Abstract. There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S1-valued function defined on the boundary of a bounded regular domain of Rn. 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 S1-valued boundary data. The union of this harmonic map and the minimal current is the natural generalization of the harmonic extension.
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Fanghua Lin, Tristan Rivière, Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents. J. Eur. Math. Soc. 1 (1999), no. 3 pp. 237–311