다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

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 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.

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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

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Ricci curvature for metric-measure spaces via optimal transport

Riemannian center of mass and mollifier smoothing

A basic problem in Riemannian geometry is the study of relations between the topological structure and the Riemannian structure of a complete, connected Riemannian manifold M of dimension n > 2. By a classical theorem of Myers [10] such a manifold is compact if the sectional curvature K of M satisfies K > a > O. More precisely, the diameter d(M) of M satisfies d(M) < zc/< 8 . After the pioneering work of Rauch [11] the following result, known as the sphere theorem, was proved first by Berger [1] in even dimensions and finally by Klingenberg [8] as stated.

A generalized sphere theorem

Since Milnor’s discovery of exotic spheres [Mi], one of the most intriguing problems in Riemannian geometry has been whether there are exotic spheres with positive curvature. It is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature [Hi]. On the other hand, there are many examples of exotic spheres with positive Ricci curvature (cf. [Ch1], [He], [Po], and [Na]) and this work recently culminated in [Wr] where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive Ricci curvature. This includes all exotic spheres in dimension 7. So far, however, no example of an exotic sphere with positive sectional curvature has been found. In fact, until now, only one example of an exotic sphere with nonnegative sectional curvature was known, the so-called Gromoll-Meyer sphere [GM] in dimension 7. As one of our main results we prove:

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Curvature and symmetry of Milnor spheres

One of the central problems in Riemannian geometry is to determine how large the classes of manifolds with positive/nonnegative sectional -, Ricci or scalar curvature are (see [Gr]). For scalar curvature the situation is fairly well understood by comparison. Special surgery constructions as in [SY, Wr] and bundle constructions as in [Na] have resulted in a large number of interesting manifolds with positive Ricci curvature. So far the only known obstructions to have positive Ricci curvature come from obstructions to have positive scalar curvature, (see [Li] and [RS]), and from the classical Bonnet-Myers Theorem, which implies that a closed manifold with positive Ricci curvature must have finite fundamental group. It is well known that among homogeneous manifolds G/H this is also a sufficient condition (see e.g. the proof of Corollary 3.5 or [Br]). In this paper we prove that this is true as well when the manifold admits an action by a compact Lie group G with orbits of codimension one.

Cohomogeneity one manifolds with positive Ricci curvature

In this paper we explore the geometry and topology of cohomogeneity one manifolds, i.e. manifolds with a group action whose principal orbits are hypersurfaces. We show that the principal group action of every principal SO(3) and SO(4) bundle over S^4 extends to a cohomogeneity one action. As a consequence we prove that every vector bundle and every sphere bundle over S^4 has a complete metric with non-negative curvature. It is well known that 15 of the 27 exotic spheres in dimension 7 can be written as S^3 bundles over S^4 in infinitely many ways, and hence we obtain infinitely many non-negatively curved metrics on these exotic spheres. A further consequence will be that there are infinitely many almost free actions by SO(3) on S^7, i.e. all isotropy groups are finite. These actions preserve the Hopf fibration S^3 -> S^7 -> S^4 but do not extend to the disc D^8. We also construct infinitely many such actions on the 15 exotic 7-spheres mentioned above.

The existence of a map conjugating two Cl-close G-actions has already been proved by Palais [5]. Palais' proof relies essentially on the fact that there exists a representation of G in an orthogonal group 0 (n) and an equivariant embedding of M in IR". The main tool in our approach is to define a "center of mass" for almost constant maps. This enables us to define a specific map conjugating the two group actions if they are Ca-close. Using this notion of center we prove in the last paragraph a differential geometric version of the theorem:

Karsten Grove

Papers

A generalized sphere theorem

Curvature and symmetry of Milnor spheres

Cohomogeneity one manifolds with positive Ricci curvature

Curvature and symmetry of Milnor spheres

How to Conjugate C1-Close Group Actions