The Jacobian and the Ginzburg-Landau energy

We show that if the Ginzburg-Landau energy $$I_{ϵ}(u_{ϵ}):=\frac{1}{|\ln ϵ|}{\int }_{\mathrm{\Omega }}\frac{1}{2}|\mathrm{\nabla }{u}_{ϵ}{|}^2+\frac{(1-|u_{ϵ}{|}^2{)}^2}{4{ϵ}^2}{u}_{ϵ}:R^m\supset \mathrm{\Omega }\to {\mathbb{R}}^{2}m\ge 2$$ is uniformly bounded for a sequence of functions $u_{ϵ}$ as $ϵ\to 0$, then the Jacobians $\{J{u}_{ϵ}\}$ are precompact in a ppropriate weak topologies. We further show that any limiting measure must be rectifiable. These results have potential applications in problems a variety of problems, including for example questions involving dynamics of vortex filaments in superfluids. (joint work with Mete Soner)