Regularity of cylindrical singular sets of mean curvature flow
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914017070546944 |
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| author | Sun, Ao Wang, Zhihan Xue, Jinxin |
| author_facet | Sun, Ao Wang, Zhihan Xue, Jinxin |
| contents | In this paper, we study the $k$-cylindrical singular set of mean curvature flow in $\mathbb R^{n+1}$ for each $1\leq k\leq n-1$. We prove that they are locally contained in a $k$-dimensional $C^{2,α}$-submanifold after removing some lower-dimensional parts. Moreover, if the $k$-cylindrical singular set is a $k$-submanifold, then its curvature is determined by the asymptotic profile of the flow at these singularities. As a byproduct, we provide a detailed asymptotic profile and graphical radius estimate at these singularities. The proof is based on a new $L^2$-distance non-concentration property that we introduced in [SWX25], modified into a relative version that allows us to modulo those low eigenmodes that are not decaying fast enough and do not contribute to the curvature of the singular set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01707 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity of cylindrical singular sets of mean curvature flow Sun, Ao Wang, Zhihan Xue, Jinxin Differential Geometry Analysis of PDEs In this paper, we study the $k$-cylindrical singular set of mean curvature flow in $\mathbb R^{n+1}$ for each $1\leq k\leq n-1$. We prove that they are locally contained in a $k$-dimensional $C^{2,α}$-submanifold after removing some lower-dimensional parts. Moreover, if the $k$-cylindrical singular set is a $k$-submanifold, then its curvature is determined by the asymptotic profile of the flow at these singularities. As a byproduct, we provide a detailed asymptotic profile and graphical radius estimate at these singularities. The proof is based on a new $L^2$-distance non-concentration property that we introduced in [SWX25], modified into a relative version that allows us to modulo those low eigenmodes that are not decaying fast enough and do not contribute to the curvature of the singular set. |
| title | Regularity of cylindrical singular sets of mean curvature flow |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2509.01707 |