Local singularities of compact multiply warped Ricci flow solutions
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913688276959232 |
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| author | Isenberg, James Knopf, Dan Ma, Zilu Sesum, Natasa |
| author_facet | Isenberg, James Knopf, Dan Ma, Zilu Sesum, Natasa |
| contents | We demonstrate that any four-dimensional shrinking Ricci soliton $(\mathcal B \times {\mathbb S^2}, g)$, where $\mathcal B$ is any two-dimensional complete noncompact surface and $g$ is a warped product metric over the base $\mathcal B$, has to be isometric to the generalized cylinder $\mathbb R^2\times\mathbb S^2$ equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products -- but not products -- and provide rigorous examples of the formation of generalized cylinder singularity models $\mathbb R^k\times\mathbb S^\ell$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local singularities of compact multiply warped Ricci flow solutions Isenberg, James Knopf, Dan Ma, Zilu Sesum, Natasa Differential Geometry 35C08, 35K59, 53C21 We demonstrate that any four-dimensional shrinking Ricci soliton $(\mathcal B \times {\mathbb S^2}, g)$, where $\mathcal B$ is any two-dimensional complete noncompact surface and $g$ is a warped product metric over the base $\mathcal B$, has to be isometric to the generalized cylinder $\mathbb R^2\times\mathbb S^2$ equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products -- but not products -- and provide rigorous examples of the formation of generalized cylinder singularity models $\mathbb R^k\times\mathbb S^\ell$. |
| title | Local singularities of compact multiply warped Ricci flow solutions |
| topic | Differential Geometry 35C08, 35K59, 53C21 |
| url | https://arxiv.org/abs/2502.08500 |