2 Updated tags.
| link
1
source | link

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 \begin{equation} \partial_t \gamma = \tau^{-\frac{1}{2}} n, \end{equation} 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 \begin{equation} \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, \end{equation} because \begin{equation} \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, \end{equation} 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. \begin{equation} \frac{\mathrm{d}}{\mathrm{d} t} \int_{\Gamma_t} \tau^{\frac{1}{2}} \mathrm{d} s = 0. \end{equation}

  • The length of the curve $\Gamma_t$ is non-increasing, i.e. \begin{equation} \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. \end{equation} In fact, using Fenchel's theorem and Gauss-Bonnet theorem, one can show that \begin{equation} \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). \end{equation} 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 $t$): \begin{equation} A_t \leq \frac{1}{\inf_{\Gamma_0} \tau^2} \left( \int_{\Gamma_0} \kappa \mathrm{d} s - 2 \pi \right). \end{equation}

  • 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.