Journal
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
Volume -, Issue -, Pages -Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/prm.2023.70
Keywords
low regularity; optimal transport; Ricci curvature bounds; synthetic geometry; tensor distributions
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We compare two standard approaches to defining lower Ricci curvature bounds for Riemannian metrics of regularity below $C<^>2$.
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.
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