Synthetic notions of Ricci flow for metric measure spaces
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912185852100608 |
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| author | Erbar, Matthias Li, Zhenhao Schultz, Timo |
| author_facet | Erbar, Matthias Li, Zhenhao Schultz, Timo |
| contents | We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of dynamic convexity and local concavity of the entropy along Wasserstein geodesics on the one hand and in terms of global and short-time asymptotic transport cost estimates for the heat flow on the other hand. We show that these properties characterise smooth (weighted) Ricci flows. Further, we investigate the relation between the different notions in the non-smooth setting of time-dependent metric measure spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07175 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Synthetic notions of Ricci flow for metric measure spaces Erbar, Matthias Li, Zhenhao Schultz, Timo Differential Geometry Metric Geometry We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of dynamic convexity and local concavity of the entropy along Wasserstein geodesics on the one hand and in terms of global and short-time asymptotic transport cost estimates for the heat flow on the other hand. We show that these properties characterise smooth (weighted) Ricci flows. Further, we investigate the relation between the different notions in the non-smooth setting of time-dependent metric measure spaces. |
| title | Synthetic notions of Ricci flow for metric measure spaces |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2501.07175 |