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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cabezas-Rivas, E., Topping, P.M.: The canonical expanding soliton and Harnack inequalities for Ricci flow. To appear, Trans. Am. Math. Soc. http://www.warwick.ac.uk/~maseq
Chow B., Chu S.-C.: A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow. Math. Res. Lett. 2, 701–718 (1995)
Chow B., Chu S.-C.: Space–time formulation of Harnack inequalities for curvature rows of hypersurfaces. J. Geom. Anal. 11, 219–231 (2001)
Chow B., Knopf D.: New Li–Yau–Hamilton inequalities for the Ricci flow via the space–time approach. J. Differ. Geom. 60, 1–54 (2002)
Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci flow. Graduate Studies in Mathematics, 77. American Mathematical Society, Providence, RI; Science Press, New York (2006)
Gromov M.: Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13, 178–215 (2003)
Hamilton R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)
Hamilton R.S.: The Harnack estimate for the Ricci flow. J. Differ. Geom. 37, 225–243 (1993)
Hamilton, R.S.: The formation of singularities in the Ricci flow. Surveys in differential geometry, vol. II (Cambridge, 1993), pp. 7–136. International Press, Cambridge (1995)
Li P., Yau S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)
Lott J.: Optimal transport and Perelman’s reduced volume. Calc. Var. Partial Differ. Equ. 36, 49–84 (2009)
McCann R.J., Topping P.M.: Ricci flow, entropy and optimal transportation. Am. J. Math. 132, 711–730 (2010)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. http://arXiv.org/abs/math/0211159v1 (2002)
Perelman, G.: Ricci flow with surgery on three-manifolds. http://arxiv.org/abs/math/0303109v1 (2003)
Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. http://arXiv.org/abs/math/0307245v1 (2003).
Sturm K.-T., von Renesse M.-K.: Transport inequalities, gradient estimates, entropy and Ricci curvature. Commun. Pure Appl. Math. 58, 923–940 (2005)
Topping, P.M.: Lectures on the Ricci flow. L.M.S. Lecture Notes Series 325 C.U.P. http://www.warwick.ac.uk/~maseq/RFnotes.html (2006)
Topping P.M.: \({\mathcal{L}}\)-optimal transportation for Ricci flow. J. Reine Angew. Math. 636, 93–122 (2009)
Villani, C.: Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Huisken.
Rights and permissions
About this article
Cite this article
Cabezas-Rivas, E., Topping, P.M. The canonical shrinking soliton associated to a Ricci flow. Calc. Var. 43, 173–184 (2012). https://doi.org/10.1007/s00526-011-0407-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-011-0407-x