Hyperbolic one-vortex at curvature minus one half ($\Phi =\frac{2z}{1+|z{|}^2},A=\frac{2(xdy-ydx)}{1+|z{|}^2}$)

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 $\pi$.