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

We consider the magnetic Ginzburg–Landau equations in R⁴ {−ε²(∇−iA)²u=[Formula presented](1−|u|²)u,ε²d^⁎dA=〈(∇−iA)u,iu〉, formally corresponding to the Euler–Lagrange equations for the energy functional E(u,A)=[Formula presented]∫R⁴|(∇−iA)u|²+ε²|dA|²+[Formula presented](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 (u_ε,A_ε) which has a zero set consisting of a smooth surface close to M×{0}⊂R⁴. Away from the latter surface we have |u_ε|→1 and u_ε(x)→[Formula presented],A_ε(x)→[Formula presented](−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.