  We prove convergence of the gradient flow of the Ginzburg-Landau energy
functional on a Riemann surface in the self-dual Bogomolny case, in Coulomb
gauge. The proof is direct and makes use of the associated nonlinear first
order differential operators (the Bogomolny operators). One aim is to
illustrate that the Bogomolny structure, which is known to be of great utility
in the static elliptic case, can also be used effectively in evolution
problems. We also identify the minimizers and minimum value of the energy when
the Bogomolny bound is not achieved (below the Bradlow limit).
