Abstract

We provide a quantitative three dimensional vortex approximation construction for the Ginzburg-Landau functional. This construction gives an approximation of vortex lines coupled to a lower bound for the energy, optimal to leading order, analogous to the two dimensional ones, and valid for the first time at the epsilon-level. These tools allow for a new approach to analyzing the behavior of global minimizers for the Ginzburg-Landau functional below and near the first critical field in three dimensions, followed in Roman (On the first critical field in the three dimensional Ginzburg-Landau model of superconductivity, 2018). In addition, they allow one to obtain an epsilon-quantitative product estimate for the study of Ginzburg-Landau dynamics.