Scalar curvature under weak limits of manifolds
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866913088630947840 |
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| author | Mazurowski, Liam Yao, Xuan |
| author_facet | Mazurowski, Liam Yao, Xuan |
| contents | We show that scalar curvature lower bounds are preserved under certain weak convergence of smooth three manifolds to a smooth limit. More precisely, suppose that $M_k$ and $M$ are smooth, closed, Riemannian three manifolds. Assume that there are smooth, surjective, $λ_k$-Lipschitz maps $f_k\colon M_k \to M$ and that $\text{Vol}(M_k)\to \text{Vol}(M)$ and $λ_k\to 1$. Then if each $M_k$ has scalar curvature bounded below by $κ$ so does $M$. This result answers questions of Gromov, Sormani, Allen, and others. The proof relies on a delicate comparison between $μ$-bubbles in $M_k$ and $μ$-bubbles in $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03136 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Scalar curvature under weak limits of manifolds Mazurowski, Liam Yao, Xuan Differential Geometry We show that scalar curvature lower bounds are preserved under certain weak convergence of smooth three manifolds to a smooth limit. More precisely, suppose that $M_k$ and $M$ are smooth, closed, Riemannian three manifolds. Assume that there are smooth, surjective, $λ_k$-Lipschitz maps $f_k\colon M_k \to M$ and that $\text{Vol}(M_k)\to \text{Vol}(M)$ and $λ_k\to 1$. Then if each $M_k$ has scalar curvature bounded below by $κ$ so does $M$. This result answers questions of Gromov, Sormani, Allen, and others. The proof relies on a delicate comparison between $μ$-bubbles in $M_k$ and $μ$-bubbles in $M$. |
| title | Scalar curvature under weak limits of manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.03136 |