Evolution flows by curvature and torsion for helical curves

This paper investigates geometric evolution equations for the family of circular helices where the normal velocity depends on curvature κ and torsion τ. We contrast the classical flow ∂ₜ γ = κⁿ N with the novel torsion-dependent flow ∂ₜ γ = τⁿ N. We show a difference in their long-term behavior: while flows by curvature lead to an infinite-time collapse towards the axis for n ≥ 1, the flow by torsion forces extinction in finite time for $n>0$. We provide analytical solutions for the radius evolution, characterize the existence of ancient solutions, and analyze the extinction time. Finally, we study the flow ∂ₜγ=(aκ+bτ)N which, under some values of a and b, provides solutions whose shape remains invariant for all t.