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

Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the European Mathematical Society Springer Journals

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

, Volume 1 (3) – Sep 1, 1999
75 pages

/lp/springer-journals/complex-ginzburg-landau-equations-in-high-dimensions-and-codimension-eegTCixvEj
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
1435-9855
DOI
10.1007/s100970050008
Publisher site
See Article on Publisher Site

### 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.

### Journal

Journal of the European Mathematical SocietySpringer Journals

Published: Sep 1, 1999