Decay of excess for the abelian Higgs model

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})\leq(1+\tau(n))\cdot2\pi\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 codimension-two analogue of Savin’s theorem. The main ingredient is an Allard-type improvement of flatness.