Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916268100026368 |
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| author | Chau, Albert Martens, Adam |
| author_facet | Chau, Albert Martens, Adam |
| contents | Lai (2021) used singular Ricci flows, introduced by Kleiner and Lott (2017), to construct a nonnegative Ricci curvature Ricci flow $g(t)$ emerging from an arbitrary 3D complete noncompact Riemannian manifold $(M^3, g_0)$ which has nonnegative Ricci curvature. We show $g(t)$ is complete for positive times provided $g_0$ satisfies a volume ratio lower bound that approaches zero at spatial infinity. Our proof combines a pseudolocality result of Lai (2021) for singular flows, together with a pseudolocality result of Hochard (2016) and Simon and Topping (2022) for nonsingular flows. We also show that the construction of complete nonnegative complex sectional curvature flows by Cabezas-Rivas and Wilking (2015) can be adapted here to show $g(t)$ is complete for positive times provided $g_0$ is a compactly supported perturbation of a nonnegative sectional curvature metric on $\mathbb{R}^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08088 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows Chau, Albert Martens, Adam Differential Geometry 53E20 Lai (2021) used singular Ricci flows, introduced by Kleiner and Lott (2017), to construct a nonnegative Ricci curvature Ricci flow $g(t)$ emerging from an arbitrary 3D complete noncompact Riemannian manifold $(M^3, g_0)$ which has nonnegative Ricci curvature. We show $g(t)$ is complete for positive times provided $g_0$ satisfies a volume ratio lower bound that approaches zero at spatial infinity. Our proof combines a pseudolocality result of Lai (2021) for singular flows, together with a pseudolocality result of Hochard (2016) and Simon and Topping (2022) for nonsingular flows. We also show that the construction of complete nonnegative complex sectional curvature flows by Cabezas-Rivas and Wilking (2015) can be adapted here to show $g(t)$ is complete for positive times provided $g_0$ is a compactly supported perturbation of a nonnegative sectional curvature metric on $\mathbb{R}^3$. |
| title | Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows |
| topic | Differential Geometry 53E20 |
| url | https://arxiv.org/abs/2307.08088 |