Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914702206959616 |
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| author | Chau, Albert Martens, Adam |
| author_facet | Chau, Albert Martens, Adam |
| contents | We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the curvature is sub-critical $L^p$ integrable, this flow converges locally smoothly to a limiting metric $g(\infty)$ on $M$ with $(M,g(\infty))$ isometric to the standard flat $\mathbb{R}^n$, which implies topological rigidity of $M$. This generalizes work of Chen, who proved analogous results for asymptotically flat manifolds. We also prove a long-time Ricci flow existence (and likewise topological rigidity) result for unbounded curvature initial data, assuming the initial data is a locally smooth limit of bounded curvature manifolds as described above. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02564 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature Chau, Albert Martens, Adam Differential Geometry 53E20 We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the curvature is sub-critical $L^p$ integrable, this flow converges locally smoothly to a limiting metric $g(\infty)$ on $M$ with $(M,g(\infty))$ isometric to the standard flat $\mathbb{R}^n$, which implies topological rigidity of $M$. This generalizes work of Chen, who proved analogous results for asymptotically flat manifolds. We also prove a long-time Ricci flow existence (and likewise topological rigidity) result for unbounded curvature initial data, assuming the initial data is a locally smooth limit of bounded curvature manifolds as described above. |
| title | Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature |
| topic | Differential Geometry 53E20 |
| url | https://arxiv.org/abs/2403.02564 |