Characterization of fixed-degree energy minimizers as chiral axisymmetric skyrmions

Determine whether the minimizers of the micromagnetic energy E with a prescribed skyrmion number, whose existence was established in the cited variational works, necessarily have the chiral axisymmetric form m_*^{(N,\gamma)}(r,\psi)=\big(\sin\theta(r)\cos(\gamma-N\psi),\sin\theta(r)\sin(\gamma-N\psi),\cos\theta(r)\big).

Background

The paper studies the two-dimensional Landau–Lifshitz–Gilbert equation near a Bloch skyrmion associated with the micromagnetic energy containing exchange, Dzyaloshinskii–Moriya, and Zeeman terms. The authors focus on chiral axisymmetric profiles, which are explicitly parameterized by a radial polar angle, an integer degree or skyrmion number N, and a helicity γ.

Earlier variational results establish the existence of energy minimizers with prescribed skyrmion number in particular settings. However, the paper states that it is unresolved whether those minimizers belong to the chiral axisymmetric class. Resolving this would connect the general variational existence theory with the explicitly analyzable axisymmetric skyrmion profiles used in the stability analysis.

References

It is not known whether the minimizers of $E$ with a given skyrmion number obtained in are of the above form~définition skyrmion_0, and this goes beyond the scope of this article.

définition skyrmion_0:

m∗(N,γ)(r,ψ)=(sin⁡(θ(r))cos⁡(γ−Nψ)sin⁡(θ(r))sin⁡(γ−Nψ)cos⁡(θ(r))),m^{(N,\gamma)}_*(r,\psi)=\begin{pmatrix} \sin\big(\theta(r)\big)\cos(\gamma -N\psi)\\ \sin\big(\theta(r)\big)\sin(\gamma -N\psi)\\ \cos\big(\theta(r)\big) \end{pmatrix} ,

— Asymptotic stability of a Bloch skyrmion for the Landau-Lifshitz-Gilbert equation  (2610.23753 - Berthoumieu et al., 16 Oct 2026) in Section 1.2, subsection “Chiral axisymmetric skyrmions”