Synthetic versus distributional lower Ricci curvature bounds

TL;DR

This paper compares synthetic and distributional approaches to defining lower Ricci curvature bounds for less regular Riemannian metrics, establishing implications between the two under certain regularity conditions.

Contribution

It demonstrates the relationship between synthetic and distributional Ricci bounds, showing how distributional bounds imply entropy bounds and vice versa under specific regularity assumptions.

Findings

Distributional bounds imply entropy bounds for $C^1$ metrics.

Entropy bounds imply distributional bounds for $C^{1,1}$ metrics with regularisation.

The equivalence holds under an additional convergence condition.

Abstract

We compare two standard approaches to defining lower Ricci curvature bounds for Riemannian metrics of regularity below $C^2$. These are, on the one hand, the synthetic definition via weak displacement convexity of entropy functionals in the framework of optimal transport, and the distributional one based on non-negativity of the Ricci-tensor in the sense of Schwartz. It turns out that distributional bounds imply entropy bounds for metrics of class $C^1$ and that the converse holds for $C^{1,1}$-metrics under an additional convergence condition on regularisations of the metric.