Refined Jacobian estimates for Ginzburg-Landau functionals

We prove various estimates that relate the Ginzburg-Landau energy E epsilon (u) = integral(Omega)vertical bar del u vertical bar(2)/2 + (vertical bar u vertical bar(2) - 1)(2)/(4 epsilon(2)) dx of a function u is an element of H-1 (Q; R-2), Omega subset of R-2, to the distance in the W--1,W-1 norm between the Jacobian J(u) = det del u and a sum of point masses. These are interpreted as quantifying the precision with which "vortices" in a function u can be located via measure-theoretic tools such as the Jacobian; and the extent to which variations in the Ginzburg-Landau energy due to translation of vortices can be detected using the Jacobian. We give examples to show that some of our estimates are close to optimal.