On solutions to the Ginzburg-Landau equations in higher dimensions

We establish a glueing theorem for the Ginzburg-Landau equations in dimension $n>2$. To this end, we consider a nondegenerate minimal submanifold of codimension 2, and construct a one-parameter family of solutions to the Ginzburg-Landau equations such that the energy density concentrates near this submanifold. The proof is based on a construction of suitable approximate solutions and the implicite function theorem.