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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.01922 |
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| _version_ | 1866913564448522240 |
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| author | Xu, Kai |
| author_facet | Xu, Kai |
| contents | Let $M$ be a closed orientable 3-manifold with scalar curvature greater than or equal to 1. If $M$ has nonvanishing second homotopy group, then it is known that the $π_2$-systole of $M$ (i.e. the minimal achievable area of homotopically nontrivial spheres) is at most $8π$. We prove the following gap theorem: if $M$ is further not a quotient of $S^2\times S^1$, then the $π_2$-systole of $M$ is no greater than an improved constant $c\approx 5.44π$. This statement follows as a new topological application of Huisken and Ilmanen's weak inverse mean curvature flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_01922 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A topological gap theorem for the $π_2$-systole of positive scalar curvature 3-manifolds Xu, Kai Differential Geometry 53C20, 53E10 Let $M$ be a closed orientable 3-manifold with scalar curvature greater than or equal to 1. If $M$ has nonvanishing second homotopy group, then it is known that the $π_2$-systole of $M$ (i.e. the minimal achievable area of homotopically nontrivial spheres) is at most $8π$. We prove the following gap theorem: if $M$ is further not a quotient of $S^2\times S^1$, then the $π_2$-systole of $M$ is no greater than an improved constant $c\approx 5.44π$. This statement follows as a new topological application of Huisken and Ilmanen's weak inverse mean curvature flow. |
| title | A topological gap theorem for the $π_2$-systole of positive scalar curvature 3-manifolds |
| topic | Differential Geometry 53C20, 53E10 |
| url | https://arxiv.org/abs/2307.01922 |