Interesting geometric flow of space curves with non-disappearing twist

Recently, when thinking about CMC surfaces, I've developed an interesting geometric flow for curves $ mathbb {R} ^ 3 $ given by
begin {equation}
partial_t gamma = tau ^ {- frac {1} {2}} n,
end {equation}

from where $ gamma $ designates the parameterization, $ tau $ is twist and $ n $ is the main normal vector. Here and beyond, we only consider closed curves and use them $ Gamma_t $ the curve given by the parameterization $ gamma ( cdot, t) $,

This river has many fascinating features. First, curves that evolve according to this motion law follow, a zero mean curvature surface! Thus, it can be used to create a minimum surface with a prescribed limit given by the initial curve.

There are several problems with this river. Most important is the term $ tau ^ {- frac {1} {2}} $ is only defined if $ tau $ is absolutely positive. At twist of the initial curve $ Gamma_0 $ is limited by a positive constant lower $ 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 geqtau (u, 0) geq C> 0,
end {equation}

there
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 + 2tau partial_s left( tau^{-frac{1}{2}} right) right) right] right) = kappa,
end {equation}

from where $ partial_s $ is the arclength derivative and $ kappa $ is the curvature. Thus, the curve can not develop a vertex (torsion point) $ tau ^ {- frac {1} {2}} $ remains clearly defined.

Ask:

  1. Is there an easy way to test or refute the existence and uniqueness of this river?

  2. Is there a similar geometric flux with torsion that has already been studied?


I end this post with a list of interesting features (I can provide evidence on request):

  • The integral of $ sqrt { tau} $ is obtained, i.
    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 $ does not stretch, i.
    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}

    You can really show that with Fennel's theorem and Gauss-Bonnet's theorem
    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}

    When the right side is finite and the river exists for $ t in[0+infty)$[0+infty)$[0+infty)$[0+infty)$, the length of $ Gamma_t $ must approach zero – the curve shrinks to a point when the time approaches infinity.

  • Area $ A_t $ the area created is limited by a constant that depends only on the shape of the initial curve $ Gamma_0 $ (it does not matter to the 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 to this motion is a shrinking helix curve that creates the helicoid surface. I do not know any analytic solution for a closed curve.