Limits of manifolds with a Kato bound on the Ricci curvature
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866913566477516800 |
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| author | Carron, Gilles Mondello, Ilaria Tewodrose, David |
| author_facet | Carron, Gilles Mondello, Ilaria Tewodrose, David |
| contents | We study the structure of Gromov-Hausdorff limits of sequences of Riemannian manifolds $\{(M_α^n,g_α)\}_{α\in A}$ whose Ricci curvature satisfies a uniform Kato bound. We first obtain Mosco convergence of the Dirichlet energies to the Cheeger energy and show that tangent cones of such limits satisfy the $\mathrm{RCD}(0,n)$ condition. When assuming a non-collapsing assumption, we introduce a new family of monotone quantities, which allows us to prove that tangent cones are also metric cones. We then show the existence of a well-defined stratification in terms of splittings of tangent cones. We finally prove volume convergence to the Hausdorff $n$-measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_05940 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Limits of manifolds with a Kato bound on the Ricci curvature Carron, Gilles Mondello, Ilaria Tewodrose, David Differential Geometry We study the structure of Gromov-Hausdorff limits of sequences of Riemannian manifolds $\{(M_α^n,g_α)\}_{α\in A}$ whose Ricci curvature satisfies a uniform Kato bound. We first obtain Mosco convergence of the Dirichlet energies to the Cheeger energy and show that tangent cones of such limits satisfy the $\mathrm{RCD}(0,n)$ condition. When assuming a non-collapsing assumption, we introduce a new family of monotone quantities, which allows us to prove that tangent cones are also metric cones. We then show the existence of a well-defined stratification in terms of splittings of tangent cones. We finally prove volume convergence to the Hausdorff $n$-measure. |
| title | Limits of manifolds with a Kato bound on the Ricci curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2102.05940 |