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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.

We study the mechanism of proving non-collapsing in the context of extrinsic curvature flows via the maximum principle in combination with a suitable two-point function in homogeneity greater than one. Our paper serves as the first step in this direction and we consider the case of a curve which is C2-close to a circle initially and which flows by a power greater than one of the curvature along its normal vector.

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CURVATURE-TORSION ENTROPY FOR TWISTED CURVES UNDER CURVE SHORTENING FLOW

Non-collapsing in homogeneity greater than one via a two-point method for a special case

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CURVATURE-TORSION ENTROPY FOR TWISTED CURVES UNDER CURVE SHORTENING FLOW

Non-collapsing in homogeneity greater than one via a two-point method for a special case