Bogomolny vortex equation

At critical coupling on a surface with conformal metric $g=\Omega dzd\bar{z}$, the vortex equations are

For $u=\log |\varphi {|}^2$ and zeros of multiplicities $N_r$ at $z_r$, the Taubes equation is

Integrating the Taubes equation on a compact surface of area $A$ gives $A\ge 4\pi N$. A nontrivial vortex with nonzero Higgs field requires the strict inequality $A>4\pi N$.