OP-00844 Open

Uniqueness of a hemispheric saddle profile on a magnetic sphere

Note: A second profile at κ = 4, which would answer this in the negative, is claimed in a Lean development on prove2.me (shivm, October 2026, https://prove2.me/theorems/2f49ead8-433f-4b5e-9fc9-c44349429863). The site is not one the registry links papers from, and the claim has not been checked here.

Submit a result

For $\kappa \ge 4$ let $$\mathcal{E}_\kappa(m) = \frac12 \int_{S^2} \Bigl(|\nabla_{S^2} m|^2 + \kappa \bigl(1 - (m(x) \cdot x)^2\bigr)\Bigr)\,dS$$ be the energy of a unit magnetization field $m : S^2 \to S^2$, and write an axisymmetric field in spherical coordinates as $$m(\theta, \varphi) = \bigl(\sin h(\theta) \cos\varphi,\ \sin h(\theta) \sin\varphi,\ \cos h(\theta)\bigr).$$ Gustafson, Meinert and Melcher reduce the critical points of $\mathcal{E}_\kappa$ in this form to the profile equation $$h'' + \cot\theta\, h' - \frac{\sin(2h)}{2 \sin^2\theta} - \frac{\kappa}{2} \sin(2h - 2\theta) = 0 \qquad (0 < \theta < \pi).$$ The conditions $$h(0) = 0, \qquad h(\pi) = 2\pi, \qquad h(\pi - \theta) = 2\pi - h(\theta)$$ specify their hemispheric class with one complete profile rotation; the corresponding maps have degree zero. They showed that $h(\theta) = 2\theta$ satisfies the equation and all three conditions when $\kappa = 4$, and that the map it defines is a saddle point of $\mathcal{E}_4$; and, by the implicit function theorem, that for every $\kappa$ in a neighbourhood of $4$ the equation has a solution with $h(0) = 0$ and $h(\pi) = 2\pi$.

Problem. Let $\kappa \ge 4$. Is there exactly one profile $h \in C^\infty([0,\pi])$, inducing a smooth map on the whole sphere, that satisfies the profile equation and the three conditions above?

Source: Gustafson, Meinert and Melcher (2025), Remark 3.18, who conjecture that the answer is yes; stated for the profile equation (2.6) by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: AP — Analysis of PDEs
Secondary: MP
Keywords: AIM Open Applied Problems, micromagnetics, spherical ferromagnets

References

@misc{gustafson2025,
  author        = {Gustafson, Stephen and Meinert, Daniel and Melcher, Christof},
  title         = {Saddle point configurations for spherical ferromagnets},
  year          = {2025},
  eprint        = {2509.05159},
  archivePrefix = {arXiv},
}
@misc{aim-spherical-ferromagnet-profile-uniqueness,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Uniqueness of a hemispheric saddle profile on a magnetic sphere}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/241-spherical-ferromagnet-profile-uniqueness.md},
  note         = {Entry 241 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-00844, "Uniqueness of a hemispheric saddle profile on a magnetic sphere", clopen.solutions.