= Bogomolny degree bound for the O3 sigma model
{c}
{title2=$E\geq4\pi|Q|$}
For $E=\tfrac12\int(|\partial_1\mathbf n|^2+|\partial_2\mathbf n|^2)$, completing the square gives $E=\tfrac12\int|\partial_1\mathbf n\pm\mathbf n\times\partial_2\mathbf n|^2\pm4\pi Q$, hence $E\geq4\pi|Q|$. Saturation yields the <Bogomolny equations> of the <sigma-model lump>. With the oriented <stereographic projection> $\mathbf n=(2\operatorname{Re}w,2\operatorname{Im}w,1-|w|^2)/(1+|w|^2)$, holomorphic rational maps have positive degree and saturate the bound; antiholomorphic maps have negative degree.
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