Time-like Ricci curvature bounds via optimal transport

Limites de courbures de Ricci temporelles via transport optimal

Andrea Mondino (1) , Fanny Bastien (2) , Hugo Béchet (2)

1 University of Oxford
2 IF - Institut Fourier

The goal of the talk is to present a recent work in collaboration with Cavalletti (SISSA) on optimal transport in Lorentzian synthetic spaces. The aim is to set up a “Lorentzian analog” of the celebrated Lott-Sturm-Villani theory of CD(K,N) metric measure spaces. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics (with respect to a suitable Lorentzian Wasserstein distance) of probability measures. The smooth Lorentzian setting was previously investigated by McCann and Mondino-Suhr. After recalling the general setting of Lorentzian synthetic spaces (including remarkable examples fitting the framework), I will discuss some basics of optimal transport theory thereof in order to define "timelike Ricci curvature bounded below and dimension bounded above'' for a (possibly non-smooth) Lorentzian space. The notion of "timelike Ricci curvature bounded below and dimension bounded above'' for a (possibly non-smooth) Lorentzian space is stable under a suitable weak convergence of Lorentzian synthetic spaces, giving a glimpse on the strength of the proposed approach. As an application of the optimal transport approach to timelike Ricci curvature lower bounds, I will discuss an extension of the Hawking's Singularity Theorem (in sharp form) to the synthetic setting.

Mots clés: optimal transport; Ricci curvature; eem2021; grenoble; curvature constraints and spaces of metrics; optimal transport

Mots clés: contraintes de courbures et espaces métriques; contraintes de courbures et espaces métriques

Domaines: Mathématiques [math]

Andrea Mondino, Fanny Bastien, Hugo Béchet. Time-like Ricci curvature bounds via optimal transport: Curvature Constraints and Spaces of Metrics. 2021. ⟨hal-03677345⟩