Calculus of Variations and Partial Differential Equations

, Volume 43, Issue 1, pp 173-184

First online:

The canonical shrinking soliton associated to a Ricci flow

  • Esther Cabezas-RivasAffiliated withMathematisches Institut, Westfälische Wilhelms-Universität Münster Email author 
  • , Peter M. ToppingAffiliated withMathematics Institute, University of Warwick

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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)