On the rectifiability of $\mathsf{CD}(K,N)$ and $\mathsf{MCP}(K,N)$ spaces with unique tangents
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918007002890240 |
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| author | Magnabosco, Mattia Mondino, Andrea Rossi, Tommaso |
| author_facet | Magnabosco, Mattia Mondino, Andrea Rossi, Tommaso |
| contents | We prove rectifiability results for $\mathsf{CD}(K,N)$ and $\mathsf{MCP}(K,N)$ metric measure spaces $(\mathsf{X},\mathsf{d},\mathfrak{m})$ with pointwise Ahlfors regular reference measure $\mathfrak{m}$ and with $\mathfrak{m}$-almost everywhere unique metric tangents. In particular, we show rectifiability if (i) $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is $\mathsf{CD}(K,N)$ for an arbitrary $N$ and has Hausdorff dimension $n<5$, or (ii) $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is $\mathsf{MCP}(K,N)$ and non-collapsed, namely it has Hausdorff dimension $N$. Our strategy is based on the failure of the $\mathsf{CD}$ condition in sub-Finsler Carnot groups, on a new result on the failure of the non-collapsed $\mathsf{MCP}$ on sub-Finsler Carnot groups, and on the recent breakthrough by Bate [Invent. Math., 230(3):995-1070, 2022]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01151 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the rectifiability of $\mathsf{CD}(K,N)$ and $\mathsf{MCP}(K,N)$ spaces with unique tangents Magnabosco, Mattia Mondino, Andrea Rossi, Tommaso Metric Geometry Differential Geometry We prove rectifiability results for $\mathsf{CD}(K,N)$ and $\mathsf{MCP}(K,N)$ metric measure spaces $(\mathsf{X},\mathsf{d},\mathfrak{m})$ with pointwise Ahlfors regular reference measure $\mathfrak{m}$ and with $\mathfrak{m}$-almost everywhere unique metric tangents. In particular, we show rectifiability if (i) $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is $\mathsf{CD}(K,N)$ for an arbitrary $N$ and has Hausdorff dimension $n<5$, or (ii) $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is $\mathsf{MCP}(K,N)$ and non-collapsed, namely it has Hausdorff dimension $N$. Our strategy is based on the failure of the $\mathsf{CD}$ condition in sub-Finsler Carnot groups, on a new result on the failure of the non-collapsed $\mathsf{MCP}$ on sub-Finsler Carnot groups, and on the recent breakthrough by Bate [Invent. Math., 230(3):995-1070, 2022]. |
| title | On the rectifiability of $\mathsf{CD}(K,N)$ and $\mathsf{MCP}(K,N)$ spaces with unique tangents |
| topic | Metric Geometry Differential Geometry |
| url | https://arxiv.org/abs/2505.01151 |