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Singularities of Curve Shortening Flow with Convex Projections

Qi Sun

TL;DR

This work establishes that closed immersed space curves in $\mathbb{R}^n$ with a one-to-one convex projection onto a plane experience Type I singularities under Curve Shortening Flow and, upon appropriate rescaling, converge to a unit circle, i.e., become asymptotically circular. The authors develop a sharp blow-up analysis by proving non-uniqueness of tangent flows (via barrier methods) and then a subsequent Allard–Almgren-type argument to recover uniqueness, enabling a confident classification of the singularity. Key steps include strengthening Type II blow-up results to show limits are multiplicity-two lines with non-unique directions, then using barrier subsolutions to force a unique tangent flow in the end; once Type I is secured, Abresch–Langer shrinkers and the convex-projection property imply the limit must be a round circle. As an application, perturbing any immersed curve in $\mathbb{R}^n$ into $\mathbb{R}^{n+2}$ to gain a convex projection yields a CSF that shrinks to a round point, aligning with a higher-codimension analogue of Huisken’s conjecture. Overall, the paper extends sharp Type I/II blow-up analysis and uniqueness results to higher codimension curves under convex-projection constraints, with implications for generic singularities and geometric evolution in higher codimension.

Abstract

We show that any 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 closed immersed curve in $\mathbb R^n$ can be perturbed to a closed immersed curve in $\mathbb R^{n+2}$ which shrinks to a round point under Curve Shortening flow.

Singularities of Curve Shortening Flow with Convex Projections

TL;DR

This work establishes that closed immersed space curves in with a one-to-one convex projection onto a plane experience Type I singularities under Curve Shortening Flow and, upon appropriate rescaling, converge to a unit circle, i.e., become asymptotically circular. The authors develop a sharp blow-up analysis by proving non-uniqueness of tangent flows (via barrier methods) and then a subsequent Allard–Almgren-type argument to recover uniqueness, enabling a confident classification of the singularity. Key steps include strengthening Type II blow-up results to show limits are multiplicity-two lines with non-unique directions, then using barrier subsolutions to force a unique tangent flow in the end; once Type I is secured, Abresch–Langer shrinkers and the convex-projection property imply the limit must be a round circle. As an application, perturbing any immersed curve in into to gain a convex projection yields a CSF that shrinks to a round point, aligning with a higher-codimension analogue of Huisken’s conjecture. Overall, the paper extends sharp Type I/II blow-up analysis and uniqueness results to higher codimension curves under convex-projection constraints, with implications for generic singularities and geometric evolution in higher codimension.

Abstract

We show that any closed immersed curve in with a one-to-one convex projection onto some -plane develops a Type~I singularity and becomes asymptotically circular under Curve Shortening flow in . As an application, we prove an analog of Huisken's conjecture for Curve Shortening flow in , showing that any closed immersed curve in can be perturbed to a closed immersed curve in which shrinks to a round point under Curve Shortening flow.
Paper Structure (39 sections, 69 theorems, 333 equations, 4 figures)

This paper contains 39 sections, 69 theorems, 333 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Main result
  5. Application
  6. Strategy of our proof
  7. Outline of the paper
  8. Known results on blow-up
  9. CSF with one-to-one convex projections
  10. Geometry of CSF with convex projections
  11. Type I singularity and compact blow-up limits
  12. Type II singularity and non-compact blow-up limits
  13. Improved blow-up results for curves with convex projections
  14. Preparations
  15. Graphicality
  16. ...and 24 more sections

Key Result

Theorem 1.4

If the initial curve $\gamma_0$ has a one-to-one convex projection onto the $xy$-plane, then CSF $\gamma(\cdot,t)$ develops a Type I singularity and becomes asymptotically circular as $t\rightarrow T$.

Figures (4)

  • Figure 1: Examples on CSF with a one-to-one convex projection
  • Figure 2: Snapshots of the evolution of a perturbation of the planar figure eight curve from different angles. Previously appeared in sun2024curve.
  • Figure 3: Points $p(\tau)$, $q(\tau)$ on $S^1_\tau\Gamma(\cdot,\tau)$.
  • Figure 4: Points $p(\tau)$, $q(\tau)$ on $S^{bis}_\tau\Gamma(\cdot,\tau)$.

Theorems & Definitions (174)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5: Perturbing immersed closed curves
  • Corollary 1.6: Perturbing a planar figure-eight
  • Definition 1.7: Following Huisken huisken1990asymptotic
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 164 more