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

  • Andrew Colinet

    University of Toronto, Canada
  • Robert Jerrard

    University of Toronto, Canada
  • Peter Sternberg

    Indiana University, Bloomington, USA
Solutions of the Ginzburg–Landau equations with vorticity concentrating near a nondegenerate geodesic cover

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Abstract

It is well-known that under suitable hypotheses, for a sequence of solutions of the (simplified) Ginzburg–Landau equations , the energy and vorticity concentrate as around a codimension stationary varifold – a (measure-theoretic) minimal surface. Much less is known about the question of whether, given a codimension 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 -dimensional closed Riemannian manifold , 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.

Cite this article

Andrew Colinet, Robert Jerrard, Peter Sternberg, Solutions of the Ginzburg–Landau equations with vorticity concentrating near a nondegenerate geodesic. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1397