Synthetic versus distributional lower Ricci curvature bounds
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866911370174267392 |
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| author | Kunzinger, Michael Oberguggenberger, Michael Vickers, James A. |
| author_facet | Kunzinger, Michael Oberguggenberger, Michael Vickers, James A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_03715 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Synthetic versus distributional lower Ricci curvature bounds Kunzinger, Michael Oberguggenberger, Michael Vickers, James A. Differential Geometry Metric Geometry 49Q22, 53C21, 46T30 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. |
| title | Synthetic versus distributional lower Ricci curvature bounds |
| topic | Differential Geometry Metric Geometry 49Q22, 53C21, 46T30 |
| url | https://arxiv.org/abs/2207.03715 |