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 $\kappa$ and torsion $\tau$. We contrast the classical flow $\partial_t \gamma = \kappa^n \mathbf{N}$ with the novel torsion-dependent flow $\partial_t \gamma = \tau^n \mathbf{N}$. We show a difference in their long-term behavior: while flows by curvature lead to an infinite-time collapse towards the axis for $n \ge 1$, 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 $\partial_t\gamma=(a\kappa+b\tau)\mathbf{N}$ which, under some values of $a$ and $b$, provides solutions whose shape remains invariant for all $t$.