Decay of excess for the abelian Higgs model

In this article we prove that entire critical points $(u,\nabla)$ of the self-dual $U(1)$-Yang-Mills-Higgs functional $E_1$, with energy $$E_1(u,\nabla;B_R):=\int_{B_R}\left[|\nabla u|^2+\frac{(1-|u|^2)^2}{4}+|F_\nabla|^2\right]\leq(2\pi+\tau(n)) \omega_{n-2}R^{n-2}$$ for all $R>0$, have unique blow-down. Moreover, we show that they are two-dimensional in ambient dimension $2\leq n\leq4$, or in any dimension $n\ge2$ assuming that $(u,\nabla)$ 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.