Skip to main content Accessibility help
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 1
  • Print publication year: 2014
  • Online publication date: August 2014

5 - Ricci flow: the foundations via optimal transportation




Since the creation of Ricci flow by Hamilton in 1982, a rich theory has been developed in order to understand the behaviour of the flow, and to analyse the singularities that may occur, and these developments have had profound applications, most famously to the Poincaré conjecture. At the heart of the theory lie a large number of a priori estimates and geometric constructions, which include most notably the Harnack estimates of Hamilton, the L-length of Perelman (in the spirit of Li-Yau), the logarithmic Sobolev inequality arising from Perelman's W-entropy, and the reduced volume of Perelman, amongst others.

The objective of these lectures is to explain this theory from the point of view of optimal transportation. As I explain in Section 5.4, Ricci flow and optimal transportation combine rather well, and we will see fundamental but elementary aspects of this when we see in Theorem 5.2 how diffusions contract under reverse-time Ricci flow. However, the key to the whole theory is to realise to which object one should apply this result: not the original Ricci flow, but a new Ricci flow derived from the original one, on a base manifold of one higher dimension, that we call the canonical soliton. In this way, essentially the entire foundational theory of Ricci flow mentioned above drops out naturally.

Throughout the lectures I emphasise the intuition; the objective is to demonstrate how one can discover the theory rather than treat it as a black box that just happens to work.

[1] M., Arnaudon, A., Coulibaly and A., Thalmaier, Horizontal diffusion in C1 path space. Séminaire de Probabilités XLIII, pp. 73-94, Lecture Notes in Mathematics 2006, Springer (2010).
[2] S., Brendle, A generalization of Hamilton's differential Harnack inequality for the Ricci flow. J. Differential Geom., 82 (2009), 207–227.
[3] S., Brendle and R., Schoen, Manifolds with 1/4-pinched curvature are space forms. J. Am. Math. Soc., 22 (2009) 287-307.
[4] E., Cabezas-Rivas and P.M., Topping, The canonical shrinking soliton associated to a Ricci flow. Calc.Var. PDE, 43 (2012) 173–184.
[5] E., Cabezas-Rivas and P.M., Topping, The canonical expanding soliton and Harnack inequalities for Ricci flow. Trans. Am. Math. Soc., 364 (2012) 3001–3021.
[6] B., Chow and S.-C., Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow. Math. Res. Lett., 2 (1995) 701-718.
[7] B., Chow and D., Knopf, New Li-Yau-Hamilton inequalities for the Ricci flow via the space-time approach. J. Differential Geom., 60 (2002), 1–54.
[8] S., Gallot, D., Hulin and J., Lafontaine, Riemannian Geometry (second edition), Springer-Verlag (1993).
[9] R.S., Hamilton, Three-manifolds with positive Ricci curvature. J. Differential Geom., 17 (1982) 255-306.
[10] R.S., Hamilton, The Harnack estimate for the Ricci flow. J. Differential Geom., 37 (1993) 225-243.
[11] R.S., Hamilton, The formation of singularities in the Ricci flow. Surveys in Differential Geometry, Vol. II (Cambridge, MA, 1993), pp. 7-136, International Press, Cambridge, MA, 1995.
[12] S., Helmensdorfer and P.M., Topping, The geometry of differential Harnack estimates. Act. Semin Theor. Spectr. Geom. [Grenoble 2011-2012], 30 (2013) 77-89.
[13] P., Li and S.-T., Yau, On the parabolic kernel of the Schrödinger operator. Acta Math., 156 (1986) 153–201.
[14] J., Lott, Optimal transport and Perelman's reduced volume. Calc. Var. Partial Dif. Equations, 36 (2009) 49-84.
[15] R.J., McCann and P.M., Topping, Ricci flow, entropy and optimal transportation. Am. J. Math., 132 (2010) 711-730.
[16] H., Nguyen, Invariant curvature cones and the Ricci flow. PhD thesis, Australian National University (2007).
[17] F., Otto and C., Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal., 173 (2000) 361-400.
[18] G., Perelman, The entropy formula for the Ricci flow and its geometric applications. (2002).
[19] G., Perelman, Ricci flow with surgery on three-manifolds. (2003).
[20] G., Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. (2003).
[21] K.-T., Sturm and M.-K., von Renesse, Transport inequalities, gradient estimates, entropy and Ricci curvature. Commun. Pure Appl. Math., 58 (2005) 923-940.
[22] P.M., Topping, Diameter control under Ricci flow. Commun. Anal. Geom., 13 (2005) 1039-1055.
[23] P.M., Topping, ‘Lectures on the Ricci flow.’ L.M.S. Lecture Note Series 325, C.U.P. (2006)
[24] P.M., Topping, ℒ-optimal transportation for Ricci flow. J. Reine Angew. Math., 636 (2009) 93-122.
[25] C., Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58 American Mathematical Society (2003).
[26] Z.-H., Zhang, On the completeness of gradient Ricci solitons. Proc. Am. Math. Soc., 137 (2009) 2755–2759.