Perelman's Entropy Functional at Type I Singularities of the Ricci Flow

TL;DR

This paper investigates the behavior of Ricci flow singularities using Perelman's entropy functional, providing new insights into the nature of blow-up limits and their entropy properties at Type I singularities.

Contribution

It offers an alternative proof that blow-up limits are non-flat gradient shrinking Ricci solitons and relates the limit W-density to the entropy of these solitons.

Findings

Blow-up limits are non-flat gradient shrinking Ricci solitons.

No entropy is lost during the blow-up process.

A relation between limit W-density and soliton entropy is established.

Abstract

We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber and Enders-M\"{u}ller-Topping that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result relates a limit W-density at a Type I singular point to the entropy of the limit gradient shrinking soliton obtained by blowing-up at this point. In particular, we show that no entropy is lost at infinity during the blow-up process.