Solutions to the Magnetic Ginzburg–Landau Equations Concentrating on Codimension-2 Minimal Submanifolds

Abstract

AbstractWe consider the magnetic Ginzburg–Landau equations in a compact manifold N$$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2{\varDelta }^Au = \frac{1}{2}(1-|u|^{2})u,\\ \varepsilon ^2d^*dA=\langle \nabla ^Au,iu\rangle . \end{array}\right. \end{aligned}$$ - ε 2 Δ A u = 1 2 ( 1 - | u | 2 ) u , ε 2 d ∗ d A = ⟨ ∇ A u , i u ⟩ . Here $$u:N\rightarrow \mathbb {C}$$ 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 $$M\subset N$$ M ⊂ N which is also oriented and non-degenerate, we construct a solution $$(u_{\varepsilon },A_{\varepsilon })$$ ( u ε , A ε ) such that $$u_\varepsilon $$ u ε has a zero set consisting of a smooth surface close to M. Away from M we have $$\begin{aligned} u_\varepsilon (x)\rightarrow \frac{z}{|z|},\quad A_\varepsilon (x)\rightarrow \frac{1}{|z|^2}(-z_2dz^1+z_1dz^2),\quad x=\exp _y(z^\beta \nu _\beta (y)) \end{aligned}$$ u ε ( x ) → z | z | , A ε ( x ) → 1 | z | 2 ( - z 2 d z 1 + z 1 d z 2 ) , x = exp y ( z β ν β ( y ) ) as $$\varepsilon \rightarrow 0$$ ε → 0 , for all sufficiently small $$z\ne 0$$ z ≠ 0 and $$y\in M$$ y ∈ M . Here, $$\{\nu _1,\nu _2\}$$ { 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 $$N=\mathbb {R}^4$$ N = R 4 and the special case of a two-dimensional minimal surface in $$\mathbb {R}^3$$ R 3 , regarded as a codimension 2 minimal submanifold in $$\mathbb {R}^4$$ R 4 , with finite total curvature and non-degenerate. We construct a solution $$(u_\varepsilon ,A_\varepsilon )$$ ( u ε , A ε ) which has a zero set consisting of a smooth 2-dimensional surface close to $$M\times \{0\}\subset \mathbb {R}^4$$ M × { 0 } ⊂ R 4 . Away from the latter surface we have $$|u_\varepsilon | \rightarrow 1$$ | u ε | → 1 and asymptotic behavior as in (1).