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

  • Carlo Mantegazza EMAIL logo and Reto MĂĽller

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 (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.

Funding source: Italian FIRB Ideas

Award Identifier / Grant number: Analysis and Beyond

Funding source: Imperial College Junior Research Fellowship

We thank Robert Haslhofer, Hans-Joachim Hein and Miles Simon for fruitful and interesting discussions. Both authors were partially supported by the Italian FIRB Ideas “Analysis and Beyond”. In addition, RM was partially financed by an Imperial College Junior Research Fellowship.

Received: 2012-12-7
Published Online: 2013-6-22
Published in Print: 2015-6-1

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