Bogomolny bound for the inverse-density Abelian Higgs model ($V\ge \pi N$)

Let $j_i=\mathrm{Im}(\bar{\Phi }{D}_i\Phi )$. The identityand completing the square give, when the current boundary integral vanishes,For flux $2\pi N>0$, the lower bound is $\pi N$, saturated by the two first-order Bogomolny equations. Division by $h$ is interpreted in the finite-energy sense at its zeros.