The Library

L-optimal transportation for Ricci flow

Tools

Topping, Peter. (2009) L-optimal transportation for Ricci flow. Journal fur die reine und angewandte Mathematik, Vol.2009 (No.636). pp. 93-122. ISSN 0075-4102

Preview

PDF
CRELLE.2009.083.pdf - Requires a PDF viewer.
Download (350Kb)

Official URL: http://dx.doi.org/10.1515/CRELLE.2009.083

Abstract

We introduce the notion of L-optimal transportation, and use it to construct a natural monotonic quantity for Ricci flow which includes a selection of other monotonicity results, including some key discoveries of Perelman [13] (both related to entropy and to L-length) and a recent result of McCann and the author [11].

Item Type:

Journal Article

Subjects:

Q Science > QA Mathematics

Divisions:

Faculty of Science > Mathematics

Library of Congress Subject Headings (LCSH):

Ricci flow

Journal or Publication Title:

Journal fur die reine und angewandte Mathematik

Publisher:

Walter de Gruyter & Co

ISSN:

0075-4102

Official Date:

November 2009

Dates:

Date	Event
November 2009	Published

Volume:

Vol.2009

Number:

No.636

Number of Pages:

Page Range:

pp. 93-122

DOI:

10.1515/CRELLE.2009.083

Status:

Peer Reviewed

Publication Status:

Published

Access rights to Published version:

Open Access

References:

[1] V. Bangert, Analytische Eigenschaften konvexer Funktionen auf Riemannschen Mannigfaltigkeiten, J. reine
angew. Math. 307 (1979), 309–324.
[2] J. D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass
transfer problem, Numer. Math. 84 (2000), 375–393.
[3] P. Bernard and B. Bu¤oni, Optimal mass transportation and Mather theory, J. Eur. Math. Soc. 9 (2007),
85–121.
[4] I. Chavel, Riemannian geometry: a modern introduction, Cambridge tracts math. 108 (1995).
[5] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo and L. Ni,
Ricci flow: Techniques and Applications: Part I: Geometric aspects, Math. Surv. Monogr. 135 (2007).
[6] D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschla¨ger, A Riemannian interpolation inequality a`
la Borel, Brascamp and Lieb, Invent. Math. 146 (2001), 219–257.
[7] D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschla¨ger, Pre´kopa-Leindler type inequalities on
Riemannian manifolds, Jacobi fields, and optimal transport, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006),
613–635.
[8] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Di¤. Geom. 17 (1982), 255–306.
[9] R. S. Hamilton, The Harnack estimate for the Ricci flow, J. Di¤. Geom. 37 (1993), 225–243.
[10] R. J. McCann, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal. 11 (2001),
589–608.
[11] R. J. McCann and P. M. Topping, Ricci flow, entropy and optimal transportation, Amer. J. Math., to appear.
[12] 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.
[13] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, http://arXiv.org/
math.DG/0211159v1, 2002.
[14] G. Perelman, Ricci flow with surgery on three-manifolds, http://arXiv.org/math.DG/0303109v1, 2003.
[15] K.-T. Sturm and M.-K. von Renesse, Transport inequalities, gradient estimates, entropy and Ricci curvature,
Comm. Pure Appl. Math. 58 (2005), 923–940.
[16] P. M. Topping, Lectures on the Ricci flow, L.M.S. Lect. notes ser. 325 (2006).
[17] C. Villani, Topics in optimal transportation, Grad. Stud. Math. 58 (2003).
[18] C. Villani, Optimal transport, old and new (Saint-Flour 2005), Version of October 2007.
[19] R. Ye, On the l-function and the reduced volume of Perelman, preprint 2004.