We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that the expectation of their normalized L-distance is non-increasing. As an immediate corollary we obtain a new proof of a recent result of Topping (J. reine angew. Math. 636 (2009), 93-122), namely that the normalized L-transportation cost between two solutions of the heat equation is non-increasing as well.
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@cite_1: e Conjecture, and more generally Thurston’s Geometrization Conjecture, using the Ricci flow approach of Hamilton. Perelman’s proofs are concise and, at times, sketchy. The purpose of these notes is to provide the details that are missing in [46] and [47], which contain Perelman’s arguments for the Geometrization Conjecture. Among other things, we cover the construction of the Ricci flow with surgery of [47]. We also discuss the long-time behavior of the Ricci flow with surgery, which is needed for the full Geometrization Conjecture. The papers of Colding and Minicozzi [23; 24] and Perelman [48], which are not covered in these notes, each provide a shortcut in the case of the Poincar´ e Conjecture. Namely, these papers show that if the initial manifold is simply-connected then the Ricci flow with surgery becomes extinct in a finite time, thereby removing the issue of the long-time behavior. Combining this claim with the proof of existence of Ricci flow with surgery gives the shortened proof in the simply-connected case. These notes are intended for readers with a solid background in geometric analysis. Good sources for background material on Ricci flow are Chow and Knopf [21], Chow, Lu and Ni [22], Hamilton [30] and Topping [60]. The notes are self-contained but are designed to be read along with [46; 47]. For the most part we follow the format of [46; 47] and use the section numbers of [46; 47] to label our sections. We have done this in
@cite_3: In this paper, we give a complete proof of the Poincare and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.
@cite_0: We show that a certain entropy-like function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume.
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@cite_13: We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.
@cite_11: We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdor limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X;d) in which the distance between two points equals the inmum of the lengths of curves joining the points. In the rest of this introduction we assume that X is a compact length space. Alexandrov gave a good notion of a length space having bounded below by K", with K a real number, in terms of the geodesic triangles in X. In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the GromovHausdor topology on compact metric spaces (modulo isometries); they form
The Ricci flow was introduced by Hamilton @cite_10 . There he effectively used it to solve the Poincar ' e conjecture for 3-manifolds with positive Ricci curvature. By following his approach, Perelman @cite_13 @cite_4 @cite_19 finally solved the Poincar ' e conjecture (see also @cite_3 @cite_1 @cite_7 ). There he used @math -functional as a crucial tool. At the same stage, he also studied the heat equation in @cite_13 in relation with the geometry of Ricci flows. It suggests that analysing the heat equation is still an efficient way to investigate geometry of the underlying space even in the time-dependent metric case. This general principle has been confirmed in recent developments in this direction. In connection with the theory of optimal transportation, McCann and Topping @cite_16 showed contraction in the @math -Wasserstein distance for the heat equation under backwards Ricci flow on a compact manifold. Topping's result @cite_11 can be regarded as an extension of it to contraction in the normalized @math -transportation cost (see @cite_0 also). By taking @math , he recovered the monotonicity of Perelman's @math -entropy, which is one of fundamental ingredients in Perelman's work.