Spaces of Curvature Bounded Below

Burago, Dmitri; Burago, Yuri; Ivanov, Sergei O.

doi:10.1090/gsm/033/10

Cited by 10 publications

(12 citation statements)

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“…Application to the geometric description of P 2 (M). Let us recall some facts about a finite-dimensional Alexandrov space Y with curvature bounded below [11,12]. Let n be the dimension of Y .…”

Section: Appendix a The Wasserstein Space As An Alexandrov Spacementioning

confidence: 99%

Ricci curvature for metric-measure spaces via optimal transport

2009

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Abstract. We define a notion of a measured length space X having nonnegative N -Ricci curvature, for N ∈ [1, ∞), or having ∞-Ricci curvature bounded below by K, for K ∈ R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P 2 (X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdorff 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 infimum 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 Gromov-Hausdorff topology on compact metric spaces (modulo isometries); they form a closed subset.In view of Alexandrov's work, it is natural to ask whether there are metric space versions of other types of Riemannian curvature, such as Ricci curvature. This question takes substance from Gromov's precompactness theorem for Riemannian manifolds with Ricci curvature bounded below by K, dimension bounded above by N and diameter bounded above by D [23, Theorem 5.3]. The precompactness indicates that there could be a notion of a length space having "Ricci curvature bounded below by K", special cases of which would be Gromov-Hausdorff limits of manifolds with lower Ricci curvature bounds. Gromov-Hausdorff limits of manifolds with Ricci curvature bounded below have been studied by various authors, notably Cheeger and Colding [15,16,17,18]. One feature of their work, along with the earlier work of Fukaya [21], is that it turns out to be useful to add an auxiliary Borel probability measure ν and consider metric-measure spaces (X, d, ν). (A compact Riemannian manifold M has a canonical measure ν given by the normalized Riemannian density.) There is a measured Gromov-Hausdorff topology on such triples (X, d, ν) (modulo isom...

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Section: Appendix a The Wasserstein Space As An Alexandrov Spacementioning

confidence: 99%

Ricci curvature for metric-measure spaces via optimal transport

2009

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“…Let us first recall two possible definitions of Positively Curved (PC) spaces in the sense of Alexandrov, referring to [10] and to [11] for other equivalent definitions and for the more general case of spaces with curvature ≥ k, k ∈ R. In the case of a smooth Riemannian manifold (M, g) equipped with the Riemannian distance d g all the local definitions are equivalent to assume that the sectional curvature of M is nonnegative (or bounded by kg, in the case of curvature ≥ k).…”

Section: Lower Curvature Bound In the Sense Of Alexandrovmentioning

confidence: 99%

Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures

2017

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We develop a full theory for the new class of Optimal EntropyTransport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a pair of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, which quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.

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“…This result, as well as its generalization to higher dimensions (see Theorem 33), is hardly surprising, given the fact that, by [43], Theorem 29, Myers' theorem holds for general Alexandrov spaces of curvature ≥ 0 > 0, and WaldBerestovskii curvature is essentially equivalent to the Rinow curvature (see [34]), hence, to the Alexandrov curvature (see, e.g., [33,Chapter 1]). Rather, we give in the special case of surfaces (manifolds) a simpler, more intuitive proof of the Burago-Gromov-Perelman extension of Myers' theorem.…”

Section: Remark 30mentioning

confidence: 97%