Scalar curvature rigidity for products of spheres and tori
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918188729499648 |
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| author | Chow, Tsz-Kiu Aaron |
| author_facet | Chow, Tsz-Kiu Aaron |
| contents | We prove Llarull-type rigidity for $S^{n-m}\times\mathbb{T}^m$ ($3\le n\le 7$, $1\le m\le n-2$). If a closed spin $(M^n,g)$ admits a degree-nonzero map to $S^{n-m}\times\mathbb{T}^m$ whose spherical projection is area non-increasing, and there exists $ψ\in C^\infty(M)$ with $-Δ_Mψ-\frac{1}{2}|D_Mψ|^2+\frac{1}{2}\big(R_M-(n-m)(n-m-1)\big)\ge0$, then $(M,g)$ is isometrically covered by $S^{n-m}\times\mathbb{R}^m$. For bands, we extend Gromov's torical inequality and obtain sharp width bounds: $\text{dist}(\partial_-M,\partial_+M)\le 2π\sqrt{n/((n+1)σ)}$ when $R_M\ge (n-m)(n-m-1)+σ$. The method combines stable weighted slicing with a spectral Dirac operator argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scalar curvature rigidity for products of spheres and tori Chow, Tsz-Kiu Aaron Differential Geometry Analysis of PDEs 53C21, 53C24, 53C27 We prove Llarull-type rigidity for $S^{n-m}\times\mathbb{T}^m$ ($3\le n\le 7$, $1\le m\le n-2$). If a closed spin $(M^n,g)$ admits a degree-nonzero map to $S^{n-m}\times\mathbb{T}^m$ whose spherical projection is area non-increasing, and there exists $ψ\in C^\infty(M)$ with $-Δ_Mψ-\frac{1}{2}|D_Mψ|^2+\frac{1}{2}\big(R_M-(n-m)(n-m-1)\big)\ge0$, then $(M,g)$ is isometrically covered by $S^{n-m}\times\mathbb{R}^m$. For bands, we extend Gromov's torical inequality and obtain sharp width bounds: $\text{dist}(\partial_-M,\partial_+M)\le 2π\sqrt{n/((n+1)σ)}$ when $R_M\ge (n-m)(n-m-1)+σ$. The method combines stable weighted slicing with a spectral Dirac operator argument. |
| title | Scalar curvature rigidity for products of spheres and tori |
| topic | Differential Geometry Analysis of PDEs 53C21, 53C24, 53C27 |
| url | https://arxiv.org/abs/2511.04407 |