Decay of excess for the abelian Higgs model

In this article we prove that entire critical points (u,∇) of the self-dual $U(1)$-Yang-Mills-Higgs functional E₁, with energy E₁(u,∇;BR):=∫BR[|∇ u|²+(1-|u|²)²/4+|F_∇|²]≤(2π+τ(n)) ωₙ₋₂Rⁿ⁻² for all $R>0$, have unique blow-down. Moreover, we show that they are two-dimensional in ambient dimension 2≤ n≤4, or in any dimension n≥2 assuming that (u,∇) is a local minimizer, thus establishing a co-dimension-two analogue of Savin's theorem. The main ingredient is an Allard-type improvement of flatness.