Calculus of Variations and Partial Differential Equations

, Volume 43, Issue 1, pp 173–184

The canonical shrinking soliton associated to a Ricci flow


DOI: 10.1007/s00526-011-0407-x

Cite this article as:
Cabezas-Rivas, E. & Topping, P.M. Calc. Var. (2012) 43: 173. doi:10.1007/s00526-011-0407-x


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)


Copyright information

© Springer-Verlag 2011

Authors and Affiliations

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