# Minimal submanifolds from the abelian Higgs model

## TL;DR

This paper analyzes the asymptotic behavior of solutions to the abelian Higgs model energy on line bundles over Riemannian manifolds, showing convergence to stationary varifolds and providing a PDE proof of Almgren's existence theorem.

## Contribution

It establishes the convergence of energy measures to stationary varifolds and constructs nontrivial critical points with uniform energy bounds, offering new insights into the geometric analysis of the model.

## Key findings

- Energy measures converge to stationary integral varifolds.
- Curvature forms converge to integral cycles.
- Constructs nontrivial solutions with bounded energy.

## Abstract

Given a Hermitian line bundle $L\to M$ over a closed, oriented Riemannian manifold $M$, we study the asymptotic behavior, as $\epsilon\to 0$, of couples $(u_\epsilon,\nabla_\epsilon)$ critical for the rescalings \begin{align*} &E_\epsilon(u,\nabla)=\int_M\Big(|\nabla u|^2+\epsilon^2|F_\nabla|^2+\frac{1}{4\epsilon^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 $\limsup_{\epsilon\to 0}E_\epsilon(u_\epsilon,\nabla_\epsilon)<\infty$, we show that the energy measures converge subsequentially to (the weight measure $\mu$ of) a stationary integral $(n-2)$-varifold. Also, we show that the $(n-2)$-currents dual to the curvature forms converge subsequentially to $2\pi\Gamma$, for an integral $(n-2)$-cycle $\Gamma$ with $|\Gamma|\le\mu$. Finally, we provide a variational construction of nontrivial critical points $(u_\epsilon,\nabla_\epsilon)$ 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.