Energy-Angular Momentum Relations for Rotating Skyrmions

Definitions and mathematical statement

The supplied record asks for an energy--angular momentum relation for skyrmions and its consequences for rotating Landau--Lifshitz motion. It does not specify the underlying spatial domain, the energy functional, the angular momentum, the admissible skyrmion class, or the evolution equation. Consequently, the following is a conditional formulation of the target, not a fully specified variational problem.

Let $\mathcal S$ denote the admissible class intended by the source, let $E$ denote its energy, and let $j(u)\geq 0$ denote the magnitude of its angular momentum. These are placeholders for definitions absent from the supplied record; in particular, no choice of angular momentum or norm is asserted. For an attainable magnitude $s\in j(\mathcal S)$, write formally \[ M(s):=\inf\{E(u):u\in\mathcal S,\ j(u)=s\}. \] This notation does not assert that the infimum is attained. The source does not specify whether $\mathcal S$ consists of configurations, stationary profiles, relative equilibria, or another class of skyrmion solutions.

Requested statement. Determine how the minimal energy $M(s)$ depends on the angular momentum magnitude $s$, and determine whether this dependence yields estimates for the rotation frequency $\omega$ of the corresponding Landau--Lifshitz solution and proves that this solution is non-static. Here non-static means that, for a solution $u$ on its time interval $I$, \[ \exists\,t_1,t_2\in I\quad u(t_1)\ne u(t_2). \] The correspondence between constrained energy minimizers and evolving solutions, the rotation action, the definition and normalization of $\omega$, and the intended range of $s$ remain unspecified. No differentiability of $M$, frequency formula, existence of minimizers, or implication from a nonzero frequency to non-static motion is asserted.

The supplied dated assessment reports that existence of localized rotating Landau--Lifshitz solutions is known, while the energy--angular momentum law and proof that the constructed motion is non-static remain open. This is supplied curation, not an independently verified mathematical or literature conclusion.

Short English statement

Determine how the least skyrmion energy varies with angular momentum magnitude, and whether that relation controls the rotation frequency $\omega$ and establishes non-static Landau--Lifshitz motion. The supplied record omits the model, functionals, admissible class, and parameter range needed for a fully precise formulation. Its open-status and existence assertions are supplied dated assessments requiring independent review.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30004486$, source problem code catalogue entry, title Energy-Angular Momentum Relations for Rotating Skyrmions. The record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied report citation: Calculus of Variations (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/22, pp. 1139--1196. https://doi.org/10.4171/owr/2020/22. The record separately gives proposed year 2020; the relationship between this year and the citation's 2021 designation is unresolved.
• Supplied literature source: Melcher--Sakellaris, Curvature stabilized skyrmions with angular momentum. https://arxiv.org/abs/1902.04881. This is attribution of the cited work, not identification of the problem's proposer.
• Supplied literature source: OWR Calculus of Variations report. https://ems.press/content/serial-article-files/46860.
• Literature assessment date: 2026-08-22. The supplied dated curation labels the problem open and reports known existence of localized rotating Landau--Lifshitz solutions, with the energy--angular momentum law and non-staticity of the constructed motion still open. These claims and the linked sources have not been independently verified here.
• The supplied registry labels the statement corrected_verified and records a manual semantic, typographic, and LaTeX review claiming self-containedness. These are source metadata; the missing mathematical specifications listed in this draft still require review.