Abstract
We show that a complete Ricci flow of bounded curvature which begins from a manifold with a Ricci lower bound, local entropy bound, and small local scale-invariant integral curvature control will have global point-wise curvature control at positive times which only depends on the initial almost Euclidean structure. As applications, we use the Ricci flows to study the diffeomorphism type of manifolds and the regularity of Gromov-Hausdorff limit of manifolds with small curvature concentration.
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