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

TL;DR

This paper studies the evolution of symmetric figure-eight curves under the curve shortening flow, showing they converge to a bowtie shape after normalization and that certain arcs approach the Grim Reaper soliton.

Contribution

It introduces a new analysis of figure-eight curves with symmetry and convexity under the flow, demonstrating convergence to a specific limiting shape and connecting to the Grim Reaper soliton.

Findings

Normalized limits of the curves converge to a bowtie shape.

Certain arcs of the evolving curves approach the Grim Reaper soliton.

The results extend understanding of curve shortening flow to symmetric figure-eight shapes.

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.