Solutions of the Ginzburg–Landau equations with vorticity concentrating near a nondegenerate geodesic

Abstract

It is well-known that under suitable hypotheses, for a sequence of solutions of the (simplified) Ginzburg–Landau equations $-\Delta u_{\epsilon }+{\epsilon }^{-2}(|u_{\epsilon }{|}^2-1)u_{\epsilon }=0$, the energy and vorticity concentrate as $\epsilon \to 0$ around a codimension $2$ stationary varifold – a (measure-theoretic) minimal surface. Much less is known about the question of whether, given a codimension $2$ minimal surface, there exists a sequence of solutions for which the given minimal surface is the limiting concentration set. The corresponding question is very well-understood for minimal hypersurfaces and the scalar Allen–Cahn equation, and for the Ginzburg–Landau equations when the minimal surface is locally area-minimizing, but otherwise quite open. We consider this question on a $3$-dimensional closed Riemannian manifold $(M,g)$, and we prove that any embedded nondegenerate closed geodesic can be realized as the asymptotic energy/ vorticity concentration set of a sequence of solutions of the Ginzburg–Landau equations.