Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications

The first goal is to set foundational results on optimal transport in Lorentzian (pre-)length spaces, including cyclical monotonicity, stability of optimal couplings and Kantorovich duality (several results are new even for smooth Lorentzian manifolds).

The second one is to give a synthetic notion of “timelike Ricci curvature bounded below and dimension bounded above” for a measured Lorentzian pre-length space using optimal transport. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics of probability measures. This notion is proved to be stable under a suitable weak convergence of measured Lorentzian pre-length spaces, giving a glimpse on the strength of the approach we propose.

The third goal is to draw applications, most notably extending volume comparisons and Hawking singularity Theorem (in sharp form) to the synthetic setting.

The framework of Lorentzian pre-length spaces includes as remarkable classes of examples: space-times endowed with a causally plain (or, more strongly, locally Lipschitz) continuous Lorentzian metric, closed cone structures, some approaches to quantum gravity.

Keywords

optimal transport, Ricci curvature, Hawking singularity theorem, non-smooth Lorentzian geometry

2010 Mathematics Subject Classification

Primary 53C23, 53C50. Secondary 49J52, 53C80, 58Z05, 83C99.

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Received 21 July 2023

Published 18 July 2024