Perelman's functional and reduced volume

TL;DR

This paper surveys recent progress on Perelman's functional, reduced volume, and reduced length, focusing on local non-collapsing results, injectivity radius bounds, and the gradient flow perspective of Ricci flow.

Contribution

It provides a comprehensive survey of recent developments related to Perelman's work on Ricci flow, including new results on non-collapsing and geometric bounds.

Findings

Establishment of local non-collapsing results for Ricci flow

Positive lower bounds on injectivity radius under blow-up analysis

Insights into the gradient flow formalism and physical motivation of Ricci flow

Abstract

In recent years, there has seen much interest and increased research activities on Perelman's paper. Section one and two of this paper aim to establish Perelman's local non-collapsing result for the Ricci flow. This will provide a positive lower bound on the injectivity radius for the Ricci flow under blow-up analysis. We also discuss the gradient flow formalism of the Ricci flow and Perelman's motivation from physics. In this paper, we survey some of the recent progress on Perelman's functional, reduced volume and reduced length.