Local control on the geometry in 3D Ricci flow

The geometry of a ball within a Riemannian manifold is coarsely controlled if it has a lower bound on its Ricci curvature and a positive lower bound on its volume. We prove that such coarse local geometric control must persist for a definite amount of time under three-dimensional Ricci flow, and leads to local $C/t$ decay of the full curvature tensor, irrespective of what is happening beyond the local region. As a by-product, our results generalise the Pseudolocality theorem of Perelman [19, §10.1 and §10.5] and Tian-Wang [25] in this dimension by not requiring the Ricci curvature to be almostpositive, and not asking the volume growth to be almost-Euclidean. Our results also have applications to the topics of starting Ricci flow with manifolds of unbounded curvature, to the use of Ricci flow as a mollifier, and to the well-posedness of Ricci flow starting with Ricci limit spaces. In [24] we use results from this paper to prove that 3D Ricci limit spaces are locally bi-Hölder equivalent to smooth manifolds, going beyond a full resolution of the conjecture of Anderson, Cheeger, Colding and Tian in this dimension.