An $\epsilon$-regularity theorem for Perelman's reduced volume

TL;DR

This paper establishes an $psilon$-regularity theorem for Perelman's reduced volume, demonstrating that near-maximal reduced volume implies lower bounds on curvature radius in Ricci flow.

Contribution

It provides a new regularity criterion linking reduced volume closeness to 1 with curvature bounds in Ricci flow, enhancing understanding of geometric regularity.

Findings

Reduced volume close to 1 implies curvature radius bounds.

Established $psilon$-regularity criterion for Ricci flow.

Improved understanding of geometric regularity in Ricci flow.

Abstract

In this article, we prove an $\epsilon$-regularity theorem for Perelman's reduced volume. We show that on a Ricci flow, if Perelman's reduced volume is close to $1$, then the curvature radius at the base point cannot be too small.