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

Abstract

We consider the magnetic Ginzburg-Landau equations in $\mathbb{R}^4$ $$ \begin{cases} -\varepsilon^2(\nabla-iA)^2u = \frac{1}{2}(1-|u|^{2})u,\\ \varepsilon^2 d^*dA = \langle(\nabla-iA)u,iu\rangle \end{cases} $$ formally corresponding to the Euler-Lagrange equations for the energy functional $$ E(u,A)=\frac{1}{2}\int_{\mathbb{R}^4}|(\nabla-iA)u|^{2}+\varepsilon^2|dA|^{2}+\frac{1}{4\varepsilon^2}(1-|u|^{2})^{2}. $$ 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 $(u_\varepsilon,A_\varepsilon)$ 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 $|u_\varepsilon| \to 1$ and $$ u_\varepsilon(x)\, \to\, \frac {z}{|z|},\quad A_\varepsilon(x)\, \to\, \frac 1{|z|^2} ( -z_2 \nu(y) + z_1 {\textbf{e}}_4), \quad x = y + z_1 \nu(y) + z_2 {\textbf{e}}_4 $$ for all sufficiently small $z\ne 0$. Here $y\in M$ and $\nu(y)$ is a unit normal vector field to $M$ in $\mathbb{R}^3$.