A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917438809964544 |
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| author | Wang, Chen Wu, Guoqiang |
| author_facet | Wang, Chen Wu, Guoqiang |
| contents | Let $(M^n, g, f)$ be an $n$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \frac{1}{2}g$.
1. If its scalar curvature is $\frac{k}{2}$, Ricci curvature is nonnegative and sectional curvature has upper bound $\frac{1}{2(k-1)}$, we prove that the Ricci shrinker is isometric to a finite quotient of $\mathbb{R}^{n-k}\times \mathbb{S}^k$.
2. If $M$ has constant scalar curvature $R=\frac{n-2}{2}$, and each level set of $f$ has vanishing Weyl curvature, we prove that it is a finite quotient of $\mathbb{R}^2\times \mathbb{S}^{n-2}$. This can be seen a generalization of Cheng-Zhou's four dimensional result \cite{Cheng-Zhou} to high dimension, since the level set of the potential function $f$ has vanishing Weyl curvature automatically when $n=4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23939 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature Wang, Chen Wu, Guoqiang Differential Geometry Analysis of PDEs Let $(M^n, g, f)$ be an $n$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \frac{1}{2}g$. 1. If its scalar curvature is $\frac{k}{2}$, Ricci curvature is nonnegative and sectional curvature has upper bound $\frac{1}{2(k-1)}$, we prove that the Ricci shrinker is isometric to a finite quotient of $\mathbb{R}^{n-k}\times \mathbb{S}^k$. 2. If $M$ has constant scalar curvature $R=\frac{n-2}{2}$, and each level set of $f$ has vanishing Weyl curvature, we prove that it is a finite quotient of $\mathbb{R}^2\times \mathbb{S}^{n-2}$. This can be seen a generalization of Cheng-Zhou's four dimensional result \cite{Cheng-Zhou} to high dimension, since the level set of the potential function $f$ has vanishing Weyl curvature automatically when $n=4$. |
| title | A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2604.23939 |