A reading guide
Ginzburg–Landau Theory
A route from planar vortices to higher-dimensional concentration, self-dual abelian Higgs fields and minimal submanifolds.
Where to start
Begin with Bethuel–Brezis–Hélein for the planar vortex picture, then study Jacobians and energy bounds before moving to higher dimensions.
For the direction closest to my work, follow self-dual abelian Higgs vortices into Pigati–Stern and Parise–Pigati–Stern, then the stability and excess-decay papers. The magnetic GL monograph is a complementary route.
Useful background: Sobolev spaces, elliptic PDE, variational methods and degree theory; differential forms and connections for abelian Higgs, and currents and varifolds for geometric limits.
Non-magnetic GL: planar vortices
In the planar non-magnetic model, the order parameter is a complex-valued map. Its modulus is driven toward one, while topological constraints force zeros around which the phase winds.
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Ginzburg–Landau Vortices
The classical starting point for two-dimensional vortices. The book develops the asymptotics of minimizers, the emergence of point defects and the renormalized energy governing their interaction. It is a great place to begin learning the theory.
Jacobians & variational convergence
The Jacobian records the vorticity in a form that survives weak convergence. It connects energy estimates for maps with measures and integral currents, and makes the link with GMT precise.
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The Jacobian and the Ginzburg–Landau Energy
A central reference for compactness of Jacobians and lower bounds for the rescaled Ginzburg–Landau energy. Read this after the planar vortex picture to understand how vorticity can be controlled without locating each zero individually.
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Variational Convergence for Functionals of Ginzburg–Landau Type
A variational convergence framework relating Ginzburg–Landau-type energies to area. The formulation in terms of integral currents explains how topological singularities of maps give rise to geometric objects of higher codimension.
η-ellipticity & higher dimensions
η-ellipticity is a small-energy principle: sufficiently small normalized energy forces the modulus to stay away from zero in a smaller region. It is a key tool for locating the concentration set as the core scale tends to zero.
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Asymptotics for the Ginzburg–Landau Equation in Arbitrary Dimensions
A foundational reference for the higher-dimensional equation at logarithmic energy scale. It develops the estimates and concentration analysis behind the passage from vortices to codimension-two geometric limits.
Magnetic Ginzburg–Landau
Introducing a gauge field changes the energy balance and allows one to study applied magnetic fields and vortex distributions. This is a useful companion to the non-magnetic theory, with its own scaling regimes.
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Vortices in the Magnetic Ginzburg–Landau Model
A systematic treatment of the magnetic model. The chapters on the vortex-ball construction and Jacobian estimates are especially useful for learning the energy methods; later chapters study applied-field regimes and vortex distributions.
Abelian Higgs: self-dual vortices
Here the focus is the self-dual U(1) Yang–Mills–Higgs model. The unknowns are a section of a Hermitian line bundle and a connection. In two dimensions, the Bogomolny decomposition leads to first-order vortex equations and a topological energy bound.
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Arbitrary N-Vortex Solutions to the First Order Ginzburg–Landau Equations
The basic existence and uniqueness result for prescribed vortex zeros in the plane, with multiplicities. This supplies the two-dimensional family of minimizers that underlies quantitative stability questions.
From abelian Higgs to minimal submanifolds
For the self-dual rescaling, bounded energy leads to a codimension-two limit. This differs from the logarithmic normalization in non-magnetic GL: the gauge field and the coupling are essential to the comparison.
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Minimal Submanifolds from the Abelian Higgs Model
Shows that, as the core scale tends to zero, uniformly energy-bounded critical points have subsequential normalized energy measures given by stationary integral varifolds of codimension two. A natural bridge from gauge-theoretic PDE to the compactness and regularity questions of GMT.
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Convergence of the Self-Dual U(1)-Yang–Mills–Higgs Energies to the (n−2)-Area Functional
Complements the critical-point picture with Γ-convergence to codimension-two area. The gauge-invariant Jacobian converges to an integral cycle, and the recovery construction realizes cycles in the prescribed homology class.
Connections with my work
The references above lead to two related questions: quantitative control near planar vortices, and improvement of flatness for higher-dimensional abelian Higgs fields.
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Quantitative Stability for Yang–Mills–Higgs Instantons in Two Dimensions
After Taubes’s classification, the next question is quantitative: does a small energy deficit control the distance to a minimizing vortex? This paper establishes a quadratic stability estimate for nearly minimizing pairs, including a corresponding result on compact surfaces.
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Decay of Excess for the Abelian Higgs Model
Building on the codimension-two limit, we prove an Allard-type improvement of flatness and apply it to entire critical points with normalized energy close to that of a multiplicity-one plane. The results include uniqueness of blow-down and two-dimensionality under the dimension or minimizing assumptions in the paper.
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New Weighted Inequalities on Two-Manifolds
Weighted Poincaré and elliptic inequalities on surfaces, motivated by estimates for a weighted Hodge decomposition. These provide analytic background for the stability questions above.