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, ${\mathsf{sTBM}}_q^{\ast }(K,N)$. 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.

arXiv comment32 pages