Singularities of Curve Shortening Flow with Convex Projections

Abstract

We show that any smooth closed immersed curve in ${\mathbb{R}}^n$ 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 ${\mathbb{R}}^n$. As an application, we prove an analog of Huisken's conjecture for Curve Shortening flow in ${\mathbb{R}}^n$, showing that any smooth closed immersed curve in ${\mathbb{R}}^n$ can be smoothly perturbed to a closed immersed curve in ${\mathbb{R}}^{n+2}$ which shrinks to a round point under Curve Shortening flow. Our proof relies on a novel contradiction argument in which Type~{II} singularities are excluded by proving both the uniqueness and non-uniqueness of the tangent flows at the singular point.

59 pages, 11 figures, minor updates