Singularities of Curve Shortening Flow with Convex Projections

2025/10/16 by Sun, Qi
#53E10 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.14863

Abstract

We show that any closed immersed curve in \mathbb Rⁿ with a one-to-one convex projection onto some 2-plane develops a Type~I singularity and becomes asymptotically circular under Curve Shortening flow in \mathbb Rⁿ. As an application, we prove an analog of Huisken's conjecture for Curve Shortening flow in \mathbb Rⁿ, showing that any closed immersed curve in \mathbb Rⁿ can be perturbed to a closed immersed curve in \mathbb Rⁿ⁺² which shrinks to a round point under Curve Shortening flow.

Related

• Curve Shortening Flow of Space Curves with Convex Projections
• Curve shortening flows on surfaces that are not convex at infinity
• The Dirichlet problem for curve shortening flow
• Huisken's Distance Comparison Principle in Higher Codimension
• Curve shortening flow on Riemann surfaces with possible ambient conic singularities