Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916134226231296 |
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| author | Borza, Samuël Magnabosco, Mattia Rossi, Tommaso Tashiro, Kenshiro |
| author_facet | Borza, Samuël Magnabosco, Mattia Rossi, Tommaso Tashiro, Kenshiro |
| contents | In this paper, we investigate the validity of synthetic curvature-dimension bounds in the sub-Finsler Heisenberg group, equipped with a positive smooth measure. Firstly, we study the measure contraction property, in short $\mathsf{MCP}$, proving that its validity depends on the norm generating the sub-Finsler structure. Indeed, we show that, if it is neither $C^1$ nor strongly convex, the associated Heisenberg group does not satisfy $\mathsf{MCP}(K,N)$ for any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. On the contrary, we prove that the sub-Finsler Heisenberg group, equipped with a $C^{1,1}$ and strongly convex norm, and with the Lebesgue measure, satisfies $\mathsf{MCP}(0,N)$ for some $N \in (1,\infty)$. Additionally, we provide a lower bound on the optimal dimensional parameter, and we also study the case of $C^1$ and strongly convex norms. Secondly, we address the validity of the curvature-dimension condition pioneered by Sturm and Lott-Villani, in short $\mathsf{CD}(K,N)$. We show that the sub-Finsler Heisenberg group, equipped with a $C^1$ and strongly convex norm, and with a positive smooth measure, does not satisfy the $\mathsf{MCP}(K,N)$ condition for any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. Combining this result with our findings regarding the measure contraction property, we conclude the failure of the $\mathsf{CD}$ condition in the Heisenberg group for every sub-Finsler structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14779 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups Borza, Samuël Magnabosco, Mattia Rossi, Tommaso Tashiro, Kenshiro Metric Geometry Differential Geometry In this paper, we investigate the validity of synthetic curvature-dimension bounds in the sub-Finsler Heisenberg group, equipped with a positive smooth measure. Firstly, we study the measure contraction property, in short $\mathsf{MCP}$, proving that its validity depends on the norm generating the sub-Finsler structure. Indeed, we show that, if it is neither $C^1$ nor strongly convex, the associated Heisenberg group does not satisfy $\mathsf{MCP}(K,N)$ for any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. On the contrary, we prove that the sub-Finsler Heisenberg group, equipped with a $C^{1,1}$ and strongly convex norm, and with the Lebesgue measure, satisfies $\mathsf{MCP}(0,N)$ for some $N \in (1,\infty)$. Additionally, we provide a lower bound on the optimal dimensional parameter, and we also study the case of $C^1$ and strongly convex norms. Secondly, we address the validity of the curvature-dimension condition pioneered by Sturm and Lott-Villani, in short $\mathsf{CD}(K,N)$. We show that the sub-Finsler Heisenberg group, equipped with a $C^1$ and strongly convex norm, and with a positive smooth measure, does not satisfy the $\mathsf{MCP}(K,N)$ condition for any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. Combining this result with our findings regarding the measure contraction property, we conclude the failure of the $\mathsf{CD}$ condition in the Heisenberg group for every sub-Finsler structure. |
| title | Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups |
| topic | Metric Geometry Differential Geometry |
| url | https://arxiv.org/abs/2402.14779 |