Scalar curvature rigidity of the four-dimensional sphere
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866914721433649152 |
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| author | Cecchini, Simone Wang, Jinmin Xie, Zhizhang Zhu, Bo |
| author_facet | Cecchini, Simone Wang, Jinmin Xie, Zhizhang Zhu, Bo |
| contents | Let $(M,g)$ be a closed connected oriented (possibly non-spin) smooth four-dimensional manifold with scalar curvature bounded below by $n(n-1)$. In this paper, we prove that if $f$ is a smooth map of non-zero degree from $(M, g)$ to the unit four-sphere, then $f$ is an isometry. Following ideas of Gromov, we use $μ$-bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case of strict inequality. Our proof of rigidity is based on the harmonic map heat flow coupled with the Ricci flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12633 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scalar curvature rigidity of the four-dimensional sphere Cecchini, Simone Wang, Jinmin Xie, Zhizhang Zhu, Bo Differential Geometry Let $(M,g)$ be a closed connected oriented (possibly non-spin) smooth four-dimensional manifold with scalar curvature bounded below by $n(n-1)$. In this paper, we prove that if $f$ is a smooth map of non-zero degree from $(M, g)$ to the unit four-sphere, then $f$ is an isometry. Following ideas of Gromov, we use $μ$-bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case of strict inequality. Our proof of rigidity is based on the harmonic map heat flow coupled with the Ricci flow. |
| title | Scalar curvature rigidity of the four-dimensional sphere |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2402.12633 |