The canonical shrinking soliton associated to a Ricci flow

Article

Abstract

To every Ricci flow on a manifold \({\mathcal{M}}\) over a time interval \({I\subset\mathbb{R}_-}\), we associate a shrinking Ricci soliton on the space–time \({\mathcal{M}\times I}\). We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the results of the second author Topping (J Reine Angew Math 636:93–122, 2009), and McCann and the second author (Am J Math 132:711–730, 2010); we briefly survey the link between these subjects.

Mathematics Subject Classification (2000)

53C44 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Mathematisches InstitutWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

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