Llarull type theorems on complete manifolds with positive scalar curvature
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911808336429056 |
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| author | Hao, Tianze Shi, Yuguang Sun, Yukai |
| author_facet | Hao, Tianze Shi, Yuguang Sun, Yukai |
| contents | In this paper, without assuming that manifolds are spin, we prove that if a compact orientable, and connected Riemannian manifold $(M^{n},g)$ with scalar curvature $R_{g}\geq 6$ admits a non-zero degree and $1$-Lipschitz map to $(\mathbb{S}^{3}\times \mathbb{T}^{n-3},g_{\mathbb{S}^{3}}+g_{\mathbb{T}^{n-3}})$, for $4\leq n\leq 7$, then $(M^{n},g)$ is locally isometric to $\mathbb{S}^{3}\times\mathbb{T}^{n-3}$. Similar results are established for noncompact cases as $(\mathbb{S}^{3}\times \mathbb{R}^{n-3},g_{\mathbb{S}^{3}}+g_{\mathbb{R}^{n-3}})$ being model spaces (see Theorem \ref{noncompactrigidity1}, Theorem \ref{noncompactrigidity2}, Theorem \ref{noncompactrigidity3}, Theorem \ref{noncompactrigidity4}). We observe that the results differ significantly when $n=4$ compared to $n\geq 5$. Our results imply that the $ε$-gap length extremality of the standard $\mathbb{S}^3$ is stable under the Riemannian product with $\mathbb{R}^m$, $1\leq m\leq 4$ (see $D_{3}$. Question in Gromov's paper \cite{Gromov2017}, p.153). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_10173 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Llarull type theorems on complete manifolds with positive scalar curvature Hao, Tianze Shi, Yuguang Sun, Yukai Differential Geometry 53C21, 53C24 In this paper, without assuming that manifolds are spin, we prove that if a compact orientable, and connected Riemannian manifold $(M^{n},g)$ with scalar curvature $R_{g}\geq 6$ admits a non-zero degree and $1$-Lipschitz map to $(\mathbb{S}^{3}\times \mathbb{T}^{n-3},g_{\mathbb{S}^{3}}+g_{\mathbb{T}^{n-3}})$, for $4\leq n\leq 7$, then $(M^{n},g)$ is locally isometric to $\mathbb{S}^{3}\times\mathbb{T}^{n-3}$. Similar results are established for noncompact cases as $(\mathbb{S}^{3}\times \mathbb{R}^{n-3},g_{\mathbb{S}^{3}}+g_{\mathbb{R}^{n-3}})$ being model spaces (see Theorem \ref{noncompactrigidity1}, Theorem \ref{noncompactrigidity2}, Theorem \ref{noncompactrigidity3}, Theorem \ref{noncompactrigidity4}). We observe that the results differ significantly when $n=4$ compared to $n\geq 5$. Our results imply that the $ε$-gap length extremality of the standard $\mathbb{S}^3$ is stable under the Riemannian product with $\mathbb{R}^m$, $1\leq m\leq 4$ (see $D_{3}$. Question in Gromov's paper \cite{Gromov2017}, p.153). |
| title | Llarull type theorems on complete manifolds with positive scalar curvature |
| topic | Differential Geometry 53C21, 53C24 |
| url | https://arxiv.org/abs/2310.10173 |