Synthetic versus distributional lower Ricci curvature bounds

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.

arXiv comment23 pages, small correction in the proof of Th. 4.3

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Cited by in corpus (2)

• The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature
• The singularity theorems of General Relativity and their low regularity extensions