A Condition Ensuring Spatial Curves Develop Type-II Singularities Under Curve Shortening Flow

2012/09/16 by Gabriel Khan, Khan, Gabriel
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows #math.DG

paper · pdf · doi:10.48550/arxiv.1209.4072

This draft replaces the second one, in which we fix some typos. The second draft replaced the first one, which contained a mistake in the calculations, reducing the generality of the theorem. Thank you to Ben Andrews for pointing this out

openalex publication_date 2012/09/16 · arxiv created 2015/12/16 · arxiv updated 2015/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide a condition for spatial curves which rules out the development of a type I singularity. The condition is that after the last time for which an inflection point develops, if the torsion is ever everywhere non-negative, the curve cannot develop a type I singularity.

Citations

• Distance comparison principle and Grayson type theorem in the three dimensional curve shortening flow

Related

• Non existence of Type II singularities for embedded and unknotted space curves
• Singularities of the area preserving curve shortening flow with a\n Neumann free boundary condition
• Type I singularities in the curve shortening flow associated to a density
• Singularities of Curve Shortening Flow with Convex Projections
• Singularities of low entropy high codimension curve shortening flow