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

TL;DR

This paper demonstrates that in a 3D Riemannian manifold, any embedded nondegenerate closed geodesic can serve as the limiting concentration set for solutions to the Ginzburg-Landau equations, linking minimal geodesics to vortex concentration.

Contribution

It establishes the existence of Ginzburg-Landau solutions concentrating near nondegenerate geodesics, extending understanding of vortex concentration beyond minimal surfaces.

Findings

Any nondegenerate closed geodesic can be realized as a vortex concentration set.

Solutions of Ginzburg-Landau equations can be constructed to concentrate near specified geodesics.

The result connects geometric properties of geodesics with vortex behavior in Ginzburg-Landau theory.

Abstract

It is well-known that under suitable hypotheses, for a sequence of solutions of the (simplified) Ginzburg-Landau equations $-\Delta u_\varepsilon +\varepsilon^{-2}(|u_\varepsilon|^2-1)u_\varepsilon = 0$, the energy and vorticity concentrate as $\varepsilon\to 0$ 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.