Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature

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 $g(\mathrm{\infty })$ on $M$ with $(M,g(\mathrm{\infty }))$ 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.