 Stereographic energy of the O3 sigma model

In the oriented stereographic projection $\varphi =(2\mathrm{Re}w,2\mathrm{Im}w,1-|w{|}^2)/(1+|w{|}^2)$, the sphere's metric is $4|dw{|}^2/(1+|w{|}^2{)}^2$. For energy $V=\int |\nabla \varphi {|}^2$, its topological degree is $N={\pi }^{-1}\int \mathrm{Im}(\overline{w_{x_1}}{w}_{x_2})/(1+|w{|}^2{)}^2$. The inequality $|w_{x_1}{|}^2+|w_{x_2}{|}^2\ge 2|\mathrm{Im}(\overline{w_{x_1}}{w}_{x_2})|$ gives $V\ge 8\pi |N|$. Holomorphic maps and antiholomorphic maps that are rational maps saturate the respective signs. The more common energy with a prefactor $1/2$ has bound $4\pi |N|$ instead.