Finite-time singularity formation for unconstrained Willmore flow
Let $\Sigma$ be a smooth closed connected orientable surface and $f_0 : \Sigma \to \mathbb{R}^3$ a smooth immersion. For an immersion $f$ with induced metric $g$, unit normal $\nu$, second fundamental form $A$, mean curvature $H = \operatorname{tr}_g A$ and trace-free part $A^\circ = A - (H/2) g$, the Willmore flow is $$\partial_t f \cdot \nu = -\bigl(\Delta_g H + H |A^\circ|^2\bigr), \qquad f(0) = f_0,$$ with compatible curvature and normal conventions and arbitrary tangential reparametrization; there are no area, volume or spontaneous-curvature terms. Up to a constant time scale it is the $L^2$-gradient flow of the Willmore energy $\mathcal{W}(f) = \frac14 \int_\Sigma H^2 \, d\mu$, which is $4\pi$ for a round sphere. The maximal smooth flow has finite lifetime $T < \infty$ if it cannot be smoothly extended past $T$, equivalently if its curvature becomes unbounded as $t \uparrow T$.
Kuwert and Schätzle proved that from every immersed sphere with $\mathcal{W}(f_0) \le 8\pi$ the flow exists for all time and converges to a round sphere. Mayer and Simonett computed an axisymmetric flow, started from a surface of revolution generated by part of a limaçon, that appears to pinch off into two tangent spheres in finite time. Blatt proved that rotationally symmetric spheres of this kind, among them some with Willmore energy arbitrarily close to $8\pi$, develop a singularity under the flow, at a finite or an infinite time.
Problem. Is there a smooth closed connected orientable surface $\Sigma$ and a smooth immersion $f_0 : \Sigma \to \mathbb{R}^3$ whose maximal smooth Willmore flow has finite lifetime $T < \infty$?
Source: Mayer and Simonett (2002), the Conjecture in §1, who conjecture that the answer is yes, after the numerical example of their §8.5.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, finite-time singularities, geometric flows, Willmore flow
References
@article{mayer2002,
author = {Mayer, Uwe F. and Simonett, Gieri},
title = {A numerical scheme for axisymmetric solutions of curvature-driven free boundary problems, with applications to the {W}illmore flow},
journal = {Interfaces Free Bound.},
volume = {4},
number = {1},
year = {2002},
pages = {89--109},
doi = {10.4171/IFB/54},
}
@article{kuwert2004,
author = {Kuwert, Ernst and Sch{\"a}tzle, Reiner},
title = {Removability of point singularities of {W}illmore surfaces},
journal = {Ann. of Math. (2)},
volume = {160},
number = {1},
year = {2004},
pages = {315--357},
doi = {10.4007/annals.2004.160.315},
}
@article{blatt2009,
author = {Blatt, Simon},
title = {A singular example for the {W}illmore flow},
journal = {Analysis (Munich)},
volume = {29},
number = {4},
year = {2009},
pages = {407--430},
doi = {10.1524/anly.2009.1017},
}
@misc{aim-willmore-finite-time-singularity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Finite-time singularity formation for unconstrained Willmore flow}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/224-willmore-finite-time-singularity.md},
note = {Entry 224 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-00655, "Finite-time singularity formation for unconstrained Willmore flow", clopen.solutions.