= Hyperbolic one-vortex at curvature minus one half
{title2=$\Phi=\frac{2z}{1+|z|^2},\quad A=\frac{2(x\,dy-y\,dx)}{1+|z|^2}$}
For metric $8|dz|^2/(1-|z|^2)^2$, these smooth fields satisfy the <Bogomolny vortex equations> with one simple zero at the origin. Their curvature coefficient is $F_{12}=4/(1+|z|^2)^2$, and their physical <magnetic field> is $(1-|z|^2)^2/[2(1+|z|^2)^2]=(1-|\Phi|^2)/2$. The <magnetic flux> is $2\pi$, and the critically coupled energy in the <conformal-surface vortex square completion> normalization is $\pi$.
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