Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimal Transport: Old and New

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.

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The entropy formula for the Ricci flow and its geometric applications

인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound

 $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$
 (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD(K, ∞). The additional real parameter N plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K, N) is equivalent to $${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $$
 and dim(M) ⩽ N. The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is compact. Condition CD(K, N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers theorem. Moreover, it implies the doubling property and local, scale-invariant Poincare inequalities on balls. In particular, it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.

/pdf/on-the-geometry-of-metric-measure-spaces-ii-aa7ov6g1d9.pdf

On the geometry of metric measure spaces. II

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 \curvature 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

/pdf/ricci-curvature-for-metric-measure-spaces-via-optimal-3t6zu9nl81.pdf

Ricci curvature for metric-measure spaces via optimal transport

In this paper and in we study the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de nite lower bound say RicMn i n In Sections and sometimes in we also assume a lower volume bound Vol B pi v In this case the sequence is said to be non collapsing If limi Vol B pi then the sequence is said to collapse It turns out that a convergent sequence is noncollapsing if and only if the limit has positive n dimensional Hausdor measure In par ticular any convergent sequence is either collapsing or noncollapsing Moreover if the sequence is collapsing it turns out that the Hausdor dimension of the limit is actually n see Sections and Our theorems on the in nitesimal structure of limit spaces have equivalent statements in terms of or implications for the structure on a small but de nite scale of manifolds with RicMn n Al though both contexts are signi cant for the most part it is the limit spaces which are emphasized here Typically the relation between corre sponding statements for manifolds and limit spaces follows directly from the continuity of the geometric quantities in question under Gromov Hausdor limits together with Gromov s compactness theorem Theorems see also Remark are examples of

https://projecteuclid.org/download/pdf_1/euclid.jdg/1214342146

On the structure of spaces with Ricci curvature bounded below. I

The basic rigidity theorems for manifolds of nonnegative or positive Ricci curvature are the "volume cone implies metric cone" theorem, the maximal diameter theorem, [Cg], and the splitting theorem, [CG]. Each asserts that if a certain geometric quantity (volume or diameter) is as large as possible relative to the pertinent lower bound on Ricci curvature, then the metric on the manifold in question is a warped product metric of a particular type. In this paper we provide quantitative generalizations of the above mentioned results. Among the applications are the splitting theorem for GromovHausdorff limit spaces X, where Mn -* X, Ricmn ? -i see [FY]. Other applications include the assertion that for complete manifolds, M', with Ricmf > 0 and Euclidean volume growth, all tangent cones at infinity are metric cones; compare [BKN], [CT], [P1]. Via resealing arguments, there are also strong consequences for the local structure of manifolds whose Ricci curvature satisfies a fixed lower bound and for their Gromov-Hausdorff limits. Some of these are announced in [CCol]; for a more detailed discussion see [CCo2], [CCo3], [CCo4]. Our work further develops and significantly extends techniques which were introduced in [Col], [Co2] and significantly extended in [Co3], in order to prove certain "stability" conjectures of Anderson-Cheeger, Gromov and Perelman. The results of [Col]-[Co3] were announced in [Co4]. We briefly review some of those results. Let dGH denote the Gromov-Hausdorff distance between metric spaces; see [GLP]. Let S' denote the unit sphere and recall that S' is the unique complete

Lower bounds on Ricci curvature and the almost rigidity of warped products

It has long been conjectured that starting at a generic smooth closed embedded surface in R 3 , the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders. That is, the only singularities of a generic ow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key to showing this conjecture is to show that shrinking spheres, cylinders, and planes are the only stable self-shrinkers under the mean curvature ow. We prove this here in all dimensions. An easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away.

https://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p07-p.pdf

Generic mean curvature flow I; generic singularities

It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. 
The key in showing this conjecture is to show that shrinking spheres, cylinders and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every other singularity than spheres and cylinders can be perturbed away.

The purpose of this paper is to give a new (integral) estimate of distances and angles on manifolds with a given lower Ricci curvature bound. This will provide us with an integral version of the Toponogov comparison triangle theorem for Ricci curvature and "almost extreme triangles" (see the earlier works [Cl] and [C2] for an analog of this when the manifold has positive Ricci curvature). Using this, we prove the Anderson-Cheeger conjecture saying that the volume is a continuous function on the space of all closed n-manifolds with Ricci curvature greater or equal to -(n - 1) equipped with the GromovHausdorff metric. We also prove Gromov's conjecture (for n 57 3) saying that an almost nonnegatively Ricci curved n-manifold with first Betti number equal to n is a torus. Further, we prove a conjecture of Anderson-Cheeger saying that an open n-manifold with nonnegative Ricci curvature whose tangent cone at infinity is in is itself in. Finally we prove a conjecture of Fukaya-Yamaguchi. We will now describe these results in more detail. Let dGH denote the Gromov-Hausdorff distance [GLP]. First we have the following result which was conjectured by Anderson-Cheeger.

Tobias H. Colding

Papers

On the structure of spaces with Ricci curvature bounded below. I

Lower bounds on Ricci curvature and the almost rigidity of warped products

Generic mean curvature flow I; generic singularities

Generic mean curvature flow I; generic singularities

Ricci curvature and volume convergence