Allard Regularity for Abelian Yang--Mills--Higgs Equation

TL;DR

This paper develops a regularity theory for solutions to the Abelian Yang--Mills--Higgs equations in the singular limit, revealing how solutions concentrate near minimal submanifolds and analyzing their geometric properties.

Contribution

It introduces a new regularity framework inspired by Allard's theory, constructing approximate solutions and analyzing their perturbations in the context of Abelian gauge theories.

Findings

Solutions concentrate along codimension-two sets as epsilon approaches zero.

Uniform Lipschitz and curvature estimates are established for the solutions.

Hölder regularity is obtained for scalar and connection components.

Abstract

We study solutions to the self-dual Abelian Yang--Mills--Higgs (YMH) equations in the singular limit $\e \to 0 $, where the associated self-dual Ginzburg--Landau type energy \begin{align*} E_\e\begin{pmatrix}u\\ A\end{pmatrix} = \int_M \left( |\nabla^A u|^2 + \e^2 |F_A|^2 + \frac{(1 - |u|^2)^2}{4\e^2} \right) \mathrm{dvol}_g \end{align*} exhibits concentration along codimension-two sets. Using techniques inspired by Allard's regularity theory, we construct approximate solutions concentrating near a minimal submanifold and analyse their perturbations via a linearised operator projected orthogonally to gauge and translational zero modes. By working in Fermi coordinates and enforcing Coulomb gauge conditions, we derive uniform Lipschitz and curvature estimates for the solutions and obtain H\"older regularity for the scalar and connection components. These results establish a geometric framework for understanding vortex sheet formation and provide a regularity theory for the limiting defect set in the context of Abelian gauge theories.