Robert L. Jerrard

University of Toronto

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

It is well-known that under various 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 of the GL equations for which the given minimal surface is the limiting energy 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 equatons. This is joint work with Andrew Colinet and Peter Sternberg.