CURVATURE-TORSION ENTROPY FOR TWISTED CURVES UNDER CURVE SHORTENING FLOW

Abstract

Abstract We study curve-shortening flow for twisted curves in $\mathbb {R}^3$ (that is, curves with nowhere vanishing curvature $\kappa $ and torsion $\tau $ ) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type-II singularity which admits sequences along which ${\tau }/{\kappa ^2} \to \infty $ . This contrasts strongly with Altschuler’s planarity theorem, which shows that ${\tau }/{\kappa } \to 0$ along any essential blow-up sequence.