1. Passing through nondegenerate singularities in mean curvature flows
- Author
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Sun, Ao, Wang, Zhihan, and Xue, Jinxin
- Subjects
Mathematics - Differential Geometry ,Mathematics - Analysis of PDEs ,Mathematics - Geometric Topology - Abstract
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., Comment: 52 pages, comments are welcome!
- Published
- 2025