Calculus of Variations and Geometric Measure Theory

C. Mantegazza - R. Müller

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

created by root on 17 May 2012
modified on 08 Jul 2022


Published Paper

Inserted: 17 may 2012
Last Updated: 8 jul 2022

Journal: J. Reine Angew. Math. (Crelle's Journal)
Volume: 703
Pages: 173-199
Year: 2015

ArXiv: 1205.4143 PDF


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.