Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow

Abstract

We study curve-shortening flow for twisted curves in ${\mathbb{R}}^3$ (i.e., 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 $\frac{\tau }{{\kappa }^2}\to \infty$. This contrasts strongly with Altschuler's planarity theorem [J. Differential Geom. (1991)], which shows that along any essential blow-up sequence, $\frac{\tau }{\kappa }\to 0$.

arXiv comment10 pages

References in corpus (3)

• Distance comparison principle and Grayson type theorem in the three dimensional curve shortening flow
• A Condition Ensuring Spatial Curves Develop Type-II Singularities Under Curve Shortening Flow
• Singularities of low entropy high codimension curve shortening flow