Synthetic versus distributional lower Ricci curvature bounds

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...

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Bibliographic Details
Published inProceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 154; no. 5; pp. 1406 - 1430
Main Authors Kunzinger, Michael, Oberguggenberger, Michael, Vickers, James A.
Format Journal Article
LanguageEnglish
Published Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.10.2024
Subjects
tensor distributions
low regularity
53C21
optimal transport
Ricci curvature bounds
46T30
49Q22
synthetic geometry
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Summary: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 regularizations of the metric.
ISSN:0308-2105
1473-7124
DOI:10.1017/prm.2023.70