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The Affine Shape of a Figure-Eight under the Curve Shortening Flow

Matei P. Coiculescu, Richard Evan Schwartz

Abstract

We consider the curve shortening flow applied to a class of figure-eight curves: those with dihedral symmetry, convex lobes, and a monotonicity assumption on the curvature. We prove that when (non-conformal) linear transformations are applied to the solution so as to keep the bounding box the unit square, the renormalized limit converges to a quadrilateral which we call a bowtie. Along the way we prove that suitably chosen arcs of our evolving curves, when suitably rescaled, converge to the Grim Reaper Soliton under the flow. Our Grim Reaper Theorem is an analogue of a theorem of S. Angenent, which is proven in the locally convex case.

The Affine Shape of a Figure-Eight under the Curve Shortening Flow

Abstract

We consider the curve shortening flow applied to a class of figure-eight curves: those with dihedral symmetry, convex lobes, and a monotonicity assumption on the curvature. We prove that when (non-conformal) linear transformations are applied to the solution so as to keep the bounding box the unit square, the renormalized limit converges to a quadrilateral which we call a bowtie. Along the way we prove that suitably chosen arcs of our evolving curves, when suitably rescaled, converge to the Grim Reaper Soliton under the flow. Our Grim Reaper Theorem is an analogue of a theorem of S. Angenent, which is proven in the locally convex case.

Paper Structure

This paper contains 18 sections, 31 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preservation of Monotonicity
  3. Strictly Parabolic Equations
  4. Evolution Equations
  5. Monotonicity
  6. Some Asymptotic Results
  7. The Grim Reaper Theorem
  8. Counting Zeros
  9. The Sine Lemma
  10. The End of the Proof
  11. Asymptotic Formulas
  12. The Y Bound
  13. The X Bound
  14. The Support Function
  15. The Parabolic Rescaling
  16. ...and 3 more sections

Key Result

Theorem 1.1

Assume that $C(0)$ is monotone. Let $J \subset (0,\pi)$ be an arbitrary closed interval. Let $\epsilon>0$ be given. For $t$ sufficiently close to $T$, we have

Theorems & Definitions (54)

  • Theorem 1.1: Grim Reaper
  • Theorem 1.2: Bowtie
  • Lemma 1.3: Migration
  • Theorem 2.1: The Maximum Principle
  • Theorem 2.2: The Sturmian Principle
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 44 more