Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909090016395264 |
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| author | Zhao, Ziyi Zhu, Xiaohua |
| author_facet | Zhao, Ziyi Zhu, Xiaohua |
| contents | Let $(M^n,g)$ $(n\ge 4)$ be a complete noncompact $κ$-noncollapsed steady Ricci soliton with $\rm{Rm}\geq 0$ and $\rm{Ric}> 0$ away from a compact set $K$ of $M$.
We prove that there is no any $(n-1)$-dimensional compact split limit Ricci flow of type I arising from the blow-down of $(M, g)$, if there is an $(n-1)$-dimensional noncompact split limit Ricci flow.
Consequently, the compact split limit ancient flows of type I and type II cannot occur simultaneously from the blow-down.
As an application, we prove that $(M^n,g)$ with $\rm{Rm}\geq 0$ must be isometric the Bryant Ricci soliton up to scaling, if there exists a sequence of rescaled Ricci flows $(M,g_{p_i}(t); p_i)$ of $(M,g)$ converges subsequently to a family of shrinking quotient cylinders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00316 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set Zhao, Ziyi Zhu, Xiaohua Differential Geometry Primary: 53E20, Secondary: 53C20, 53C25, 58J05 Let $(M^n,g)$ $(n\ge 4)$ be a complete noncompact $κ$-noncollapsed steady Ricci soliton with $\rm{Rm}\geq 0$ and $\rm{Ric}> 0$ away from a compact set $K$ of $M$. We prove that there is no any $(n-1)$-dimensional compact split limit Ricci flow of type I arising from the blow-down of $(M, g)$, if there is an $(n-1)$-dimensional noncompact split limit Ricci flow. Consequently, the compact split limit ancient flows of type I and type II cannot occur simultaneously from the blow-down. As an application, we prove that $(M^n,g)$ with $\rm{Rm}\geq 0$ must be isometric the Bryant Ricci soliton up to scaling, if there exists a sequence of rescaled Ricci flows $(M,g_{p_i}(t); p_i)$ of $(M,g)$ converges subsequently to a family of shrinking quotient cylinders. |
| title | Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set |
| topic | Differential Geometry Primary: 53E20, Secondary: 53C20, 53C25, 58J05 |
| url | https://arxiv.org/abs/2402.00316 |