Legendrian curve shortening flow in $\mathbb{R}^{3}$

TL;DR

This paper investigates the evolution of figure-eight curves in the plane under curve shortening flow, demonstrating conditions under which they shrink to a point, inspired by Legendrian flow in three dimensions.

Contribution

It extends the study of curve shortening flow to figure-eight curves with specific symmetry and curvature conditions, providing new insights into their singularity formation.

Findings

Figure-eight curves shrink to a point under certain conditions

Symmetry and curvature constraints influence the flow behavior

First singular time corresponds to complete shrinking of the curve

Abstract

Motivated by Legendrian curve shortening flows in $\mathbb{R}^{3}$, we study the curve shortening flow of figure-eight curves in the plane. We show that, under some symmetry and curvature conditions, a figure-eight curve will shrink to a point at the first singular time.