Small curvature concentration and Ricci flow smoothing

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. As applications, we obtain under similar assumptions a compactness result and a gap theorem for complete noncompact manifolds with Ric ≥ 0.

Ricci curvature integrals, local functionals, and the Ricci flow

“…Actually, many important results based on integral Ricci lower bound were established by the work of Petersen-Wei [32] [33], D. Yang [45] [46] [47], S. Gallot [18], etc. The Ricci flow behavior with initial data satisfying Ricci or other curvature's integral pinching conditions was also studied by many people, e.g., see [43] [44] [6]. Under an appropriate integral Ricci curvature condition(cf.…”