On the volume measure of non-smooth spaces with Ricci curvature bounded below
We prove that, given an $RCD^{*}(K,N)$-space $(X,d,m)$, then it is possible to $m$-essentially cover $X$ by measurable subsets $(R_{i})_{i\in \mathbb{N}}$ with the following property: for each $i$ there exists $k_{i} \in \mathbb{N}\cap [1,N]$ such that $m\llcorner R_{i}$ is absolutely continuous wit...
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| Format: | Journal article |
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Scuola Normale Superiore
2018
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| _version_ | 1797106311312179200 |
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| author | Kell, M Mondino, A |
| author_facet | Kell, M Mondino, A |
| author_sort | Kell, M |
| collection | OXFORD |
| description | We prove that, given an $RCD^{*}(K,N)$-space $(X,d,m)$, then it is possible to $m$-essentially cover $X$ by measurable subsets $(R_{i})_{i\in \mathbb{N}}$ with the following property: for each $i$ there exists $k_{i} \in \mathbb{N}\cap [1,N]$ such that $m\llcorner R_{i}$ is absolutely continuous with respect to the $k_{i}$-dimensional Hausdorff measure. We also show that a Lipschitz differentiability space which is bi-Lipschitz embeddable into a euclidean space is rectifiable as a metric measure space, and we conclude with an application to Alexandrov spaces. |
| first_indexed | 2024-03-07T06:59:57Z |
| format | Journal article |
| id | oxford-uuid:ff66c446-2c9d-423b-8f8f-c551dae93472 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:59:57Z |
| publishDate | 2018 |
| publisher | Scuola Normale Superiore |
| record_format | dspace |
| spelling | oxford-uuid:ff66c446-2c9d-423b-8f8f-c551dae934722022-03-27T13:44:35ZOn the volume measure of non-smooth spaces with Ricci curvature bounded belowJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ff66c446-2c9d-423b-8f8f-c551dae93472Symplectic Elements at OxfordScuola Normale Superiore2018Kell, MMondino, AWe prove that, given an $RCD^{*}(K,N)$-space $(X,d,m)$, then it is possible to $m$-essentially cover $X$ by measurable subsets $(R_{i})_{i\in \mathbb{N}}$ with the following property: for each $i$ there exists $k_{i} \in \mathbb{N}\cap [1,N]$ such that $m\llcorner R_{i}$ is absolutely continuous with respect to the $k_{i}$-dimensional Hausdorff measure. We also show that a Lipschitz differentiability space which is bi-Lipschitz embeddable into a euclidean space is rectifiable as a metric measure space, and we conclude with an application to Alexandrov spaces. |
| spellingShingle | Kell, M Mondino, A On the volume measure of non-smooth spaces with Ricci curvature bounded below |
| title | On the volume measure of non-smooth spaces with Ricci curvature bounded below |
| title_full | On the volume measure of non-smooth spaces with Ricci curvature bounded below |
| title_fullStr | On the volume measure of non-smooth spaces with Ricci curvature bounded below |
| title_full_unstemmed | On the volume measure of non-smooth spaces with Ricci curvature bounded below |
| title_short | On the volume measure of non-smooth spaces with Ricci curvature bounded below |
| title_sort | on the volume measure of non smooth spaces with ricci curvature bounded below |
| work_keys_str_mv | AT kellm onthevolumemeasureofnonsmoothspaceswithriccicurvatureboundedbelow AT mondinoa onthevolumemeasureofnonsmoothspaceswithriccicurvatureboundedbelow |