## RICCI FLOW, ENTROPY AND OPTIMAL TRANSPORTATION

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Citations: | 12 - 1 self |

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@MISC{Mccann_ricciflow,,

author = {J. Mccann and Peter M. Topping},

title = {RICCI FLOW, ENTROPY AND OPTIMAL TRANSPORTATION},

year = {}

}

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### Abstract

Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented n-dimensional manifold M. Suppose two families of normalized n-forms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ∆g(τ). If these n-forms represent two evolving distributions of particles over M, the minimum root-mean-square distance W2(ω(τ), ˜ω(τ), τ) to transport the particles of ω(τ) onto those of ˜ω(τ) is shown to be non-increasing as a function of τ, without sign conditions on the curvature of (M, g(τ)). Moreover, this contractivity property is shown to characterize supersolutions to the Ricci flow.

### Citations

454 | The entropy formula for the Ricci flow and its geometric applications, preprint, available at arXive:math.DG/0211159
- Perelman
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Citation Context ...attained by some joint probability π0. The structure of the minimising π0 will be recalled in the proofs below; it is not generally smooth. Following a construction from Perelman’s work on Ricci flow =-=[23]-=-, [29, Chapter 6], let ω(x, τ) evolve under the heat equation ∂ω (5) = ∆g(τ)ω, ∂τ where ∆g is the connection Laplacian with respect to g. This evolution preserves the total mass: d dτ ∫ M A ω = 0, and... |

396 |
Three-manifolds with positive Ricci curvature
- Hamilton
- 1982
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Citation Context ...kwards Ricci flow equation ∂g (1) = 2 Ric(g) ∂τ where Ric(g) is the Ricci tensor of g. Given terminal data g(τ2), such a family can always be constructed for τ1 sufficiently close to τ2 (see Hamilton =-=[12]-=-, DeTurck [10], [29, Ch. 5]). The geodesic distance d(x, y, τ) between two points x, y ∈ M, with respect to g(τ), evolves according to the formula ∣ ∣ (2) d 2 (x, y, τ) = inf σ(0)=x,σ(1)=y ∫ 1 0 dσ ∣ ... |

225 | The geometry of dissipative evolution equations: The porous medium equation
- Otto
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Citation Context ...ch build on the Riemannian adaptation by Otto & Villani [22] and Cordero-Erausquin, McCann & Schmuckenschläger [8] [9], of Jordan, Kinderlehrer & Otto’s gradient flow formulation of the dynamics [14] =-=[21]-=- from Euclidean space and McCann’s displacement convexity [19]. In the Euclidean context, W2-contractivity of the heat evolution was also established by Ambrosio, Gigli & Savaré [1] and Carrillo, McCa... |

164 |
Gradient flows in metric spaces and in the space of probability measures
- Ambrosio, Gigli, et al.
- 2005
(Show Context)
Citation Context ..., y, τ) between two points x, y ∈ M, with respect to g(τ), evolves according to the formula ∣ ∣ (2) d 2 (x, y, τ) = inf σ(0)=x,σ(1)=y ∫ 1 0 dσ ∣ ds ∣ where the infimum is taken over smooth curves σ : =-=[0, 1]-=- → M joining x to y. Similarly, given two Borel probability measures ν and ˜ν on M, the 2-Wasserstein distance W2(ν, ˜ν, τ) between them evolves according to its definition ∫ d 2 (x, y, τ) dπ(x, y). (... |

146 | Riemannian Geometry: A Modern Introduction - Chavel - 2006 |

132 | The variational formulation of the Fokker-Planck Equation
- Jordan, Kinderlehrer, et al.
- 1998
(Show Context)
Citation Context ..., which build on the Riemannian adaptation by Otto & Villani [22] and Cordero-Erausquin, McCann & Schmuckenschläger [8] [9], of Jordan, Kinderlehrer & Otto’s gradient flow formulation of the dynamics =-=[14]-=- [21] from Euclidean space and McCann’s displacement convexity [19]. In the Euclidean context, W2-contractivity of the heat evolution was also established by Ambrosio, Gigli & Savaré [1] and Carrillo,... |

