 Angular integral in the rational map approximation

For a degree-$N$ rational map, the angular Jacobian of a rational map has average $N$, so the Cauchy-Schwarz inequality gives $\mathcal{I}\ge N^2$. In dimensionless Skyrme model units the radial energy within the rational-map ansatz isFirst minimize $\mathcal{I}$ over admissible maps, then minimize over profiles with $f(0)=\pi$, $f(\infty )=0$. This gives a restricted variational approximation and an upper bound on the unrestricted minimum in that topological sector, not an exact multi-Skyrmion solution. Houghton, Manton and Sutcliffe developed this construction in arxiv.org/abs/hep-th/9705151.