Refined jacobian estimates for Ginzburg-Landau functional

We prove various estimates that relate the Ginzburg-Landau energy E _ε (u) = ∫_Ω | ∇_u| ²/2 + (|u|² - 1)²/(4ε²)dx of a function u ε H¹(Ωℝ²), Ω ∪ ℝ², 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.