में बचाया:
| मुख्य लेखकों: | , |
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| स्वरूप: | Preprint |
| प्रकाशित: |
2025
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| विषय: | |
| ऑनलाइन पहुंच: | https://arxiv.org/abs/2505.09681 |
| टैग: |
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| _version_ | 1866912424008876032 |
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| author | Borza, Samuël Rizzi, Luca |
| author_facet | Borza, Samuël Rizzi, Luca |
| contents | We prove that the synthetic Ricci curvature lower bound known as the measure contraction property (MCP) can fail in sub-Riemannian geometry. This may happen beyond step two, if the distance function is not Lipschitz in charts, and it already occurs in fundamental examples such as the Martinet and Engel structures.
Central to our analysis are new results, of independent interest, on the stability of the local MCP under quotients by isometric group actions for general metric measure spaces, developed under a weaker variant of the essential non-branching condition which, in contrast with the classical one, is implied by the minimizing Sard property in sub-Riemannian geometry.
As an application, we find sub-Riemannian structures with pre-medium-fat distribution that do not satisfy the MCP, answering a question raised in [L. Rifford, J. Éc. polytech. Math. 2023]. Finally, and quite unexpectedly, we show that ideal sub-Riemannian structures can fail the MCP, and this actually happens generically for rank greater than 3 and high dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09681 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Failure of the measure contraction property via quotients in higher-step sub-Riemannian structures Borza, Samuël Rizzi, Luca Differential Geometry Metric Geometry Optimization and Control 53C17, 53C21, 49Q22 We prove that the synthetic Ricci curvature lower bound known as the measure contraction property (MCP) can fail in sub-Riemannian geometry. This may happen beyond step two, if the distance function is not Lipschitz in charts, and it already occurs in fundamental examples such as the Martinet and Engel structures. Central to our analysis are new results, of independent interest, on the stability of the local MCP under quotients by isometric group actions for general metric measure spaces, developed under a weaker variant of the essential non-branching condition which, in contrast with the classical one, is implied by the minimizing Sard property in sub-Riemannian geometry. As an application, we find sub-Riemannian structures with pre-medium-fat distribution that do not satisfy the MCP, answering a question raised in [L. Rifford, J. Éc. polytech. Math. 2023]. Finally, and quite unexpectedly, we show that ideal sub-Riemannian structures can fail the MCP, and this actually happens generically for rank greater than 3 and high dimension. |
| title | Failure of the measure contraction property via quotients in higher-step sub-Riemannian structures |
| topic | Differential Geometry Metric Geometry Optimization and Control 53C17, 53C21, 49Q22 |
| url | https://arxiv.org/abs/2505.09681 |