Invariant Measures and Lower Ricci Curvature Bounds
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Invariant Measures and Lower Ricci Curvature Bounds Jaime Santos-Rodr´ıguez1 Received: 29 October 2018 / Accepted: 25 June 2019 / © Springer Nature B.V. 2019
Abstract Given a metric measure space (X, d, m) that satisfies the Riemannian Curvature Dimension condition, RCD ∗ (K, N ), and a compact subgroup of isometries G ≤ I so(X) we prove that there exists a G−invariant measure, mG equivalent to m such that (X, d, mG ) is still a RCD ∗ (K, N ) space. We also obtain applications to Lie group actions on RCD ∗ (K, N ) spaces. We look at homogeneous spaces, symmetric spaces and obtain dimensional gaps for closed subgroups of isometries. Keywords Ricci curvature dimension · Metric measure space · Isometry group Mathematics Subject Classification (2010) 53C23 · 53C21
1 Introduction and Statement of Results Synthetic Ricci lower curvature bounds were introduced in the seminal papers of LottVillani [32] and Sturm [38, 39]. This synthetic definition, known as Curvature-Dimension condition or CD(K, N ) is, broadly speaking, the convexity of certain functionals (called entropies) along geodesics on the space of probability measures. In [14] Erbar, Kuwada and Sturm defined another notion of Curvature-Dimension called Entropic Curvature Dimension condition, CD e (K, N ). Under some technical assumptions there is equivalence of different notions of synthetic Ricci curvature bounds. For example, we have that on infinitessimally Hilbertian spaces, CD e (K, N ) is equivalent to RCD ∗ (K, N ) (see Theorem 7 in [14]). One thing to notice is that unlike the case of synthetic sectional curvature a metric structure alone is not sufficient to talk about Ricci curvature. A reference measure is also
The author was supported by research grants MTM2014-57769-C3-3-P, and MTM2017-85934-C3-2-P (MINECO) and ICMAT Severo Ochoa Project SEV-2015-0554-17-1 (MINECO). Jaime Santos-Rodr´ıguez
[email protected] 1
Department of Mathematics, Universidad Aut´onoma de Madrid and ICMAT CSIC-UAM-UCM-UC3M, Madrid, Spain
J. Santos-Rodr´ıguez
required, that is we will work with metric measure spaces (X, d, m). It is of interest then to see how the measure m and the metric d interact with one another. In [24] Guijarro and the author, and independently Sosa in [37], studied the isometry group of RCD ∗ (K, N ) spaces. In both of these papers it is proved that the set of fixed points of an isometry must have zero m−measure. The main result of this article makes the connection between the metric and the measure deeper. More precisely we have: Theorem A Let (X, d, m) be an RCD ∗ (K, N ) space and, G ≤ Iso(X) a compact subgroup. Then there exists a G−invariant measure mG , equivalent to m, and such that (X, d, mG ) is an RCD ∗ (K, N ) space. The strategy for finding the desired measure is the following: In [28] Kell studied properties of optimal couplings with first marginal absolutely continuous with respect to m. He obtained a measure rigidity result which states that any two essentially nonbranching, qualitatively non-degenerate measures must be mutually
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