Solutions of the Ginzburg–Landau equations concentrating on codimension‐2 minimal submanifolds

Abstract

AbstractWe consider the magnetic Ginzburg–Landau equations on a closed manifold formally corresponding to the Euler–Lagrange equations for the energy functional where and is a 1‐form on . Given a codimension‐2 minimal submanifold , which is also oriented and non‐degenerate, we construct solutions such that has a zero set consisting of a smooth surface close to . Away from , we have as , for all sufficiently small and . Here, is a normal frame for in . This improves a recent result by De Philippis and Pigati (2022), who built a solution for which the concentration phenomenon holds in an energy, measure‐theoretical sense.