Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

Abstract. There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S^1 -valued function defined on the boundary of a bounded regular domain of R^n . When such extensions do not exist, we use the Ginzburg–Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg–Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology class induced from the S^1 -valued boundary data. The union of this harmonic map and the minimal current is the natural generalization of the harmonic extension.