## Abstract

We consider the magnetic Ginzburg–Landau equations in R^{4} {−ε^{2}(∇−iA)^{2}u=[Formula presented](1−|u|^{2})u,ε^{2}d^{⁎}dA=〈(∇−iA)u,iu〉, formally corresponding to the Euler–Lagrange equations for the energy functional E(u,A)=[Formula presented]∫R^{4}|(∇−iA)u|^{2}+ε^{2}|dA|^{2}+[Formula presented](1−|u|^{2})^{2}. Here u:R^{4}→C, A:R^{4}→R^{4} and d denotes the exterior derivative acting on the one-form dual to A. Given a minimal surface M^{2} in R^{3} with finite total curvature and non-degenerate, we construct a solution (u_{ε},A_{ε}) which has a zero set consisting of a smooth surface close to M×{0}⊂R^{4}. Away from the latter surface we have |u_{ε}|→1 and u_{ε}(x)→[Formula presented],A_{ε}(x)→[Formula presented](−z_{2}ν(y)+z_{1}e_{4}),x=y+z_{1}ν(y)+z_{2}e_{4} for all sufficiently small z≠0. Here y∈M and ν(y) is a unit normal vector field to M in R^{3}.

Original language | English |
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Article number | 109365 |

Journal | Advances in Mathematics |

Volume | 435 |

Issue number | Part A |

Early online date | 23 Oct 2023 |

DOIs | |

Publication status | Published - 15 Dec 2023 |

## Keywords

- Concentration
- Ginzburg-Landau
- Higher codimension
- Minimal surfaces
- Yang-Mills-Higgs

## ASJC Scopus subject areas

- General Mathematics