Abstract
We prove that the Abresch-Gromoll inequality holds on infinitesimally Hilbertian $CD(K,N)$ spaces in the same form as the one available on smooth Riemannian manifolds.
Highlights
E(x) ≤ 2h(x), so that the interest of the Abresch-Gromoll inequality relies on the explicit expression of fK,N which for arbitrary K ≤ 0 grants lim h↓0
Sturm [20, 21] on one side and Lott-Villani [17] on the other proposed a synthetic definition of Ricci curvature bounds on metric measure spaces, giving a meaning to the statement ‘the space (X, d, m) has Ricci curvature bounded from below by K and dimension bounded from above by N ’, in short: (X, d, m) is a CD(K, N ) space
It has been soon realized that the class of CD(K, N ) metric measure spaces includes objects that are not Riemannian in nature: a result by Cordero-Erausquin, Sturm and Villani ensures that if we endow Rd with the Lebesgue measure and the distance coming from a norm, we always obtain a CD(0, d) space, regardless of the choice of the norm
Summary
Throughout all the paper (X, d) will be a complete and separable metric space. (X, d) is said to be proper whenever all closed and bounded sets are compact. A curve γ ∈ C([0, 1], X) is said to be absolutely continuous if there exists a function f ∈ L1([0, 1]) such that t d(γt, γs) ≤ f (r) dr, ∀t, s ∈ [0, 1], t < s. The set of absolutely continuous curves from [0, 1] to X will be denoted by AC([0, 1], X). If the function f in (1.1) belongs to Lq([0, 1]), q ∈ [1, ∞], γ is said to be qabsolutely continuous, and ACq([0, 1], X) is the corresponding set of q-absolutely continuous curves. Minimizers in (1.3) are called optimal geodesic plans and the set of all such minimizers will be denoted by OptGeo(μ, ν)
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