The Jacobian and the Ginzburg-Landau energy

We show that if the Ginzburg-Landau energy $$I_\epsilon (u_\epsilon) := \frac{1}{|\ln \epsilon |} \int_\Omega \frac{1}{2} |\nabla u_\epsilon |^2 + \frac{(1-|u_\epsilon |^2 )^2 }{4\epsilon^2} \ \ \ u_\epsilon :R^m \supset \Omega \to \mathbb{R}^2 \ \ \ m \geq 2$$ is uniformly bounded for a sequence of functions $u_\epsilon$ as $\epsilon \to 0$, then the Jacobians $\{ Ju_\epsilon \}$ 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)