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

TL;DR

This paper establishes the equivalence between timelike Ricci curvature conditions and the timelike Brunn-Minkowski inequality in synthetic Lorentzian spaces, extending previous smooth spacetime results to a non-smooth setting.

Contribution

It introduces the strong $q$-timelike Brunn-Minkowski condition and proves its equivalence to various $q$-timelike curvature dimension conditions in synthetic Lorentzian geometry.

Findings

Equivalence between $ ext{TCD}_q(K,N)$ and $ ext{TBM}_q(K,N^+)$ in non-branching spaces.

Equivalence between $ ext{TCD}_q^e(K,N)$ and the reduced $ ext{sTBM}_q^*(K,N)$ condition.

Extension of Ricci curvature and Brunn-Minkowski inequality equivalence to non-smooth 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.