High Codimension Curve Shortening Flow with Free Boundary

TL;DR

This paper investigates the behavior of high codimension curve shortening flows with free boundaries, establishing curvature estimates, analyzing singularities, and characterizing flow convergence or singularity formation under entropy constraints.

Contribution

It introduces new curvature and higher-derivative estimates for free boundary flows in high codimension, and classifies singularities using entropy and blow-up analysis.

Findings

Type I boundary singularities resemble shrinking semicircles.

Type II blow-ups are Grim Reaper translators, ruled out under entropy bounds.

Flow converges to orthogonal chords or develops semicircle singularities.

Abstract

We study curve shortening flow in high codimension for arcs with free boundary meeting a fixed smooth barrier orthogonally. We prove dilation-invariant curvature and higher-derivative estimates up to the boundary using a Stahl-type localised maximum principle and an adapted cut-off. Using a reflected Gaussian entropy and blow-up analysis, Type I boundary singularities yield a shrinking semicircle model after reflection. Type II blow-ups give a Grim Reaper translator, which is ruled out under a free-boundary entropy bound $<2$. Hence in the low-entropy regime the flow either converges to the orthogonal chord or has only semicircle boundary singularities.