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| Main Authors: | , |
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| Format: | Preprint |
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2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2004.09762 |
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| _version_ | 1866912170640408576 |
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| author | Huang, Shaosai Wang, Bing |
| author_facet | Huang, Shaosai Wang, Bing |
| contents | The Colding-Gromov gap theorem asserts that an almost non-negatively Ricci curved manifold with unit diameter and maximal first Betti number is homeomorphic to the flat torus. In this paper, we prove a parametrized version of this theorem, in the context of collapsing Riemannian manifolds with Ricci curvature bounded below: if a closed manifold with Ricci curvature uniformly bounded below is Gromov-Hausdorff close to a (lower dimensional) manifold with bounded geometry, and has the difference of their first Betti numbers equal to the dimensional difference, then it is diffeomorphic to a torus bundle over the one with bounded geometry. We rely on two novel technical tools: the first is an effective control of the spreading of minimal geodesics with initial data parallel transported along a short geodesic segment, and the second is a Ricci flow smoothing result for certain collapsing initial data with Ricci curvature bounded below. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_09762 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Rigidity of the first Betti number via Ricci flow smoothing Huang, Shaosai Wang, Bing Differential Geometry 53C The Colding-Gromov gap theorem asserts that an almost non-negatively Ricci curved manifold with unit diameter and maximal first Betti number is homeomorphic to the flat torus. In this paper, we prove a parametrized version of this theorem, in the context of collapsing Riemannian manifolds with Ricci curvature bounded below: if a closed manifold with Ricci curvature uniformly bounded below is Gromov-Hausdorff close to a (lower dimensional) manifold with bounded geometry, and has the difference of their first Betti numbers equal to the dimensional difference, then it is diffeomorphic to a torus bundle over the one with bounded geometry. We rely on two novel technical tools: the first is an effective control of the spreading of minimal geodesics with initial data parallel transported along a short geodesic segment, and the second is a Ricci flow smoothing result for certain collapsing initial data with Ricci curvature bounded below. |
| title | Rigidity of the first Betti number via Ricci flow smoothing |
| topic | Differential Geometry 53C |
| url | https://arxiv.org/abs/2004.09762 |