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We show that a complete Ricci flow of bounded curvature which begins from amanifold with a Ricci lower bound, local entropy bound, and small localscale-invariant integral curvature control will have global point-wisecurvature control at positive times. As applications, we obtain under similarassumptions a compactness result and a gap theorem for complete noncompactmanifolds with nonnegative Ricci Curvature.

Small curvature concentration and Ricci flow smoothing