Decay of excess for the abelian Higgs model

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