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| Format: | Preprint |
| Published: |
2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1909.12265 |
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| _version_ | 1866909687332470784 |
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| author | Huang, Hong |
| author_facet | Huang, Hong |
| contents | We prove the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or the total space of an orbifiber bundle over $\mathbb{S}^1$ or $\mathcal{I}$ with generic fiber diffeomorphic to $\mathbb{S}^{n-1}/Γ$ such that the total space admits a metric with positive isotropic curvature, where $Γ$ is a finite subgroup of $O(n)$ acting freely on $\mathbb{S}^{n-1}$, and $\mathcal{I}$ is the one dimensional closed orbifold with two singular points both with local group $\mathbb{Z}_2$ and with $|\mathcal{I}|$ a closed interval, or a connected sum of a finite number of such manifolds. This extends a recent work of Brendle, and implies a conjecture of Schoen and a conjecture of Gromov in dimensions $n\geq 12$. The proof uses Ricci flow with surgery on compact orbifolds with isolated singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_12265 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Compact manifolds of dimension $n\geq 12$ with positive isotropic curvature Huang, Hong Differential Geometry We prove the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or the total space of an orbifiber bundle over $\mathbb{S}^1$ or $\mathcal{I}$ with generic fiber diffeomorphic to $\mathbb{S}^{n-1}/Γ$ such that the total space admits a metric with positive isotropic curvature, where $Γ$ is a finite subgroup of $O(n)$ acting freely on $\mathbb{S}^{n-1}$, and $\mathcal{I}$ is the one dimensional closed orbifold with two singular points both with local group $\mathbb{Z}_2$ and with $|\mathcal{I}|$ a closed interval, or a connected sum of a finite number of such manifolds. This extends a recent work of Brendle, and implies a conjecture of Schoen and a conjecture of Gromov in dimensions $n\geq 12$. The proof uses Ricci flow with surgery on compact orbifolds with isolated singularities. |
| title | Compact manifolds of dimension $n\geq 12$ with positive isotropic curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/1909.12265 |