The curve-shortening flow is $$ \frac{\partial C}{\partial t} = \kappa n $$ where $\kappa$ is the curvature, and $n$ is the unit normal vector. For a smooth Jordan curve $C\subset\mathbb R^2$ (closed curve without crossings), this flow has many nice properties, e.g. the only singularity happens when the curve collapses to a point.

I am curious to learn whether the following "one-sided version": $$ \frac{\partial C}{\partial t} = f(\kappa) n $$ has similar nice properties. Here, $f:\mathbb R\to \mathbb R_{\ge 0}$ is a $C^\infty$ function satisfying $f(x) =0, \forall x\le 0$ and $f(x)=x, \forall x\ge 1$ and $f'(x)>0, \forall x> 0$.

**Precise question:**
Let $C\subset \mathbb R^2$ be a non-compact simple curve which is periodic in the sense that it is invariant under the translation $(x,y)\mapsto (x+1,y)$. Is the above "one-sided curve-shortening flow" applied to $C$ defined for all times? and does it converge to a horizontal line in $\mathbb R^2$?

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