Improved lower bounds for Ginzburg-Landau energies via mass displacement

We prove some improved estimates for the Ginzburg-Landau energy (with or without a magnetic field) in two dimensions, relating the asymptotic energy of an arbitrary configuration to its vortices and their degrees, with possibly unbounded numbers of vortices. The method is based on a localization of the "ball construction method" combined with a mass displacement idea which allows to compensate for negative errors in the ball construction estimates by energy "displaced" from close by. Under good conditions, our main estimate allows to get a lower bound on the energy which includes a finite order "renormalized energy" of vortex interaction, up to the best possible precision, i.e., with only a o(1) error per vortex, and is complemented by local compactness results on the vortices. Besides being used crucially in a forthcoming paper, our result can serve to provide lower bounds for weighted Ginzburg-Landau energies.