Poincaré inequality for one forms on four manifolds with bounded Ricci curvature
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929638734823424 |
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| author | Honda, Shouhei Mondino, Andrea |
| author_facet | Honda, Shouhei Mondino, Andrea |
| contents | In this short note, we provide a quantitative global Poincaré inequality for one forms on a closed Riemannian four manifold, in terms of an upper bound on the diameter, a positive lower bound on the volume, and a two-sided bound on Ricci curvature. This seems to be the first non-trivial result giving such an inequality without any higher curvature assumptions. The proof is based on a Hodge theoretic result on orbifolds, a comparison for fundamental groups, and a spectral convergence with respect to Gromov-Hausdorff convergence, via a degeneration result to orbifolds by Anderson. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_19168 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Poincaré inequality for one forms on four manifolds with bounded Ricci curvature Honda, Shouhei Mondino, Andrea Differential Geometry In this short note, we provide a quantitative global Poincaré inequality for one forms on a closed Riemannian four manifold, in terms of an upper bound on the diameter, a positive lower bound on the volume, and a two-sided bound on Ricci curvature. This seems to be the first non-trivial result giving such an inequality without any higher curvature assumptions. The proof is based on a Hodge theoretic result on orbifolds, a comparison for fundamental groups, and a spectral convergence with respect to Gromov-Hausdorff convergence, via a degeneration result to orbifolds by Anderson. |
| title | Poincaré inequality for one forms on four manifolds with bounded Ricci curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2405.19168 |