Notes on the Cheeger and Colding version of the Reifenberg theorem for metric spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929392128622592 |
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| author | Gigli, Nicola Violo, Ivan Yuri |
| author_facet | Gigli, Nicola Violo, Ivan Yuri |
| contents | The classical Reifenberg's theorem says that a set which is sufficiently well approximated by planes uniformly at all scales is a topological Hölder manifold. Remarkably, this generalizes to metric spaces, where the approximation by planes is replaced by the Gromov-Hausdorff distance. This fact was shown by Cheeger and Colding in an appendix of one of their celebrated works on Ricci limit spaces [8]. Given the recent interest around this statement in the growing field of analysis in metric spaces, in this note we provide a self contained and detailed proof of the Cheeger and Colding result. Our presentation substantially expands the arguments in [8] and makes explicit all the relevant estimates and constructions. As a byproduct we also shows a biLipschitz version of this result which, even if folklore among experts, was not present in the literature. This work is an extract from the doctoral dissertation of the second author. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_13771 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Notes on the Cheeger and Colding version of the Reifenberg theorem for metric spaces Gigli, Nicola Violo, Ivan Yuri Metric Geometry Differential Geometry The classical Reifenberg's theorem says that a set which is sufficiently well approximated by planes uniformly at all scales is a topological Hölder manifold. Remarkably, this generalizes to metric spaces, where the approximation by planes is replaced by the Gromov-Hausdorff distance. This fact was shown by Cheeger and Colding in an appendix of one of their celebrated works on Ricci limit spaces [8]. Given the recent interest around this statement in the growing field of analysis in metric spaces, in this note we provide a self contained and detailed proof of the Cheeger and Colding result. Our presentation substantially expands the arguments in [8] and makes explicit all the relevant estimates and constructions. As a byproduct we also shows a biLipschitz version of this result which, even if folklore among experts, was not present in the literature. This work is an extract from the doctoral dissertation of the second author. |
| title | Notes on the Cheeger and Colding version of the Reifenberg theorem for metric spaces |
| topic | Metric Geometry Differential Geometry |
| url | https://arxiv.org/abs/2406.13771 |