Local control on the geometry in 3D Ricci flow

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 lea...

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Hauptverfasser: Simon, Miles, Topping, Peter M
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Mathematics - Differential Geometry
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description 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.
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Mathematics - Differential Geometry
title Local control on the geometry in 3D Ricci flow
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