= Stereographic energy of the O3 sigma model
{title2=$V=4\int\frac{|\nabla w|^2}{(1+|w|^2)^2}\,d^2x$}
In the oriented <stereographic projection> $\phi=(2\operatorname{Re}w,2\operatorname{Im}w,1-|w|^2)/(1+|w|^2)$, the sphere's metric is $4|dw|^2/(1+|w|^2)^2$. For energy $V=\int|\nabla\phi|^2$, its <topological degree> is $N=\pi^{-1}\int\operatorname{Im}(\overline{w_{x_1}}w_{x_2})/(1+|w|^2)^2$. The inequality $|w_{x_1}|^2+|w_{x_2}|^2\geq2|\operatorname{Im}(\overline{w_{x_1}}w_{x_2})|$ gives $V\geq8\pi|N|$. <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.
Back to article page
