Huisken's Distance Comparison Principle in Higher Codimension

2025/09/20 by Qi Sun, Sun, Qi
Computer Science · #53E10 #Advanced Clustering Algorithms Research #Analysis of PDEs (math.AP) #Bayesian Methods and Mixture Models #Differential Geometry (math.DG) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2509.16823

openalex publication_date 2025/09/20 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

We establish a variant of Huisken's distance comparison principle for reflection symmetric immersed Curve Shortening flow in \mathbb Rⁿ,n≥2. As an application, we show that certain symmetric Curve Shortening flow with a one-to-one convex projection develops a Type~I singularity and becomes asymptotically circular.

Citations

• Curve Shortening Flow of Space Curves with Convex Projections
• Sharp distance comparison for curve shortening flow on the round sphere
• A distance comparison principle for curve shortening flow with free boundary
• Uniqueness of compact tangent flows in Mean Curvature Flow
• Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem

Related

• Distance comparison principle and Grayson type theorem in the three dimensional curve shortening flow
• A distance comparison principle for curve flows with a global forcing term
• Singularities of Curve Shortening Flow with Convex Projections
• A distance comparison principle for curve shortening flow with free boundary
• A short proof of monotonicity formula for curve shortening flow in 3D