I will demonstrate radial symmetry and uniqueness of magnetic
skyrmions as global minimizers of the simplest local micromagnetic
energy. The strategy is to show that any radial critical point with
strictly decreasing out-of-plane component is the unique minimizer
in the class of general degree-one maps with the prescribed limit at
infinity. To this end, I identify a class of null Lagrangians that
is sufficiently flexible to contain a calibration, giving a uniform
lower bound for the energy that is uniquely saturated on the
critical point. The arguments provide a blueprint for proofs of
symmetry and uniqueness of minimizers in vectorial variational
problems under topological constraints.
This is joint work with Giovanni Di Fratta, Cyrill Muratov, and Valeriy Slastikov