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

Technical Summary: Allard Regularity for Abelian Yang–Mills–Higgs Equation

Problem Statement
This paper investigates the regularity and geometric structure of solutions to the self-dual Abelian Yang–Mills–Higgs (YMH) equations on Riemannian manifolds in the singular limit $\varepsilon \to 0$. The study focuses on the energy functional:
$$E_\varepsilon(u, A) = \int_M \left( |\nabla_A u|^2 + \varepsilon^2 |F_A|^2 + \frac{(1-|u|^2)^2}{4\varepsilon^2} \right) d\text{vol}_g$$
where $u$ is a section of a Hermitian line bundle and $A$ is a $U(1)$ connection. In this limit, critical points of $E_\varepsilon$ are expected to concentrate along codimension-two sets (vortex sheets). While previous work by Pigati–Stern established the convergence of these critical points to stationary varifolds and the $\Gamma$-convergence of the energy to the codimension-two area functional, a refined regularity theory for the nodal sets (the zero sets of the Higgs field $u$) analogous to Allard's regularity theory for minimal surfaces had not been fully developed for this gauge-theoretic setting. The primary challenge lies in the infinite-dimensional gauge invariance of the system, which complicates the ellipticity of the linearized operator.

Methodology
The authors adapt Allard's regularity theory to the Abelian YMH context through a multi-step analytical framework:

1. Approximate Solutions and Fermi Coordinates: The authors construct approximate solutions by scaling classical degree-one vortices in $\mathbb{R}^2$ and localizing them around a prescribed minimal submanifold $M$ using Fermi coordinates. The error of this ansatz is controlled via a linearized analysis.
2. Gauge Fixing and Orthogonality: To handle the non-ellipticity caused by gauge invariance, the authors enforce the Coulomb gauge condition ($d^*A = 0$) and project perturbations orthogonally to both gauge and translational zero modes. This allows the decomposition of the linearized operator into an elliptic component and a gauge component.
3. Linearized Analysis: The perturbation $\Phi = \hat{U} - \tilde{U}_0$ (where $\hat{U}$ is the rescaled solution and $\tilde{U}_0$ is the approximate solution) is analyzed. The authors derive equations for the difference and utilize Schauder estimates on the elliptic part of the linearized operator.
4. Rigidity and Excess Decay: The proof relies heavily on rigidity results for solutions with unit density. Specifically, the authors utilize a rigidity theorem (Theorem 2.5, based on Badran–del Pino and De Philippis–Halavati–Pigati) which states that entire critical points with energy close to $2\pi$ in dimensions $n+2 \leq 4$ (or for minimizers) must be two-dimensional, essentially pullbacks of the standard planar vortex.
5. Iterative Regularity Improvement: By combining Lipschitz bounds (derived from excess decay arguments) with the linearized estimates, the authors establish a bootstrapping mechanism to improve regularity from Lipschitz to $C^{2,\alpha}$.

Key Contributions and Results

• Theorem 1.1 (Improved Regularity for Nodal Sets): Assuming the nodal sets $M_\varepsilon = \{u_\varepsilon = 0\}$ converge graphically to a minimal submanifold $M$ with a uniform Lipschitz bound, the authors prove that $M_\varepsilon$ are graphs with uniformly bounded $C^{2,\alpha}$ norms. This establishes that the convergence is not merely weak but occurs in a strong $C^{2,\alpha}$ topology, provided the initial Lipschitz regularity holds.
• Theorem 1.2 (Allard-Type Regularity): Under the condition that the ambient dimension is $n+2 \leq 4$ or the solution is minimizing, and assuming the energy density is close to the minimal value ($2\pi$) and the solution satisfies the Coulomb gauge, the nodal sets are shown to be $C^{2,\alpha}$ graphs. This result extends Allard's regularity theory to the Abelian YMH model in low dimensions.
• Curvature Estimates: The paper derives uniform curvature bounds for the nodal sets, showing that the mean curvature of the limiting defect set is well-controlled.
• Technical Framework: The authors provide a robust analytic framework for handling the linearized YMH operator by isolating geometric content from gauge degrees of freedom, specifically through the decomposition $S'_\varepsilon = L_\varepsilon - \Theta_W \Theta^*_{\varepsilon, W}$.

Significance and Claims
The paper claims to establish a geometric framework for understanding vortex sheet formation in Abelian gauge theories. By proving that the nodal sets of the Higgs field converge to minimal submanifolds with $C^{2,\alpha}$ regularity (under specific dimensional or minimality constraints), the work bridges the gap between the weak convergence results of geometric measure theory and the strong regularity known for the Allen–Cahn equation.

The authors note that the extension of this Allard-type regularity to general ambient dimensions is contingent upon a rigidity statement for solutions with unit density at infinity (the "Gibbon's Conjecture"). Once this conjecture is resolved, the procedure used in the paper suggests that local Lipschitz regularity can be improved to $C^{2,\alpha}$ in higher dimensions as well. The results refine earlier work on Ginzburg–Landau and Abelian YMH models, offering precise estimates for the discrepancy from exact solutions and providing a foundation for analyzing vortex concentration in higher codimensions.