Metric Measure Spaces and Synthetic Ricci Bounds -- Fundamental Concepts and Recent Developments
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910421497151488 |
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| author | Sturm, Karl-Theodor |
| author_facet | Sturm, Karl-Theodor |
| contents | Metric measure spaces with synthetic Ricci bounds have attracted great interest in recent years, accompanied by spectacular breakthroughs and deep new insights. In this survey, I will provide a brief introduction to the concept of lower Ricci bounds as introduced by Lott-Villani and myself, and illustrate some of its geometric, analytic and probabilistic consequences, among them Li-Yau estimates, coupling properties for Brownian motions, sharp functional and isoperimetric inequalities, rigidity results, and structural properties like rectifiability and rectifiability of the boundary. In particular, I will explain its crucial interplay with the heat flow and its link to the curvaturedimension condition formulated in functional-analytic terms by Bakry-Èmery. This equivalence between the Lagrangian and the Eulerian approach then will be further explored in various recent research directions: i) time-dependent Ricci bounds which provide a link to (super-) Ricci flows for singular spaces, ii) second order calculus, upper Ricci bounds, and transformation formulas, iii) distribution-valued Ricci bounds which e.g. allow singular effects of non-convex boundaries to be taken into account. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15755 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metric Measure Spaces and Synthetic Ricci Bounds -- Fundamental Concepts and Recent Developments Sturm, Karl-Theodor Metric Geometry Metric measure spaces with synthetic Ricci bounds have attracted great interest in recent years, accompanied by spectacular breakthroughs and deep new insights. In this survey, I will provide a brief introduction to the concept of lower Ricci bounds as introduced by Lott-Villani and myself, and illustrate some of its geometric, analytic and probabilistic consequences, among them Li-Yau estimates, coupling properties for Brownian motions, sharp functional and isoperimetric inequalities, rigidity results, and structural properties like rectifiability and rectifiability of the boundary. In particular, I will explain its crucial interplay with the heat flow and its link to the curvaturedimension condition formulated in functional-analytic terms by Bakry-Èmery. This equivalence between the Lagrangian and the Eulerian approach then will be further explored in various recent research directions: i) time-dependent Ricci bounds which provide a link to (super-) Ricci flows for singular spaces, ii) second order calculus, upper Ricci bounds, and transformation formulas, iii) distribution-valued Ricci bounds which e.g. allow singular effects of non-convex boundaries to be taken into account. |
| title | Metric Measure Spaces and Synthetic Ricci Bounds -- Fundamental Concepts and Recent Developments |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2404.15755 |