CD meets CAT

Abstract

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.

Infinitesimal Hilbertianity of Locally $$\mathrm{CAT}(\kappa )$$-Spaces

“…To the best of our knowledge, this manuscript contains the first result about the structure of Sobolev functions on CAT(κ) spaces. (ii) A particular case of Theorem 1.1 has been obtained in the recent paper [30] by Kapovitch and Ketterer. There the authors consider a metric measure space (X, d, m) which is a CD(K , N ) space in the sense of Lott-Sturm-Villani ( [32,42,43]) when seen as a metric measure space and a CAT(κ) space when seen as metric space.…”

On gluing Alexandrov spaces with lower Ricci curvature bounds

“…Remark 2.5. Since RCD(K, N ) and RCD * (K, N ) spaces are essentially non-branching, the two conditions are equivalent provided m is finite (compare with Remark 2.15 in [KK17].…”

Rigidity of mean convex subsets in non-negatively curved RCD spaces and stability of mean curvature bounds

“…For a brief overview on the history of this definition we refer the reader to the preliminary section of [KK20].…”