Precompactness of domains with lower Ricci curvature bound under Gromov-Hausdorff topology
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915515107115008 |
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| author | Xu, Shicheng |
| author_facet | Xu, Shicheng |
| contents | Based on a quantitative version of the classical Hopf-Rinow theorem in terms of the doubling property, we prove new precompactness principles in the (pointed) Gromov-Hausdorff topology for domains in (maybe incomplete) Riemannian manifolds with a lower Ricci curvature bound, which are applicable to those with weak regularities considered in PDE theory, and the covering spaces of balls naturally appear in the study of local geometry and topology of manifolds with lower curvature bounds. All the new principles are more general than those earlier known for manifolds with smooth boundary, and improves those for manifolds with non-smooth boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_05140 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Precompactness of domains with lower Ricci curvature bound under Gromov-Hausdorff topology Xu, Shicheng Differential Geometry Metric Geometry 53C21, 53C23, 54A20 Based on a quantitative version of the classical Hopf-Rinow theorem in terms of the doubling property, we prove new precompactness principles in the (pointed) Gromov-Hausdorff topology for domains in (maybe incomplete) Riemannian manifolds with a lower Ricci curvature bound, which are applicable to those with weak regularities considered in PDE theory, and the covering spaces of balls naturally appear in the study of local geometry and topology of manifolds with lower curvature bounds. All the new principles are more general than those earlier known for manifolds with smooth boundary, and improves those for manifolds with non-smooth boundary. |
| title | Precompactness of domains with lower Ricci curvature bound under Gromov-Hausdorff topology |
| topic | Differential Geometry Metric Geometry 53C21, 53C23, 54A20 |
| url | https://arxiv.org/abs/2311.05140 |