119 | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev ineqaulity
- Otto, Villani
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Citation Context ...ablished assuming Ric ≥ 0: see e.g. Sturm & von Renesse [27], and the subsequent works of Lott & Villani [16] [17] and Sturm [24] [25] [26], which build on the Riemannian adaptation by Otto & Villani =-=[22]-=- and Cordero-Erausquin, McCann & Schmuckenschläger [8] [9], of Jordan, Kinderlehrer & Otto’s gradient flow formulation of the dynamics [14] [21] from Euclidean space and McCann’s displacement convexit... |

116 |
A computational fluid mechanics solution to the MongeKantorovich mass transfer problem
- Benamou, Brenier
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Citation Context ...epest descent of the Boltzmann-Shannon entropy ∫ (10) E(u) = (log u) u dV M with respect to 2-Wasserstein distance. In Euclidean space, (displacement) convexity of the entropy [19] along W2-geodesics =-=[3]-=- allowed Otto [21] to quantify rates of convergence to the heat kernel. This displacement convexity extends to Ricci non-negative manifolds [8] as conjectured by Otto & Villani [22], and actually char... |

110 | The formation of singularities in the Ricci flow - Hamilton - 1995 |

110 |
A convexity principle for interacting gases
- McCann
- 1997
(Show Context)
Citation Context ...nd Cordero-Erausquin, McCann & Schmuckenschläger [8] [9], of Jordan, Kinderlehrer & Otto’s gradient flow formulation of the dynamics [14] [21] from Euclidean space and McCann’s displacement convexity =-=[19]-=-. In the Euclidean context, W2-contractivity of the heat evolution was also established by Ambrosio, Gigli & Savaré [1] and Carrillo, McCann & Villani [6]. The connection between entropy, Ricci curvat... |

84 |
On the geometry of metric measure spaces
- Sturm
(Show Context)
Citation Context ...f the ordinary heat flow in a stationary metric, which can be established assuming Ric ≥ 0: see e.g. Sturm & von Renesse [27], and the subsequent works of Lott & Villani [16] [17] and Sturm [24] [25] =-=[26]-=-, which build on the Riemannian adaptation by Otto & Villani [22] and Cordero-Erausquin, McCann & Schmuckenschläger [8] [9], of Jordan, Kinderlehrer & Otto’s gradient flow formulation of the dynamics ... |

80 | Ricci curvature for metric-measure spaces via optimal transport. Preprint (2004), available online via http://www.umpa.ens-lyon.fr/∼cvillani
- Lott, Villani
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Citation Context ... 2-Wasserstein contractivity of the ordinary heat flow in a stationary metric, which can be established assuming Ric ≥ 0: see e.g. Sturm & von Renesse [27], and the subsequent works of Lott & Villani =-=[16]-=- [17] and Sturm [24] [25] [26], which build on the Riemannian adaptation by Otto & Villani [22] and Cordero-Erausquin, McCann & Schmuckenschläger [8] [9], of Jordan, Kinderlehrer & Otto’s gradient flo... |

72 |
Polar factorization of maps on Riemannian manifolds
- McCann
(Show Context)
Citation Context ... x ∣ yProof. (Lemma 8.) Before beginning the proof, we highlight a few implicit assertions within the statement of the lemma. First, we have defined π0 as the minimiser; uniqueness here follows from =-=[20]-=- because ω and ˜ω are smooth, and thus ν and ˜ν do not charge sets of zero volume. Second, the semiconvexity of E(u(s)) and the smoothness of u(0) and u(1) tacitly imply that u(s) ∈ L log L for each s... |

65 | M.: Diffusions hypercontractives. In: Séminaire de Probabilités XIX, number 1123 - Bakry, Emery - 1985 |

