Singularities of low entropy high codimension curve shortening flow

Abstract: We consider curve shortening flow of arbitrary codimension in an Euclidean background. We show that, close to a singularity, the flow is asymptotically planar, paralleling Altschuler's work in the case of space curves, and analyse the blow-up limits of the flow. Using these results, we then prove that the curve shortening flow of initial curves with an entropy bound converges to a round point in finite time.