Good geodesics satisfying the timelike curvature-dimension condition

Abstract

Let $(M,\mathsf{d},\mathfrak{m},\ll ,\le ,\tau )$ be a causally closed, $\mathcal{K}$-globally hyperbolic, regular measured Lorentzian geodesic space satisfying the weak timelike curvature-dimension condition ${\mathrm{wTCD}}_p^e(K,N)$ in the sense of Cavalletti and Mondino. We prove the existence of geodesics of probability measures on $M$ which satisfy the entropic semiconvexity inequality defining ${\mathrm{wTCD}}_p^e(K,N)$ and whose densities with respect to $\mathfrak{m}$ are additionally uniformly $L^{\infty }$ in time. This holds apart from any nonbranching assumption. We also discuss similar results under the timelike measure-contraction property.

arXiv comment29 pages. Minor expositionary changes. Assumption of regularity has been added