The equivalence between timelike Ricci curvature and the timelike Brunn Minkowski inequality on synthetic Lorentzian spaces

Abstract

We introduce the strong $q$-timelike Brunn-Minkowski condition $\mathsf{sTBM}_q(K,N)$ on synthetic Lorentzian spaces, for $0<q<1$. We show that, in the timelike $q$-essentially non-branching setting, the $q$-timelike curvature dimension condition $\mathsf{TCD}_q(K,N)$ is equivalent to $\mathsf{TBM}_q(K,N^+)$, and that the entropic $q$-timelike curvature dimension condition $\mathsf{TCD}_q^e(K,N)$ is equivalent to the reduced $\mathsf{sTBM}$ condition, $\mathsf{sTBM}_q^*(K,N)$. This extends, to a non-smooth setting, our earlier work in proving the equivalence between Ricci curvature and the Brunn-Minkowski inequality on $C^2$ spacetimes.