56 | Deforming metrics in the direction of their Ricci tensors. In ‘Collected papers on Ricci flow.’ Edited by
- DeTurck
- 2003
(Show Context)
Citation Context ...low equation ∂g (1) = 2 Ric(g) ∂τ where Ric(g) is the Ricci tensor of g. Given terminal data g(τ2), such a family can always be constructed for τ1 sufficiently close to τ2 (see Hamilton [12], DeTurck =-=[10]-=-, [29, Ch. 5]). The geodesic distance d(x, y, τ) between two points x, y ∈ M, with respect to g(τ), evolves according to the formula ∣ ∣ (2) d 2 (x, y, τ) = inf σ(0)=x,σ(1)=y ∫ 1 0 dσ ∣ ds ∣ where the... |

55 | A Riemannian interpolation inequality à la Borell, Brascamp and
- Cordero-Erausquin, McCann, et al.
(Show Context)
Citation Context ... [27], and the subsequent works of Lott & Villani [16] [17] and Sturm [24] [25] [26], which build on the Riemannian adaptation by Otto & Villani [22] and Cordero-Erausquin, McCann & Schmuckenschläger =-=[8]-=- [9], of Jordan, Kinderlehrer & Otto’s gradient flow formulation of the dynamics [14] [21] from Euclidean space and McCann’s displacement convexity [19]. In the Euclidean context, W2-contractivity of ... |

54 | Contractions in the 2-Wasserstein length space and thermalization of granular media
- Carrillo, McCann, et al.
(Show Context)
Citation Context ...an space and McCann’s displacement convexity [19]. In the Euclidean context, W2-contractivity of the heat evolution was also established by Ambrosio, Gigli & Savaré [1] and Carrillo, McCann & Villani =-=[6]-=-. The connection between entropy, Ricci curvature, and convergence of diffusion dates back at least to Bakry & Emery [2]. We remark that in our Ricci flow setting, no sign condition on the Ricci curva... |

53 |
Gradient flows
- Ambrosio, Gigli, et al.
(Show Context)
Citation Context ...) between two points x, y ∈ M, with respect to g(τ), evolves according to the formula (2) d 2 (x, y, τ) = inf σ(0)=x,σ(1)=y ∫ 1 ∣ 0 dσ ds 2 ∣ ds, g(τ) where the infimum is taken over smooth curves σ: =-=[0, 1]-=- → M joining x to y. Similarly, given two Borel probability measures ν and ˜ν on M, the 2-Wasserstein distance W2(ν, ˜ν, τ) between them evolves according to its definition (3) W 2 2(ν, ˜ν, τ) = ∫ inf... |

53 |
Renesse. Transport inequalities, gradient estimates, entropy, and Ricci curvature
- Sturm, Von
- 2005
(Show Context)
Citation Context ...defined by (3). This result should be compared to 2-Wasserstein contractivity of the ordinary heat flow in a stationary metric, which can be established assuming Ric ≥ 0: see e.g. Sturm & von Renesse =-=[27]-=-, and the subsequent works of Lott & Villani [16] [17] and Sturm [24] [25] [26], which build on the Riemannian adaptation by Otto & Villani [22] and Cordero-Erausquin, McCann & Schmuckenschläger [8] [... |

32 | Lectures on the Ricci flow - Topping |

16 | Weak curvature conditions and functional inequalities
- Lott, Villani
(Show Context)
Citation Context ...sserstein contractivity of the ordinary heat flow in a stationary metric, which can be established assuming Ric ≥ 0: see e.g. Sturm & von Renesse [27], and the subsequent works of Lott & Villani [16] =-=[17]-=- and Sturm [24] [25] [26], which build on the Riemannian adaptation by Otto & Villani [22] and Cordero-Erausquin, McCann & Schmuckenschläger [8] [9], of Jordan, Kinderlehrer & Otto’s gradient flow for... |

15 | Optimal transport and Perelman’s reduced volume
- Lott
- 2009
(Show Context)
Citation Context ...nt also allows one to recover essentially all of the monotonic quantities that Perelman introduced in [23] to study finite-time singularities of Ricci flow – see [30] and the subsequent paper of Lott =-=[18]-=- from 2008 where Perelman’s reduced volume was shown also to arise this way. That latter paper [18] also includes a new rigorous proof of the 2-Wasserstein contractivity on Ricci flows that makes up p... |

