Entire solutions to 4 dimensional Ginzburg–Landau equations and codimension 2 minimal submanifolds

Abstract

We consider the magnetic Ginzburg–Landau equations in R4{−ε2(∇−iA)2u=12(1−|u|2)u,ε2d⁎dA=〈(∇−iA)u,iu〉, formally corresponding to the Euler–Lagrange equations for the energy functionalE(u,A)=12∫R4|(∇−iA)u|2+ε2|dA|2+14ε2(1−|u|2)2. Here u:R4→C, A:R4→R4 and d denotes the exterior derivative acting on the one-form dual to A. Given a minimal surface M2 in R3 with finite total curvature and non-degenerate, we construct a solution (uε,Aε) which has a zero set consisting of a smooth surface close to M×{0}⊂R4. Away from the latter surface we have |uε|→1 anduε(x)→z|z|,Aε(x)→1|z|2(−z2ν(y)+z1e4),x=y+z1ν(y)+z2e4 for all sufficiently small z≠0. Here y∈M and ν(y) is a unit normal vector field to M in R3.