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%0 Thesis
%A Schroeder, Helene
%T Symmetry of Skyrmions on the sphere
%I RWTH Aachen University
%V Dissertation
%C Aachen
%M RWTH-2024-00899
%P 1 Online-Ressource : Illustrationen
%D 2024
%Z Veröffentlicht auf dem Publikationsserver der RWTH Aachen University
%Z Dissertation, RWTH Aachen University, 2024
%X We study Skyrmions on the Sphere which arise as critical points of a magnetic energy involving only exchange energy and easy - normal anisotropy. Due to the curved nature of the sphere, these two terms suffice to stabilize critical points against the scaling invariance of the Dirichlet term. Furthermore, the normal anisotropy breaks the invariance of this term under individual rotations of the domain and target sphere, leaving only invariance under joint rotations. The goal of this thesis is to understand the effect that this remaining invariance has on the symmetry of minimizers and critical points. First, we focus on axisymmetric Skyrmions on the sphere, which are themselves invariant under joint rotations around a given axis. We use standard methods to show existence and regularity of minimizers in this symmetry class and then exploit the symmetry to study their shape in more detail. A fine analysis of the energy density and other energy arguments lead to the proof of several properties. We also give some estimates for the case of a high anisotropy parameter. Secondly, we investigate the minimality of these Skyrmions in a broader class. We find that the Hessian associated to the magnetic energy is positive semidefinite and identify the elements of its kernel. Under the assumption of strict convexity within the axisymmetric class, we deduce local minimality of axisymmetric Skyrmions up to invariances of the energy. Finally, we construct non-trivial periodic solutions for the Landau-Lifshitz equation associated to the magnetic energy functional. For this, we consider the minimization of the energy under a constraint on the angular momentum which enforces symmetry breaking. We show that constrained minimizers solve an Euler-Lagrange equation with Lagrange multiplier ω and employ a Łojasiewicz inequality for the magnetic energy to confirm ω ≠ 0.
%F PUB:(DE-HGF)11
%9 Dissertation / PhD Thesis
%R 10.18154/RWTH-2024-00899
%U https://publications.rwth-aachen.de/record/977795