Compactness results for static and dynamic chiral skyrmions near the conformal limit

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We examine lower order perturbations of the harmonic map problem from $$\mathbb {R}^2$$ to $$\mathbb {S}^2$$ including chiral interaction in form of a helicity term that prefers modulation, and a potential term that enables decay to a uniform background state. Energy functionals of this type arise in the context of magnetic systems without inversion symmetry. In the almost conformal regime, where these perturbations are weighted with a small parameter, we examine the existence of relative minimizers in a non-trivial homotopy class, so-called chiral skyrmions, strong compactness of almost minimizers, and their asymptotic limit. Finally we examine dynamic stability and compactness of almost minimizers in the context of the Landau–Lifshitz–Gilbert equation including spin-transfer torques arising from the interaction with an external current.