Good geodesics satisfying the timelike curvature-dimension condition

TL;DR

This paper proves the existence of special geodesics of probability measures in Lorentzian spaces satisfying a weak curvature-dimension condition, without requiring nonbranching assumptions, and discusses related measure-contraction properties.

Contribution

It establishes the existence of entropically semiconvex geodesics with bounded densities in Lorentzian spaces under weak curvature conditions, extending previous results without nonbranching assumptions.

Findings

Existence of geodesics satisfying the entropic semiconvexity inequality.

Geodesics have densities uniformly bounded in time.

Results extend to spaces satisfying the timelike measure-contraction property.

Abstract

Let $(M,\mathsf{d},\mathfrak{m},\ll,\leq,\tau)$ be a causally closed, $\mathscr{K}$-globally hyperbolic, regular measured Lorentzian geodesic space satisfying the weak timelike curvature-dimension condition $\smash{\mathrm{wTCD}_p^e(K,N)}$ 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 $\smash{\mathrm{wTCD}_p^e(K,N)}$ 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.