Geodesic
Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents
Journal of the European Mathematical Society, Tome 1 (1999) no. 3, pp. 237-311

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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.
DOI : 10.1007/s100970050008
Classification : 35-XX, 49-XX, 53-XX, 58-XX
Keywords: 
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     author = {Fanghua Lin and Tristan Rivi\`ere},
     title = {Complex {Ginzburg-Landau} equations in high dimensions and codimension two area minimizing currents},
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     pages = {237--311},
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     volume = {1},
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     year = {1999},
     doi = {10.1007/s100970050008},
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Fanghua Lin; Tristan Rivière. Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents. Journal of the European Mathematical Society, Tome 1 (1999) no. 3, pp. 237-311. doi: 10.1007/s100970050008

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