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

Abstract

We consider the magnetic Ginzburg-Landau equations in a compact manifold $N$ $$\{\begin{array}{l}-{\epsilon }^2{\Delta }^{A}u=\frac{1}{2}(1-|u{|}^2)u,\\ {\epsilon }^2{d}^{\ast }dA=⟨{\nabla }^{A}u,iu⟩\end{array}$$ formally corresponding to the Euler-Lagrange equations for the energy functional $$E(u,A)=\frac{1}{2}{\int }_N{\epsilon }^2|{\nabla }^{A}u{|}^2+{\epsilon }^4|dA{|}^2+\frac{1}{4}(1-|u{|}^2{)}^2.$$ 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 $(u_{\epsilon },A_{\epsilon })$ such that $u_{\epsilon }$ has a zero set consisting of a smooth surface close to $M$. Away from $M$ we have $$u_{\epsilon }(x)\to \frac{z}{|z|},A_{\epsilon }(x)\to \frac{1}{|z{|}^2}(-z^{2}d{z}^1+z^{1}d{z}^2),x={\exp }_y(z^{\beta }{\nu }_{\beta }(y)).$$ as $\epsilon \to 0$, 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.

25 pages, 3 figures

References in corpus (2)

• Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates
• Serrin's Overdetermined Problem and Constant Mean Curvature Surfaces