Solutions to the Magnetic Ginzburg–Landau Equations Concentrating on Codimension-2 Minimal Submanifolds

We consider the magnetic Ginzburg–Landau equations in a compact manifold N {-ε2ΔAu=12(1-|u|2)u,ε2d∗dA=⟨∇Au,iu⟩. Here u: N→ C and A is a 1-form on N. We discuss some recent results on the construction of solutions exhibiting concentration phenomena near prescribed minimal, codimension 2 submanifolds corresponding to the vortex set of the solution. Given a codimension-2 minimal submanifold M⊂ N which is also oriented and non-degenerate, we construct a solution (u _ε, A _ε) such that u _ε has a zero set consisting of a smooth surface close to M. Away from M we have uε(x)→z|z|,Aε(x)→1|z|2(-z2dz1+z1dz2),x=expy(zβνβ(y)) as ε→ 0 , for all sufficiently small z≠ 0 and y∈ M . Here, { ν ₁, ν ₂} is a normal frame for M in N. These results improve, by giving precise quantitative information, a recent construction by De Philippis and Pigati (arXiv:2205.12389, 2022) who built solutions for which the concentration phenomenon holds in an energy, measure-theoretical sense. In addition, we consider the non-compact case N= R ⁴ and the special case of a two-dimensional minimal surface in R ³ , regarded as a codimension 2 minimal submanifold in R ⁴ , with finite total curvature and non-degenerate. We construct a solution (u _ε, A _ε) which has a zero set consisting of a smooth 2-dimensional surface close to M× { 0 } ⊂ R ⁴ . Away from the latter surface we have | u _ε| → 1 and asymptotic behavior as in (1).