Let mathbf f: R/2 pilongrightarrow C be a smooth unit-speed simple closed curve, and let D denote the space of deformations satisfying conditions (1)--(4). Smooth-in-time unfolding conjecture. There exists a continuous function mathbf h:[0,1]longrightarrow D satisfying (1)--(4), such that mathbf h(t)(x) is a smooth function of x for every t∈[0,1]. This asks that a smooth initial curve remain smooth at every time during the deformation and is stated as an open strengthening of the main theorem.

Smooth-in-time unfolding conjecture for smooth simple closed curves

At least 17 years old · documented by 2008

Let $\mathbf f:\mathbb R/2\pi\longrightarrow\mathbb C$ be a smooth unit-speed simple closed curve, and let $\mathcal D$ denote the space of deformations satisfying conditions (1)--(4). Smooth-in-time unfolding conjecture. There exists a continuous function

\mathbf h:[0,1]\longrightarrow\mathcal D

satisfying (1)--(4), such that $\mathbf h(t)(x)$ is a smooth function of $x$ for every $t\in[0,1]$ . This asks that a smooth initial curve remain smooth at every time during the deformation and is stated as an open strengthening of the main theorem.

References

Primary source

John Pardon, “On the unfolding of simple closed curves”, arXiv:0809.1404 (2008).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Open

The stronger deformation problem remains open: the known theorem unfolds every closed loop, but does not keep smooth loops smooth throughout.

The conjecture, stated as Conjecture $6.2$ in a 2008 paper, asks whether every smooth unit-speed simple closed curve admits a continuous deformation satisfying the required conditions while remaining smooth in the spatial variable at every time. It is an open strengthening of the paper's main unfolding theorem.

Known results

Every rectifiable simple closed plane curve admits a continuous, unit-speed, expansive deformation into a convex curve; this theorem does not guarantee smoothness at every time.

Current status (as of September 2026): the weaker expansive-deformation theorem is known, while the smooth-in-time strengthening remains open; no public proof, counterexample, or claimed resolution was found.

Sources

No solutions have been posted yet.