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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.13183 |
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| _version_ | 1866909791384764416 |
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| author | Chen, Zhengnan |
| author_facet | Chen, Zhengnan |
| contents | For all dimensions $n\geq5$, let $(M,g,f)$ be a $n-$dimensional shrinking gradient Ricci soliton with strictly positive isotropic curvature (PIC). Suppose furthermore that $\nabla^2f$ is $2-$nonnegative and the curvature tensor is WPIC1 at some point $\bar{x}\in M$. Then $(M,g)$ must be a quotient of either $S^{n}$ or $S^{n-1}\times\mathbb{R}$. Our result partially extends the classification result for 4-dimensional PIC shrinking Ricci solitons established in [LNW16] to highter dimensions. Combining the pinching estimates deduced in [Chen24] we also extend the result in [CL23] to dimensions $n\geq9$. Namely that a complete ancient solution to the Ricci flow of dimension $n\geq9$ with uniformly PIC must be weakly PIC2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13183 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions Chen, Zhengnan Differential Geometry For all dimensions $n\geq5$, let $(M,g,f)$ be a $n-$dimensional shrinking gradient Ricci soliton with strictly positive isotropic curvature (PIC). Suppose furthermore that $\nabla^2f$ is $2-$nonnegative and the curvature tensor is WPIC1 at some point $\bar{x}\in M$. Then $(M,g)$ must be a quotient of either $S^{n}$ or $S^{n-1}\times\mathbb{R}$. Our result partially extends the classification result for 4-dimensional PIC shrinking Ricci solitons established in [LNW16] to highter dimensions. Combining the pinching estimates deduced in [Chen24] we also extend the result in [CL23] to dimensions $n\geq9$. Namely that a complete ancient solution to the Ricci flow of dimension $n\geq9$ with uniformly PIC must be weakly PIC2. |
| title | On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2509.13183 |