Local control on the geometry in 3D Ricci flow
Differential Geometry
2017-07-03 v2 Analysis of PDEs
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 and Tian-Wang in this dimension by not requiring the Ricci curvature to be almost-positive, and not asking the volume growth to be almost-Euclidean.
Cite
@article{arxiv.1611.06137,
title = {Local control on the geometry in 3D Ricci flow},
author = {Miles Simon and Peter M. Topping},
journal= {arXiv preprint arXiv:1611.06137},
year = {2017}
}
Comments
Minor updates in order to integrate with arXiv:1706.09490 prior to being submitted