Dimension constraints in some problems involving intermediate curvature
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913522893455360 |
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| author | Xu, Kai |
| author_facet | Xu, Kai |
| contents | In arXiv:2207.08617 [math.DG] Brendle-Hirsch-Johne proved that $T^m\times S^{n-m}$ does not admit metrics with positive $m$-intermediate curvature when $n\leq 7$. Chu-Kwong-Lee showed in arXiv:2208.12240 [math.DG] a corresponding rigidity statement when $n\leq 5$. In this paper, we show the sharpness of the dimension constraints by giving concrete counterexamples in $n\geq 7$ and extending the rigidity result to $n=6$. Concerning uniformly positive intermediate curvature, we show that simply-connected manifolds with dimension $\leq 5$ and bi-Ricci curvature $\geq 1$ have finite Urysohn 1-width. Counterexamples are constructed in dimension $\geq 6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_02730 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dimension constraints in some problems involving intermediate curvature Xu, Kai Differential Geometry In arXiv:2207.08617 [math.DG] Brendle-Hirsch-Johne proved that $T^m\times S^{n-m}$ does not admit metrics with positive $m$-intermediate curvature when $n\leq 7$. Chu-Kwong-Lee showed in arXiv:2208.12240 [math.DG] a corresponding rigidity statement when $n\leq 5$. In this paper, we show the sharpness of the dimension constraints by giving concrete counterexamples in $n\geq 7$ and extending the rigidity result to $n=6$. Concerning uniformly positive intermediate curvature, we show that simply-connected manifolds with dimension $\leq 5$ and bi-Ricci curvature $\geq 1$ have finite Urysohn 1-width. Counterexamples are constructed in dimension $\geq 6$. |
| title | Dimension constraints in some problems involving intermediate curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2301.02730 |