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Synthetic versus distributional lower Ricci curvature bounds
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 154 / Issue 5 / October 2024
- Published online by Cambridge University Press:
- 23 August 2023, pp. 1406-1430
- Print publication:
- October 2024
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- Article
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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.
