Stationary Solutions of the Curvature Preserving Flow on Space Curves

TL;DR

This paper investigates stationary solutions of a geometric flow on space curves with positive torsion, analyzing their stability and explicitly characterizing solutions when curvature is constant.

Contribution

It provides a complete classification of stationary solutions for the flow and establishes their linear stability in the case of constant curvature and torsion.

Findings

Classified all stationary solutions with constant curvature.

Proved linear stability of helical solutions.

Derived explicit formulas for torsion linearization.

Abstract

We study a geometric flow on curves, immersed in $\mathbb{R}^3$, that have strictly positive torsion. The evolution equation is given by $$X_{t}=\frac{1}{\sqrt{\tau}} \textbf{B}$$ where $\tau$ is the torsion and $\textbf{B}$ is the unit binormal vector. In the case of constant curvature, we find all of the stationary solutions and linearize the PDE for torsion around stationary solutions admitting an explicit formula. Afterwards, we prove the $L^2(\mathbb{R})$ linear stability of the stationary solutions corresponding to helices with constant curvature and constant torsion.