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

TL;DR

This paper constructs special entire solutions to the magnetic Ginzburg-Landau equations in four-dimensional space, linking their zero sets to minimal submanifolds and exploring their energy-minimizing properties.

Contribution

It introduces new entire solutions in 4D, with zero sets near minimal submanifolds and saddle solutions with specific zero set structures, advancing understanding of the solution space.

Findings

Existence of solutions with zero sets close to minimal submanifolds.

Construction of saddle solutions with zero sets as two vertical planes.

Solutions are believed to be energy minimizers within their class.

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.