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The geometry of a ball within a Riemannian manifold is coarsely controlled ifit has a lower bound on its Ricci curvature and a positive lower bound on itsvolume. We prove that such coarse local geometric control must persist for adefinite amount of time under three-dimensional Ricci flow, and leads to localC/t decay of the full curvature tensor, irrespective of what is happeningbeyond the local region.  As a by-product, our results generalise the Pseudolocality theorem ofPerelman and Tian-Wang in this dimension by not requiring the Ricci curvatureto be almost-positive, and not asking the volume growth to be almost-Euclidean.

Local control on the geometry in 3D Ricci flow