Improved geodesics for the reduced curvature-dimension condition in branching metric spaces

TL;DR

This paper demonstrates that in metric measure spaces satisfying the reduced curvature-dimension condition, one can construct geodesics in the Wasserstein space with convexity properties and bounded densities, enhancing understanding of geometric analysis in such spaces.

Contribution

It establishes the existence of geodesics with convexity and density bounds in spaces satisfying the CD*(K,N) condition, extending previous results in metric measure geometry.

Findings

Existence of geodesics satisfying the convexity inequality at all intermediate times.

Measures along these geodesics have explicit upper bounds on their densities.

Results depend on curvature, dimension, and support diameter bounds.

Abstract

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of CD*(K,N) also for intermediate times and in addition the measures along these geodesics have an upper-bound on their densities. This upper-bound depends on the bounds for the densities of the end-point measures, the lower-bound K for the Ricci-curvature, the upper-bound N for the dimension, and on the diameter of the union of the supports of the end-point measures.