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# Interesting geometric flow of space curves with non-vanishing torsion

Recently, while thinking about CMC surfaces, I came up with an interesting geometric flow for curves in $$\mathbb{R}^3$$ given by $$$$\partial_t \gamma = \tau^{-\frac{1}{2}} n,$$$$ where $$\gamma$$ denotes the parametrization, $$\tau$$ is torsion and $$n$$ is the principal normal vector. Here and after, we consider only closed curves and we use $$\Gamma_t$$ to denote the curve given by the parametrization $$\gamma(\cdot, t)$$.

This flow has many intriguing properties. First of all, curves evolving according to this motion law trace out a zero mean curvature surface! Thus, it might be used for generating minimal surface with a prescribed boundary given by the initial curve.

There are several problems with this flow. Most importantly, the term $$\tau^{-\frac{1}{2}}$$ is defined only when $$\tau$$ is strictly positive. However, when torsion of the initial curve $$\Gamma_0$$ is lower bounded by some positive constant $$C$$, we get $$$$\tau(u, t) = \left( \sqrt{\tau(u, 0)} + \int_{0}^{t} \kappa(u, \bar{t}) \mathrm{d} \bar{t} \right)^2 \geq \tau(u, 0) \geq C > 0,$$$$ because $$$$\partial_t \sqrt{\tau} = \tfrac{1}{2} \tau^{-\frac{1}{2}} \partial_t \tau = \tfrac{1}{2} \tau^{-\frac{1}{2}} \left( 2 \tau^{\frac{1}{2}} \kappa + \partial_s \left[ \tfrac{1}{\kappa} \left( \tau^{-\frac{1}{2}} \partial_s \tau + 2\tau \partial_s \left( \tau^{-\frac{1}{2}} \right) \right) \right] \right) = \kappa,$$$$ where $$\partial_s$$ is the arclength derivative and $$\kappa$$ is the curvature. Thus the curve cannot develop a vertex (point of vanishing torsion) and $$\tau^{-\frac{1}{2}}$$ remains well-defined.

Questions:

1. Is there any simple way to proof or disproof the existence and uniqueness of this flow?

2. Is there any similar geometric flow involving torsion that has been already studied?

I will end this post with a list of interesting properties (I can provide proofs upon request):

• The integral of $$\sqrt{\tau}$$ is preserved, i.e. $$$$\frac{\mathrm{d}}{\mathrm{d} t} \int_{\Gamma_t} \tau^{\frac{1}{2}} \mathrm{d} s = 0.$$$$

• The length of the curve $$\Gamma_t$$ is non-increasing, i.e. $$$$\frac{\mathrm{d}}{\mathrm{d} t} \int_{\Gamma_t} \mathrm{d} s = - \int_{\Gamma_t} \kappa \tau^{-\frac{1}{2}} \mathrm{d} s \leq 0.$$$$ In fact, using Fenchel's theorem and Gauss-Bonnet theorem, one can show that $$$$\int_{0}^{t} \left( \int_{\Gamma_\bar{t}} \mathrm{d} s \right) \mathrm{d} \bar{t} \leq \frac{1}{\inf_{\Gamma_0} \tau^{\frac{3}{2}}} \left( \int_{\Gamma_0} \kappa \mathrm{d} s -2\pi \right).$$$$ If the right-hand side is finite and the flow exists for $$t \in [0,+\infty)$$, the length of $$\Gamma_t$$ must approach zero - the curve shrinks to a point as time approaches infinity.

• Area $$A_t$$ of the generated surface is bounded by a constant which depends only on the shape of the initial curve $$\Gamma_0$$ (it does not depend on time $$t$$): $$$$A_t \leq \frac{1}{\inf_{\Gamma_0} \tau^2} \left( \int_{\Gamma_0} \kappa \mathrm{d} s - 2 \pi \right).$$$$

• A simple analytical solution for this motion is a shrinking helix curve, which generates the helicoid surface. I do not know any analytical solution for a closed curve.