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.

arXiv commentOur paper was accepted to the Journal of Differential Geometry, and this version is the final version we sent to the journal. It agrees with the previous version except that we corrected a few typos and also shortened the proof of the Migration Lemma in the last chapter