Abstract
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.
Citation
Miles Simon. Peter M. Topping. "Local control on the geometry in 3D Ricci flow." J. Differential Geom. 122 (3) 467 - 518, November 2022. https://doi.org/10.4310/jdg/1675712996
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