Solutions of the Ginzburg-Landau equations concentrating on codimension-2 minimal submanifolds

TL;DR

This paper constructs solutions to the Ginzburg-Landau equations in a compact manifold that concentrate along a codimension-2 minimal submanifold, providing a geometric description of the zero set and asymptotic behavior.

Contribution

It introduces a method to explicitly construct solutions concentrating on minimal submanifolds, advancing understanding of vortex-like solutions in geometric PDEs.

Findings

Solutions concentrate near minimal submanifolds as epsilon approaches zero.

Zero set of solutions forms a smooth surface close to the minimal submanifold.

Asymptotic behavior of solutions matches prescribed geometric limits.

Abstract

We consider the magnetic Ginzburg-Landau equations in a compact manifold $N$ $$ \begin{cases} -\varepsilon^2 \Delta^{A} u=\frac{1}{2}(1-|u|^2)u,\\ \varepsilon^2 d^*dA=\langle\nabla^A u,iu\rangle \end{cases} $$ formally corresponding to the Euler-Lagrange equations for the energy functional $$ E(u,A)=\frac{1}{2}\int_{N}\varepsilon^2|\nabla^Au|^{2}+\varepsilon^4|dA|^{2}+\frac{1}{4}(1-|u|^{2})^{2}. $$ Here $u:N\to \mathbb{C}$ and $A$ is a 1-form on $N$. Given a codimension-2 minimal submanifold $M\subset N$ which is also oriented and non-degenerate, we construct a solution $(u_\varepsilon,A_\varepsilon)$ such that $u_\varepsilon$ has a zero set consisting of a smooth surface close to $M$. Away from $M$ we have $$ u_\varepsilon(x)\to\frac{z}{|z|},\quad A_\varepsilon(x)\to\frac{1}{|z|^2}(-z^2dz^1+z^1dz^2),\quad x=\exp_y(z^\beta\nu_\beta(y)). $$ as $\varepsilon\to 0$, for all sufficiently small $z\ne 0$ and $y\in M$. Here, $\{\nu_1,\nu_2\}$ is a normal frame for $M$ in $N$. This improves a recent result by De Philippis and Pigati who built a solution for which the concentration phenomenon holds in an energy, measure-theoretical sense.