Passing through nondegenerate singularities in mean curvature flows
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866929689360072704 |
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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 properties of nondegenerate cylindrical singularities of mean curvature flow. We prove they are isolated in spacetime and provide a complete description of the geometry and topology change of the flow passing through the singularities. Particularly, the topology change agrees with the level sets change near a critical point of a Morse function, which is the same as performing surgery. The proof is based on a new $L^2$-distance monotonicity formula, which allows us to derive a discrete almost monotonicity of the ``decay order", a discrete mean curvature flow analog to Almgren's frequency function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_16678 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Passing through nondegenerate singularities in mean curvature flows Sun, Ao Wang, Zhihan Xue, Jinxin Differential Geometry Analysis of PDEs Geometric Topology In this paper, we study the properties of nondegenerate cylindrical singularities of mean curvature flow. We prove they are isolated in spacetime and provide a complete description of the geometry and topology change of the flow passing through the singularities. Particularly, the topology change agrees with the level sets change near a critical point of a Morse function, which is the same as performing surgery. The proof is based on a new $L^2$-distance monotonicity formula, which allows us to derive a discrete almost monotonicity of the ``decay order", a discrete mean curvature flow analog to Almgren's frequency function. |
| title | Passing through nondegenerate singularities in mean curvature flows |
| topic | Differential Geometry Analysis of PDEs Geometric Topology |
| url | https://arxiv.org/abs/2501.16678 |