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 (2010) and Enders-Müller-Topping (2011) 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.

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-Mueller-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.

Perelman's entropy functional at type I singularities of the Ricci flow

MANTEGAZZA, Carlo Maria;
2015

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-Mueller-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.
File in questo prodotto:
File Dimensione Formato  
Wdensity.pdf

Accesso chiuso

Tipologia: Altro materiale allegato
Licenza: Non pubblico
Dimensione 455.99 kB
Formato Adobe PDF
455.99 kB Adobe PDF   Richiedi una copia
Crelle2013.pdf

accesso aperto

Descrizione: post-print full text
Tipologia: Altro materiale allegato
Licenza: Non pubblico
Dimensione 354.44 kB
Formato Adobe PDF
354.44 kB Adobe PDF

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/4908
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 12
  • ???jsp.display-item.citation.isi??? 14
social impact