Local control on the geometry in 3D Ricci flow

TL;DR

This paper proves that local geometric bounds in 3D Riemannian manifolds persist under Ricci flow, ensuring curvature decay and generalizing pseudolocality without strict curvature or volume conditions.

Contribution

It establishes local control of geometry under Ricci flow in three dimensions without near-positivity or Euclidean volume assumptions, extending previous pseudolocality results.

Findings

Local geometric bounds persist over time under Ricci flow.

Curvature decays at a rate of C/t locally.

Results generalize pseudolocality theorem without strict curvature or volume conditions.

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.