The canonical shrinking soliton associated to a Ricci flow

To every Ricci flow on a manifoldMover a time interval I subset of R-, we associate a shrinking Ricci soliton on the space-time M x 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.