Contractibility of the space of Jordan curves

Question

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 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.

Answer 1

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.

Answer 2

Anton's construction depends continuously on the curve but does not seem quite canonical, in the sense that smoothings of two congruent curves may not be congruent. We can ensure that this will be the case by refining the argument as follows.

Given a Jordan curve $J$ in $\textbf{R}^2$, let $C$ be the smallest circle which encloses it, and $C_\epsilon$ be the outer parallel circle at some fixed distance $\epsilon>0$. Then the region between $J$ and $C_\epsilon$ forms a topological annulus $A$. There exists a unique annulus $A'$ of the form $r\leq |z|\leq 1$ which is conformal to $A$ (e.g. see Thm 2.7.3 in Conformal Maps and Geometry by D. Beliaev). Let $f\colon A'\to A$ be a conformal map. Restricting $f$ to circles $|z|=t$ yields a homotopy between $J$ and $C_\epsilon$ which depends only on the congruence class of $J$ (the choice of $f$ does not matter because the only conformal maps $A'\to A'$ are rotations, or an inversion).

Addendum (8/8/2023): Igor Belegradek and I just finished a paper where we show that the space of Jordan domains in any surface of constant curvature admits a strong deformation retraction onto the space of round disks, and this deformation is equivariant under isometries of the surface. The proof is based on a point selection theorem from the interior of domains which allows us to apply Riemann's mapping theorem, or more precisely Caratheodory's kernel theorem, to a family of domains, as discussed in a related post.