## Abstract

We consider the magnetic Ginzburg–Landau equations in a compact manifold N {-ε2ΔAu=12(1-|u|2)u,ε2d∗dA=⟨∇Au,iu⟩. Here u: N→ C and A is a 1-form on N. We discuss some recent results on the construction of solutions exhibiting concentration phenomena near prescribed minimal, codimension 2 submanifolds corresponding to the vortex set of the solution. Given a codimension-2 minimal submanifold M⊂ N which is also oriented and non-degenerate, we construct a solution (u
_{ε}, A
_{ε}) such that u
_{ε} has a zero set consisting of a smooth surface close to M. Away from M we have uε(x)→z|z|,Aε(x)→1|z|2(-z2dz1+z1dz2),x=expy(zβνβ(y)) as ε→ 0 , for all sufficiently small z≠ 0 and y∈ M . Here, { ν
_{1}, ν
_{2}} is a normal frame for M in N. These results improve, by giving precise quantitative information, a recent construction by De Philippis and Pigati (arXiv:2205.12389, 2022) who built solutions for which the concentration phenomenon holds in an energy, measure-theoretical sense. In addition, we consider the non-compact case N= R
^{4} and the special case of a two-dimensional minimal surface in R
^{3} , regarded as a codimension 2 minimal submanifold in R
^{4} , with finite total curvature and non-degenerate. We construct a solution (u
_{ε}, A
_{ε}) which has a zero set consisting of a smooth 2-dimensional surface close to M× { 0 } ⊂ R
^{4} . Away from the latter surface we have | u
_{ε}| → 1 and asymptotic behavior as in (1).

Original language | English |
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Number of pages | 18 |

Journal | Vietnam Journal of Mathematics |

Early online date | 29 Jan 2024 |

DOIs | |

Publication status | Published - 29 Jan 2024 |

### Funding

The authors have been supported by Royal Society Research Professorship RP-R1-180114, United Kingdom.

Funders | Funder number |
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Royal Society | RP-R1-180114 |

## Keywords

- Ginzburg–Landau
- Minimal submanifolds
- Yang–mills–higgs

## ASJC Scopus subject areas

- General Mathematics