Title: Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow
Authors: Gabriel Khan
Categories: math.DG math.AP
Comments: 10 pages
MSC-class: 53E10 53A04
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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, $\frac{\tau}{\kappa} \to 0$.
\\ ( https://arxiv.org/abs/2305.07171 , 94kb)