Timelike Ricci bounds for low regularity spacetimes by optimal transport (Q6592226)

The groundbreaking discoveries in [\textit{R. J. McCann}, Camb. J. Math. 8, No. 3, 609--681 (2020; Zbl 1454.53058); \textit{A. Mondino} and \textit{S. Suhr}, J. Eur. Math. Soc. (JEMS) 25, No. 3, 933--994 (2023; Zbl 1536.83003)] have led to the synthetic theory of TCD (Time-like Curvature-Dimension) and TMCP (Time-like Measure-Contraction Property) spaces via the Boltzmann entropy and later via the Rényi entropy [\textit{M. Braun}, J. Math. Pures Appl. (9) 177, 46--128 (2023; Zbl 1521.53049)] in the framework of measured Lorentzian spaces [A. Mondino and S. Suhr, loc. cit.; \textit{M. Kunzinger} and \textit{C. Sämann}, Ann. Global Anal. Geom. 54, No. 3, 399--447 (2018; Zbl 1501.53057)].\N\NThis paper aims to provide a first link of this novel synthetic viewpoint to the more customary analytic approach to\N\[\N\mathrm{Ric}_{g}\geq K\text{ in all time-like directions}\N\]\Nfor Lorentzian metrics of low regularity by distribution theory. The main result of this paper is the following theorem.\N\NTheorem. Assume \(g\)\ to be \(C^{1}\), and suppose it satisfies the distributional strong energy condition. Then for every \(p\in\left( 0,1\right) \)\ and every compactly supported, \(\mathfrak{m}\)-absolutely continuous mass distributions \(\mu_{0},\mu_{1}\in\mathcal{P}\left( \mathcal{M}\right) \) (the space of Borel probability measures on \(\mathcal{M}\)) such that \(x\ll_{g}y\) for \ every \(x\in\mathrm{spt\,}\mu_{0}\)\ and every \(y\in\mathrm{spt\,}\mu_{1}\), there exists a time-like proper-time parametrized \(l_{g,p}\)-geodestic \(\left( \mu_{t}\right) _{t\in\left[ 0,1\right] }\)\ connecting \(\mu_{0}\)\ to \(\mu_{1}\)\ such that every \(t\in\left[ 0,1\right] \)\ satisfies\N\[\N\mathcal{S}_{g}^{n}\left( \mu_{t}\right) \leq\left( 1-t\right) \mathcal{S}_{g}^{n}\left( \mu_{0}\right) +t\mathcal{S}_{g}^{n}\left( \mu_{1}\right).\N\]\N\NFrom this theorem, time-like geometric inequalities such as the Brunn-Minkowski inequality [\textit{H. Brunn}, Diss. München (1887; JFM 19.0615.01); \textit{L. Lusternik}, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 1935, No. 3, 55--58 (1935; JFM 61.0760.03); \textit{L. Lusternik}, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 1935, No. 3, 55--58 (1935; Zbl 0012.27203)], the Bishop-Gromov inequality [\textit{R. L. Bishop} and \textit{R. J. Crittenden}, Geometry of manifolds. New York and London: Academic Press (1964; Zbl 0132.16003), Corollary 4, p.256] and the Bonnet-Myers inequality [\textit{S. B. Myers}, Duke Math. J. 8, 401--404 (1941; JFM 67.0673.01); \textit{S. B. Myers}, Duke Math. J. 8, 401--404 (1941; Zbl 0025.22704)] are inferred in sharp form under the same assumptions on \(g\)\ as in the above theorem but without any time-like non-branching assumption.