Rényi's entropy on Lorentzian spaces. Timelike curvature-dimension conditions

Abstract

For a Lorentzian space measured by m in the sense of Kunzinger, Sämann, Cavalletti, and Mondino, we introduce and study synthetic notions of timelike lower Ricci curvature bounds by K∈R and upper dimension bounds by N∈[1,∞), namely the timelike curvature-dimension conditions TCDp(K,N) and TCDp⁎(K,N) in weak and strong forms, where p∈(0,1), and the timelike measure-contraction properties TMCP(K,N) and TMCP⁎(K,N). These are formulated by convexity properties of the Rényi entropy with respect to m along ℓp-geodesics of probability measures.We show many features of these notions, including their compatibility with the smooth setting, sharp geometric inequalities, stability, equivalence of the named weak and strong versions, local-to-global properties, and uniqueness of chronological ℓp-optimal couplings and chronological ℓp-geodesics. We also prove the equivalence of TCDp⁎(K,N) and TMCP⁎(K,N) to their respective entropic counterparts in the sense of Cavalletti and Mondino.Some of these results are obtained under timelike p-essential nonbranching, a concept which is a priori weaker than timelike nonbranching.