Hyperbolic vortex

At the integrable curvature scale on the Poincare disc, the vortex Taubes equation reduces to the Liouville equation. Holomorphic self-maps of the disc then generate explicit vortex solutions.

For the disc metric $8|dz{|}^2/(1-|z{|}^2{)}^2$ and a holomorphic map $g:D\to D$, the Higgs magnitudesolves the critically coupled vortex equations. Choosing $g(z)=z^{N+1}$ gives a radial vortex of winding $N$.