= Commuting-current obstruction in a two-dimensional Skyrme model
{title2=$[R_1,R_2]=0,\quad\partial_iR_i=0\ \Longrightarrow\ R_i=0$}
For a smooth finite-energy static planar <Skyrme model> field, <Derrick scaling> gives zero quartic energy, hence $[R_1,R_2]=0$ pointwise. The quartic first variation then vanishes, leaving the sigma-model equation $\partial_iR_i=0$. The right-current <Maurer-Cartan equation> gives $\partial_1R_2-\partial_2R_1=[R_1,R_2]=0$. Each current component is therefore harmonic on the plane and square-integrable. The mean-value estimate on expanding disks makes each component zero, so $U$ is constant. This excludes smooth finite-energy planar static solutions of this potential-free SU(2) model, without misapplying the same conclusion to a pure curved-target sigma model.
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