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

Abstract

We consider the magnetic Ginzburg-Landau equations in ${\mathbb{R}}^4$ $$\{\begin{array}{l}-{\epsilon }^2(\nabla -iA{)}^{2}u=\frac{1}{2}(1-|u{|}^2)u,\\ {\epsilon }^2{d}^{\ast }dA=⟨(\nabla -iA)u,iu⟩\end{array}$$ formally corresponding to the Euler-Lagrange equations for the energy functional $$E(u,A)=\frac{1}{2}{\int }_{{\mathbb{R}}^4}|(\nabla -iA)u{|}^2+{\epsilon }^2|dA{|}^2+\frac{1}{4{\epsilon }^2}(1-|u{|}^2{)}^2.$$ Here $u:{\mathbb{R}}^4\to \mathbb{C}$, $A:{\mathbb{R}}^4\to {\mathbb{R}}^4$ and $d$ denotes the exterior derivative acting on the one-form dual to $A$. Given a 2-dimensional minimal surface $M$ in ${\mathbb{R}}^3$ with finite total curvature and non-degenerate, we construct a solution $(u_{\epsilon },A_{\epsilon })$ which has a zero set consisting of a smooth 2-dimensional surface close to $M\times \{0\}\subset {\mathbb{R}}^4$. Away from the latter surface we have $|u_{\epsilon }|\to 1$ and $$u_{\epsilon }(x)\to \frac{z}{|z|},A_{\epsilon }(x)\to \frac{1}{|z{|}^2}(-z_2\nu (y)+z_1{\text{e}}_4),x=y+z_1\nu (y)+z_2{\text{e}}_4$$ for all sufficiently small $z\ne 0$. Here $y\in M$ and $\nu (y)$ is a unit normal vector field to $M$ in ${\mathbb{R}}^3$.