Entire solutions of the magnetic Ginzburg-Landau equation in $\mathbb{R}^4$

Abstract

We construct entire solutions of the magnetic Ginzburg-Landau equations in dimension 4 using Lyapunov-Schmidt reduction. The zero set of these solutions are close to the minimal submanifolds studied by Arezzo-Pacard\cite{Arezzo}. We also show the existence of a saddle type solution to the equations, whose zero set consists of two vertical planes in $\mathbb{R}^4$. These two types of solutions are believed to be energy minimizers of the corresponding energy functional and lie in the same connect component of the moduli space of entire solutions.