Giga’s embedded curve-diffusion conjecture
Let $\gamma \colon S^1 \times [0, T) \to \mathbb{R}^2$ be the maximal smooth evolution of a smooth closed regular curve by curve diffusion, $$\partial_t \gamma \cdot \nu = -\partial_s^2 \kappa,$$ where $s$ is arclength, $\nu$ is a chosen unit normal and $\kappa$ is the signed curvature in the compatible convention; tangential velocity only reparametrizes the curve. Embeddedness is not preserved: Giga and Ito gave a simple closed curve that develops a self-intersection in finite time. Wheeler proved that curves with total curvature $2\pi$, positive enclosed area, isoperimetric ratio close to one and small normalised $L^2$ oscillation of curvature evolve for all time and converge exponentially fast to a round circle, and that every flow with $T = \infty$ converges exponentially fast in the smooth topology to a round circle, possibly multiply covered.
Problem. If $\gamma(\cdot, t)$ is an embedding for every $0 \le t < T$, must $T = \infty$? Equivalently, must every evolution that becomes singular in finite time have lost embeddedness at an earlier regular time?
Source: Giga (around 2010), as recorded by Wheeler (2022), who calls it Giga's conjecture.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, curve diffusion flow, embedded curves, geometric evolution equations
References
@article{wheeler2022,
author = {Wheeler, Glen},
title = {Convergence for global curve diffusion flows},
journal = {Math. Eng.},
volume = {4},
number = {1},
year = {2022},
pages = {1--13},
doi = {10.3934/mine.2022001},
eprint = {2004.08494},
archivePrefix = {arXiv},
}
@article{wheeler2013,
author = {Wheeler, Glen},
title = {On the curve diffusion flow of closed plane curves},
journal = {Ann. Mat. Pura Appl. (4)},
volume = {192},
number = {5},
year = {2013},
pages = {931--950},
doi = {10.1007/s10231-012-0253-2},
eprint = {1201.3735},
archivePrefix = {arXiv},
}
@article{giga1998,
author = {Giga, Yoshikazu and Ito, Kazuo},
title = {On pinching of curves moved by surface diffusion},
journal = {Comm. Appl. Anal.},
volume = {2},
number = {3},
year = {1998},
pages = {393--406},
}
@misc{aim-embedded-curve-diffusion,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Giga’s embedded curve-diffusion conjecture}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/223-embedded-curve-diffusion.md},
note = {Entry 223 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00589, "Giga’s embedded curve-diffusion conjecture", clopen.solutions.