# 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.

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.

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