14 | Riemannian Geometry. Second Edition - Gallot, Hulin, et al. - 1990 |

10 | Prékopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields and optimal
- Cordero-Erausquin, McCann, et al.
(Show Context)
Citation Context ... ∈ K). By working directly with the definition of Jacobi fields, one can estimate, for x ∈ K, ∂ (20) 2 ( ) 2 ( ) ∣ ϕ 1 ∂ϕ dσ dσ ∣∣∣σ(x,F (s, x) ≥ (s, x) + Ric , ∂s2 n ∂s ds ds (x),s) as in Lemma 6 of =-=[9]-=- (c.f. [13, §17] or [11, (4.18)], say). One deduces first from this a lower bound for ∂2ϕ ∂s2 (s, x), uniformly in s ∈ [0, 1] and x ∈ K. (We are using the boundedness of the diameter of (M, g) here to... |

10 | An inequality for a functional of probability distributions and its application to Kac’s one-dimensional model of a Maxwellian gas. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 27 - Tanaka - 1973 |

10 | L-optimal transportation for Ricci flow
- Topping
(Show Context)
Citation Context ...e Free University (Berlin) where parts of this work were performed. Added in proof: Since this work appeared in preprint form in 2006, there have been several developments which we briefly survey. In =-=[30]-=- from 2007, a notion of L-optimal transportation was introduced and a contractivity result was proved that generalises the 2-Wasserstein contractivity on Ricci flows in this paper. That viewpoint also... |

5 | The Canonical Shrinking Soliton associated to a Ricci flow
- Cabezas-Rivas, Topping
- 2008
(Show Context)
Citation Context ...at makes up part of this paper. Finally it turns out to be fruitful to extend the results of this paper to 1-Wasserstein contractivity. The extension to this situation was made by Tom Ilmanen, and in =-=[4]-=- a new space-time Ricci soliton construction was made which was inspired by an attempt to reconcile this 1-Wasserstein contractivity with the L-Wasserstein contractivity results from [30]. 2. properti... |

2 |
Generalized ricci curvature bounds and convergence of metric measure spaces
- Sturm
(Show Context)
Citation Context ...activity of the ordinary heat flow in a stationary metric, which can be established assuming Ric ≥ 0: see e.g. Sturm & von Renesse [27], and the subsequent works of Lott & Villani [16] [17] and Sturm =-=[24]-=- [25] [26], which build on the Riemannian adaptation by Otto & Villani [22] and Cordero-Erausquin, McCann & Schmuckenschläger [8] [9], of Jordan, Kinderlehrer & Otto’s gradient flow formulation of the... |

2 |
A curvature-dimension condition for metric measure spaces
- Sturm
- 2006
(Show Context)
Citation Context ...ity of the ordinary heat flow in a stationary metric, which can be established assuming Ric ≥ 0: see e.g. Sturm & von Renesse [27], and the subsequent works of Lott & Villani [16] [17] and Sturm [24] =-=[25]-=- [26], which build on the Riemannian adaptation by Otto & Villani [22] and Cordero-Erausquin, McCann & Schmuckenschläger [8] [9], of Jordan, Kinderlehrer & Otto’s gradient flow formulation of the dyna... |

2 | Lectures on the Ricci flow, London Mathematical Society lecture note series 325 - Topping - 2006 |

1 | Fokker-Planck dynamics and entropies for normalized Ricci flow
- Carfora
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Citation Context ...inks between optimal transportation and Ricci flow, and has formally derived a result similar to our Corollary 1 [15]. We are also grateful to him for pointing out related developments due to Carfora =-=[5]-=-, and for comments on this work. This research was supported in part by Natural Sciences and Engineering Research Council of Canada Grant 217006-03, United States National Science Foundation Grant DMS... |

1 |
Talk at UCSB
- Lott
- 2005
(Show Context)
Citation Context ...f this work. John Lott has informed us that he has independently been considering the links between optimal transportation and Ricci flow, and has formally derived a result similar to our Corollary 1 =-=[15]-=-. We are also grateful to him for pointing out related developments due to Carfora [5], and for comments on this work. This research was supported in part by Natural Sciences and Engineering Research ... |