Non-branching geodesics and optimal maps in strong $$CD(K,\infty )$$ C D ( K , ∞ ) -spaces

We prove that if (X, d, m) is an essentially non-branching metric measure space with m(X) = 1, having Ricci curvature bounded from below by K and dimension bounded above by N ∈ (1, ∞), understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequalityà la Lévy-Gromov holds true. Measure theoretic rigidity is also obtained. model isoperimetric profile I CD K,N,D such that if a Riemannian manifold with density verifying CD(K, N ) has diameter at most D > 0, then the isoperimetric profile function of the weighted manifold is bounded from below by I CD K,N,D . After the works of Cordero-Erausquin-McCann-Schmuckenshläger [25], Otto-Villani [42] and von Renesse-Sturm [47], it was realized that the CD(K, ∞) condition in the smooth setting may be equivalently formulated synthetically as a certain convexity property of an entropy functional along W 2 Wasserstein geodesics (associated to L 2 -Optimal-Transport). This idea led Lott-Villani [36] and Sturm [51, 52], to propose a successful (and compatible with the classical one) synthetic definition of CD(K, N ) for a general (complete, separable) metric space (X, d) endowed with a (locally-finite Borel) reference measure m ("metricmeasure space", or m.m.s.); the theory of m.m.s.'s verifying CD(K, N ) has then extensively developed leading to a rich and fruitful approach to the geometry of m.m.s.'s by means of 3,4,27,26,5,38,29,33,13]. See also [1] for a recent account on the topic. Building on the work by Klartag [34] and the localization paradigm developed by Payne-Weinberger [44], Gromov-Milman [31] and Kannan-Lovász-Simonovits [32], the first author with Mondino [19] managed to extend Lévy-Gromov-Milman isoperimetric inequality to the class of essentially non-branching (see Section 2 for the definition) m.m.s.'s verifying CD(K, N ) with m(X) = 1; in particular [19] proves that

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“…In the paper we will only consider essentially non-branching spaces, let us recall their definition (introduced in [46]).…”

Section: Backgroundsmentioning

confidence: 99%

Isoperimetric inequality under Measure-Contraction property

Cavalletti

Santarcangelo

2019

Journal of Functional Analysis

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“…Recall that an mms is essentially non-branching and satisfies the (CD-)CD * -condition if and only if it satisfies the strong (CD-)CD * -condition; for these results and definitions see [5,30]. The CD-, CD * -, and MCP-conditions allow for non-Riemannian geometries which include, but are not restricted to, Finsler manifolds.…”

Section: Moreover If Iso(x) Is a Lie Group Then Iso M (X) Is So As Wellmentioning

confidence: 99%

The Isometry Group of an RCD ∗ Space is Lie

Sosa

2017

Potential Anal

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We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requirements. The conditions are satisfied by RCD * -spaces and, under extra assumptions, by CD-spaces, CD * -spaces, and MCP-spaces. However, we show that the MCP-condition by itself is not enough to guarantee a smooth behavior of these automorphism groups.

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“…More precisely, since each condition implies CD(K, ∞), together with infinitesimally Hilbertianness (X, d, m) satisfies the condition RCD(K, ∞) in the sense of [AGS14b]. Therefore, the Boltzmann-Shanon entropy is even strongly K-convex, and hence (X, d, m) is essentially non-branchning by [RS14]. Then, first we know that CD * (K, N ) is equivalent to CD e (K, N ) by [EKS15].…”

Section: 2mentioning

confidence: 99%

CD meets CAT

Kapovitch

Ketterer

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We show that if a noncollapsed CD(K, n) space X with n ≥ 2 has curvature bounded above by κ in the sense of Alexandrov then K ≤ (n − 1)κ and X is an Alexandrov space of curvature bounded below by K − κ(n − 2). We also show that if a CD(K, n) space Y with finite n has curvature bounded above then it is infinitesimally Hilbertian.Theorem 1.1. Let n ≥ 2 be a natural number and let (X, d, H n ) be a complete metric measure space which is CBA(κ) (has curvature bounded above by κ in the sense of Alexandrov) and satisfies CD(K, n). Then κ(n − 1) ≥ K, and (X, d) is an Alexandrov space of curvature bounded below by K − κ(n − 2). In particular X is infinitesimally Hilbertian.Examples given by manifolds of constant sectional curvature show that the lower curvature bound provided by this theorem is optimal.Theorem 1.1 shows that X has two sided curvature bounds in Alexandrov sense. By work of Alexandrov, Berestovsky and Nikolaev (see [BN93]) this immediately gives the following corollary:The space of m-absolutely continuous probability measures in P 2 (X) is denoted by P 2 (X, m). The Shanon-Boltzmann entropy of a metric measure space (X, d, m) is defined as Ent m : P 2 b (X) → (−∞, ∞], Ent m (µ) = log ρdµ if µ = ρm and (ρ log ρ) + is m-integrable, and ∞ otherwise. Note Dom Ent m ⊂ P 2 (X, m), and Ent m : P 2 b (X) → (−∞, ∞] is lower semicontinuous. By Jensen's inequality one has Ent m µ ≥ − log m(supp µ) if m(supp µ) < ∞.Definition 2.4 ([Stu06a, LV09, EKS15]). A metric measure space (X, d, m) satisfies the curvaturedimension condition CD(K, ∞) for K ∈ R if Ent m is weakly K-convex.A metric measure space (X, d, m) satisfies the entropic curvature-dimension condition CD e (K, N ) for K ∈ R and N ∈ (0, ∞) if Ent m is weakly (K, N )-convex.

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Non-branching geodesics and optimal maps in strong $$CD(K,\infty )$$ C D ( K , ∞ ) -spaces

Cited by 105 publications

References 20 publications

Isoperimetric inequality under Measure-Contraction property

Isoperimetric inequality under Measure-Contraction property

The Isometry Group of an RCD ∗ Space is Lie

CD meets CAT

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