolds. We also prove a long-time
Ricci flow existence (and likewise topological rigidity) result for unbounded
curvature initial data, assuming the initial data is a locally smooth limit of
bounded curvature manifolds as described above.
