Thin-tube flows around arbitrary closed curves

Construct, for every closed C^1 curve in Euclidean space, a mean curvature flow whose initial hypersurface is the boundary of a tubular neighborhood of the curve and whose singular set is the curve itself.

Background

The marriage ring is presented as an example of a mean curvature flow whose singular set is a curve and whose singularities are modeled by a one-axis cylinder. The authors formulate a broader conjectural existence problem for flows collapsing onto an arbitrary prescribed closed C1 curve.

References

Conjecturally, for any closed $C1$-curve $\gamma$, there exists a mean curvature flow ${M_t}{t\in[0,t_0)}$ such that $M{0}$ is the boundary of a tubular neighbourhood of $\gamma$, and $M_t$ collapses to a singular set $\gamma$.

— Mean curvature flows with cylindrical singularities II: stability and genericity  (2609.19390 - Sun et al., 16 Sep 2026) in Section 1, paragraph discussing the marriage ring and thin tubes

Any smooth curve $\gamma\subsetR3$ can be the singular set of a $($mean convex$)$ mean curvature flow.

— Mean curvature flows with cylindrical singularities II: stability and genericity  (2609.19390 - Sun et al., 16 Sep 2026) in Conjecture (Colding–Minicozzi), Section 1