= Conformal-surface vortex square completion
{title2=$V\geq\pi|N|,\quad B=\Omega^{-1}F_{12}$}
For metric $g=\Omega(dx^2+dy^2)$, set $F_{12}=\partial_xA_y-\partial_yA_x$ and use the physical <magnetic field> $B=F_{12}/\Omega$. At critical coupling, define $j_i=\operatorname{Im}(\bar\Phi D_i\Phi)$. The identity $|D_x\Phi|^2+|D_y\Phi|^2=|(D_x+iD_y)\Phi|^2+F_{12}|\Phi|^2+\partial_xj_y-\partial_yj_x$ completes the energy into nonnegative squares plus half the <magnetic flux> and half the current boundary integral. For unit-magnitude vacuum boundary data, the combined boundary contribution is $\tfrac12\oint(A+j)=\pi N$, since $j=d\arg\Phi-A$ there. In the more restrictive covariantly decaying sector the current integral separately vanishes and the flux is $2\pi N$. Equality for positive $N$ requires $(D_x+iD_y)\Phi=0$ and $B=(1-|\Phi|^2)/2$. The factor $\Omega$ appears instead if the symbol $B$ denotes the coordinate component $F_{12}$.
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