Arnold-Thom conjecture for the arrival time of surfaces

2024/05/29 by Tang-Kai Lee, Lee, Tang-Kai, Jingze Zhu +1 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2405.19064

openalex publication_date 2024/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Following Łojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. We prove Łojasiewicz's theorem and Arnold's conjecture in the context of arrival time functions for mean curvature flows in \mathbb Rⁿ⁺¹ with neck or non-degenerate cylindrical singularities. In particular, we prove the conjectures for all mean convex mean curvature flows of surfaces, including the cases when the arrival time functions are not C². The results also apply to mean curvature flows starting from two-spheres or generic closed surfaces.

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