Minimal submanifolds from the abelian Higgs model

Abstract

Given a Hermitian line bundle $L\to M$ over a closed, oriented Riemannian manifold $M$, we study the asymptotic behavior, as $ε\to 0$, of couples $(u_ε,\nabla_ε)$ critical for the rescalings \begin{align*} &E_ε(u,\nabla)=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{1}{4ε^2}(1-|u|^2)^2\Big) \end{align*} of the self-dual Yang-Mills-Higgs energy, where $u$ is a section of $L$ and $\nabla$ is a Hermitian connection on $L$ with curvature $F_{\nabla}$. Under the natural assumption $\limsup_{ε\to 0}E_ε(u_ε,\nabla_ε)<\infty$, we show that the energy measures converge subsequentially to (the weight measure $μ$ of) a stationary integral $(n-2)$-varifold. Also, we show that the $(n-2)$-currents dual to the curvature forms converge subsequentially to $2πΓ$, for an integral $(n-2)$-cycle $Γ$ with $|Γ|\leμ$. Finally, we provide a variational construction of nontrivial critical points $(u_ε,\nabla_ε)$ on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren's existence result of (nontrivial) stationary integral $(n-2)$-varifolds in an arbitrary closed Riemannian manifold.