Refined Jacobian estimates for Ginzburg-Landau functionals

We prove various estimates that relate the Ginzburg-Landau energy E e (u) =∫Ω|∇u| 2 /2+(|u| 2 -1)2/(4e 2 ) dx of a function u ∈ H 1 (Ω; R 2 ), Ω ⊂R 2 , to the distance in the W -1,1 norm between the Jacobian J(u) = det ∇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.