Rigidity of area non-increasing maps
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914691418161152 |
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| author | Lee, Man-Chun Tam, Luen-Fai Wan, Jingbo |
| author_facet | Lee, Man-Chun Tam, Luen-Fai Wan, Jingbo |
| contents | In this work, we consider the area non-increasing map between manifolds with positive curvature. By exploring the strong maximum principle along the graphical mean curvature flow, we show that an area non-increasing map between certain positively curved manifolds is either homotopy trivial, Riemannian submersion, local isometry or isometric immersion. This implies that an area non-increasing self map of $\mathbb{CP}^n$, $n\ge 2$ is either homotopically trivial or is an isometry. This confirms a speculation of Tsai-Tsui-Wang. We also use Brendle's sphere Theorem and mean curvature flow coupled with Ricci flow to establish related results on manifolds with positive $1$-isotropic curvature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_10940 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rigidity of area non-increasing maps Lee, Man-Chun Tam, Luen-Fai Wan, Jingbo Differential Geometry Analysis of PDEs Geometric Topology In this work, we consider the area non-increasing map between manifolds with positive curvature. By exploring the strong maximum principle along the graphical mean curvature flow, we show that an area non-increasing map between certain positively curved manifolds is either homotopy trivial, Riemannian submersion, local isometry or isometric immersion. This implies that an area non-increasing self map of $\mathbb{CP}^n$, $n\ge 2$ is either homotopically trivial or is an isometry. This confirms a speculation of Tsai-Tsui-Wang. We also use Brendle's sphere Theorem and mean curvature flow coupled with Ricci flow to establish related results on manifolds with positive $1$-isotropic curvature. |
| title | Rigidity of area non-increasing maps |
| topic | Differential Geometry Analysis of PDEs Geometric Topology |
| url | https://arxiv.org/abs/2312.10940 |