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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.25116 |
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| _version_ | 1866911714419671040 |
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| author | Wang, Changliang Wang, Zhixin |
| author_facet | Wang, Changliang Wang, Zhixin |
| contents | We study the Gromov--Sormani MinA scalar curvature compactness conjecture for warped product metrics on $\mathbb{S}^2\times\mathbb{S}^1$ of the form introduced by Kazaras-Xu in \cite{KazarasXu2023} as follows: \[ g_i=φ_i^{-2}h_i+φ_i^2dξ^2, \qquad h_i=dr^2+u_i^2(r)dθ^2. \] Assuming nonnegative scalar curvature, a uniform volume upper bound, and a positive lower bound for the areas of closed minimal surfaces, we prove a uniform diameter bound for the base surfaces $(\mathbb{S}^2,h_i)$. Based on this key estimate, we further obtain compactness of the base warping functions $u_i$ and local and global estimates for the fiber warping functions $φ_i$.
After passing to a subsequence, the metrics converge in $L^p$, for every finite $p$, to a limit metric $g_\infty$. %on the regular region. We also obtain Gromov--Hausdorff and Sormani--Wenger intrinsic flat subconvergence, and prove that $g_\infty$ has nonnegative scalar curvature in the distributional sense of Lee--LeFloch. Thus the Gromov--Sormani scalar curvature compactness conjecture is verified for this warped product class. Finally, we construct a $C^{1,α}$ example illustrating the subtlety of volume-limit tests for nonnegative scalar curvature in low regularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_25116 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Scalar Curvature Compactness for Warped Products on $\mathbb{S}^2\times\mathbb{S}^1$ with Varying Base Metrics Wang, Changliang Wang, Zhixin Differential Geometry We study the Gromov--Sormani MinA scalar curvature compactness conjecture for warped product metrics on $\mathbb{S}^2\times\mathbb{S}^1$ of the form introduced by Kazaras-Xu in \cite{KazarasXu2023} as follows: \[ g_i=φ_i^{-2}h_i+φ_i^2dξ^2, \qquad h_i=dr^2+u_i^2(r)dθ^2. \] Assuming nonnegative scalar curvature, a uniform volume upper bound, and a positive lower bound for the areas of closed minimal surfaces, we prove a uniform diameter bound for the base surfaces $(\mathbb{S}^2,h_i)$. Based on this key estimate, we further obtain compactness of the base warping functions $u_i$ and local and global estimates for the fiber warping functions $φ_i$. After passing to a subsequence, the metrics converge in $L^p$, for every finite $p$, to a limit metric $g_\infty$. %on the regular region. We also obtain Gromov--Hausdorff and Sormani--Wenger intrinsic flat subconvergence, and prove that $g_\infty$ has nonnegative scalar curvature in the distributional sense of Lee--LeFloch. Thus the Gromov--Sormani scalar curvature compactness conjecture is verified for this warped product class. Finally, we construct a $C^{1,α}$ example illustrating the subtlety of volume-limit tests for nonnegative scalar curvature in low regularity. |
| title | Scalar Curvature Compactness for Warped Products on $\mathbb{S}^2\times\mathbb{S}^1$ with Varying Base Metrics |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.25116 |