Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.


If the curves are smooth one can do this via curve shortening flow/rescaling. I think this approach even works for rectifiable curves, by a [paper](https://link.springer.com/article/10.1007/s00039-013-0248-1) of Lauer. But I do not know a reference for the general topological case. In the smooth case, is there a way to do this without using flows?


By the space of Jordan curves here I mean one-to-one continuous maps from $\textbf{S}^1$ to $\textbf{R}^2$, modulo homeomorphisms of $\textbf{S}^1$. So two Jordan curves are close provided that they admit parameterizations which are pointwise close.



Contractibility of the space of Jordan curves
It seems that the answer is "yes".


Let us identify $\mathbb{R}^2$ with $\mathbb{S}^2\setminus\{n\}$.
Each Jordan curve $\gamma$ bounds a disc containing $n$.
This disc admits a conformal parametrization by the unit disc $\mathbb{D}$ such that its center goes to $n$.
This parametrization is unique up to rotation of $\mathbb{D}$.
In particular, the image $\gamma\_r$ of the circle of radius $r$ in $\mathbb{D}$ is completely determined by $\gamma$.
Note that there is a homotopy from that sends $\gamma$ to $\gamma\_{1/2}$.
The continuity at $t=1$ follows from Thm 15 (VIII, §81) in "Automorphic Functions" by Lester R Ford.


So the question is reduced to the smooth case, and you know it already.


