Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Which shapes can appear in a Curve Shortening Flow Singularity?

Sigurd Angenent, Evan Patrick Davis, Ellie DeCleene, Paige Ellingson, Ziheng Feng, Edgar Gevorgyan, Aris Lemmenes, Alex Moon, Tyler Joseph Tommasi, Yamin Zhou

Abstract

We study possible tangles that can occur in singularities of solutions to plane Curve Shortening Flow. We exhibit solutions in which more complicated tangles with more than one self-intersection disappear into a singular point. It seems that there are many examples of this kind and that a complete classification presents a problem similar to the problem of classifying all knots in $\mathbb R^3$. As a particular example, we introduce the so-called $n$-loop curves, which generalize Matt Grayson's Figure-Eight curve, and we conjecture a generalization of the Coiculescu-Schwarz asymptotic bow-tie result, namely, a vanishing $n$-loop, when rescaled anisotropically to fit a square bounding box, converges to a "squeezed bow-tie," i.e. the curve $\{(x, y) : |x|\leq 1, y=\pm x^{n-1}\}\cup\{(\pm 1, y) : |y|\leq 1\}$. As evidence in support of the conjecture, we provide a formal asymptotic analysis on one hand, and a numerical simulation for the cases $n=3$ and $n=4$ on the other.

Which shapes can appear in a Curve Shortening Flow Singularity?

Abstract

We study possible tangles that can occur in singularities of solutions to plane Curve Shortening Flow. We exhibit solutions in which more complicated tangles with more than one self-intersection disappear into a singular point. It seems that there are many examples of this kind and that a complete classification presents a problem similar to the problem of classifying all knots in . As a particular example, we introduce the so-called -loop curves, which generalize Matt Grayson's Figure-Eight curve, and we conjecture a generalization of the Coiculescu-Schwarz asymptotic bow-tie result, namely, a vanishing -loop, when rescaled anisotropically to fit a square bounding box, converges to a "squeezed bow-tie," i.e. the curve . As evidence in support of the conjecture, we provide a formal asymptotic analysis on one hand, and a numerical simulation for the cases and on the other.
Paper Structure (28 sections, 74 equations, 14 figures)

This paper contains 28 sections, 74 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Existence and uniqueness
  4. The Gage-Hamilton-Grayson theorem for embedded curves
  5. Oaks' Theorem for immersed curves
  6. Two typical examples
  7. The Symmetric Figure Eight
  8. The Symmetric Cardioid
  9. The rate at which loops and eyes lose area
  10. Loops and eyes
  11. The area of a loop or eye
  12. Three intersections simultaneously vanishing into a singular point
  13. Other singularity types
  14. Flat knots and Tangles
  15. $n$-loop curves
  16. ...and 13 more sections

Figures (14)

  • Figure 1: According to Oaks' theorem Oaks1994 any solution $\gamma(t, \cdot)$ to Curve Shortening Flow that becomes singular at some finite time $T>0$ has a limit curve $\gamma(T, \cdot)$, which has finitely many singular points $P_1, \dots , P_N$. Right before the singularity happens, the curve $\gamma(t, \cdot)$ has a self-intersection arbitrarily close to each singular point. In this figure above, the two loops in the initial curve contract at the same time, each resulting in one of the singular points $P_1, P_2$. A curious possibility that Oaks' theorem leaves open is the existence of "hairs" such as the arc $P_1Q$ in the limit curve $\gamma(T, \cdot)$.
  • Figure 2: Grayson's Symmetric Figure-Eight.
  • Figure 3: The Symmetric Cardioid.
  • Figure 4: A loop with a convex corner, one with a concave corner, and an eye
  • Figure 5: Constructing a solution for which three self-intersections vanish into one singularity.
  • ...and 9 more figures

Theorems & Definitions (3)

  • proof
  • proof
  • proof